Equidistribution and Counting Under Equilibrium States in Negative Curvature and Trees: Applications to Non-Archimedean

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Progress in Mathematics 329

Anne Broise-Alamichel Jouni Parkkonen Frédéric Paulin

Equidistribution and Counting Under Equilibrium States in Negative Curvature and Trees Applications to Non-Archimedean Diophantine Approximation

Progress in Mathematics Volume 329

Series Editors Antoine Chambert-Loir, Université Paris-Diderot, Paris, France Jiang-Hua Lu, The University of Hong Kong, Hong Kong SAR, China Michael Ruzhansky, Imperial College, London, UK Yuri Tschinkel, Courant Institute of Mathematical Sciences, New York, USA

More information about this series at http://www.springer.com/series/4848

Anne Broise-Alamichel • Jouni Parkkonen Frédéric Paulin

Equidistribution and Counting Under Equilibrium States in Negative Curvature and Trees Applications to Non-Archimedean Diophantine Approximation Appendix by Jérôme Buzzi (Université Paris-Saclay, Orsay, France)

Anne Broise-Alamichel Laboratoire de mathématique d’Orsay Université Paris-Saclay Orsay, France

Jouni Parkkonen Department of Mathematics and Statistics University of Jyväskylä Jyväskylä, Finland

Frédéric Paulin Laboratoire de mathématique d’Orsay Université Paris-Saclay Orsay, France

ISSN 0743-1643 ISSN 2296-505X (electronic) Progress in Mathematics ISBN 978-3-030-18314-1 ISBN 978-3-030-18315-8 (eBook) https://doi.org/10.1007/978-3-030-18315-8 Mathematics Subject Classification (2010): 37D40, 37A25, 53C22, 20E08, 11R58, 37D35, 52A55, 37B10, 60J10, 11R11, 11E25 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Contents

1

Introduction 1.1 Geometric and dynamical tools . . . . . . . 1.2 The distribution of common perpendiculars 1.3 Counting in weighted graphs of groups . . . 1.4 Selected arithmetic applications . . . . . . 1.5 General notation . . . . . . . . . . . . . . .

Part I 2

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Geometry and Dynamics in Negative Curvature

Negatively Curved Geometry 2.1 Background on CAT(−1) spaces . . . . . . . . . 2.2 Generalised geodesic lines . . . . . . . . . . . . . 2.3 The unit tangent bundle . . . . . . . . . . . . . . 2.4 Normal bundles and dynamical neighbourhoods 2.5 Creating common perpendiculars . . . . . . . . . 2.6 Metric and simplicial trees, and graphs of groups Discrete-time geodesic flow on trees . . . . . . . Cross-ratios of ends of trees . . . . . . . . . . . . Bass–Serre’s graphs of groups . . . . . . . . . . .

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Potentials, Critical Exponents, and Gibbs Cocycles 3.1 Background on (uniformly local) Hölder continuity 3.2 Potentials . . . . . . . . . . . . . . . . . . . . . . . 3.3 Poincaré series and critical exponents . . . . . . . 3.4 Gibbs cocycles . . . . . . . . . . . . . . . . . . . . 3.5 Systems of conductances on trees and generalised electrical networks . . . . . . . . . . .

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vi

Contents

4 Patterson–Sullivan and Bowen–Margulis Measures with Potential on CAT(−1) Spaces 4.1 Patterson densities . . . . . . . . . . . . . . . . . . . 4.2 Gibbs measures . . . . . . . . . . . . . . . . . . . . . The Gibbs property of Gibbs measures . . . . . . . The Hopf–Tsuji–Sullivan–Roblin theorem . . . . . . On the finiteness of Gibbs measures . . . . . . . . . Bowen–Margulis measure computations in locally symmetric spaces . . . . . . . . . . . . . . On the cohomological invariance of Gibbs measures 4.3 Patterson densities for simplicial trees . . . . . . . . 4.4 Gibbs measures for metric and simplicial trees . . .

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Symbolic Dynamics of Geodesic Flows on Trees 5.1 Two-sided topological Markov shifts . . . . . . . . . . . 5.2 Coding discrete-time geodesic flows on simplicial trees . 5.3 Coding continuous-time geodesic flows on metric trees . 5.4 The variational principle for metric and simplicial trees

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111 112 125 132

6

Random Walks on Weighted Graphs of Groups 6.1 Laplacian operators on weighted graphs of groups . . . . . . . . . 141 6.2 Patterson densities as harmonic measures for simplicial trees . . . 147

7

Skinning Measures with Potential on CAT(−1) Spaces 7.1 Skinning measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 7.2 Equivariant families of convex subsets and their skinning measures . . . . . . . . . . . . . . . . . . . . . . . . 165

8

Explicit Measure Computations for Simplicial Trees and Graphs of Groups 8.1 Computations of Bowen–Margulis measures for simplicial trees . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 8.2 Computations of skinning measures for simplicial trees . . . . . . . 175

9

Rate 9.1 9.2 9.3

Part II

of Mixing for the Geodesic Flow Rate of mixing for Riemannian manifolds . . . . . . . . . . . . . . 181 Rate of mixing for simplicial trees . . . . . . . . . . . . . . . . . . 182 Rate of mixing for metric trees . . . . . . . . . . . . . . . . . . . . 194

Geometric Equidistribution and Counting

10 Equidistribution of Equidistant Level Sets to Gibbs Measures 10.1 A general equidistribution result . . . . . . . . . . . . . . . . . . . 207

Contents

vii

10.2 Rate of equidistribution of equidistant level sets for manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 10.3 Equidistribution of equidistant level sets on simplicial graphs and random walks on graphs of groups . . . . . . . . . . . . . . . . 215 10.4 Rate of equidistribution for metric and simplicial trees . . . . . . . 220 11 Equidistribution of Common Perpendicular Arcs 11.1 Part I of the proof of Theorem 11.1: The common part . . . 11.2 Part II of the proof of Theorem 11.1: The metric tree case . . 11.3 Part III of the proof of Theorem 11.1: The manifold case . . 11.4 Equidistribution of common perpendiculars in simplicial trees 12 Equidistribution and Counting of Common Perpendiculars in Quotient Spaces 12.1 Multiplicities and counting functions in Riemannian orbifolds 12.2 Common perpendiculars in Riemannian orbifolds . . . . . . . 12.3 Error terms for equidistribution and counting for Riemannian orbifolds . . . . . . . . . . . . . . . . . . . . . . 12.4 Equidistribution and counting for quotient simplicial and metric trees . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Counting for simplicial graphs of groups . . . . . . . . . . . . 12.6 Error terms for equidistribution and counting for metric and simplicial graphs of groups . . . . . . . . . . . . . . . . . . .

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228 230 234 242

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13 Geometric Applications 13.1 Orbit counting in conjugacy classes for groups acting on trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 13.2 Equidistribution and counting of closed orbits on metric and simplicial graphs (of groups) . . . . . . . . . . . . . . . . . . . . . 291

Part III

Arithmetic Applications

14 Fields with Discrete Valuations 14.1 Local fields and valuations . . . . . . . . . . . . . . . . . . . . . . 299 14.2 Global function fields . . . . . . . . . . . . . . . . . . . . . . . . . 301 15 Bruhat–Tits Trees and Modular Groups 15.1 Bruhat–Tits trees . . . . . . . . . . . . . . . . . . . . . . 15.2 Modular graphs of groups . . . . . . . . . . . . . . . . . . 15.3 Computations of measures for Bruhat–Tits trees . . . . . 15.4 Exponential decay of correlation and error terms for arithmetic quotients of Bruhat–Tits trees . . . . . . . 15.5 Geometrically finite lattices with infinite Bowen–Margulis measure . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . 307 . . . . . 311 . . . . . 313 . . . . . 318 . . . . . 325

viii

Contents

16 Equidistribution and Counting of Rational Points in Completed Function Fields 16.1 Counting and equidistribution of non-Archimedean Farey fractions . . . . . . . . . . . . . . . . . . . 329 16.2 Mertens’s formula in function fields . . . . . . . . . . . . . . . . . 338 17 Equidistribution and Counting of Quadratic Irrational Points in Non-Archimedean Local Fields 17.1 Counting and equidistribution of loxodromic fixed points . . . . . 342 17.2 Counting and equidistribution of quadratic irrationals in positive characteristic . . . . . . . . . . . . . . . . . . . . . . . . 346 17.3 Counting and equidistribution of quadratic irrationals in Qp . . . 354 18 Equidistribution and Counting of Cross-ratios 18.1 Counting and equidistribution of cross-ratios of loxodromic fixed points . . . . . . . . . . . . . . . . . . . . . . . . 362 18.2 Counting and equidistribution of cross-ratios of quadratic irrationals . . . . . . . . . . . . . . . . . . . . . . . . . . 368 19 Equidistribution and Counting of Integral Representations by Quadratic Norm Forms . . . . . . . . . . . . . . . . . . . . . . . . . . 371 Appendix by J. Buzzi: A Weak Gibbs Measure is the Unique Equilibrium A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 A.2 Proof of the main result, Theorem A.4 . . . . . . . . . . . . . . . . 381 List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409

Chapter 1

Introduction In this book, we study equidistribution and counting problems concerning locally geodesic arcs in negatively curved spaces endowed with potentials, and we deduce, from the special case of tree quotients, various arithmetic applications to equidistribution and counting problems in non-Archimedean local fields. For several decades, tools in ergodic theory and dynamical systems have been used to obtain geometric equidistribution and counting results on manifolds, both inspired by and with applications to arithmetic and number-theoretic problems, in particular in Diophantine approximation. Especially pioneered by Margulis, this field has produced a huge corpus of works, by Bowen, Cosentino, Clozel, Dani, Einseidler, Eskin, Gorodnik, Ghosh, Guivarc’h, Kim, Kleinbock, Kontorovich, Lindenstraus, Margulis, McMullen, Michel, Mohammadi, Mozes, Nevo, Oh, Pollicott, Roblin, Shah, Sharp, Sullivan, Ullmo, Weiss, and the last two authors, just to mention a few contributors. We refer to the surveys [Bab2, Oh, PaP16, PaP17c] and we will explain in more detail in this introduction the relations of our work to previous works. In this text, we consider geometric equidistribution and counting problems weighted with a potential function in quotient spaces of CAT(−1) spaces by discrete groups of isometries. The CAT(−1) spaces form a huge class of metric spaces that contains (but is not restricted to) metric trees, hyperbolic buildings and simply connected complete Riemannian manifolds with sectional curvature bounded above by −1. In Chapter 2, we review some basic properties of these spaces and we refer to [BridH] for more details. Although some of the equidistribution and counting results with potentials on negatively curved manifolds are known,1 as well as some analogous results on CAT(−1) spaces without potential,2 bringing together these two aspects and producing new results and applications is one of the goals of this book. 1 See, 2 See,

for instance, [PauPS]. for instance, [Rob2].

© Springer Nature Switzerland AG 2019 A. Broise-Alamichel et al., Equidistribution and Counting Under Equilibrium States in Negative Curvature and Trees, Progress in Mathematics 329, https://doi.org/10.1007/978-3-030-18315-8_1

1

2

Chapter 1. Introduction

We extend the theory of Patterson–Sullivan, Bowen–Margulis, and skinning measures to CAT(−1) spaces with potentials, with a special emphasis on trees endowed with a system of conductances. We prove that under natural nondegeneracy, mixing, and finiteness assumptions, the pushforward under the geodesic flow of the skinning measure of properly immersed locally convex closed subsets of locally CAT(−1) spaces equidistributes to the Gibbs measure, generalising the main result of [PaP14a]. We also prove that the (appropriate generalisations of) the initial and terminal tangent vectors of the common perpendiculars to any two properly immersed locally convex closed subsets jointly equidistribute to the skinning measures when the lengths of the common perpendiculars tend to +∞. This result is then used to prove asymptotic results on weighted counting functions of common perpendiculars whose lengths tend to +∞. Common perpendiculars have been studied, in various particular cases, sometimes not explicitly, by Basmajian, Bridgeman, Bridgeman–Kahn, Eskin–McMullen, Herrmann, Huber, Kontorovich– Oh, Margulis, Martin–McKee–Wambach, Meyerhoff, Mirzakhani, Oh–Shah, Pollicott, Roblin, Shah, the last two authors and many others. See the comments after Theorem 1.5 below, and the survey [PaP16] for references. In Part III of this book, we apply the geometric results obtained for trees to deduce arithmetic applications in non-Archimedean local fields. In particular, we prove equidistribution and counting results for rationals and quadratic irrationals in any completion of any function field over a finite field. Let us now describe more precisely the content of this book, restricted to special cases for the sake of the exposition.

1.1

Geometric and dynamical tools

Let Y be a geodesically complete connected proper locally CAT(−1) space (or good orbispace), that is nonelementary.3 In this introduction, we will mainly concentrate on the cases in which Y is either a metric graph (or graph of finite groups in the sense of Bass and Serre; see [Ser3]) or a Riemannian manifold (or good orbifold) of dimension at least 2 with sectional curvature at most −1. Let G Y be the space of locally geodesic lines of Y , on which the geodesic flow (gt )t∈R acts by real translations on the source. When Y is a simplicial4 graph (of finite groups), we consider the discrete-time geodesic flow (gt )t∈Z ; see Section 2.6. If Y is a Riemannian manifold, then G Y is naturally identified with the unit tangent bundle T 1 Y by the map that associates to a locally geodesic line its tangent vector at time 0. In general, we define T 1 Y as the space of germs at time 0 of locally geodesic lines in Y , and G Y maps onto T 1 Y with possibly uncountable fibers. 3 That 4 That

is, whose fundamental group is not virtually nilpotent. is, if its edges all have length 1.

1.1. Geometric and dynamical tools

3

Let F : T 1 Y → R be a continuous map, called a potential, which plays the same role in the construction of Gibbs measures/equilibrium states as the energy function in Bowen’s treatment of the thermodynamic formalism of symbolic dynamical systems in [Bowe3, Sect. 1]. We define in Section 3.3 the critical exponent δF associated with F , which describes the logarithmic growth of an orbit of the fundamental group on the universal cover of Y weighted by the (lifted) potential F , and which coincides with the classical critical exponent when F = 0. When Y is a metric graph, we associate in Section 3.5 a potential Fc to a system of conductances c (that is, a map from the set of edges of Y to R), in such a way that the correspondence c 7→ Fc is bijective at the level of cohomology classes, and we denote δFc by δc . In this introduction, we assume that F is bounded and that δF is finite and positive in order to simplify the statements. We say that the pair (Y, F ) satisfies the HC property if the integral of F on compact locally geodesic segments of Y varies in a Hölder-continuous way on its extremities (see Definition 3.13). The pairs that have the HC property include complete Riemannian manifolds with pinched sectional curvature at most −1 and H¨ older-continuous potentials, and metric graphs with any potential. This HC property is a new technical idea compared to [PauPS] that allows the extensions to our very general framework. See also [ConLT], under the very strong assumption that Y is compact. In Chapter 4, building on the works of [Rob2]5 when F = 0 and of [PauPS]6 when Y is a Riemannian manifold, we generalise, to locally CAT(−1) spaces Y endowed with a potential F satisfying the HC property, the construction and basic properties of the Patterson densities at infinity of the universal cover of Y associated with F and the Gibbs measure mF on G Y associated with F . Using the Patterson–Sullivan–Bowen–Margulis approach, the Patterson densities are limits of renormalised measures on the orbit points of the fundamental group on the universal cover of Y , weighted by the potential, and the Gibbs measures on G Y are local products of Patterson densities on the endpoints of the geodesic line, with the Lebesgue measure on the time parameter, weighted by the Gibbs cocycle defined by the potential. Generalising a result of [CoP2], we prove in Section 6.2 that when Y is a regular simplicial graph and c is an antireversible system of conductances, then the Patterson measures, normalised to be probability measures, are harmonic measures (or hitting measures) on the boundary at infinity of the universal cover of Y for a transient random walk on the vertices, whose transition probabilities are constructed using the total masses of the Patterson measures. 5 This 6 This

reference builds on the works of Patterson, Sullivan, Coornaert, Burger–Mozes,. . . reference builds on the works of Ledrappier [Led], Hamenst¨ adt, Coud` ene, Mohsen.

4


Gibbs measures were first introduced in statistical mechanics, and are naturally associated via the thermodynamic formalism7 with symbolic dynamics. We prove in Section 4.2 that our Gibbs measures satisfy a Gibbs property analogous to the one in symbolic dynamics. If F = 0, the Gibbs measure mF is the Bowen– Margulis measure mBM . If Y is a compact Riemannian manifold and F is the d t strong unstable Jacobian v 7→ − dt ln Jac g (v), then mF is the Liout=0 W − (v) ville measure and δF = 0 (see [PauPS, Chap. 7] for more general assumptions on Y ). Thus, one interesting aspect of Gibbs measures is that they form a natural family of measures invariant under the geodesic flow that interpolates between the Liouville measure and the Bowen–Margulis measure (which in variable curvature are in general not in the same measure class). Another interesting point is that such measures are plentiful: a recent result of Belarif [Bel] proves that when Y is a geometrically finite Riemannian manifold with pinched negative curvature and topologically mixing geodesic flow, the finite and mixing Gibbs measures associated with bounded Hölder-continuous potentials are, once normalised, dense (for the weak-star topology) in the whole space of probability measures invariant under the geodesic flow. The Gibbs measures are remarkable measures for CAT(−1) spaces endowed with potentials due to their unique ergodic-theoretic properties. Let (Z, (φt )t∈R ) be a topological space endowed with a continuous one-parameter group of homeomorphisms and let ψ : Z → R be a bounded continuous map. Let M be the set of Borel probability measures on Z invariant under the flow (φt )t∈R . Let hm (φ1 ) be the (metric) entropy of the geodesic flow with respect to m ∈ M . The metric pressure for ψ of a measure m ∈ M and the pressure of ψ are respectively Z Pψ (m) = hm (φ1 ) +

ψ dm Z

and

Pψ = sup Pψ (m) . m∈M

An element m ∈ M is an equilibrium state for ψ if the least upper bound defining Pψ is attained on m. Let F ] : G Y → R be the composition of the canonical map G Y → T 1 Y with F , and note that F ] = F if Y is a Riemannian manifold. When F = 0 and Y is a Riemannian manifold whose sectional curvatures and first derivatives are bounded, by [OtaP, Theo. 2], the pressure PF coincides with the entropy of the geodesic flow, it is equal to the critical exponent of the fundamental group of Y , and the Bowen–Margulis measure mF = mBM , normalised to be a probability measure, is the measure of maximal entropy. When Y is a Riemannian manifold whose sectional curvatures and first derivatives are bounded and F is Hölder-continuous, by [PauPS, Theo. 6.1], we have PF = δF . If furthermore the Gibbs measure mF is finite and normalised to be a probability measure, then mF is an equilibrium state for F . 7 See,

for instance, [Rue3, Kel, Zin].

1.1. Geometric and dynamical tools

5

In Section 5.4, we prove an analogue of these results for the potential F ] when Y is a metric graph of groups. The case in which Y is a finite simplicial graph8 is classical by the work of Bowen [Bowe3], since it reduces to arguments of subshifts of finite type (see, for instance, [CoP1]). When Y is a compact9 locally CAT(−1) space,10 a complete statement about existence, uniqueness, and Gibbs property of equilibrium states for any Hölder-continuous potential is given in [ConLT]. Theorem 1.1 (The variational principle for metric graphs of groups). Assume that Y is a metric graph of finite groups, with a positive lower bound and finite upper bound on the lengths of edges. If the critical exponent δF is finite and if the Gibbs measure mF is finite, then PF ] = δF and the Gibbs measure normalised to be a probability measure is the unique equilibrium state for F ] . The main tool is a natural coding of the discrete-time geodesic flow by a topological Markov shift (see Section 5.1). This coding is delicate when the vertex stabilisers are nontrivial, in particular since it does not satisfy in general the Markovian property of dependence only on the immediate past (see Section 5.2). We then apply results of Buzzi and Sarig in symbolic dynamics over a countable alphabet (see the Appendix, written by J. Buzzi) and suspension techniques introduced in Section 5.3. See also [Kemp]. Let Y be any geodesically complete connected proper locally CAT(−1) space, and let D be any connected proper nonempty properly immersed11 closed locally convex subset of Y . In Chapter 7, we generalise for nonconstant potentials on Y + − the construction of the skinning measures σD and σD on the outer and inner unit normal bundles of D in Y . We refer to Section 2.4 for the appropriate definition of the outer and inner unit normal bundles of D when the boundary of D is not + − smooth. We construct these measures σD and σD as the induced measures in Y of pushforwards of the Patterson densities associated with the potential F to the outer and inner unit normal bundles of the lift of D in the universal cover of Y . This construction generalises the one in [PaP14a] when F = 0, which itself generalises the one in [OhS1, OhS2] when Y is a Riemannian manifold with constant curvature and the lift of D is a ball, a horoball, or a totally geodesic submanifold. In Section 10.1, we prove the following result on the equidistribution of equidistant hypersurfaces in CAT(−1) spaces. This result is a generalisation of [PaP14a, Thm. 1] (valid in Riemannian manifolds with zero potential), which itself generalised those in [Mar2, EM, PaP12] when Y has constant curvature, F = 0, and D is a ball, a horoball, or a totally geodesic submanifold. See also [Rob2] when Y is a CAT(−1) space, F = 0, and D is a ball or a horoball. 8 That

is, a finite graph of trivial groups with edge lengths 1. is a very strong assumption that we do not want to make in this text. 10 This is not in the orbifold sense hence this excludes, for instance, the case of graphs of groups with some nontrivial vertex stabiliser. 11 By definition, D is the image in Y , by the universal covering map, of a proper nonempty closed convex subset of the universal cover of Y , whose family of images under the universal covering group is locally finite. 9 This

6


Theorem 1.2. Let Y, D be as above, and let F be a potential of Y satisfying the HC property. Assume that the Gibbs measure mF on G Y is finite and mixing for + the geodesic flow (gt )t∈R , and that the skinning measure σD is finite and nonzero. + t Then, as t tends to +∞, the pushforwards (g )∗ σD of the skinning measure of D by the geodesic flow weak-star converge to the Gibbs measure mF (after normalisation as probability measures). We prove in Theorem 10.4 an analogue of Theorem 1.2 for the discrete-time geodesic flow on simplicial graphs and, more generally, simplicial graphs of groups. As a special case, we recover known results on nonbacktracking simple random walks on regular graphs. The equidistribution of the pushforward of the skinning measure of a subgraph is a weighted version of the following classical result, see, for instance, [AloBLS], which under further assumptions on the spectral properties on the graph gives precise rates of convergence. Corollary 1.3. Let Y be a finite regular graph that is not bipartite. Let Y0 be a nonempty connected proper subgraph. Then the nth vertex of the nonbacktracking simple random walk on Y starting transversally to Y0 converges in distribution to the uniform distribution as n → +∞. See Chapter 10 for more details and for the extensions to nonzero potential and to graphs of groups, as well as Section 10.4 for error terms.

1.2

The distribution of common perpendiculars

Let D− and D+ be connected proper nonempty properly immersed locally convex closed subsets of Y . A common perpendicular from D− to D+ is a locally geodesic path in Y starting perpendicularly from D− and arriving perpendicularly to D+ .12 We denote the length of a common perpendicular α from D− to D+ by λ(α), and its initial and terminal unit tangent vectors by vα− and vα+ . In b the general CAT(−1) case, vα± are two different parametrisations (by ∓[0, λ(α)]) of α, considered as elements of the space G Y of generalised locally geodesic lines in Y ; see [BartL] or Section 2.2. For all t > 0, we denote by Perp(D− , D+ , t) the set of common perpendiculars from D− to D+ with length at most t (considered with multiplicities), and we define the counting function with weights by R X ND− , D+ , F (t) = e αF , α∈Perp(D − , D + , t)

R R λ(α) where α F = 0 F (gt vα− ) dt.13 We refer to Section 12.1 for the definition of the multiplicities in the manifold case, which are equal to 1 if D− and D+ are embedded and disjoint. Higher multiplicities for common perpendiculars α can 12 See 13 See

Section 2.5 for explanations when the boundary of D− or D+ is not smooth. Section 3.2 for precisions when Y is a general CAT(−1)-space.

1.2. The distribution of common perpendiculars

7

occur, for instance, when D− is a nonsimple closed geodesic and the initial point of α is a multiple point of D− . Let Perp(D− , D+ ) be the set of all common perpendiculars from D− to + D (considered with multiplicities). The family (λ(α))α∈Perp(D− , D+ ) is called the marked ortholength spectrum from D− to D+ . The set of lengths (with multiplicities) of elements of Perp(D− , D+ ) is called the ortholength spectrum of D− , D+ . This second set was introduced by Basmajian [Basm] (under the name “full orthogonal spectrum”) when M has constant curvature, and D− and D+ are disjoint or equal embedded totally geodesic hypersurfaces or embedded horospherical cusp neighbourhoods or embedded balls. We refer to the paper [BridK], which proves that the ortholength spectrum with D− = D+ = ∂M determines the volume of a compact hyperbolic manifold M with totally geodesic boundary (see also [Cal] and [MasaM]). We prove in Chapter 12 that the critical exponent δF of F is the exponential growth rate of ND− , D+ , F (t), and we give an asymptotic formula of the form ND− , D+ , F (t) ∼ c eδF t as t → +∞, with error term estimates in appropriate situations. The constants c that will appear in such asymptotic formulas will be explicit, in terms of the measures naturally associated with the (normalised) potential F : the Gibbs measure mF and the skinning measures of D− and D+ . When F = 0 and Y is a Riemannian manifold with pinched sectional curvature and finite and mixing Bowen–Margulis measure, the asymptotics of the counting function ND− , D+ , 0 (t) are described in [PaP17b, Thm. 1]. The only restriction on the two convex sets D± is that their skinning measures be finite. Here, we generalise that result by allowing for nonzero potential and more general CAT(−1) spaces than just manifolds. The counting function ND− , D+ , 0 (t) has been studied in negatively curved manifolds since the 1950s and in a number of more recent works, sometimes in a different guise. A number of special cases (all with F = 0 and covered by the results of [PaP17b]) were known: • D− and D+ are reduced to points, by, for instance, [Hub2], [Mar1], and [Rob2]. • D− and D+ are horoballs, by [BeHP], [HeP3], [Cos], and [Rob2] without an explicit form of the constant in the asymptotic expression. • D− is a point and D+ is a totally geodesic submanifold, by [Herr], [EM] and [OhS3] in constant curvature. • D− is a point and D+ is a horoball, by [Kon] and [KonO] in constant curvature, and [Kim] in rank-one symmetric spaces. • D− is a horoball and D+ is a totally geodesic submanifold, by [OhS1] and [PaP12] in constant curvature. • D− and D+ are (properly immersed) locally geodesic lines in constant curvature and dimension 3, by [Pol2]. We refer to the survey [PaP16] for more details on the manifold case.

8


When X is a compact metric or simplicial graph and D± are points, the asymptotics of ND− , D+ , 0 (t) as t → +∞ are treated in [Gui], as well as [Rob2]. Under the same setting, see also the work of Kiro–Smilansky–Smilansky announced in [KiSS] for a counting result of paths (not assumed to be locally geodesic) in finite metric graphs with rationally independent edge lengths and vanishing potential. The proofs of the asymptotic results on the counting function ND− , D+ , F are based on the following simultaneous equidistribution result, which shows that the initial and terminal tangent vectors of the common perpendiculars equidistribute to the skinning measures of D− and D+ . We denote the unit Dirac mass at a point z by ∆z and the total mass of any measure m by kmk. Theorem 1.4. Assume that Y is a nonelementary Riemannian manifold with pinched sectional curvature at most −1 or a metric graph. Let F : T 1 Y → R be a potential, with finite and positive critical exponent δF , which is bounded and H¨ older-continuous when Y is a manifold. Let D± be as above. Assume that the b mixing b Gibbs measure mF is finite and for the geodesic flow. For the weak-star convergence of measures on G Y × G Y , we have R X + − lim δF kmF k e−δF t e α F ∆vα− ⊗ ∆vα+ = σD − ⊗ σD + . t→+∞

α∈Perp(D − , D + , t)

There is a similar statement for nonbipartite simplicial graphs and for more general graphs of groups on which the discrete-time geodesic flow is mixing for the Gibbs measure; see the end of Chapter 11 and Section 12.4. Again, the results can then be interpreted in terms of nonbacktracking random walks. In Section 12.2, we deduce our counting results for common perpendiculars between the subsets D− and D+ from the above simultaneous equidistribution theorem. Theorem 1.5. (1) Let Y, F, D± be as in Theorem 1.4. Assume that the Gibbs measure mF is finite and mixing for the continuous-time geodesic flow and that the + − skinning measures σD − and σD + are finite and nonzero. Then, as s → +∞, ND− , D+ , F (s) ∼

+ − δF s kσD − k kσD + k e . kmF k δF

(2) If Y is a finite nonbipartite simplicial graph, then ND− , D+ ,F (n) ∼

+ − eδF kσD − k kσD + k eδF n . (eδF − 1) kmF k

Assertion (1) of Theorem 1.5 is valid when Y is a good orbifold instead of a manifold or a metric graph of finite groups instead of a metric graph (for the appropriate notion of multiplicities), and when D− and D+ are replaced by locally finite families. See Section 12.4 for generalisations of Assertion (2) of Theorem 1.5

1.2. The distribution of common perpendiculars

9

to (possibly infinite) simplicial graphs of finite groups and Sections 12.3 and 12.6 for error terms. We avoid any compactness assumption on Y ; we assume only that the Gibbs measure mF of F is finite and that it is mixing for the geodesic flow. By Babillot’s theorem [Bab1], if the length spectrum of Y is not contained in a discrete subgroup of R, then mF is mixing if finite. If Y is a Riemannian manifold, this condition is satisfied, for instance, if the limit set of a fundamental group of Y is not totally disconnected; see, for instance, [Dal1, Dal2]. When Y is a metric graph, Babillot’s mixing condition is in particular satisfied if the lengths of the edges of Y are rationally independent. As in [PaP17b], we have very weak finiteness and curvature assumptions on the space and the convex subsets we consider. Furthermore, the space Y is no longer required to be a manifold, and we extend the theory to nonconstant weights using equilibrium states. Such a weighted counting has been used only in the orbitcounting problem in manifolds with pinched negative curvature in [PauPS]. The approach is based on ideas from Margulis’ thesis to use the mixing of the geodesic flow. Our skinning measures are much more general than the Patterson measures appearing in earlier works. As in [PaP17b], we push simultaneously the unit normal vectors to the two convex sets D− and D+ in opposite directions. Classically, an important characterisation of the Bowen–Margulis measure on closed negatively curved Riemannian manifolds (F = 0) is that it coincides with the weak-star limit of properly normalised sums of Lebesgue measures supported on periodic orbits. The result was extended to CAT(−1) spaces with zero potential in [Rob2] and to Gibbs measures in the manifold case in [PauPS, Theo. 9.11]. As a corollary of the simultaneous equidistribution result Theorem 1.4, we obtain a weighted version for simplicial and metric graphs of groups. The following is a simplified version of such a result for Gibbs measures of metric graphs. Let Per0 (t) be the set of prime periodic orbits of the geodesic flow on Y . Let λ(g) denote the length of a closed orbit g. Let Lg be the Lebesgue measure along g and let Lg (F ) be the period of g for the potential F . Theorem 1.6. Assume that Y is a finite metric graph, that the critical exponent δF is positive, and that the Gibbs measure mF is mixing for the continuous-time geodesic flow. As t → +∞, the measures δF eδF t

X g∈Per0 (t)

converge to

mF kmF k

eLg (F ) Lg

and

δF t eδF t

X g∈Per0 (t)

eLg (F )

Lg λ(g)

for the weak-star convergence of measures.

See Section 13.2 for the proof of the full result and for a similar statement for (possibly infinite) simplicial graphs of finite groups. As a corollary, we obtain counting results of simple loops in metric and simplicial graphs, generalising results of [ParP], [Gui].

10


Corollary 1.7. Assume that Y is a finite metric graph whose vertices have degrees at least 3, such that the critical exponent δF is positive. (1) If the Gibbs measure is mixing for the continuous-time geodesic flow, then X g∈Per0 (t)

eLg (F ) ∼

eδF t δF t

as t → +∞. (2) If Y is simplicial and if the Gibbs measure is mixing for the discrete-time geodesic flow, then X eδF eδF t eLg (F ) ∼ δF e −1 t 0 g∈Per (t)

as t → +∞. When error bounds are known for the mixing property of the continuoustime or discrete-time geodesic flow on G Y , we obtain corresponding error terms in the equidistribution result of Theorem 1.2 generalising [PaP14a, Thm. 20] (where F = 0) and in the approximation of the counting function ND− , D+ , 0 by the expression introduced in Theorem 1.5. In the manifold case, see [KM1], [Clo], [Dol1], [Sto], [Live], [GLP], and Section 12.3 for definitions and precise references. Here is an example of such a result in the manifold case. Theorem 1.8. Assume that Y is a compact Riemannian manifold and mF is exponentially mixing under the geodesic flow for the H¨ older regularity, or that Y is a locally symmetric space, the boundary of D± is smooth, mF is finite, smooth, and exponentially mixing under the geodesic flow for the Sobolev regularity. Assume that the strong stable/unstable ball masses by the conditionals of mF are H¨ older-continuous in their radius. + − (1) As t tends to +∞, the pushforwards (gt )∗ σD by − of the skinning measure of D the geodesic flow equidistribute towards the Gibbs measure mF (after normalisation as probability measures) with exponential speed. + − (2) If the skinning measures σD − and σD + are finite and nonzero, there exists κ > 0 such that as t → +∞,

ND− , D+ , F (t) =

+ − kσD − k kσD + k eδF t 1 + O(e−κt ) . δF kmF k

See Section 12.3 for a discussion of the assumptions and the dependence of O(·) on the data. Similar (sometimes more precise) error estimates were known earlier for the counting function in special cases of D± in constant-curvature geometrically finite manifolds (often in small dimension) through the work of Huber, Selberg, Patterson, Lax and Phillips [LaxP], Cosentino [Cos], Kontorovich and Oh [KonO], Lee and Oh [LeO].

1.3. Counting in weighted graphs of groups

11

When Y is a finite-volume hyperbolic manifold and the potential F is constant 0, the Gibbs measure is proportional to the Liouville measure, and the skinning measures of totally geodesic submanifolds, balls, and horoballs are proportional to the induced Riemannian measures of the unit normal bundles of their boundaries. In this situation, there are very explicit forms of the counting results in finite-volume hyperbolic manifolds; see [PaP17b, Cor. 21], [PaP16]. These results are extended to complex (resp. quaternionic) hyperbolic space in [PaP17a] (resp. [PaP20]). As an example of this result, if D− and D+ are closed geodesics of Y of lengths `− and `+ , respectively, then the number N (s) = ND− , D+ , 0 (s) of common perpendiculars (counted with multiplicity) from D− to D+ of length at most s satisfies, as s → +∞, n

N (s) ∼

1.3

2 π 2 −1 Γ( n−1 `− `+ (n−1)s 2 ) e . n n−2 2 (n − 1)Γ( 2 ) Vol(Y )

(1.1)

Counting in weighted graphs of groups

From now on in this introduction, we consider only metric or simplicial graphs or graphs of groups. Let Y be a connected finite graph with set of vertices V Y and set of edges EY (see [Ser3] for the conventions). We assume that the degree of the graph at each vertex is at least 3. Let λ : EY → ]0, +∞[ with λ(e) = λ(e) for every e ∈ EY be an edge length map, let Y = |Y|λ be the geometric realisation of Y in which the geometric realisation of every edge e ∈ EY has length λ(e), and let c : EY → R be a map, called a (logarithmic) system of conductances in the analogy between graphs and electrical networks; see, for instance, [Zem]. Let Y± be proper nonempty subgraphs of Y. For every t > 0, we denote by Perp(Y− , Y+ , t) the set of edge paths α = (e1 , . . . , ek ) in Y without backtracking, Pk Pk of length λ(α) = i=1 λ(ei ) at most t, of conductance c(α) = i=1 c(ei ), starting from a vertex of Y− but not by an edge of Y− , ending at a vertex of Y+ but not by an edge of Y+ . Let NY− ,Y+ (t) =

X

ec(α)

α∈Perp(Y− ,Y+ , t)

be the number of paths without backtracking from Y− to Y+ of length at most t, counted with weights defined by the system of conductances. Recall that a real number x is Diophantine if it is badly approximable by rational numbers, that is, if there exist α, β > 0 such that |x − pq | > α q −β for all p, q ∈ Z with q > 0. We obtain the following result, which is a very simplified version of our results for the sake of this introduction.

12


Theorem 1.9. (1) If Y has two cycles whose ratio of lengths is Diophantine, then there exists C > 0 such that for every k ∈ N − {0}, as t → +∞, NY− ,Y+ (t) = C eδc t 1 + O(t−k ) . (2) If λ ≡ 1, then there exist C 0 , κ > 0 such that as n ∈ N tends to +∞, NY− ,Y+ (n) = C 0 eδc n 1 + O(e−κ n ) . Note that the Diophantine assumption on Y in Theorem 1.9 (1) is standard in the theory of quantum graphs (see, for instance, [BerK]). The constants C = CY± , c, λ > 0 and C 0 = CY0 ± , c > 0 in the above asymptotic formulas are explicit. When c ≡ 0 and λ ≡ 1, the constants can often be determined concretely, as indicated in the two examples below.14 Among the ingredients in these computations are the explicit expressions of the total mass of many Bowen– Margulis measures and skinning measures obtained in Chapter 8. See Sections 12.4, 12.5, and 12.6 for generalisations of Theorem 1.9 when the graphs Y± are not embedded in Y, and for versions in (possibly infinite) metric graphs of finite groups. In particular, Assertion (2) of Theorem 1.9 remains valid if Y is the quotient of a uniform simplicial tree by a geometrically finite lattice in the sense of [Pau4], such as (see[Lub1]) an arithmetic lattice in PGL2 over a non-Archimedean local field. Recall that a locally finite metric tree X is uniform if it admits a discrete and cocompact group of isometries, and that a lattice Γ of X is a lattice in the locally compact group of isometries of X preserving without edge inversions the simplicial structure. We refer, for instance, to [BasK, BasL] for uncountably many examples of tree lattices. Example 1.10. (1) When Y is a (q + 1)-regular finite graph with constant edge length map λ ≡ 1 and vanishing system of conductances c ≡ 0, then δc = ln q, and if furthermore Y+ and Y− are vertices, then (see Equation (12.11)) C0 =

q+1 . (q − 1) Card(V Y)

(2) When Y is biregular of degrees p + 1 and q + 1 with p, q > 2, when λ ≡ 1 and √ c ≡ 0, then δc = ln pq , and if furthermore the subgraphs Y± are simple cycles of lengths L± , then (see Equation (12.12)) the number of common perpendiculars of even length at most 2N from Y− to Y+ as N → +∞ is asymptotic to (p + q) L− L+ (pq)N +1 . 2 (pq − 1) Card(EY) 14 See

Section 12.4 for more examples.

1.3. Counting in weighted graphs of groups

13

The main tool in order to obtain the error terms in Theorem 1.9 and its more general versions is to study the error terms in the mixing property of the geodesic flow. Using the already mentioned coding (given in Section 5.2) of the discretetime geodesic flow by a two-sided topological Markov shift, classical reduction to one-sided topological Markov shift, and results of Young [You1] on the decay of correlations for Young towers with exponentially small tails, we in particular obtain the following simple criteria for the exponential decay of correlation of the discrete-time geodesic flow, where we assume Y to be only locally finite (and maybe not finite). See Theorem 9.1 for the complete result. Theorem 1.11. Assume that the Gibbs measure mF is finite and mixing for the discrete-time geodesic flow on Y. Assume, moreover, that there exist a finite subset E of V Y and C 0 , κ0 > 0 such that for all n ∈ N, we have 0 mF {` ∈ G Y : `(0) ∈ E and ∀ k ∈ {1, . . . , n}, `(k) ∈ / E} 6 C 0 e−κ n . Then the discrete-time geodesic flow has exponential decay of H¨ older correlations for mF . The assumption of having exponentially small mass of geodesic lines that have a big return time to a given finite subset of V Y is in particular satisfied (see Section 9.2) if Y is the quotient of a uniform simplicial tree by a geometrically finite lattice,15 such as an arithmetic lattice in PGL2 over a non-Archimedean local field, see [Lub1], but also by many other examples of Y. This statement corrects the mistake in [Kwo], as indicated in its erratum. These results allow us to prove in Section 9.3, under Diophantine assumptions, the rapid mixing property for the continuous-time geodesic flow, which leads to Assertion (1) of Theorem 1.9, see Section 12.6. The proof uses suspension techniques due to Dolgopyat [Dol2] when Y is a compact metric tree, and to Melbourne [Mel1] otherwise. As a corollary of the general version of the counting result Theorem 1.5, we have the following asymptotic for the orbital counting function in conjugacy classes for groups acting on trees. Let X be a metric tree. Given x0 ∈ X and a nontrivial conjugacy class K in a discrete group Γ of isometries of X, we consider the counting function NK, x0 (t) = Card{γ ∈ K : d(x0 , γx0 ) 6 t} , introduced by Huber [Hub1] when X is replaced by the real hyperbolic plane and Γ is a lattice. We refer to [PaP15] for many results on the asymptotic growth of such orbital counting functions in conjugacy classes when X is replaced by a finitely generated group with a word metric or a complete simply connected pinched negatively curved Riemannian manifold. See also [ChaP, ArCT, Pol3]. 15 See,

for instance, [Pau4].

14


Theorem 1.12. Let X be a uniform metric tree with vertices of degree > 3, let δ be the Hausdorff dimension of its space of ends, let Γ be a discrete group of isometries of X, let x0 be a vertex of X with trivial stabiliser in Γ, and let K be the conjugacy class in Γ of a loxodromic element γ0 in Γ with translation length λ(γ0 ) in X. (1) If the metric graph Γ\X is compact and has two cycles whose ratio of lengths is Diophantine, then there exists C > 0 such that for every k ∈ N − {0}, as t → +∞, δ NK, x0 (t) = C e 2 t 1 + O(t−k ) . (2) If X is simplicial nonbipartite and Γ is a geometrically finite lattice of X, then there exist C 0 , κ > 0 such that as n ∈ N tends to +∞, NK, x0 (n) = C 0 eδ b(n−λ(γ0 ))/2c 1 + O(e−κ n ) . We refer to Theorem 13.1 for a more general version, including a version with a system of conductances in the counting function, and when K is elliptic. When Γ\X is compact and Γ is torsion free,16 Assertion (1) of this result is due to Kenison and Sharp [KeS], who proved it using transfer operator techniques for suspensions of subshifts of finite type. Up to strengthening the Diophantine assumption, using work of Melbourne [Mel1] on the decay of correlations of suspensions of Young towers, we are able to extend Assertion (1) to all geometrically finite lattices Γ of X in Section 13.1. 0 The constants C = CK,x0 and C 0 = CK,x are explicit. For instance, in 0 Assertion (2), if X is the geometric realisation of a regular simplicial tree X of degree q + 1, if x0 is a vertex of X, if Vol(Γ\\X) =

X [x]∈Γ\V X

1 |Γx |

is the volume17 of the quotient graph of groups Γ\\X, then C0 =

λ(γ0 ) , [ZΓ (γ0 ) : γ0Z ] Vol(Γ\\X)

where ZΓ (γ0 ) is the centraliser of γ0 in Γ. When furthermore Γ is torsion free, γ0 is not a proper power and Γ\X is finite, since δ = ln q, we get that there exists κ0 > 0 such that NK, x0 (n) =

0 λ(γ0 ) q b(n−λ(γ0 ))/2c + O(q (1−κ ) n/2 ) Card(Γ\X)

as n ∈ N tends to +∞, thus recovering the result of [Dou], who used the spectral theory of the discrete Laplacian. 16 In

particular, Γ then has the very restricted structure of a free group. for instance, [BasK, BasL].

17 See,

1.4. Selected arithmetic applications

1.4

15

Selected arithmetic applications

We end this introduction by giving a sample of our arithmetic applications (see Part III of this book) of the ergodic and dynamical results on the discrete-time geodesic flow on simplicial trees described in Part II of this book, as summarised above. Our equidistribution and counting results of common perpendiculars between subtrees indeed produce equidistribution and counting results of rationals and quadratic irrationals in non-Archimedean local fields. We refer to [BrPP] for an announcement of the results of Part III, with a presentation different from the one in this introduction. To motivate what follows, consider R = Z the ring of integers, K = Q its b = R the completion of Q for the usual Archimedean absolute field of fractions, K value | · |, and HaarKb the Lebesgue measure of R (which is the Haar measure of the additive topological group R normalised so that HaarKb ([0, 1]) = 1). The following equidistribution result of rationals, due to Neville [Nev], is a b for the weak-star convergence quantitative statement on the density of K in K: b as s → +∞, we have of measures on K, X π −2 lim s ∆ pq = HaarKb . s→+∞ 6 p,q∈R : pR+qR=R, |q|6s

b →C Furthermore, there exists ` ∈ N such that for every smooth function ψ : K with compact support, there is an error term in the above equidistribution claim evaluated on ψ of the form O(s(ln s)kψk` ), where kψk` is the Sobolev norm of ψ. The following counting result due to Mertens on the asymptotic behaviour of the average of Euler’s totient function ϕ : k 7→ Card(R/kR)× follows from the above equidistribution result: n X k=1

ϕ(k) =

3 2 n + O(n ln n) . π

See [PaP14b] for an approach using methods similar to those in this text, and, for instance, [HaW, Thm. 330] for a more traditional proof, as well as [Walf] for a better error term. Let us now switch to a non-Archimedean setting, restricting to positive characteristic in this introduction. See Part III for analogous applications in characteristic zero. Let Fq be a finite field of order q. Let R = Fq [Y ] be the ring of polynomials in one variable Y with coefficients in Fq . Let K = Fq (Y ) be the field of rational fractions in Y with coefficients in Fq , which is the field of fractions of R. Let b = Fq ((Y −1 )) be the field of formal Laurent series in the variable Y −1 with K coefficients in Fq , which is the completion of K for the (ultrametric) absolute

16


P value | Q | = q deg P −deg Q . Let O = Fq [[Y −1 ]] be the ring of formal power series in b for this Y −1 with coefficients in Fq , which is the ball of center 0 and radius 1 in K absolute value. b is locally compact, and we endow the additive group K b with Note that K the Haar measure HaarKb normalised so that HaarKb (O) = 1. The following results extend (with appropriate constants) when K is replaced by any function field of b any completion of K; see Part III. a nonsingular projective curve over Fq and K 18 b gives an anaThe following equidistribution result of elements of K in K logue of Neville’s equidistribution result for function fields. Note that when G = GL2 (R), we have (P, Q) ∈ G(1, 0) if and only if hP, Qi = R. We denote by Hx the stabiliser of any element x of any set endowed with any action of any group H.

Theorem 1.13. Let G be any finite-index subgroup of GL2 (R). For the weak-star b we have convergence of measures on K, lim

t→+∞

(q + 1) [GL2 (R) : G] q −2 t (q − 1) q 2 [GL2 (R)(1,0) : G(1,0) ]

X

∆ P = HaarKb . Q

(P,Q)∈G(1,0), deg Q6t

We emphasise the fact that we are not assuming G to be a congruence subgroup of GL2 (R). This is made possible by our geometric and ergodic methods. The following variation of this result is more interesting when the class number of the function field K is larger than 1 (see Corollary 16.7 in Chapter 16). Theorem 1.14. Let m be a nonzero fractional ideal of R with norm N(m). For the b we have weak-star convergence of measures on K, lim

t→+∞

q+1 s−2 (q − 1) q 2

X

∆ xy = HaarKb .

(x,y)∈m×m N(m)−1 N(y)6s, Rx+Ry=m

b is a quadratic irrational over K,19 let ασ be the Galois conjugate If α ∈ K of α,20 let tr(α) = α + ασ and n(α) = αασ , and let h(α) =

1 . |α − ασ |

This is an appropriate complexity for quadratic irrationals in a given orbit by homographies under PGL2 (R). See Section 17.2 and, for instance, [HeP4, §6] for motivations and results. Note that although there are only finitely many orbits by homographies of PGL2 (R) on K (and exactly one in the particular case of this 18 See

Theorem 16.4 in Chapter 16 for a more general version. means that α does not belong to K and satisfies a quadratic equation with coefficients

19 This

in K. 20 This means that ασ is the other root in K b of the irreducible quadratic polynomial over K defining α.

1.4. Selected arithmetic applications

17

introduction), there are infinitely many orbits of PGL2 (R) in the set of quadratic b over K. The following result gives in particular that every orbit irrationals in K b as the complexity of quadratic irrationals under PGL2 (R) equidistributes in K tends to infinity. See Theorem 17.6 in Section 17.2 for a more general version. We b on P1 (K) b =K b ∪ {∞ = [1 : 0]}. denote by · the action by homographies of GL2 (K) b be a Theorem 1.15. Let G be a finite-index subgroup of GL2 (R). Let α0 ∈ K b we quadratic irrational over K. For the weak-star convergence of measures on K, have X (ln q) (q + 1) m0 [GL2 (R) : G] −1 lim s ∆α = HaarKb , 2 3 s→+∞ 2 q (q − 1) ln | tr g0 | α∈G·α0 , h(α)6s

where g0 ∈ G fixes α0 with | tr g0 | > 1, and m0 is the index of g0Z in Gα0 . Another equidistribution result of an orbit of quadratic irrationals under PGL2 (R) is obtained by taking another complexity, constructed using cross-ratios with a fixed quadratic irrational. We denote by [a, b, c, d] = (c−a)(d−b) (c−b)(d−a) the crossb If α, β ∈ K b are two quadratic irrationals over ratio of four distinct elements in K. K such that α ∈ / {β, β σ },21 let hβ (α) = max{|[α, β, β σ , ασ ]|, |[ασ , β, β σ , α]|} , which is also an appropriate complexity when α varies in a given orbit of quadratic irrationals by homographies under PGL2 (R). See Section 18.1 and, for instance, [PaP14b, §4] for motivations and results in the Archimedean case. b be Theorem 1.16. Let G be a finite-index subgroup of GL2 (R). Let α0 , β ∈ K two quadratic irrationals over K. For the weak-star convergence of measures on b − {β, β σ }, we have, with g0 and m0 as in the statement of Theorem 1.15, K lim

s→+∞

(ln q) (q + 1) m0 [GL2 (R) : G] −1 s 2 q 2 (q − 1)3 |β − β σ | ln | tr g0 | =

X

∆α

α∈G·α0 , hβ (α)6s

d HaarKb (z) . |z − β| |z − β σ |

The fact that the measure towards which we have an equidistribution is only absolutely continuous with respect to the Haar measure is explained by the invariance of α 7→ hβ (α) under the (infinite) stabiliser of β in PGL2 (R). See Theorem 18.4 in Section 18.1 for a more general version. The last statement of this introduction is an equidistribution result for the integral representations of quadratic norm forms (x, y) 7→ n(x − yα) 21 See

Section 18.1 when this condition is not satisfied.

18


b is a quadratic irrational over K. See Theorem 19.1 on K × K, where α ∈ K in Section 19 for a more general version, and, for instance, [PaP14b, §5.3] for motivations and results in the Archimedean case. There is an extensive bibliography on the integral representation of norm forms and more generally decomposable forms over function fields; see, for instance, [Sch1, Maso1, Gyo, Maso2]. These references mostly consider higher degrees, with an algebraically closed ground field of characteristic 0, instead of Fq . b be a Theorem 1.17. Let G be a finite-index subgroup of GL2 (R) and let β ∈ K quadratic irrational over K. For the weak-star convergence of measures on the b − {β, β σ }, we have space K lim

s→+∞ q 2

(q + 1) [GL2 (Rv ) : G] s−1 (q − 1) [GL2 (Rv )(1,0) : G(1,0) ]

=

X

∆ xy

(x,y)∈G(1,0), |x2 −xy tr(β)+y 2 n(β)|6s

d HaarKb (z) . |z − β| |z − β σ |

Furthermore, we have error estimates in the arithmetic applications: there exists κ > 0 such that for every locally constant function with compact support b → C in Theorems 1.13, 1.14, and 1.15, or ψ : K b − {β, β σ } → C in Theorems ψ:K 1.16 and 1.17, there are error terms in the above equidistribution claims evaluated on ψ of the form O(s−κ ), where s = q t in Theorem 1.13, with for each result an explicit control on the test function ψ involving only some norm of ψ, see in particular Section 15.4. The link between the geometry described in the first part of this introduction and the above arithmetic statements is provided by the Bruhat–Tits tree of b see [Ser3] and Section 15.1 for background. We refer to Part III for (PGL2 , K), more general arithmetic applications. Acknowledgment. This work was partially supported by NSF grants no 093207800 and DMS-1440140, while the third author was in residence at MSRI, Berkeley CA, during the spring 2015 and fall 2016 semesters. The second author thanks Université Paris-Sud, Forschungsinstitut f¨ ur Mathematik of ETH Z¨ urich, and Vilho, Yrj¨ o ja Kalle V¨ ais¨ alän rahasto for their support during the preparation of this work. This research was supported by the CNRS PICS n0 6950 “Equidistribution et comptage en courbure négative et applications arithmétiques.” We thank, for interesting discussions on this text, Y. Benoist, J. Buzzi (for his help in Sections 5.2 and 9.2, and for kindly agreeing to insert the Appendix used in Section 5.4), N. Curien, S. Mozes, M. Pollicott (for his help in Section 9.3), R. Sharp, and J.-B. Bost (for his help in Sections 14.2 and 16.1). We especially thank O. Sarig for his help in Section 9.2: in a long email, he explained to us how to prove Theorem 9.2.

1.5. General notation

1.5

19

General notation

In this preamble, we introduce some general notation that will be used throughout the book. We recommend the use of the List of Symbols (mostly in alphabetical order by the first letter) and the Index for easy references to the definitions in the text. Let A be a subset of a set E. We denote by 1A : E → {0, 1} the characteristic (or indicator) function of A: 1A (x) = 1 if x ∈ A, and 1A (x) = 0 otherwise. We denote by c A = E − A the complementary subset of A in E. We denote by bxc = sup{n ∈ N : n 6 x} the lower integral part of any x ∈ R and by dxe = inf{n ∈ N : x 6 n} its upper integral part. We denote by ln the natural logarithm (with ln(e) = 1). We denote by Card(E) or by |E| the order of a finite set E. We denote by kµk the total mass of a finite positive measure µ. If (X, A ) and (Y, B) are measurable spaces, f : X → Y a measurable map, and µ a measure on X, we denote by f∗ µ the image measure of µ by f , with f∗ µ(B) = µ(f −1 (B)) for every B ∈ B. If (X, d) is a metric space, then B(x, r) is the closed ball with center x ∈ X and radius r > 0. For every subset A of a metric space and for every > 0, we denote by N A the closed -neighbourhood of A, and by convention N0 A = A. We denote by N− A the set of points of A at distance at least from the complement of A. Given a topological space Z, we denote by Cc (Z) the vector space of continuous maps from Z to R with compact support. ∗

Given a locally compact topological space Z, we denote by * the weak-star ∗ convergence of (Borel, positive) measures on Z: we have µn * µ if and only if limn→+∞ µn (f ) = µ(f ) for every f ∈ Cc (Z). The negative part of a real-valued map f is f − = max{0, −f }. We denote by ∆x the unit Dirac mass at a point x in any measurable space. Finally, the symbol right at the end of a statement indicates that this statement will not be given a proof, either since a reference is given or since it is an immediate consequence of previous statements.

Part I Geometry and Dynamics in Negative Curvature

Chapter 2

Negatively Curved Geometry 2.1

Background on CAT(−1) spaces

Let X be a geodesically complete proper CAT(−1) space, let x0 ∈ X be an arbitrary basepoint, and let Γ be a nonelementary discrete group of isometries of X. We refer, for example, to [BridH] for the relevant terminology, proofs, and complements on these notions. In this section, we recall some definitions and notation for the sake of completeness. A metric space is proper if its closed balls are compact. A geodesic in a metric space X 0 is an isometric map c from an interval I of R into X 0 .1 A metric space X 0 is geodesic if for all x, y ∈ X 0 , there exists a geodesic segment c : [a, b] → X 0 from x = c(a) to y = c(b). A geodesic metric space X 0 is geodesically complete (or has extendible geodesics) if every isometric map from an interval in R to X 0 extends to at least one isometric map from R to X 0 . A comparison triangle of a triple of points (x, y, z) in a metric space X 0 is a (unique up to isometry) triple of points (x, y, z) in the real hyperbolic plane H2R (with constant sectional curvature −1) such that d(x, y) = d(x, y), d(y, z) = d(y, z), and d(z, x) = d(z, x). y

y p

p x

x q

q z

z X0

H2R

1 We

say that c is a geodesic segment if I is compact, a geodesic ray if I is a half-infinite interval, and a geodesic line if I = R.


23

24

Chapter 2. Negatively Curved Geometry

A metric space X 0 is CAT(−1) if it is geodesic and if for every triple of points (x, y, z) in X 0 , for all geodesic segments a, b respectively from x to y and from x to z, and for all points p, q in the image of a, b respectively, if (x, y, z) is a comparison triangle of (x, y, z), if p (resp. q) is the point on the geodesic segment from x to y (resp. z) at distance d(x, p) (resp. d(x, q)) from x, then d(p, q) 6 d(p, q). We will put a special emphasis on the case that X is a (proper, geodesically complete) R-tree, that is, a uniquely arcwise connected geodesic metric space. In the introduction, we denoted by Y the geodesically complete proper locally CAT(−1) good orbispace Γ\X; see, for instance, [GH, Ch. 11] for the terminology. Two geodesic rays ρ, ρ0 : [0, +∞[ → X are asymptotic if their images are at finite Hausdorff distance, or equivalently if there exists a ∈ R such that limt→+∞ d(ρ(t), ρ0 (t + a)) = 0. We denote by ∂∞ X the space at infinity of X, which consists of the asymptotic classes of geodesic rays in X, and we endow it with the quotient topology of the compact-open topology. It coincides with the space of (Freudenthal’s) ends of X when X is an R-tree. Let x, y ∈ X and ξ, ξ 0 ∈ ∂∞ X. We denote by [x, y] = [y, x] the unique image of a geodesic segment from x to y. We denote by [x, ξ[ the image of the unique geodesic ray ρ : [0, +∞[ → X in the asymptotic class ξ with ρ(0) = x, and we say that ρ starts from x and ends at ξ. We denote by ]ξ, ξ 0 [ = ]ξ 0 , ξ[ the unique image of a geodesic line ` : R → X with t 7→ `(t) and t 7→ `(−t) in the asymptotic classes ξ and ξ 0 respectively, and we say that ` starts from ξ 0 and ends at ξ. We endow the disjoint union X ∪ ∂∞ X with the unique metrisable compact topology (independent of the basepoint x0 ) inducing the above topologies on X and ∂∞ X, such that a sequence (yi )i∈N in X converges to ξ ∈ ∂∞ X if and only if limi→+∞ d(x0 , yi ) = +∞ and, with ci : [0, d(x0 , yi )] → X the geodesic segment from x0 to yi and ρ : [0, +∞[ → X the geodesic ray from x0 to ξ, we have limi→+∞ d(ci (t), ρ(t)) = 0 for every t > 0. We denote by Isom(X) the isometry group of X, and we endow it with the compact-open topology. Its action on X uniquely extends to a continuous action on X ∪ ∂∞ X. We say that a discrete subgroup Γ0 of Isom(X) is nonelementary if it does not fix a point or an unordered pair of points in X ∪ ∂∞ X. We denote by ΛΓ the limit set of Γ, which is the set of accumulation points in ∂∞ X of any orbit of Γ in X. It is the smallest closed nonempty Γ-invariant subset of ∂∞ X. A subset D of X ∪ ∂∞ X is convex if for all u, v ∈ D, the image of the unique geodesic segment, ray or line from u to v is contained in D. We denote by C ΛΓ the convex hull in X of ΛΓ, which is the intersection of the closed convex subsets of X ∪ ∂∞ X containing ΛΓ. When X is an R-tree, then a subset D of X is convex if and only if it is connected, and we will call it a subtree. In particular, if X is an R-tree, then C ΛΓ is equal to the union of the geodesic lines between pairs of distinct points in ΛΓ, since this union is connected and contained in C ΛΓ. A point ξ ∈ ∂∞ X is called a conical limit point of Γ if there exists a sequence of orbit points of x0 under Γ converging to ξ while staying at bounded distance

2.1. Background on CAT( 1) spaces

25

from a geodesic ray ending at ξ. The set of conical limit points of Γ is the conical limit set Λc Γ of Γ. A point p ∈ ΛΓ is a bounded parabolic limit point of Γ if its stabiliser Γp in Γ acts properly discontinuously with compact quotient on ΛΓ − {p}. The discrete nonelementary group of isometries Γ of X is said to be geometrically finite if every element of ΛΓ is either a conical limit point or a bounded parabolic limit point of Γ. See, for instance, [Bowd], as well as [Pau4] when X is an R-tree, and [DaSU] for a very interesting study of equivalent conditions in even greater generality. For all x ∈ X ∪ ∂∞ X and A ⊂ X, the shadow of A seen from x is the subset Ox A of ∂∞ X consisting of the endpoints towards +∞ of the geodesic rays starting at x and meeting A if x ∈ X, and of the geodesic lines starting at x and meeting A if x ∈ ∂∞ X. The translation length of an isometry γ ∈ Isom(X) is λ(γ) = inf d(x, γx) . x∈X

An element γ ∈ Isom(X) is elliptic if it fixes a point in X, and then λ(γ) = 0. An element γ ∈ Isom(X) is parabolic if it is not elliptic and fixes a unique point in ∂∞ X, and then λ(γ) = 0. An element γ ∈ Isom(X) is loxodromic if λ(γ) > 0, and then Axγ = {x ∈ X : d(x, γx) = λ(γ)} is (the image of) a geodesic line in X, called the translation axis of γ. In particular, the restriction of γ to Axγ is conjugated, by any isometry between Axγ and R, to a translation of R of the form t 7→ t ± λ(γ). For all β ∈ Isom(X) and n ∈ Z − {0}, we have Axβγβ −1 = β Axγ ,

λ(βγβ −1 ) = λ(γ), Axγ n = Axγ ,

λ(γ n ) = |n|λ(γ) . (2.1)

A loxodromic element γ ∈ Isom(X) has exactly two fixed points γ− , γ+ in the space X ∪ ∂∞ X, with γ− ∈ ∂∞ X its repulsive fixed point2 and γ+ ∈ ∂∞ X its attractive fixed point.3 We will need the following well-known lemma later on. An element of Γ is primitive in Γ if there exist no γ0 ∈ Γ and k ∈ N − {0, 1} such that γ = γ0k . Note that there might exist a primitive loxodromic element γ in Γ, whose translation length is not minimal among the translation lengths of the loxodromic elements γ 0 ∈ Γ with Axγ = Axγ 0 .4 Lemma 2.1. (1) For every loxodromic element γ ∈ Γ, there exist k 0 ∈ N − {0} 0 and a primitive loxodromic element γ1 ∈ Γ such that γ = γ1k , and there exist every x ∈ X ∪ (∂∞ X − {γ+ }), we have limn→+∞ γ −n x = γ− . every x ∈ X ∪ (∂∞ X − {γ− }), we have limn→+∞ γ +n x = γ+ . 4 For instance, if X is the real hyperbolic plane H2 and if Γ contains an orientation-preserving R loxodromic element γ such that the stabiliser in Γ of Axγ is generated by γ and by the symmetry n σ with respect to Axγ , then γ 2 σ is primitive for all n ∈ Z. 2 For 3 For

26


k ∈ N − {0}, a primitive loxodromic element γ0 ∈ Γ whose translation length is minimal among the translation lengths of the loxodromic elements γ 0 ∈ Γ with Axγ = Axγ 0 , and an element γ 0 ∈ Γ pointwise fixing Axγ such that γ = γ0k γ 0 . (2) For every compact subset K of X, for all A > 0 and r > 0, there exists L > 0 such that for all loxodromic elements γ, α ∈ Γ, if Axγ meets K, if λ(α) = λ(γ) 6 A, and if Axγ and Axα have segments of length at least L at Hausdorff distance at most r, then Axα = Axγ . (3) For every compact subset K of X and for every A > 0, there exists N ∈ N such that for every loxodromic element γ ∈ Γ whose translation axis meets K, the cardinality of the set of loxodromic elements α ∈ Γ with Axα = Axγ and λ(α) 6 A is at most N . Proof. (1) If γ ∈ Γ is loxodromic, then the group of restrictions to Axγ of the elements of Γ preserving Axγ is conjugated, by any isometry between Axγ and R, to a discrete group of isometries Λ of R. Since replacing γ by an element of Γ having a power at least 2 equal to γ strictly decreases the translation length, then by Equation (2.1), the first claim of (1) holds. The normal subgroup of Λ consisting of translations is isomorphic to Z, generated by the conjugate of the restriction to Axγ of an element γ0 ∈ Γ. Any such element has minimal translation length, hence is primitive, since if there exist γ1 ∈ Γ and n ∈ N − {0, 1} with γ0 = γ1n , by Equation (2.1) we would have Axγ1 = Axγ0 and λ(γ1 ) = n1 λ(γ0 ) < λ(γ0 ). There exists k ∈ Z − {0} such that the restrictions of γ and γ0k to Axγ coincide. Hence γ 0 = γ0−k γ pointwise fixes Axγ , and up to replacing γ0 by its inverse, Assertion (1) of Lemma 2.1 holds. (2) Since the action of Γ on X is properly discontinuous, there exists N = N (K, A, r) > 1 such that for every loxodromic element γ ∈ Γ whose translation axis meets K and whose translation length is at most A, for every x ∈ Axγ , the cardinality of {β ∈ Γ : d(x, βx) 6 2r} is at most N . Let L = AN . For every loxodromic element α ∈ Γ with λ(α) = λ(γ) 6 A, assume that [x, y] and [x0 , y 0 ] are segments in Axγ and Axα respectively, with length exactly L such that d(x, x0 ), d(y, y 0 ) 6 r. We may assume, up to replacing γ and α by their inverses, that γ translates from x towards y and α translates from x0 towards y 0 . In particular, for k = 0, . . . , N , we have d(α−k γ k x, x) 6 d(γ k x, αk x0 ) + d(x0 , x) 6 2r, since γ k x and αk x0 are respectively the points at distance kλ(γ) 6 kA 6 L from x and x0 on the segments [x, y] and [x0 , y 0 ]. Hence by the definition of N , there 0 0 0 0 exists k 6= k 0 such that α−k γ k = α−k γ k . Therefore γ k−k = αk−k , which implies by Equation (2.1) that Axγ = Axα . (3) Since the action of Γ on X is properly discontinuous, there exists N 0 ∈ N such that the cardinality of the stabiliser in Γ of a point of K is at most N 0 , and there exists η > 0 such that λ(γ) > η for every loxodromic element γ ∈ Γ whose translation axis meets K. Let us fix such an element γ ∈ Γ. We may assume that its translation length is minimal among the translation lengths of the loxodromic

2.1. Background on CAT( 1) spaces

27

elements γ 0 ∈ Γ with Axγ = Axγ 0 . Then as seen in the proof of Assertion (1), for every loxodromic element α ∈ Γ with Axα = Axγ , there exist k ∈ Z − {0} and α0 ∈ Γ pointwise fixing Axγ such that α = γ k α0 . Thus if λ(α) 6 A, then A A 0 |k| = λ(α) λ(γ) 6 η , and there are at most N = N (2d η e + 1) such elements α. For every x ∈ X, recall that the Gromov–Bourdon visual distance dx on ∂∞ X seen from x (see [Bou]) is defined by 1

dx (ξ, η) = lim e 2 (d(ξt , ηt )−d(x, ξt )−d(x, ηt )) , t→+∞

(2.2)

where ξ, η ∈ ∂∞ X and t 7→ ξt , ηt are any geodesic rays ending at ξ, η respectively. If X is an R-tree, if ξ, η ∈ ∂∞ X are distinct, if p ∈ X is such that [x, p] = [x, ξ[∩[x, η[, then dx (ξ, η) = e−d(x, p) .

(2.3)

For all x ∈ X, ξ, η ∈ ∂∞ X, and γ ∈ Isom(X), we have dγx (γξ, γη) = dx (ξ, η) . By the triangle inequality, for all x, y ∈ X and ξ, η ∈ ∂∞ X, we have e−d(x, y) 6

dx (ξ, η) 6 ed(x, y) . dy (ξ, η)

(2.4)

In particular, the identity map from (∂∞ X, dx ) to (∂∞ X, dy ) is a bilipschitz homeomorphism. Under our assumptions, (∂∞ X, dx0 ) is hence a compact metric space, on which Isom(X) acts by bilipschitz homeomorphisms. The following well-known result compares shadows of balls to balls for the visual distance. Lemma 2.2. For every geodesic ray ρ in X starting from x ∈ X and ending at ξ ∈ ∂∞ X, for all R > 0 and t ∈ ]R, +∞[ , we have Bdx (ξ, R e−t ) ⊂ Ox B(ρ(t), R) ⊂ Bdx (ξ, eR e−t ) . Proof. In order to prove the left inclusion, we adapt the proof of the left inclusion in [HeP2, Lem. 3.1] (which uses only the CAT(−1) property). Let us fix a point ξ 0 ∈ Bdx (ξ, R e−t ) − {ξ}, let ρ0 be the geodesic ray from x to ξ 0 , and let p be the closest point to w = ρ(t) on the image of ρ0 . For every s > t, let y = ρ(s) and z = ρ0 (s). Let (x, y, z) be a comparison triangle in H2R of (x, y, z), and let θ ∈ [0, π] be its angle at x. Let w be the point on [x, y] at distance t from x, and let p be its orthogonal projection on [x, z]. By the CAT(−1) property, we have d(w, p) 6 d(w, p) .

28


z = ρ0 (s) p x

θ w = ρ(t) H2R

y = ρ(s)

By the hyperbolic sine rule for right-angled triangles in H2R , we have sin θ =

sinh d(w, p) and sinh t

sin

sinh 12 d(y, z) sinh 12 d(y, z) θ = = . 2 sinh d(x, y) sinh s

Hence d(w, p) 6 d(w, p) 6

ed(x,w) 1 θ sinh d(w, p) = et sin θ 6 et sin . 2 sinh d(x, w) 2 2

Since lim sin

s→+∞

1 θ = lim e 2 d(y,z)−s = dx (ξ, ξ 0 ) 6 R e−t , 2 s→+∞

we hence have d(w, p) 6 R, so that ξ 0 ∈ Ox B(w, R), as wanted. In order to prove the right inclusion in Lemma 2.2, let ξ 0 ∈ Ox B(ρ(t), R) and let ρ be the geodesic ray from x to ξ 0 . The closest point p to ρ(t) on the image of ρ0 satisfies d(p, ρ(t)) 6 R, hence d(x, p) > t − R. Therefore, for t0 large enough, d(ρ0 (t0 ), p) 6 t0 − (t − R), and 0

1

dx (ξ 0 , ξ) 6 lim sup e 2

d(ρ(t0 ), ρ(t))+d(ρ(t), p)+d(p, ρ0 (t0 )) −t0

t0 →+∞

1

6 0 lim e 2

(t0 −t)+R+(t0 −t+R) −t0

t →+∞

= eR e−t .

Therefore ξ 0 ∈ Bdx (ξ, eR e−t ), as wanted.

The Busemann cocycle of X is the map β : ∂∞ X × X × X → R defined by (ξ, x, y) 7→ βξ (x, y) = lim d(ξt , x) − d(ξt , y) , t→+∞

where t 7→ ξt is any geodesic ray ending at ξ. If X is an R-tree, if p ∈ X is such that [x, ξ[ ∩ [y, ξ[ = [p, ξ[ , then βξ (x, y) = d(x, p) − d(y, p) .

(2.5)

2.2. Generalised geodesic lines

29

The triangle inequality gives immediately the upper bound | βξ (x, y) | 6 d(x, y) .

(2.6)

The horosphere with center ξ ∈ ∂∞ X through x ∈ X is {y ∈ X : βξ (x, y) = 0}, and {y ∈ X : βξ (x, y) 6 0} is the (closed) horoball centered at ξ bounded by this horosphere. Horoballs are nonempty proper closed (strictly) convex subsets of X. Given a horoball H and t > 0, we denote by H [t] = {x ∈ H : d(x, ∂H ) > t} the horoball contained in H (hence centered at the same point at infinity as H ) whose boundary is at distance t from the boundary of H .

2.2

Generalised geodesic lines b

Let G X be the space of 1-Lipschitz maps w : R → X that are isometric on a closed interval and locally constant outside it.5 This space was introduced by Bartels and b in [BartL], to which we refer for the following basic properties. The elements L¨ uck b of X. Any geodesic segment or ray of G X are called the generalised geodesic lines of X will be considered as an element of G X, by extending it continuously to R b outside its domain of definition. as locally constant b We endow G X with the distance d = dG X defined by b Z +∞ ∀ w, w0 ∈ G X,

d(w, w0 ) =

d(w(t), w0 (t)) e−2|t| dt .

(2.7)

−∞

b

The group Isom(X) acts isometrically on G X by postcomposition. The bdistance db induces the topology of uniform convergence on compact subsets on G X, and b G X is a proper metric space. t b The geodesic flow (g )t∈R on G X is the one-parameter group bof homeomorphisms of the space G X defined by gt w : s 7→ w(s + t) for all w ∈ G X and t ∈ R. It commutes with the action of Isom(X). If w is isometric exactly on the interval I, then g−t w is isometric exactly on the interval t + I. The footpoint projection is the Isom(X)-equivariant 21 -H¨ b b older-continuous6 b G X. bThe antipodal map π :b G X → X defined by π(w) = w(0) for all w ∈ b map of G X is the Isom(X)-equivariant isometric map ι : G X → G X defined by ιw : s 7→ w(−s) for all w ∈ G X, which satisfies ι ◦ gt = g−t ◦ ι for every t ∈ R and π ◦ ι = π.

5 That

is, constant on each complementary component. Section 3.1 for the definition of the (locally uniform) H¨ older continuity used in this book, and Proposition 3.2 for a proof of this claim. 6 See

30


b

The positive and negative endpoint maps are the continuous maps from G X to X ∪ ∂∞ X defined by w 7→ w± = lim w(t) . t→±∞

b G X of geodesic lines in X is the b Isom(X)-invariant closed metric The space subspace of G X consisting of the elements ` ∈ G X with `± ∈ ∂∞ X. Note that the distances on G X considered in [BartL] and [PauPS] are topologically equivalent to, although slightly different from, the restriction to G X of the distance defined in Equation (2.7). The factor e−2|t| in this equation, sufficient for dealing with Hölder 2 √ continuity issues, is replacedb by e−t / π in [PauPS] and by e−|t| /2 in [BartL]. Note that for all w ∈ G X and s ∈ R, we have d(w, gs w) 6 |s| ,

(2.8)

with equality if w ∈ G X. We will also consider the Isom(X)-invariant closed subspaces b G± X = {w ∈ G X : w± ∈ ∂∞ X} ,

and their Isom(X)-invariant closed subspaces G±, 0 X consisting of the elements ρ ∈ G± X that are isometric exactly on ±[0, +∞[. The subspaces G X and G± X satisfy G− X ∩ G+ X = G X and they are invarib maps G± bX ant under the geodesic flow. The antipodal map ι preserves G X, and b G∓, 0 X. bWe denote again by ι : Γ\ G X → Γ\ G X to G∓ X as well as G±, 0 X to t t and by (g )t∈R with g : Γ\ G X → Γ\ G X the quotient maps of ι and gt , for every b t ∈ R. Let w ∈ G X be isometric exactly on an interval I of R. If I is compact, then w is a (generalised) geodesic segment, and if I = ]−∞, a] or I = [a, +∞[ for some a ∈ R, then w is a (generalised) (negative or positive) geodesic ray in X. Every geodesic line w b ∈ G X such that w| b I = w|I is an extension of w. Note that w b is an extension of w if and only if γ w b is an extension of γw for every γ ∈ Isom(X), if b if and only if gs w and only if ιw b is an extension of ιw, and b is an extension of gs w for every s ∈ R. For any subset Ω0 of G X and any subset A of R, let Ω0 |A = {w|A : w ∈ Ω0 } . Remark 2.3. Let (ì )i∈N be a sequence of generalised geodesic lines such that + [t− i , ti ] is the maximal segment on which ì is isometric. Let (si )i∈N be a sequence in R such that t± i − si → ±∞ as i → +∞ and ì (si ) stays in a compact subset of X. Then d(ì , G X) → 0 as i → +∞. Furthermore, if (si )i∈N is bounded, then up to extracting a subsequence, (ì )i∈N converges to an element in G X. This conceptually important observation explains how it is conceivable that long common perpendicular segments may equidistribute towards measures supported on geodesic lines. See Chapter 11 for further developments of these ideas.

2.3. The unit tangent bundle

2.3

31

The unit tangent bundle

In this book, we define the unit tangent bundle T 1 X of X as the space of germs at 0 of the geodesic lines in X. It is the quotient space T 1 X = G X/ ∼, where ` ∼ `0 if and only if there exists > 0 such that `|[−,] = `0 |[−,] . The canonical projection from G X to T 1 X will be denoted by ` 7→ v` . When X is a Riemannian manifold, the spaces G X and T 1 X canonically identify with the usual unit tangent bundle of X, but in general, the map ` 7→ v` has infinite fibers. We endow T 1 X with the quotient distance d = dT 1 X of the distance of G X, defined by ∀ v, v 0 ∈ T 1 X,

dT 1 X (v, v 0 ) =

inf

`, `0 ∈G X : v=v` , v 0 =v`0

d(`, `0 ) .

(2.9)

It is easy to check that this distance is indeed Hausdorff, hence that T 1 X is locally compact, and that it induces on T 1 X the quotient topology of the compact-open topology of G X. The map ` 7→ v` is 1-Lipschitz. The action of Isom(X) on G X induces an isometric action of Isom(X) on T 1 X. The antipodal map and the footpoint projection restricted to G X respectively induce an Isom(X)-equivariant isometric map ι : T 1 X → T 1 X and an Isom(X)-equivariant 21 -Hölder-continuous map π : T 1 X → X called the antipodal map and footpoint projection of T 1 X. The canonical projection from G X to T 1 X is Isom(X)-equivariant and commutes with the antipodal map: for all γ ∈ Isom(X) and ` ∈ G X, we have γv` = vγ` , ιv` = vι` , and π(v` ) = π(`). We denote again by ι : Γ\T 1 X → Γ\T 1 X the quotient map of ι. 2 Let ∂∞ X be the subset of ∂∞ X × ∂∞ X that consists of pairs of distinct points at infinity of X. Hopf ’s parametrisation of G X is the homeomorphism that 2 identifies G X with ∂∞ X × R, by the map ` 7→ (`− , `+ , t), where t is the signed distance from the closest point to the basepoint x0 on the geodesic line ` to `(0).7 We have gs (`− , `+ , t) = (`− , `+ , t + s) for all s ∈ R, and for all γ ∈ Γ, we have γ(`− , `+ , t) = (γ`− , γ`+ , t + tγ, `− , `+ ), where tγ, `− , `+ ∈ R depends only on γ, `− , and `+ . In Hopf’s parametrisation, the restriction of the antipodal map to G X is the map (`− , `+ , t) 7→ (`+ , `− , −t).

The strong stable leaf of w ∈ G+ X is W + (w) = ` ∈ G X : lim d(`(t), w(t)) = 0 , t→+∞

and the strong unstable leaf of w ∈ G− X is W − (w) = ιW + (ιw) = ` ∈ G X : lim d(`(t), w(t)) = 0 . t→−∞

7 More

precisely, `(t) is the closest point to x0 on `.

32


∂∞ X

w ∈ G− X

w− = `−

w 0 ∈ G+ X

0 w0 (t) w+ = `0+

`0 (t)

HB+ (w0 )

w(−t) `(−t) `(0)

`0 (0)

H− (w)

`0 ∈ W + (w0 ) −

` ∈ W (w) `0−

`+

For every w ∈ G± X, let dW ± (w) be Hamenst¨ adt’s distance on W ± (w) defined 8 0 ± as follows: for all `, ` ∈ W (w), let 1

0

dW ± (w) (`, `0 ) = lim e 2 d(`(∓t), ` (∓t))−t . t→+∞

The above limits exist, and Hamenstädt’s distances are distances inducing the original topology on W ± (w). For all `, `0 ∈ W ± (w) and γ ∈ Isom(X), we have γW ± (w) = W ± (γw) and dW ± (γw) (γ`, γ`0 ) = dW ± (w) (`, `0 ) = dW ∓ (ιw) (ι`, ι`0 ) . Furthermore, for every s ∈ R, we have gs W ± (w) = W ± (gs w) and for all `, `0 ∈ W ± (w), dW ± (gs w) (gs `, gs `0 ) = e∓s dW ± (w) (`, `0 ) .

(2.10)

If X is an R-tree, and if for all w ∈ G+ X and `, `0 ∈ W + (w), [s, +∞[ is the maximal interval on which ` and `0 agree, then dW + (w) (`, `0 ) = es . The following lemma compares the distance in G X with Hamenstädt’s distance for two geodesic lines in the same strong (un)stable leaf. Lemma 2.4. There exists a universal constant c > 0 such that for all w ∈ G± X and `, `0 ∈ W ± (w), we have d(`, `0 ) 6 c dW ± (w) (`, `0 ) 8 See

and

[HeP1, Appendix] and compare with [Ham1].

d(π(`), π(`0 )) 6 dW ± (w) (`, `0 ) .


33

Proof. We could refer to [PaP14a, Lem. 3] (see also [PauPS, Lem. 2.4]) for a proof of the first result. Note that the distance on G X considered in these references is slightly different from the one in this book, hence we give a full proof for the sake of completeness. We assume that w ∈ G+ X, the proof when w ∈ G− X is similar. `(t) ` p p0

w

`(S)

`+ = `− = w+ =1 `0 (S)

`0

∂∞ X

`0 (t) Let `, `0 ∈ W + (w). We may assume that ` 6= `0 . By the convexity properties of the distance in X, the map from R to R defined by t 7→ d(`(t), `0 (t)) is decreasing, with image ]0, +∞[ . Let S ∈ R be such that d(`(S), `0 (S)) = 1. For every t 6 S, let p and p0 be the closest point projections of `(S) and `0 (S) on the geodesic segment [`(t), `0 (t)]. We have d(p, `(S)), d(p0 , `0 (S)) 6 1 by comparison. Hence, by convexity and the triangle inequality, d(`(t), `0 (t)) > d(`(t), p) + d(p0 , `0 (t)) > d(`(t), `(S)) − 1 + d(`0 (t), `0 (S)) − 1 = 2(S − t − 1) . Thus by the definition of the Hamenstädt distance dW + (w) , we have dW + (w) (`, `0 ) > eS−1 .

(2.11)

By the triangle inequality, if t 6 S, then d(`(t), `0 (t)) 6 d(`(t), `(S)) + d(`(S), `0 (S)) + d(`0 (S), `0 (t)) = 2(S − t) + 1 . Since X is CAT(−1), if t > S, we have by comparison d(`(t), `0 (t)) 6 eS−t sinh d(`(S), `0 (S)) = (sinh 1) eS−t . Therefore, by the definition of the distance d on G X (see Equation (2.7)), Z S Z +∞ d(`, `0 ) 6 (2(S − t) + 1) e−2|t| dt + (sinh 1) eS−t e−2|t| dt = O(eS ) . −∞

S

The first inequality of Lemma 2.4 hence follows from Equation (2.11). The second one is proved in [PauPS, Lem. 2.4], and we again give a proof only for the sake of completeness. Let w ∈ G− X and `, `0 ∈ W − (w). Let x = π(`), x0 = π(`0 ), and ρ = dW − (w) (`, `0 ). Consider the ideal triangle ∆ with vertices `+ , `0+ , and `− = `0− (see the picture below on the left). Let p ∈ `(R), p0 ∈ `0 (R), and q ∈ ]`+ , `0+ [ be the tangency points of the unique triple of pairwise tangent horospheres centered at the vertices of ∆: β`− (p, p0 ) = 0, β`+ (p, q) = 0, and β`0+ (p0 , q) = 0. By the definition of the Hamenstädt distance, we have p = `(− ln ρ).

34


1/ρ

x

`+ − ln ρ

p ` q

− ln ρ p0

x0

`0

x x0

1 `− =

`0−

p0

p q

`0+ − 12

0

1 2

Consider the ideal triangle ∆ in the hyperbolic upper half-plane H2R , with vertices − 12 , 12 and ∞ (see the above picture on the right). Let p = (− 12 , 1), p0 = ( 12 , 1), and q = (0, 12 ) be the pairwise tangency points of horospheres centered at the vertices of ∆. Let x and x0 be the points at algebraic (hyperbolic) distance − ln ρ from p and p0 , respectively, on the upwards oriented vertical lines through them. By comparison, we have d(x, x0 ) 6 d(x, x0 ) 6 1/e− ln ρ = ρ. Let H be a horoball in X, centered at ξ ∈ ∂∞ X. The strong stable leaves W + (w) are equal for all geodesic rays w starting at time t = 0 from a point of ∂H and converging to ξ. Using the homeomorphism ` 7→ `− from W + (w) to ∂∞ X −{ξ}, Hamenstädt’s distance on W + (w) defines a distance dH on ∂∞ X −{ξ} that we also call Hamenst¨ adt’s distance. For all `, `0 ∈ W + (w), we have dH (`− , `+ ) = dW + (w) (`, `0 ) , and for all η, η 0 ∈ ∂∞ X − {ξ}, we have 1

dH (η, η 0 ) = lim e 2 d(`η (−t), `η0 (−t))−t , t→+∞

(2.12)

where `η , `η0 are the geodesic lines starting from η, η 0 respectively, ending at ξ, and passing through the boundary of H at time t = 0. Note that for every t > 0, if H [t] is the horoball contained in H whose boundary is at distance t from the boundary of H , then we have dH [t] = e−t dH .

(2.13)

Let w ∈ G± X and η 0 > 0. We define B ± (w, η 0 ) as the set of ` ∈ W ± (w) such that there exists an extension w b ∈ G X of w with dW ± (w) (`, w) b < η 0 . In ± 0 particular, B (w, η ) contains all the extensions of w, and is the union of the open balls centered at the extensions of w, of radius η 0 , for Hamenstädt’s distance on W ± (w).


35

∂∞ X

+ = w +

π(B + (w, η ))

w w(η) w(0)

+ π(Vw, η, η )

w(−η)

∈

+ Vw,η,η

−

∈ B + (w, η) −

The union over t ∈ R of the images under gt of the strong stable leaf of w ∈ G+ X is the stable leaf [ W 0+ (w) = gt W + (w) t∈R

of w, which consists of the elements ` ∈ G X with `+ = w+ . Similarly, the unstable leaf of w ∈ G− X, [ W 0− (w) = gt W − (w) , t∈R

consists of the elements ` ∈ G X with `− = w− . Note that the (strong) (un)stable leaves are subsets of the space of geodesic lines G X. The (un)stable leaves are invariant under the geodesic flow, and for all w ∈ G± X and γ ∈ Isom(X), we have ι W 0± (w) = W 0∓ (ι w) and γW 0± (w) = W 0± (γw) . The unstable horosphere H− (w) of w ∈ G− X is the horosphere in X centered at w− and passing through w(0) b for any extension w b ∈ G X of w (see the picture above the definition of Hamenstädt’s distance). The stable horosphere H+ (w) of w ∈ G+ X is the horosphere in X centered at w+ and passing through w(0) b for any extension w b ∈ G X of w. These horospheres H± (w) do not depend on the chosen extensions w b of w ∈ G± X. The unstable horoball HB− (w) of w ∈ G− X and stable horoball HB+ (w) of w ∈ G+ X are the horoballs bounded by these horospheres. Note that π(W ± (w)) = H± (w) (2.14) for every w ∈ G± X, and that w(0) belongs to H± (w) if and only if w is isometric at least on ±[0, +∞[ .

36

2.4


Normal bundles and dynamical neighbourhoods

In this section, adapting [PaP17b, §2.2] to the present context, we define spaces of geodesic rays that generalise the unit normal bundles of submanifolds of negatively curved Riemannian manifolds. When X is a manifold, these normal bundles are submanifolds of the unit tangent bundle of X, which identifies with G X. In general and in particular in trees, it is essential to use geodesic rays to define normal bundles, and not geodesic lines. Let D be a nonempty proper9 closed convex subset in X. We denote by ∂D its boundary in X and by ∂∞ D its set of points at infinity. Let PD : X ∪ (∂∞ X − ∂∞ D) → D be the (continuous) closest point map to D, defined on ξ ∈ ∂∞ X − ∂∞ D by setting PD (ξ) to be the unique point in D that minimises the function y 7→ βξ (y, x0 ) from 1 D to R. The outer unit normal bundle ∂+ D of (the boundary of) D is 1 ∂+ D = {ρ ∈ G+, 0 X : PD (ρ+ ) = ρ(0)} . 1 The inner unit normal bundle ∂− D of (the boundary of) D is 1 1 ∂− D = ι∂+ D = {ρ ∈ G−, 0 X : PD (ρ− ) = ρ(0)}. 1 1 Note that ∂+ D and ∂− D are spaces of geodesic rays. If X is a smooth manifold, then these spaces have a natural identification with subsets of G X, because every geodesic ray is the restriction of a unique geodesic line. But this does not hold in general.

∂∞ X D ρ0 (0) = PD (ρ0+ )

ρ(0) = PD (ρ− )

+ 0 1 ρ0 = PD (ρ+ ) ∈ ∂+ D

ρ−

9 That

is, different from X.

− 1 ρ = PD (ρ− ) ∈ ∂− D

ρ0+

2.4. Normal bundles and dynamical neighbourhoods

37

Remark 2.5. Since X is assumed to be proper with extendible geodesics, we have 1 π(∂± D) = ∂D .

To see this, let x ∈ ∂D and let (xk )k∈N be a sequence of points in the complement 1 of D converging to x. For all k ∈ N, let ρk ∈ ∂+ D be a geodesic ray with ρk (0) = PD (xk ) and such that the image of ρk contains xk . Since the closest point map does not increase distances, the sequence (PD (xk ))k∈N converges to x. Since X is proper, the space ∂∞ X is compact and the sequence ((ρk )+ )k∈N has a subsequence that converges to a point ξ ∈ ∂∞ X. The claim follows from the continuity of the closest point map. The possible failure of this equality when X is not proper is easy to see. For example, let X be the R-tree constructed by starting with the Euclidean line D = R and attaching a copy of the half-line [0, +∞[ at each x ∈ D such that 1 x > 0. Then 0 ∈ ∂D − π(∂± D). 1 The restriction of the endpoint map ρ 7→ ρ± to ∂± D is a homeomorphism ± onto its image ∂∞ X − ∂∞ D. We denote its inverse map by PD (see the above 1 picture): for every ρ ∈ ∂± D, we have ± ρ = PD (ρ± ) . ± 1 1 Note that PD = π ◦ PD . For every isometry γ of X, we have ∂± (γD) = γ ∂± D ± ± 1 and PγD ◦ γ = γ ◦ PD . In particular, ∂± D is invariant under the isometries of X that preserve D. For every w ∈ G± X, we have a canonical homeomorphism 1 Nw± : W ± (w) → ∂∓ HB± (w) ,

which associates to each geodesic line ` ∈ W ± (w) the unique geodesic ray ρ ∈ 1 ∂∓ HB± (w) such that `∓ = ρ∓ , or equivalently, such that `(t) = ρ(t) for every ± t ∈ R with ∓t > 0. It is easy to check that Nγw ◦γ = γ ◦Nw± for every γ ∈ Isom(X). We define UD± = {` ∈ G X : `± ∈ / ∂∞ D} .

(2.15)

Note that UD± is an open subset of G X, invariant under the geodesic flow. We ± have UγD = γUD± for every isometry γ of X, and in particular, UD± is invariant under the isometries of X preserving D. Define ± 1 fD : UD± → ∂± D

as the composition mapping of the continuous endpoint map ` 7→ `± from UD± ± 1 onto ∂∞ X − ∂∞ D and the homeomorphism PD from ∂∞ X − ∂∞ D to ∂± D (see ± ± the picture below). The continuous map fD takes ` ∈ UD to the unique element + 1 1 ρ ∈ ∂± D such that ρ± = `± . The fiber of ρ ∈ ∂+ D for fD is exactly the stable

38


− 1 leaf W 0+ (ρ), and the fiber of ρ ∈ ∂− D for fD is the unstable leaf W 0− (ρ). For all γ ∈ Isom(X) and t ∈ R, we have ± ± ± ± fγD ◦ γ = γ ◦ fD and fD ◦ g t = fD .

(2.16)

∂∞ X D ρ(0) = PD (`+ )

`−

+ + 1 ρ = fD (`) = PD (`+ ) ∈ ∂+ D

`

` + = ρ+ Let w ∈ G± X and η, η 0 > 0. We define (see the picture after Equation (2.13)) the dynamical (η, η 0 )-neighbourhood of w by ± Vw, η, η 0 =

[

gs B ± (w, η 0 ) .

(2.17)

s∈ ]−η, η [ + Example 2.6. If X is an R-tree, w ∈ G+ X, and η < ln η 0 , then Vw, η, η 0 , which is s the set of g ` where s ∈ ]−η, η[ and ` ∈ G X is such that there exists an extension w b of w with inf{t ∈ R : `(t) = w(t)} b 6 ln η 0 , is as in the following picture.

π(B + (w, η 0 )) w(η) b

w(−η) b w(0) b

w(ln b η0 )

w

w+ = w b+

+ π(Vw, η, η 0 )

± ∓ Clearly, B ± (w, η 0 ) = ιB ∓ (ιw, η 0 ), and hence we have Vw, η, η 0 = ιVιw, η, η 0 . Furthermore, for every s ∈ R, ± ± gs B ± (w, η 0 ) = B ± (gs w, e∓s η 0 ) , whence gs Vw, η, η 0 = Vgs w, η, e∓s η 0 .

(2.18)

2.4. Normal bundles and dynamical neighbourhoods

39

± ± For every γ ∈ Isom(X), we have γB ± (w,η 0 ) = B ± (γw,η 0 ) and γVw,η,η 0 = Vγw,η,η 0 . ± 0 s 0 The map from ]−η, η[ × B ± (w, η 0 ) to Vw, η, η 0 defined by (s, ` ) 7→ g ` is a homeomorphism. For all subsets Ω− of G+ X and Ω+ of G− X, let

Vη,±η0 (Ω∓ ) =

[

± Vw, η, η 0 ,

(2.19)

w∈Ω∓

which we call the dynamical neighbourhoods of Ω∓ . Note that they are subsets of G X, not of G± X. The families (Vη,±η0 (Ω∓ ))η,η0 >0 are nondecreasing in η and in η 0 . For every γ ∈ Isom(X), we have γVη,±η0 (Ω∓ ) = Vη,±η0 (γΩ∓ ), and for every t > 0, we have g±t Vη,±η0 (Ω∓ ) = Vη,±e−t η0 (g±t Ω∓ ) . (2.20) Note that [

1 Vη,±η0 (∂± D) = UD± ,

η , η 0 >0

T 1 and that η, η0 >0 Vη,±η0 (∂± D) is the set of all extensions in G X of the elements of ± 1 ∓ 1 ∂± D. Assume that Ω is a subset of ∂± D. The restriction of fD to Vη,±η0 (Ω∓ ) is ± ∓ ∓ a continuous map onto Ω , with fiber over w ∈ Ω the open subset Vw, η, η 0 of 0± W (w). We will need the following elementary lemma in Section 10.4. Lemma 2.7. There exists a universal constant c0 > 0 such that for every w ∈ G+ X + that is isometric on [sw , +∞[ and every ` ∈ Vw, η, η 0 , we have d(`, w) 6 c0 (η + η 0 + esw ) . Proof. By Equation (2.17) and by the definition of B + (w, η 0 ) in Section 2.3, there exist s ∈ ]−η, +η[ and an extension w b ∈ G X of w such that gs ` ∈ W + (w) and s 0 + dW (w) (g `, w) b 6 η . By Equation (2.8), we have d(`, gs `) 6 |s| 6 η. By Lemma 2.4,bwe have d(gs `, w) b 6 c dW + (w) (gs `, w) b 6 c η 0 . By the definition of the distance on G X (see Equation (2.7)), we have Z d(w, b w) 6

sw

|sw − t| e−2|t| dt = O(esw ) .

−∞

Therefore the result follows from the triangle inequality d(`, w) 6 d(`, gs `) + d(gs `, w) b + d(w, b w) .

40


2.5

Creating common perpendiculars

Let D− and D+ be two nonempty proper closed convex subsets of X, where X is as in the beginning of Section 2.1. A geodesic arc α : [0, T ] → X, where T > 0, is 1 a common perpendicular of length T from D− to D+ if there exists w∓ ∈ ∂± D∓ − −T + such that w |[0, T ] = g w |[0, T ] = α. Since X is CAT(−1), this geodesic arc α is the unique shortest geodesic segment from a point of D− to a point of D+ . There exists a common perpendicular from D− to D+ if and only if the closures of D− and D+ in X ∪ ∂∞ X are disjoint. When X is an R-tree, then two closed subtrees of X have a common perpendicular if and only if they are nonempty and disjoint. One of the aims of this book is to count orbits of common perpendiculars between two equivariant families of closed convex subsets of X. The crucial remark is that two nonempty proper closed convex subsets D− and D+ of X have a common perpendicular α of length a givenb T > 0 if and only if the subsets 1 1 gT /2 ∂+ D− T T and g−T /2 ∂− D+ T T of G X intersect. This intersection then [− 2 ,

2

]

[− 2 , 2 ]

consists of the common perpendicular from D− to D+ reparametrised by [− T2 , T2 ]. As a controlled perturbation of this remark, we now give an effective creation result of common perpendiculars in R-trees. It has a version satisfied for X in the generality of Section 2.1; see the end of this section. Lemma 2.8. Assume that X is an R-tree. For all R > 1, η ∈ ]0, 1], and t > 2 ln R + 4, for all nonempty closed connected subsets D− , D+ in X, and for every 1 1 D+ ), there exist s ∈ ]−2η, 2η[ geodesic line ` ∈ gt/2 Vη,+R (∂+ D− ) ∩ g−t/2 Vη,−R (∂− − + and a common perpendicular e c from D to D such that • the length of e c is t + s, ± • the endpoint of e c in D∓ is w∓ (0), where w∓ = fD ∓ (`), • the footpoint `(0) of ` lies on e c, and n t t o max d w− , `(0) , d w+ − , `(0) 6η. 2 2 `( 2t + s+ )

`(− 2t − s− ) w− (0) D−

`(0)

x− 6 log R

w− ( 2t )

x+ w

+

(− 2t )

w+ (0) 6 log R

D+

Proof. Let R, η, t, D± , ` be as in the statement. By the definition of the sets 1 1 D± ), there exist geodesic rays w± ∈ ∂∓ D± , geodesic lines w b± ∈ G X Vη,∓R (∂∓ ± ± ∓ extending w , and s ∈ ]−η, +η[ such that `± = (w )± and t

∓

dW ± (w∓ ) (g∓ 2 ∓s `, w b∓ ) 6 R .

2.6. Metric and simplicial trees, and graphs of groups

41

Let x± be the closest point to w± (0) on `. By the definition of Hamenstädt’s distances, we have t d(w± (0), x± ) = d ` ± ± s± , x± 6 ln R , 2 and in particular, x± = w± (0) if and only if `(± 2t ± s± ) = w± (0). Since t > 2 ln R + 4 and |s± | 6 2η 6 2, the points `(− 2t − s− ), x− , `(0), x+ , `( 2t + s+ ) are in this order on `. In particular, the segment [w− (0), x− ] ∪ [x− , x+ ] ∪ [x+ , w+ (0)] is a nontrivial geodesic segment from a point of D− to a point of D+ that meets D∓ only at an endpoint. Hence, D− and D+ are disjoint, and [w− (0), w+ (0)] is the image of the common perpendicular from D− to D+ . Let s = s− + s+ . The length of e c is ( 2t + s+ ) − (− 2t − s− ) = t + s. The ± ∓ point `(0) lies on e c, we have w = fD∓ (`), and the endpoints of e c are w± (0). Furthermore, t t t ∓ ∓ ∓ ∓ d w ± , `(0) = d w ± , w (0) − d `(0), ` ∓ ∓ s 2 2 2 = |s∓ | 6 η .

When X is as in the beginning of Section 2.1, the statement and the proof of the following analog of Lemma 2.8 is slightly more technical. We refer to [PaP17b, Lem. 7] for a proof in the Riemannian case, and we leave the extension to the reader, since we will not need it in this book. Lemma 2.9. Let X be as in the beginning of Section 2.1. For every R > 0, there exist t0 , c0 > 0 such that for all η ∈ ]0, 1] and all t ∈ [t0 , +∞[ , for all nonempty closed convex subsets D− , D+ in X, and for every geodesic line 1 1 w ∈ gt/2 Vη,+R (∂+ D− ) ∩ g−t/2 Vη,−R (∂− D+ ), there exist s ∈ ] − 2η, 2η[ and a com− + mon perpendicular e c from D to D such that t

t

• the length of e c is contained in [t + s − c0 e− 2 , t + s + c0 e− 2 ], ± ∓ • if w = fD∓ (w) and if p± is the endpoint of e c in D± , then d(π(w± ), p± ) 6 − 2t c0 e , t • the footpoint π(w) of w is at distance at most c0 e− 2 from a point of e c, and t

t

t

max{ d(π(g 2 w− ), π(w)), d(π(g− 2 w+ ), π(w)) } 6 η + c0 e− 2 .

2.6

Metric and simplicial trees, and graphs of groups

Metric and simplicial trees and graphs of groups are important examples throughout this book. In this section, we recall the definitions and basic properties of these objects. Using Serre’s definitions in [Ser3, §2.1], a graph X is the data (V X, EX, o, t, )

42


consisting of two sets V X and EX, called the set of vertices and the set of edges of X, two maps o, t : EX → V X, and a fixed-point-free involution e 7→ e of EX such that t(e) = o(e) for every e ∈ EX. The elements o(e), t(e), and e are called the initial vertex, the terminal vertex, and the opposite edge of an edge e ∈ EX. The quotient of EX by the involution e 7→ e is called the set of nonoriented edges of X. Recall that a connected graph is bipartite if it is endowed with a partition of its set of vertices into two nonempty subsets such that no edge has both its initial and terminal vertices in one of these subsets. The degree of a vertex x ∈ V X in a graph X is the cardinality of the set {e ∈ EX : o(e) = x}. For all j, k ∈ N, a graph X is k-regular if the degree of each vertex x ∈ V X is k, and it is (j, k)-biregular if it is bipartite with the elements of the partition of its vertices into two subsets having degree j and k respectively. A metric graph (X, λ) is an ordered pair consisting of a graph X and a map λ : EX → ]0, +∞[ with a positive lower bound10 such that λ(e) = λ(e), called its edge length map. A simplicial graph X is a metric graph whose edge length map is constant equal to 1. The topological realisation of a graph X is the topological space obtained from the family (Ie )e∈EX of closed unit intervals Ie for every e ∈ EX by the finest equivalence relation that identifies intervals corresponding to an edge and its opposite edge by the map t 7→ 1 − t and identifies the origins of the intervals Ie1 and Ie2 if and only if o(e1 ) = o(e2 ), see [Ser3, Sec. 2.1]. The geometric realisation of a metric tree (X, λ) is the topological realisation of X endowed with the maximal geodesic metric that gives length λ(e) to the topological realisation of each edge e ∈ EX, and we denote it by X = |X|λ . We identify V X with its image in X. The metric space X determines (X, λ) up to subdivisions of edges; hence we will often not make a strict distinction between X and (X, λ). In particular, we will refer to convex subsets of (X, λ) as convex subsets of X, etc. If X is a tree, the metric space X is an R-tree; hence it is a CAT(−1) space. Since λ is bounded from below by a positive constant, the R-tree X is geodesically complete if and only if X is not reduced to one vertex and has no terminal vertex (that is, no vertex of degree 1). We will denote by Aut(X, λ), and Aut X in the simplicial case, the group of edge-preserving isometries of X that have no inversions.11 Since the edge length map has a positive lower bound, the metric space X is proper if and only if X is locally finite. In this case, the nonelementary discrete subgroups Γ of isometries of X we will consider will always be edge-preserving and without inversion. If Γ 10 This

assumption, though not necessary at this stage, will be used repeatedly in this book; hence we prefer to add it to the definition. 11 An automorphism g of a graph has an inversion if there exists an edge e of the graph such that ge = e. The assumption that the elements of Aut(X) have no inversion ensures that for every subgroup Γ0 of Aut(X), the quotient map Γ0 \EX → Γ0 \EX of e 7→ e is still a fixed-point-free involution, so that with the quotient maps Γ0 \EX → Γ0 \V X defined by o and t, we do have a quotient graph structure Γ0 \X.


43

is a subgroup of Aut(X, λ), we will again denote by λ : Γ\EX → ]0, +∞[ the map induced by λ : EX → ]0, +∞[ . A locally finite metric tree (X0 , λ) is uniform if there exists some discrete subgroup Γ0 of Aut(X0 , λ) such that Γ0 \X0 is a finite graph. See [BasK, BasL] for characterisations of this property in the case of simplicial trees. b Discrete-time geodesic flow on trees. Let X be a locally finite simplicial tree. The space of generalised discrete geodesic lines of X is the locally compact space G X of 1-Lipschitz mappings w from R to the geometric realisation X = |X|1 that are isometric on a closed interval with endpoints in Z ∪ {−∞, +∞} and locally b b ∈ V X (or equivalently w(Z) ⊂ V X). Note that constant outside it, such that w(0) b b G X is hence a proper subset of G X, unless X is reduced to one vertex. b By restriction to G X, or intersection with G X, of the bobjects defined in Sections 2.2 and 2.4 for G X, we define the distance d on G X, the subspaces G± X, G X, G±,0 X, the strong stable/unstable leaves W ± (w) of w ∈ G± X and their Hamenst¨ adt distances dW ± (w) , the stable/unstable leaves W 0± (w) of w ∈ G± X, 1 the outer and inner unit normal bundles ∂± D of a nonempty proper simplicial 1 subtree D of X, the dynamical neighbourhoods Vη,±η0 (Ω∓ ) of subsets Ω∓ of ∂± D as well as the fibrations 1 fD± : UD± = {` ∈ G X : `± ∈ / ∂∞ |D|1 } → ∂± D, 1 whose fiber over ρ ∈ ∂± D is W 0± (ρ). Note that some definitions actually simplify when one considers generalised discrete geodesic lines. For instance, for all w ∈ ± G± X, η 0 > 0, and η ∈ ]0, 1[ , the dynamical neighbourhood Vw, η, η 0 is equal to ± 0 B (w, η ), and is hence independent of η ∈ ]0, 1[ . Besides the map π : G X → V X defined as in the continuous case by ` 7→ `(0), we have another natural map T π : G X → EX, which associates to ` the edge e with o(e) = `(0) and t(e) = `(1). This map is equivariant under the group of automorphisms (without inversions) Aut(X) of X, and for every subgroup Γ of Aut(X) we also denote by T π : Γ\G X → Γ\EX its quotient map. If X has no terminal vertex, then for every e ∈ EX, let

∂e X = {`+ : ` ∈ G X, T π(`) = e} be the set of points at infinity of the geodesic rays whose initial (oriented) edge is equal to e. Given x0 ∈ V X, the discrete Hopf parametrisation now identifies G X with 2 ∂∞ X × Z by the map ` 7→ (`− , `+ , t), where t ∈ Z is the signed distance from the closest vertex to the basepoint x0 on the geodesic b line ` to the vertex `(0). t b The discrete-time geodesic b flow (g )t∈Z on G X is the one-(discrete-)parameter

group of homeomorphisms of G X consisting of (the restriction to G X of) the b integral time maps of the continuous-time geodesic flow of the geometric realisation t of X: we have g w : s 7→ w(s + t) for all w ∈ G X and t ∈ Z.

44


Cross-ratios of ends of trees. Let X be a locally finite simplicial tree, with geometric realisation X = |X|1 . Recall12 that if (ξ1 , ξ2 , ξ3 , ξ4 ) is an ordered quadruple of distinct points in ∂∞ X, then their (logarithmic) cross-ratio is 1 d(x1 , x4 ) − d(x4 , x3 ) + d(x3 , x2 ) − d(x2 , x1 ) . xi →ξi , xi ∈V X 2 (2.21) A similar definition is valid for general CAT(−1) spaces, but we will need only the case of simplicial trees in this book. If x and y are the closest points on the geodesic line ]ξ1 , ξ3 [ to ξ2 and ξ4 respectively, then [[ξ1 , ξ2 , ξ3 , ξ4 ]] = d(x, y) [[ξ1 , ξ2 , ξ3 , ξ4 ]] =

lim

if ξ1 , x, y, ξ3 are in this order on ]ξ1 , ξ3 [ and [[ξ1 , ξ2 , ξ3 , ξ4 ]] = −d(x, y) otherwise. In particular, [[ξ1 , ξ2 , ξ3 , ξ4 ]] = 0 if the geodesic lines ]ξ1 , ξ3 [ and ]ξ2 , ξ4 [ are disjoint. ξ2 ξ1

− x

ξ2 + +

ξ4

y − ξ4

ξ3 ξ1

x=y

ξ3

We have the following properties: • [[ξ1 , ξ2 , ξ3 , ξ4 ]] = [[ξ4 , ξ3 , ξ2 , ξ1 ]] = −[[ξ3 , ξ2 , ξ1 , ξ4 ]]. • the cross-ratio is continuous, and even locally constant on the space of pairwise distinct quadruples of elements of ∂∞ X, • if γ is a loxodromic element of Aut(X), with repulsive and attractive fixed points γ− and γ+ in ∂∞ X respectively, then for every ξ ∈ ∂∞ X − {γ− , γ+ }, the translation length of γ satisfies λ(γ) = [[γ− , ξ, γξ, γ+ ]] . Bass–Serre’s graphs of groups. Recall (see, for instance, [Ser3, BasL]) that a graph of groups (Y, G∗ ) consists of • • • •

a graph Y, which is connected unless otherwise stated, a group Gv for every vertex v ∈ V Y, a group Ge for every edge e ∈ EY such that Ge = Ge , an injective group morphism ρe : Ge → Gt(e) for every edge e ∈ EY.

12 See [Pau2], as well as [Ota] in the case of Riemannian manifolds, and note that the convention on the order varies in the literature.


45

We will identify Ge with its image in Gt(e) by ρe , unless the meaning is not clear (which might be the case, for instance, if o(e) = t(e)). We refer to [Ser3, BasL]13 for the definition of the Bass–Serre tree T (Y, G∗ ) of (Y, G∗ ), of its fundamental group π1 (Y, G∗ ) when a basepoint in V Y is chosen, and of the simplicial action of π1 (Y, G∗ ) on T (Y, G∗ ). Note that the fundamental group of (Y, G∗ ) does not always act faithfully on its Bass–Serre tree T (Y, G∗ ), that is, the kernel of its action might be nontrivial. A subgraph of subgroups of (Y, G∗ ) is a graph of groups (Y0 , G0∗ ), where • • • •

Y0 is a subgraph of Y, for every v ∈ V Y0 , the group G0v is a subgroup of Gv , for every e ∈ EY0 , the group G0e is a subgroup of Ge , the monomorphism ρ0e : G0e → G0t(e) is the restriction to G0e of the monomorphism ρe : Ge → Gt(e) , and G0t(e) ∩ ρe (Ge ) = ρ0e (G0e ) .

This condition, first introduced in [Bass, Cor. 1.14], is equivalent to the injectivity of the natural map G0t(e) /ρ0e (G0e ) → Gt(e) /ρe (Ge ) for every e ∈ EY. It implies by [Bass, 2.15] when the underlying basepoint is chosen in Y0 that • the fundamental group Γ0 = π1 (Y0 , G0∗ ) of (Y0 , G0∗ ) injects into the fundamental group Γ = π1 (Y, G∗ ) of (Y, G∗ ), • the Bass–Serre tree X0 of (Y0 , G0∗ ) injects in an equivariant way into the Bass–Serre tree X of (Y, G∗ ) so that the stabiliser of X0 in Γ is Γ0 , and • the map (Γ0 \X0 ) → (Γ\X) induced by the inclusion map X0 → X by taking the quotient is injective: ∀ γ ∈ Γ, ∀ z ∈ V X0 ∪ EX0 , if γz ∈ V X0 ∪ EX0 , then ∃ γ 0 ∈ Γ0 ,

γ 0 z = γz . (2.22)

The edge-indexed graph (Y, i) of the graph of groups (Y, G∗ ) is the graph Y endowed with the map i : EY → N − {0} defined by i(e) = [Go(e) : Ge ] (see, for instance, [BasK, BasL]). In Section 12.4, we will consider metric graphs of groups (Y, G∗ , λ), which are graphs of groups endowed with an edge length function λ : EY → ]0, +∞[ (with λ(e) = λ(e) for every e ∈ EY). Example 2.10. The main examples of graphs of groups that we will consider in this book are the following ones. Let X be a simplicial tree and let Γ be a subgroup of Aut(X). The quotient graph of groups Γ\\X is the following graph of groups (Y, G∗ ). Its underlying graph Y is the quotient graph Γ\X. Fix a lift ze ∈ V X ∪ EX e, and fix an element for every z ∈ V Y ∪ EY. For every e ∈ EY, assume that ee = e 13 Though

see Example 2.10 for the main example encountered in this book.

46


g = t(e ge ∈ Γ such that ge t(e) e). For every y ∈ V Y∪EY, take as Gy the stabiliser Γye in Γ of the fixed lift ye. Take as monomorphism ρe : Ge → Gt(e) the restriction to Γee of the conjugation γ 7→ ge−1 γge by ge−1 . Note that Γ\\X has finite vertex groups if X is locally finite and Γ is discrete. For every choice of basepoint in V Y, there exist a group isomorphism θ : π1 (Y, G∗ ) → Γ and a θ-equivariant simplicial isomorphism from the Bass–Serre tree T (Y, G∗ ) to X (see, for instance, [Ser3, BasL]). The volume form of a graph of finite groups (Y, G∗ ) is the measure vol(Y, G∗ ) on the discrete set V Y such that for every y ∈ V Y, 1 , |Gy |

vol(Y, G∗ ) ({y}) =

where |Gy | is the order of the finite group Gy . Its total mass, called the volume of (Y, G∗ ), is X 1 Vol(Y, G∗ ) = k vol(Y, G∗ ) k = . |Gy | y∈V Y

2

2

We denote by L (Y, G∗ ) = L (V Y, vol(Y,G∗ ) ) the complex Hilbert space of square integrable maps V Y → C for this measure vol(Y,G∗ ) , by f 7→ kf k2 its norm and by (f, g) 7→ hf, gi2 its (antilinear on the right) scalar product. Let Z L20 (Y, G∗ ) = f ∈ L2 (Y, G∗ ) : f d vol(Y, G∗ ) = 0 . When Vol(Y, G∗ ) is finite, L20 (Y, G∗ ) is the orthogonal subspace to the constant functions. We also consider an (edge-)volume form Tvol(Y, G∗ ) on the discrete set EY such that for every e ∈ EY, Tvol(Y, G∗ ) ({e}) =

1 , |Ge |

with total mass TVol(Y, G∗ ) = kTvol(Y, G∗ ) k =

X e∈EY

1 . |Ge |

The (edge-)volume form of a metric graph of groups (Y, G∗ , λ) is given by Tvol(Y, G∗ ,λ) =

ds |Ge |

on each edge e of Y parametrised by its arc length s, so that its total mass is TVol(Y, G∗ , λ) = kTvol(Y, G∗ ,λ) k =

X λ(e) . |Ge |

e∈EY

For λ ≡ 1, this total mass agrees with that of the discrete definition above.


47

Remark 2.11. Note that TVol(Y, G∗ ) = Card(EY) when the edge groups are trivial. We have X 1 X 1 X X deg(e |Gy | y) TVol(Y, G∗ ) = = = , |Ge | |Gy | |Ge | |Gy | e∈EY

y∈V Y

e∈EY, o(e)=y

y∈V Y

where ye is any lift of y in the Bass–Serre tree of (Y, G∗ ). In particular, if X is a uniform simplicial tree and Γ is a discrete subgroup of Aut X, then the finiteness of Vol(Γ\\X) and that of TVol(Γ\\X) are equivalent. Defining the volume form on y) V Y by {y} 7→ deg(e |Gy | sometimes makes formulas simpler, but we will follow the convention that occurs in the classical references (see, for instance, [BasL]). If the Bass–Serre tree of (Y, G∗ ) is (q + 1)-regular, then π∗ TvolY, G∗ = (q + 1) volY, G∗

and TVol(Y, G∗ ) = (q + 1) Vol(Y, G∗ ) . (2.23)

We say that a discrete group of (inversion-free) automorphisms Γ of a locally finite metric or simplicial tree (X, λ) is a (tree) lattice of (X, λ) if the quotient graph of groups Γ\\X has finite volume: Vol(Γ\\X) < +∞ . This implies by [BasK, Prop. 4.5]14 that Γ is a lattice15 in the locally compact group Aut(X, λ) (hence that Aut(X, λ) is unimodular; see, for instance, [Rag, Chap. I, Rem. 1.9]), the converse being true, for instance, if (X, λ) is uniform. If Γ is a uniform lattice of (X, λ), that is, if Γ is a discrete subgroup of Aut(X, λ) and if the quotient graph Γ\X is finite, then Γ is clearly a (tree) lattice of (X, λ). A graph of finite groups (Y, G∗ ) is a cuspidal ray if Y is a simplicial ray such that the homomorphisms Gen → Go(en ) are surjective for its sequence of consecutive edges (ei )i∈N oriented towards the unique end of Y. By [Pau3], a discrete group Γ0 of Aut(X) (hence of Isom(|X|1 )) is geometrically finite if and only if it is nonelementary and if the quotient graph of groups by Γ0 of its minimal nonempty invariant subtree is the union of a finite graph of groups and a finite number of cuspidal rays attached to the finite graph at their finite endpoints. Remark 2.12. If X is a locally finite simplicial tree and if Γ0 is a geometrically finite discrete group of Aut(X) such that the convex hull of its limit set C ΛΓ0 is (the geometric realisation of) a uniform tree, then Γ0 is a tree lattice of the simplicial tree C ΛΓ0 . Proof. Since C ΛΓ0 is uniform, there is a uniform upper bound on the length of an edge path in C ΛΓ0 that injects in Γ0 \C ΛΓ0 and such that the stabiliser of each 14 Using

the fact that Aut(X, λ) is a closed subgroup of Aut(X). that a lattice in a locally compact group G is a discrete subgroup Γ0 of G such that the left quotient space Γ0 \G admits a probability measure invariant under translations on the right by G. 15 Recall

48


edge of this edge path is equal to the stabilisers of both endpoints of this edge. It is hence easy to see that the volume of each of the (finitely many) cuspidal rays in Γ0 \\C ΛΓ0 is finite, by a geometric series argument. Hence the volume of Γ0 \\C ΛΓ0 is finite. Note that in contrast to the case of Riemannian manifolds, there are many more (tree) lattices than there are geometrically finite (tree) lattices, even in regular trees; see, for instance, [BasL]. In Part III of this book, we will consider simplicial graphs of groups that arise from the arithmetic of non-Archimedean local fields. We say that a discrete group Γ of (inversion-free) automorphisms of a simplicial tree X is algebraic if b (a finite extension of Qp for some there exist a non-Archimedean local field K prime p or the field of formal Laurent series over a finite field) and a connected b of K-rank b semisimple algebraic group G with finite center defined over K, one, such that X identifies with the Bruhat–Tits tree of G in such a way that Γ identifies b If Γ is algebraic, then Γ is with a lattice of the locally compact group G(K). geometrically finite by [Lub1]. Note that X is then bipartite; see Section 2 of [Lub1] for a discussion and references. See Sections 14 and 15.1 for more details, and their following sections for arithmetic applications arising from algebraic lattices.

Chapter 3

Potentials, Critical Exponents, and Gibbs Cocycles Let X be a geodesically complete proper CAT(−1) space, let x0 ∈ X be an arbitrary basepoint, and let Γ be a nonelementary discrete group of isometries of X. In this chapter, we define potentials on T 1 X, which are new data on X in addition to its geometry. We introduce the fundamental tools associated with potentials, and we give some of their basic properties. The development follows [PauPS] with modifications to fit the present more general context. In Section 3.5, given a simplicial or metric tree (X, λ), with geometric realisation X, we introduce a natural method to associate a (Γ-invariant) potential Fec : T 1 X → R to a Γ-invariant function e c : EX → R defined on the set of edges of X, which we call a system of conductances on X. This construction gives an asymmetric generalisation of electric networks.

3.1

Background on (uniformly local) Hölder continuity

In this preliminary section, we recall the notion of Hölder continuity we will use in this book, which needs to be defined appropriately in order to deal with noncompactness issues. Hölder continuity will be used on the one hand for potentials when X is a Riemannian manifold in Section 3.2, and on the other hand for error term estimates in Chapters 9, 10, and 11. As in [PauPS], we will use the following uniformly local definition of Höldercontinuous maps. Let E and E 0 be two metric spaces, and let α ∈ ]0, 1]. A map f : E → E 0 is © Springer Nature Switzerland AG 2019 A. Broise-Alamichel et al., Equidistribution and Counting Under Equilibrium States in Negative Curvature and Trees, Progress in Mathematics 329, https://doi.org/10.1007/978-3-030-18315-8_3

49

50

Chapter 3. Potentials, Critical Exponents, and Gibbs Cocycles

• α-H¨ older-continuous if there exist c, > 0 such that for all x, y ∈ E with d(x, y) 6 , we have d(f (x), f (y)) 6 c d(x, y)α . • locally α-H¨ older-continuous if for every x ∈ E, there exists a neighbourhood U of x such that the restriction of f to U is α-Hölder-continuous; • H¨ older-continuous (respectively locally H¨ older-continuous) if there exists an element α ∈ ]0, 1] such that f is α-Hölder-continuous (respectively locally α-H¨ older-continuous); • Lipschitz if it is 1-Hölder-continuous and locally Lipschitz if it is locally 1H¨ older-continuous. Let E be a set. Two distances d and d0 on E are (uniformly locally) H¨ olderequivalent if the identity map from (E, d) to (E, d0 ) and the identity map from (E, d0 ) to (E, d) are Hölder-continuous. This is an equivalence relation on the set of distances on E, and a H¨ older structure on E is the choice of such an equivalence class. For a map between two metric spaces, to be Hölder-continuous depends only on the H¨ older structures on the source and target spaces. Let E and E 0 be two metric spaces. We say that a map f : E → E 0 has • at most linear growth if there exist a, b > 0 such that d(f (x), f (y)) 6 a d(x, y) + b for all x, y ∈ E , • subexponential growth if for every a > 0, there exists b > 0 such that d(f (x), f (y)) 6 b ea d(x, y) for all x, y ∈ E . Remark 3.1. When E is a geodesic space, a consequence of the (uniformly local) H¨ older-continuous property of f : E → E 0 is that f then has at most linear growth: the definition implies that d(f (x), f (y)) 6 c α−1 d(x, y) + c α for all x, y in X, by subdividing the geodesic segment in E from x to y into d(x,y) segments of equal lengths at most and using the triangle inequality in E 0 . b The following result (due to Bartels–L¨ uck [BartL] with ab different distance : G X → X is indeed on G X) proves in particular that the footpoint projection π b 1 older-continuous, as claimed in Section 2.2. Recall that G X is endowed with 2 -H¨ the distance d defined by Equation (2.7). b Proposition 3.2. For every t ∈ R, the map from G X to X defined by ` 7→ `(t) is 1 older-continuous. 2 -H¨

3.1. Background on (uniformly local) H¨ older continuity

51

b

Proof. Let `, ` ∈ G X be such that d(`, `0 ) 6 1. Assume that t > 0; otherwise R t+ Rt the argument is similar, replacing t by t− . For every > 0, we have by the triangle inequality Z t+ 0 d(`, ` ) > d(`(s), `0 (s)) e−2s ds > d(`(t), `0 (t)) − 2 e−2t−2 . 0

t 0

If d(`(t), ` (t)) > 4, let = 1, so that d(`, `0 ) >

e−2t−2 2

d(`(t), `0 (t)), whence 1

d(`(t), `0 (t)) 6 2 e2t+2 d(`, `0 ) 2 . If d(`(t), `0 (t)) 6 4, let = 14 d(`(t), `0 (t)) 6 1. The first centered formula gives d(`, `0 ) > 18 d(`(t), `0 (t))2 e−2t−2 , so that √ 1 d(`(t), `0 (t)) 6 2 2 et+1 d(`, `0 ) 2 . When X is an R-tree, the regularity property of the footpoint projection π : G X → X is stronger than the one given by Proposition 3.2 (see Lemma 3.4 (2)). The results below, which will be needed in Sections 10.4 and 12.6, say that not only are the evaluation maps ` 7→ `(t) 12 -Hölder-continuous, but so are the endpoint maps ` 7→ `± . Since we will need these facts only in the tree case, we prove them only when X is an R-tree, and we begin with a simplicial version of it. Lemma 3.3. Let X be a simplicial tree. There are universal constants 0 > 0, c0 > 1 such that for all ∈ ]0, 0 [ and ` ∈ G X, the ball Bd (`, ) is contained in √ {`0 ∈ G X : `0 (0) = `(0), `0± ∈ Bd`(0) (`± , c0 )} √ and contains {`0 ∈ G X : `0 (0) = `(0), `0± ∈ Bd`(0) (`± , c10 )}. In particular, the endpoint maps ` 7→ `± from G X to ∂∞ X are 12 -H¨ oldercontinuous. Proof. If `, `0 ∈ G X have distinct footpoints, then d(`(0), `0 (0)) > 1, so that R1 d(`(t), `0 (t)) > 12 if |t| 6 14 , so that d(`, `0 ) > −4 1 12 e−2|t| = 0 > 0. 4

Conversely, assume that `, `0 ∈ G X have equal footpoints, so that they coincide on [−N, N 0 ] for some N, N 0 ∈ N ∪ {+∞}. By the definition of the visual distances (see Equation (2.3)), we have d`(0) (`+ , `0+ ) = e−N

0

b

and similarly = e . By the definition of the distance on G X (see Equation (2.7)), we have, by an easy change of variables, Z +∞ Z −N d(`, `0 ) = 2 |t − N 0 | e−2t dt + 2 | − N − t| e2t dt d`(0) (`− , `0− )

−N

N0

−∞ 0

= (e−2N + e−2N )

Z 0

The result follows.

+∞

2 u e−2u du =

1 −2N 0 (e + e−2N ) . 2

52


Let us now give a (more technical) version of this lemma for R-trees, also proving that the footpoint projection is Lipschitz. If a and b are positive functions of some parameters, we write a b if there exists a universal constant C > 0 such that C1 b 6 a 6 C b. Lemma 3.4. Assume that X is an R-tree. (1) There exists a universal constant c1 > 0 such that for all `, `0 ∈ G X, if d(`, `0 ) 6 c1 , then `0 (0) ∈ `(R), the intersection `(R) ∩ `0 (R) is not reduced to a point, the orientations of ` and `0 coincide on this intersection, and d(`, `0 ) d`(0) (`− , `0− )2 + d`(0) (`+ , `0+ )2 + d(`(0), `0 (0)) . (2) The footpoint map π : G X → X defined by ` 7→ `(0) is (uniformly locally) Lipschitz. (3) There are universal constants 0 > 0, c0 > 1 such that for all ∈ ]0, 0 [ and ` ∈ G X, the ball Bd (`, ) in G X is contained in √ {`0 ∈ G X : `0 (0) ∈ `(R), d(`0 (0), `(0)) 6 c0 , `0± ∈ Bd`(0) (`± , c0 )} and contains 1 1 √ 0 0 0 0 ` ∈ G X : ` (0) ∈ `(R), d(` (0), `(0)) 6 , `± ∈ Bd`(0) `± , . c0 c0 (4) The endpoint maps ` 7→ `± from G X to ∂∞ X are 12 -H¨ older-continuous. Proof. Note that Assertion (2) follows from Assertion (1) and that Assertion (4) follows from Assertion (3). (1) Let `, `0 ∈ G X. If `0 (0) ∈ / `(R), then `0 (t) ∈ / `(R) for all t > 0 or `0 (t) ∈ / `(R) for all t 6 0, since X is an R-tree. In the first case, we hence have d(`(t), `0 (t)) > t for R +∞ all t > 0; thus d(`, `0 ) is at least c2 = 0 te−2 t dt = 41 > 0. The same estimate holds in the second case. By symmetry, if `(0) ∈ / `0 (R), then d(`, `0 ) > c2 > 0. This argument furthermore shows that if the geodesic segment (or ray or line) `(R) ∩ `0 (R) is reduced to a point, then d(`, `0 ) is at least c2 > 0. If d(`0 (0), `(0)) > 1, then d(`(t), `0 (t)) > 12 for |t| 6 41 ; thus 0

d(`, ` ) >

Z

1 4

− 14

1 −2|t| e dt , 2

which is a positive universal constant. If d(`0 (0), `(0)) 6 1, if `(R) ∩ `0 (R) contains `0 (0) and is not reduced to a point, but if the orientations of ` and `0 do not coincide on this intersection, then d(`(t), `0 (t)) > 2t − d(`(0), `0 (0)) > 2t − 1


R +∞

for all t > 1, so that d(`, `0 ) is at least constant.

1

53

(2t − 1) e−2t dt, which is a positive

Assume now that `0 (0) ∈ `(R) and `(0) ∈ `0 (R), that d(`0 (0), `(0)) 6 1, that `(R) ∩ `0 (R) is not reduced to a point, and that the orientations of ` and `0 coincide on this intersection. Then there exists s ∈ R such that `0 (0) = `(s), so that |s| = d(`(0), `0 (0)) 6 1. Assume, for instance, that s > 0, the other case being treated similarly. Then there exist S, S 0 > 0 maximal such that `0 (t) = `(t + s) for all t ∈ [−S, S 0 ]. We use the conventions that S = +∞ if `0− = `− , that S 0 = +∞ if `0+ = `+ , and that e−∞ = 0. Since `(0) ∈ `0 (R), we have −S + s 6 0. `0 (−∞)

`0 (+∞) `0 (0)

`0 (−S)

`(−∞)

`(−S + s)

`0 (S 0 )

`(+∞)

`(S 0 + s)

`(0) `(s)

By the definition of the visual distances (see Equation (2.3)), we have, for t big enough, d`(0) (`+ , `0+ ) = e−d(`(0), `(S

0

+s))

Similarly d`(0) (`− , `0− ) e−S . As can be seen in the above picture, we   −2t − 2S + s s d(`(t), `0 (t)) =  2t − 2S 0 − s b

= e−S

0

−s

0

e−S .

have if t 6 −S, if − S 6 t 6 S 0 + s, if t > S 0 + s .

By the definition of the distance on G X (see Equation (2.7)), by easy changes of variables, assuming that at least one of S, S 0 is at least 1 for the last line (otherwise the previous line shows that d(`, `0 ) is larger than a positive constant), we have 0

Z

−S

(−2t − 2S + s) e

d(`, ` ) =

−∞ Z +∞

+

−2|t|

Z

S 0 +s

s e−2|t| dt

dt + −S

(2t − 2S 0 − s) e−2t dt

S 0 +s

1 = e−2S+s 2

Z

+∞ −u

ue

Z

S 0 +s

du + s −S

s

e−2|t| dt +

1 −2S 0 −s e 2

Z

+∞

u e−u du

s

0

e−2S + e−2S + s . Assertion (1) of Lemma 3.4 follows. (3) The first inclusion in Assertion (3) follows easily from Assertion (1). The second inclusion follows from the argument of its proof and the fact that if d`(0) (`+ , `0+ ),

54


d`(0) (`− , `− ), and d(`(0), `0 (0)) are at most some small positive constant, then `(R) ∩ `0 (R) contains `0 (0) and is not reduced to a point. When X is a Riemannian manifold with pinched sectional curvature, the following result says that the Hölder structure defined by the distance d given in Equation (2.7) on the unit tangent bundle of X, identified with G X and T 1 X as explained previously, is the usual one. Recall that Sasaki’s metric on T 1 X is the Riemannian metric induced on the submanifold T 1 X by the canonical Riemannian metric on the tangent bundle T X, such that for every v ∈ T X, if Tv T X = Vv ⊕ Hv is the direct sum decomposition defined by the Levi-Civita connection of the Riemannian manifold X, then • the direct sum Vv ⊕ Hv is orthogonal for Sasaki’s metric, • the canonical isomorphism Vv ' Tπ(v) X is an isometry when Vv is endowed with Sasaki’s scalar product, • the restriction of the tangent map of the footpoint projection T π : Hv → Tπ(v) X is an isometry, when Hv is endowed with Sasaki’s scalar product. Proposition 3.5. When X is a Riemannian manifold with pinched sectional curvature, the following distances on T 1 X are H¨ older-equivalent: (1) the Riemannian distance on T 1 X defined by Sasaki’s metric, (2) the distance δ1 on T 1 X defined, for all `, `0 ∈ T 1 X, by ( δ1 (`, `0 ) = exp − sup t > 0 :

sup d(`(s), `0 (s)) 6 1

)! ,

s∈[−t,t]

with the convention δ1 (`, `0 ) = 1 if d(`(0), `0 (0)) > 1 and δ1 (`, `0 ) = 0 if ` = `0 , (3) the distance δ2 on T 1 X defined, for all `, `0 ∈ T 1 X, by δ2 (`, `0 ) = sup d(`(s), `0 (s)) , s∈[0,1]

(4) the distance d defined by Equation (2.7). Note that we could replace the interval [0, 1] in the definition of δ2 by any interval [a, b] with a < b. Proof. The fact that the first three distances on T 1 X are Hölder-equivalent is already known, see, for instance, [Bal, p. 70]. Let us hence prove that d and δ2 are Hölder-equivalent. By the convexity of the distance function between two geodesic segments in a CAT(−1) space, we have δ2 (`, `0 ) = max{d(`(0), `0 (0)), d(`(1), `0 (1))} . Hence by Proposition 3.2 applied twice (with t = 0 and t = 1), there exists c > 0 1 such that if d(`, `0 ) 6 1, then δ2 (`, `0 ) 6 c d(`, `0 ) 2 . Therefore the identity map from (T 1 X, d) to (T 1 X, δ2 ) is Hölder-continuous.


55

Let us prove conversely that there exist two constants c01 > 1 and c02 ∈ ]0, 1] such that for all `, `0 ∈ G X, if δ2 (`, `0 ) 6 21 , then 0

d(`, `0 ) 6 c01 δ2 (`, `0 )c2 .

(3.1)

Let x = `(0), y = `0 (0), z = `(1), z 0 = `0 (1). Note that d(y, z) > 12 by the triangle inequality, since d(x, y) 6 δ2 (`, `0 ) 6 12 and d(x, z) = 1. Let `00 be the geodesic line through y and z, oriented from y to z with `00 (0) = y. Let us prove 0 c0 that d(`, `00 ) 6 21 d(x, y)c2 for appropriate constants c01 and c02 . A similar argument proves that d(`00 , `0 ) 6 gives Equation (3.1).

c01 2

0

d(z, z 0 )c2 , and the triangle inequality for the distance d

y

`0

s

y

x

z0

0

`00 `

z

If d(y, z) 6 d(x, z), let x0 ∈ [x, z] be such d(x0 , z) = d(y, z) > 21 . We have s = d(x, x0 ) = d(x, z) − d(x0 , z) = d(x, z) − d(y, z) 6 d(x, y) by the triangle inequality, and d(x0 , y) 6 d(x, y) by convexity. By Equation (2.8), we have d(`, `00 ) 6 d(`, gs `) + d(gs `, `00 ) 6 s + d(gs `, `00 ) 6 d(x, y) + d(gs `, `00 ) . If d(y, z) > d(x, z), let y 0 ∈ [y, z] be such d(y 0 , z) = d(x, z) = 1 > 12 (see the above picture). We similarly have d(x, y 0 ) 6 d(x, y) and d(`, `00 ) 6 d(x, y) + d(gs `00 , `00 ) if s = d(y, y 0 ). We hence have only to prove that for all x, y, z ∈ X with 0 < d(x, y) 6 12 and d(x, z) = d(y, z) > 12 , if u and v are the unit tangent vectors at x and y 00 respectively pointing to z, then d(u, v) 6 c001 d(x, y)c2 for appropriate constants c001 p 1 1 00 and c2 . This follows from the following lemma with t = 0, since 2 > 2 d(x, y). We will need its more general version in the proof of Lemma 3.15. Lemma 3.6. There exist two constants c8 > 1 and c3 ∈ ]0, 1] p such that for all x, y, z ∈ X with 0 < d(x, y) 6 1 and ρ = d(x, z) = d(y, z) > 12 d(x, y), for every t ∈ [0, ρ], if u and v are the unit tangent vectors at x and y respectively pointing to z, then d(gt u, gt v) 6 c8 d(x, y)c3 (e−t + e2t−2ρ ) . (3.2) Proof. For every s ∈ R, let xs = π(gs u) and ys = π(gs v), so that x0 = x, y0 = y, and xρ = yρ = z. Let −r < r0 in R be such that d(x−r , y−r ) = d(xr0 , yr0 ) = 1, which exist, since x 6= y. We have r > 0 by convexity, since d(x0 , y0 ) 6 1, and r0 > ρ.


56

y−r y = y0

yr0

v

=1

u x−r

θ

z = xρ = yρ

=1

x = x0 xr 0

Claim 1: There exists a constant c3 ∈ ]0, 1] depending only on the lower bound of the curvature of X such that 0

e−r 6 2 d(x, y)c3 and e−r 6 d(x, y)c3 e−ρ . Proof. Let a > 1 be such that the sectional curvature of X is at least −a2 . Consider the comparison triangle (x−r , y−r , z), in the real hyperbolic space 1 2 2 a HR with constant sectional curvature −a , to the triple of points (x−r , y−r , z) in X. Let θ be its angle at z, and let x, y be the points corresponding to x, y (so that we have d(x, z) = d(y, z) = ρ). in

1 a

r

y−r

y 1

x−r

ρ z

θ ρ

x r

1 a

H2R

Since the geodesic triangles in X are less pinched than the geodesic triangles H2R , we have d(x, y) 6 d(x, y). By the hyperbolic sine rule in a1 H2R , we have sinh a2 d(x−r , y−r ) sinh a2 θ = = sin sinh(a(r + ρ)) 2 sinh a d(x−r , z) sinh a2 d(x, y) sinh a2 d(x, y) 6 = . sinh(a ρ) sinh a d(x, z)

Since the map t 7→ (ln 2)t + ln(1 − 2t ) is nonnegative on [0, 1], and since p ρ> d(x, y), we have √ 1 p e2aρ > eln 2 d(x,y) > . 1 − 12 d(x, y) p aρ −aρ Hence sinh(aρ) = e −e > 14 eaρ d(x, y). Therefore, since d(x, y) 6 1, 2 p p 2 sinh a2 d(x, y) 4 sinh(aρ) / d(x, y) eaρ −ar e = ar+aρ 6 6 p 6 2 d(x, y) , a e 2 sinh(ar + aρ) d(x, y) sinh 2 1 2

1

so that the first assertion of Claim 1 follows, since 2 a 6 2, with c3 =

1 2a .

In order to prove the second assertion of Claim 1, let θ be the angle at z between the unit tangent vectors gρ u and gρ v (see the picture before the statement of Claim 1). Since the geodesic triangles in X are less pinched than their


57

comparison triangles in a1 H2R , and again by the hyperbolic sine rule in have sinh a2 d(xr0 , yr0 ) sinh a2 θ = sin > . 2 sinh(a(r0 − ρ)) sinh a d(xr0 , z)

1 a

H2R , we

Since the geodesic triangles in X are more pinchedpthan their comparison triangles in H2R , since d(x, y) 6 1 and since sinh ρ > ρ > 12 d(x, y), we have sinh 12 d(x, y) (sinh 12 )d(x, y) θ 1 p sin 6 6 6 2 (sinh ) d(x, y) . 2 sinh d(x, z) sinh ρ 2 1 sinh 1 a Let c4 = sinh a2 , which is positive and strictly less than 1, since a > 1. Then 2

0

e−ar =

e−aρ

1 e−aρ −aρ sinh 2 d(x, y) 6 e 2 sinh(a(r0 − ρ)) 2 sinh a2 sinh ρ p d(x, y) ,

6

ear0 −aρ

6 e−aρ c4 a

(3.3)

so that the second assertion of Claim 1 follows, since c4 6 1, again with 1 c3 = 2a . Let us remark that since d(x, y) 6 1, Equation (3.3) also gives that r0 − ρ > − ln c4 > 0 . Since θ is also the angle between the unit tangent vectors −gρ u and −gρ v, a similar argument gives that sinh 12 d(xr0 , yr0 ) sinh a2 d(x, y) sinh 12 θ a d(x, y) = > sin > > . sinh(r0 − ρ) sinh d(xr0 , z) 2 2 sinh(a ρ) sinh a d(x, z) Let c5 = have 0

2 sinh 12 a(1−c24 )

er −ρ 6

> 0. Since the map t 7→

sinh(t) et

is nondecreasing on R, we hence

2 sinh 12 e− ln c4 2 c5 sinh(r0 − ρ) 6 sinh(a ρ) 6 eaρ . sinh(− ln c4 ) 1 − c24 a d(x, y) d(x, y)

Therefore 1 + r0 6 (a + 1)ρ + (1 + ln c5 ) − ln d(x, y) ,

(3.4)

a formula that will be useful later on. Let t ∈ [0, ρ]. By the definition of the distance d on G X (see Equation (2.7)), we have Z +∞ Z +∞ d(gt u, gt v) = d(π(gs gt u), π(gs gt v)) e−2|s| ds = d(xs , ys ) e−2|s−t| ds . −∞

−∞

(3.5) We subdivide the integral

R +∞ −∞

as

R −r −∞

+

R0 −r

+

Rρ 0

+

R r0 ρ

+

R +∞ r0

.

58


Claim 2: We have Z

−r

d(xs , ys ) e−2|s−t| ds 6 2 e−2t d(x, y)c3 .

I1 = −∞

Proof. By the triangle inequality, for every s ∈ [r, +∞[ , we have d(x−s , y−s ) 6 d(x−s , x−r ) + d(x−r , y−r ) + d(y−r , y−s ) 6 2s + 1 . Hence Z I1 =

+∞

d(x−s , y−s ) e

−2(s+t)

ds 6 e

r

−2t

Z

+∞

(2s + 1) e−2s ds 6 e−2t e−r ,

r

and the result follows from the first assertion of Claim 1.

Claim 3: We have Z

0

I2 =

d(xs , ys ) e−2|s−t| ds 6 2 (sinh 1) e−2t d(x, y)c3 .

−r

Proof. Recall that since X is CAT(−1), it follows by an easy exercise in hyperbolic geometry (see, for instance, [PauPS, Lem. 2.5 (i)]), that for all x0 , y 0 , z 0 in X such that d(x0 , z 0 ) = d(y 0 , z 0 ), for every t0 ∈ [0, d(x0 , z 0 )], if x0t (respectively yt0 ) is the point on [x0 , z 0 ] (respectively [y 0 , z 0 ]) at distance t0 from x0 (respectively y 0 ), then 0

d(x0t , yt0 ) 6 e−t sinh d(x0 , y 0 ) .

(3.6)

Hence for all s ∈ [0, r], we have d(x−s , y−s ) 6 e−(r−s) sinh d(x−r , y−r ) = e−r+s sinh 1 . Thus Z

r

d(x−s , y−s ) e−2(s+t) ds Z +∞ 6 (sinh 1) e−2t e−r+s e−2s ds = (sinh 1) e−2t e−r ,

I2 =

0

0

and the result also follows from the first assertion of Claim 1. Claim 4: There exists a universal constant c6 > 0 such that Z ρ I3 = d(xs , ys ) e−2|s−t| ds 6 c6 e−t d(x, y) . 0

Proof. By Equation (3.6) and since d(x, y) 6 1, for every s ∈ [0, ρ], we have d(xs , ys ) 6 e−s sinh d(x, y) 6 (sinh 1) e−s d(x, y) .


59

Therefore Z t Z ρ I3 = d(xs , ys ) e−2(t−s) ds + d(xs , ys ) e−2(s−t) ds 0 t Z t Z +∞ 4 sinh 1 6 (sinh 1) d(x, y) e−2t es ds + e2t e−3s ds 6 d(x, y) e−t . 3 0 t Claim 5: We have r0

Z

d(xs , ys ) e−2|s−t| ds 6 (sinh 1) e2t−2ρ d(x, y)c3 .

I4 = ρ

Proof. By Equation (3.6), for every s ∈ [ρ, r0 ], we have 0

0

d(xs , ys ) 6 e−(r −s) sinh d(xr0 , yr0 ) = (sinh 1) e−r +s . Hence Z I4 =

r0

d(xs , ys ) e−2(s−t) ds 6 (sinh 1) e2t e−r

ρ

0

Z

r0

0

e−s ds 6 (sinh 1) e2t−ρ e−r .

ρ

The result follows from the second assertion of Claim 1.

Claim 6: There exists a constant c7 > 0 depending only on the lower bound of the curvature of X such that Z +∞ I5 = d(xs , ys ) e−2|s−t| ds 6 c7 e2t−2ρ d(x, y)c3 . r0

Proof. By the triangle inequality, for every s ∈ [r0 , +∞[ , we have d(xs , ys ) 6 d(xs , xr0 ) + d(xr0 , yr0 ) + d(yr0 , ys ) 6 2s + 1 . Hence Z

+∞

I5 = r0

d(xs , ys ) e−2(s−t) ds 6 e2t

Z

+∞

0

(2s + 1) e−2s ds = e2t (1 + r0 )e−2r .

r0

p By Equation (3.3), since sinh ρ > ρ > 21 d(x, y), c4 6 1 and c3 = d(x, y) 2/a 0 e−2r 6 e−2ρ c24 6 e−2ρ d(x, y)2c3 . 2 sinh ρ

1 2a ,

we have

Since c 6 ec for every c > 0 and d(x, y) 6 1, we have − ln d(x, y) 6 c13 d(x, y)−c3 . Hence by Equation (3.4), we have ρ 1 2c3 c3 I5 6 e2t−2ρ (a + 1)c24 d(x, y)2/a + (1 + ln c ) d(x, y) + d(x, y) . 5 c3 (2 sinh ρ)2/a Assuming, as we may, that a > 2, the map ρ0 7→ Since c3 =

1 2a

6

2 a,

this proves Claim 6.

ρ0 (sinh ρ0 )2/a

is bounded on [0, +∞[ .

60


Since d(x, y) 6 1, it follows from Equation (3.5) and from Claims 2 to 6 that there exists a constant c8 > 0 depending only on the lower bound of the curvature of X such that d(gt u, gt v) = I1 + I2 + I3 + I4 + I5 6 c8 d(x, y)c3 (e−t + e2t−2ρ ) . This proves Lemma 3.6, hence concludes the proof of Proposition 3.5.

Let D be a nonempty proper closed convex subset of X. The regularity ± 1 property in the Riemannian manifold case of the fibrations fD : UD± → ∂± D defined in Section 2.4, which will be needed in Section 12.3, is the Hölder continuity one, as proved in the following lemma (see also [PaP17b, Lem. 6]). Lemma 3.7. Assume that X is a Riemannian manifold with pinched negative cur± vature. The maps fD are H¨ older-continuous on any set of elements ` ∈ UD± such ± that d(π(`), π(fD (`))) is bounded. + − Proof. We prove the result for fD , the one for fD follows similarly. We will use the H¨ older-equivalent distances δ1 and δ2 , defined in the statement of Proposition 3.5 on the unit tangent bundle of X identified with G X, as explained previously.

v 0

D

v w

w0

T

v+ = w+

x y0

y x0 0 0 v+ = w+

+ Let v, v 0 ∈ T 1 X be such that d(v(0), v 0 (0)) 6 1, let w = fD (v) and w0 = 0 Let T = sup{t > 0 : sups∈[0,t] d(v(s), v (s)) 6 1}, so that δ1 (v, v 0 ) > e−T . 0 We may assume that T is finite; otherwise, v+ = v+ , whence w = w0 . Let x = v(T ) 0 0 0 and x = v (T ), which satisfy d(x, x ) 6 1. Let y (respectively y 0 ) be the closest point to x (respectively x0 ) on the geodesic ray defined by w (respectively w0 ). By convexity, since d(v(0), w(0)) and d(v 0 (0), w0 (0)) are bounded by a constant 0 0 c > 0 and since v+ = w+ , v+ = w+ , we have d(x, y) 6 c and d(x0 , y 0 ) 6 c. By the triangle inequality, we have d(y, y 0 ) 6 2c + 1, d(y, w(1)) > T − 2c − 1, and d(y 0 , w0 (1)) > T − 2c − 1. By convexity, and since closest point maps exponentially decrease the distances, there exists a constant c0 > 0 such that + 0 fD (v ).

δ2 (w, w0 ) = d(w(1), w0 (1)) 6 c0 d(y, y 0 ) e−(T −2c−1) 6 c0 (2c + 1) e2c+1 δ1 (v, v 0 ) . The result follows.

When X is an R-tree, we have a stronger version of Lemma 3.7, which will be needed in Section 12.6. Lemma 3.8. Assume that X is an R-tree. Let η, R > 0 be such that η 6 1 6 ln R, and let D be a nonempty closed convex subset of X. Then the restriction to ± 1 the dynamical neighbourhood Vη,±R (∂± D) of the fibration fD is (uniformly locally) Lipschitz, with constants independent of η.


61

1 Proof. We assume, for instance, that ± = +. Let `, `0 ∈ Vη,+R (∂+ D) and let w = + + 0 0 fD (`), w = fD (` ). + 1 1 Since the fiber over ρ ∈ ∂+ D of the restriction to Vη,+R (∂+ D) of fD is Vρ,+η, R ,1 0 there exist s, s0 ∈ ] − η, η[ such that gs ` ∈ B + (w, R) and gs `0 ∈ B + (w0 , R), so 0 that gs `(t) = w(t) and gs `0 (t) = w0 (t) for all t > ln R by the definition of the Hamenst¨ adt balls. Up to permuting ` and `0 , we assume that s0 > s.

`0 ` `(0)

s0 6 η `0 (0) ` w w0

6 ln R

`0

By (the proof of) Lemma 3.4 (1), there exists a constant cR > 0 depending only on R such that if d(`, `0 ) 6 cR and s00 = d(`(0), `0 (0)), then s00 = s0 −s and the 0 geodesic lines gs ` and gs `0 coincide at least on [− ln R − 1, ln R + 1] . In particular, we have, since |s|, |s0 | 6 η 6 1, w(ln R) = `(s + ln R) = `0 (s0 + ln R) = w0 (ln R) . Since the origin of w is the closest point on D to any point of w([0, +∞[), we hence have that w(t) = w0 (t) for all t ∈ [0, ln R]. Therefore (using Equation (2.8) for the last inequality), 0

Z

+∞ 0

d(w, w ) =

d(w(t), w (t)) e

−2 t

Z

ln R

= e2s

+∞

dt =

0

d(gs `(t), gs `0 (t)) e−2 t dt

ln R

Z

+∞

00

00

d(`(u), gs `0 (u)) e−2 u du 6 e2s d(`, gs `0 )

ln R+s

6e

2s

= e2s

00 d(`, `0 ) + d(`0 , gs `0 ) 6 e2s (d(`, `0 ) + s00 ) d(`, `0 ) + d(`(0), `0 (0)) ,

so that the result follows from Lemma 3.4 (2). Note that when X is (the geometric realisation of) a simplicial tree, we have s = s0 = s00 = 0, and the above computations simplify to give d(w, w0 ) 6 d(`, `0 ). 1 See

the end of Section 2.4.

62


For every metric space Z and for every α ∈ ]0, 1], the H¨ older norm of a bounded α-H¨ older-continuous function f : Z → R is kf kα = kf k∞ + kf k0α , where kf k0α =

sup x, y ∈ Z 0 1, b line that is isometric exactly on [0, T ]. and let α ∈ G X be a generalised geodesic For every generalised geodesic line ρ ∈ G X that is isometric exactly on [0, +∞[ such that ρ|[0,T ] = α|[0,T ] , we have d(α, ρ) =

e−2 T 1 if PD (ξ) 6= PD (ξ 0 ), e 2 dD (ξ, ξ 0 ) = dx (ξ, ξ 0 ) = e−d(x, y) 6 1 if PD (ξ) = PD (ξ 0 ) = x   and [x, ξ[ ∩ [x, ξ 0 [ = [x, y] . In particular, although in general it is not an actual distance on its whole domain (∂∞ X − ∂∞ D)2 , the map dD is locally a distance, and we can define with the standard formula the β-Hölder continuity of maps with values in (∂∞ X −∂∞ D, dD ) and the β-H¨ older norm of a function defined on (∂∞ X − ∂∞ D, dD ). In the next result, we endow ∂∞ X − ∂∞ D with the distance-like map dD . Proposition 3.10. Let X be a locally finite simplicial tree without terminal vertices, and let D be a proper nonempty simplicial subtree of X. Let X = |X|1 and 1 D = |D|1 be their geometric realisations. The homeomorphism ∂ + from ∂+ D onto older-continuous, and for all β ∈ ]0, 1] (∂∞ X − ∂∞ D) defined by ρ 7→ ρ+ is 12 -H¨ 1 and ψ ∈ Cbβ (∂∞ X − ∂∞ D), the map ψ ◦ ∂ + : ∂+ D → R is bounded and β2 -H¨ older4 See 5 See

Equation (2.2). Equation (2.12).

64


continuous, with β kψ ◦ ∂ + k β 6 1 + 2 2 +1 kψkβ . 2

1 Proof. Let us prove that for every ρ, ρ0 ∈ ∂+ D, if d(ρ, ρ0 ) 6 1, then ρ(0) = ρ0 (0), and √ 1 dD (ρ+ , ρ0+ ) = 2 d(ρ, ρ0 ) 2 . (3.9)

This proves that the map ∂ + is 12 -Hölder-continuous. We may assume that ρ 6= ρ0 . 1 Let ρ, ρ0 ∈ ∂+ D. If ρ(0) 6= ρ0 (0), then the images of ρ and ρ0 are disjoint and their connecting segment in the tree X joins ρ(0) and ρ0 (0); hence for every t ∈ [0, +∞[ , we have d(ρ(t), ρ0 (t)) = d(ρ(0), ρ0 (0)) + d(ρ(t), ρ(0)) + d(ρ0 (t), ρ0 (0)) > 1 + 2 t . Thus d(ρ, ρ0 ) =

Z

0

d(ρ(0), ρ0 (0)) e2 t dt +

−∞ Z 0

>

+∞

Z

d(ρ(t), ρ0 (t)) e−2 t dt

0

e2 t dt +

+∞

Z

−∞

(1 + 2t) e−2 t dt > 2

Z

0

+∞

e−2 t dt = 1 .

0

0

Assume that x = ρ(0) = ρ (0) and let n be the length of the intersection of ρ and ρ0 . Then dD (ρ+ , ρ0+ ) = dx (ρ+ , ρ0+ ) = lim

0

1

t→+∞

e 2 d(ρ(t), ρ (t))−t = e−n .

0

Furthermore, since ρ(t) = ρ (t) for t 6 n and d(ρ(t), ρ0 (t)) = 2(t − n) otherwise, we have, using the change of variables u = 2(t − n), Z +∞ Z +∞ du e−2 n 0 −2 t −2 n d(ρ, ρ ) = 2 (t − n) e dt = e u e−u = . 2 2 n 0 This proves Equation (3.9). Let β ∈ ]0, 1] and ψ ∈ Cbβ (∂∞ X − ∂∞ D). We have kψ ◦ ∂ + k∞ = kψk∞ , since + ∂ is a homeomorphism, and, by Equation (3.9), kψ ◦ ∂ + k0β = 2

|ψ ◦ ∂ + (ρ) − ψ ◦ ∂ + (ρ0 )|

sup

β

d(ρ, ρ0 ) 2

1 D, 0 0. If furthermore E 0 = R and f is bounded, then sup x,y∈E, x6=y

|f (x) − f (y)| |f (x) − f (y)| 2 = sup 6 α kf k∞ . α d(x, y)α d(x, y) x,y∈E, d(x, y)>

For all ∈ ]0, 1] and β > 0, we denote by Cb lc, β (E) the real vector space6 of -locally constant functions f : E → R endowed with the lc-norm of exponent β defined by kf k lc, β = −β kf k∞ . The above remark proves that if β ∈ ]0, 1], then kf kβ 6 3kf k lc, β , so that the inclusion map from Cb lc, β (E) into Cbβ (E) is continuous.

3.2

Potentials

In this book, a potential for Γ is a continuous Γ-invariant function Fe : T 1 X → R. The quotient function F : Γ\T 1 X → R of Fe is called a potential on Γ\T 1 X. Precomposing by the canonical projection G X → T 1 X, the function Fe defines a continuous Γ-invariant function from G X to R, also denoted by Fe, by Fe(`) = Fe(v` ) for every ` ∈ G X. 6 Note

that a linear combination of -locally constant functions is again an -locally constant function.

66


For all x, y ∈ X and every geodesic line ` ∈ G X such that `(0) = x and `(d(x, y)) = y, let Z y Z d(x,y) Fe = Fe(vgt ` ) dt . x

0

Note that for all t ∈ ]0, d(x, y)[ , the germ vgt ` is independent of the choice of such Ry a line `; hence x Fe does not depend on the extension ` of the geodesic segment [x, y]. The following properties are easy to check using the Γ-invariance of Fe and the basic properties of integrals: For all γ ∈ Γ, Z γy Z y e F = Fe ; (3.10) γx

x

for the antipodal map ι, x

Z

y

Z

Fe ◦ ι ;

Fe = y

(3.11)

x

and for all z ∈ [x, y], Z

y

Z

z

Fe = x

Z Fe +

x

y

Fe .

(3.12)

z

The period of a loxodromic isometry γ of X for the potential Fe is Z γx PerF (γ) = Fe x

for every x in the translation axis of γ. Note that for all α ∈ Γ and n ∈ N − {0}, we have PerF (αγα−1 ) = PerF (γ), PerF (γ n ) = n PerF (γ) and PerF (γ −1 ) = PerF ◦ι (γ) . (3.13) In trees, we have the following Lipschitz-type control on the integrals of the potentials along segments. Lemma 3.12. When Fe is constant or when X is an R-tree, for all x, x0 , y, y 0 ∈ X, we have Z y0 Z y Fe − Fe 6 d(x, x0 ) sup |Fe| + d(y, y 0 ) sup |Fe| . x

x0

π −1 ([x, x0 ])

π −1 ([y, y 0 ])

Proof. When Fe is constant, the result follows from the triangle inequality. Assume that X is an R-tree. Consider the case x = x0 . Let z ∈ X be such that [x, z] = [x, y] ∩ [x, y 0 ]. Using Equation (3.12) and the fact that d(y, z) + d(z, y 0 ) = d(y, y 0 ), the claim follows. The general case follows by combining this case x = x0 and a similar estimate for the case y = y 0 .

3.2. Potentials

67

Some form of uniform Hölder-type control of the potential, analogous to the Lipschitz-type one in the previous lemma, will be crucial throughout the present work. The following Definition 3.13 formalises this (weaker) assumption. Definition 3.13. The triple (X,Γ, Fe) satisfies the HC property (H¨ older-type control) e if F has subexponential growth when X is not an R-tree and if there exist κ1 > 0 and κ2 ∈ ]0, 1] such that for all x, y, x0 , y 0 ∈ X with d(x, x0 ), d(y, y 0 ) 6 1, we have Z

y

x

Z Fe −

y0

x0

e F 6 κ1 + 2 max |Fe| d(x, x0 )κ2 π −1 (B(x, 1)∪B(x0 , 1)) + κ1 + 2 −1 max 0 |Fe| d(y, y 0 )κ2 . π

(HC)

(B(y, 1)∪B(y , 1))

As mentioned to the authors by S. Tapie and B. Schapira, the HC property is a strengthening of the Bowen property of a potential, introduced by Bowen for diffeomorphisms and Franco for flows (see [Bowe2, p. 9]), and used in nonuniformly hyperbolic dynamics for instance by Climenga–Thompson and Burns–Climenga– Fisher–Thompson. By Equation (3.11), (X, Γ, Fe ◦ ι) satisfies the HC property if and only if (X, Γ, Fe) does. By the triangle inequality | d(x, y) − d(x0 , y 0 ) | 6 d(x, x0 ) + d(y, y 0 ), for every κ ∈ R, the triple (X, Γ, Fe + κ) satisfies the HC property (up to changing the constant κ1 ) if and only if (X, Γ, Fe) does. When X is assumed to be a Riemannian manifold with pinched sectional curvature, requiring the potentials to be Hölder-continuous as in [PauPS] is sufficient for having the HC property, as we will see below. Proposition 3.14. The triple (X, Γ, Fe) satisfies the HC property if one of the following conditions is satisfied: • • •

Fe is constant, X is an R-tree, X is a Riemannian manifold with pinched sectional curvature and Fe is H¨ older-continuous.

Proof. The first two cases are treated in Lemma 3.12, and we may take for them κ1 = 0 and κ2 = 1 in the definition of the HC property. The claim for Riemannian manifolds follows from the property of at most linear growth of H¨ older-continuous maps (see Remark 3.1) and from the following lemma, so that the constants κ1 > 0 and κ2 ∈ ]0, 1] of the HC property depend only on the H¨ older continuity constants of Fe and on the bounds on the sectional curvature of X. Lemma 3.15. If X is a Riemannian manifold with pinched sectional curvature and Fe is H¨ older-continuous, there exist two constants c1 > 0 and c2 ∈ ]0, 1] such that

68


for all x, y, z in X with d(x, y) 6 1, we have Z z Z z 1 Fe − Fe 6 c1 d(x, y)c2 + 2 d(x, y) 2 x

y

max

π −1 (B(x, 1)∪B(y, 1))

|Fe| .

The constants c1 and c2 depend only on the H¨ older continuity constants of Fe and the bounds on the sectional curvature of X. This lemma is similar to the second claim in [PauPS, Lem. 3.2], but the proof of this claim (and more precisely the proof of [PauPS, Lem. 2.3] used in the proof of [PauPS, Lem. 3.2]), which involves a different distance d on G X, does not extend with the present definition of d. Proof. By symmetry, we may assume that d(x, z) > d(y, z). The result is true if x = y; hence we assume that x 6= y. Let x0 be the point on [x, z] at distance d(y, z) from z. y x

x0

z p

The closest point p of y on [x, z] lies in [x0 , z] by convexity. Hence p d(x, x0 ) 6 d(x, p) 6 d(x, y) 6 d(x, y) 6 1 , since closest point maps do not increase distances and d(x, y) 6 1. Therefore Z x0 p Fe 6 d(x, x0 ) max |Fe| 6 d(x, y) max |Fe| . (3.14) −1 0 −1 π

x

Since

Rz x

Fe =

R x0

π

(B(x, 1))

Fe + x0 Fe (see Equation (3.12)), we have Z z Z z Z z Z Z z Fe − Fe 6 Fe − Fe +

x

x

([x, x ])

Rz

y

x0

y

x0

x

Fe .

(3.15)

p Assume first that d(y, z) = d(x0 , z) 6 12 d(x, y). We have Z z Z z Z x0 Z z Z z e e e e F − F 6 F + F + Fe x y x0 y x p 0 e e 6 d(x , z) −1max | F | + d(y, z) max | F | + d(x, y) −1max |Fe| π ([x0 , z]) π −1 ([y, z]) π (B(x, 1)) p 6 2 d(x, y) −1 max |Fe| , π

(B(x, 1)∪B(y, 1))

and Lemma 3.15 follows, for all c1 > 0 and c2 ∈ ]0, 1]. p Now assume that d(y, z) > 12 d(x, y). Since the distance function from a given point to a point varying on a geodesic line is convex, we have d(x0 , y) 6

3.2. Potentials

69

d(x, y). By Equations (3.14) and (3.15), we may therefore assume that x = x0 and prove that Z z Z z e F− Fe 6 c1 d(x, y)c2 x

y

for appropriate constants c1 , c2 . Since X is a Riemannian manifold, we identify G X and T 1 X with the usual unit tangent bundle of X as explained previously. Let u (respectively v) be the unit tangent pvector at x (respectively y) pointing towards z. Let ρ = d(x, z) = d(y, z) > 21 d(x, y) and t ∈ [0, ρ]. We apply Lemma 3.6, whose hypotheses are indeed satisfied. Since t ∈ [0, ρ] and d(x, y) 6 1, the term on the right-hand side of Equation (3.2) is bounded by 2 c8 . Since Fe is Hölder-continuous, let c > 0 and α ∈ ]0, 1] be the H¨ older continuity constants such that |Fe(u0 ) − Fe(v 0 )| 6 c d(u0 , v 0 )α for all 0 0 u , v ∈ T 1 X such that d(u0 , v 0 ) 6 2 c8 . Then, by Lemma 3.6, Z z Z ρ Z z Z ρ t t e e e e F − F = F (g u) − F (g v) dt 6 c d(gt u, gt v)α dt x

y

0

6 c c8 α d(x, y)αc3

0

Z 0

+∞

e−αt dt +

Z

ρ

e2αt−2αρ dt

−∞

3c α = c8 d(x, y)αc3 . 2α 3c α This concludes the proof of Lemma 3.15 with c2 = αc3 and c1 = 2α c8 , hence completes the proof of Proposition 3.14 in the Riemannian manifold case. f is a Riemannian manifold, then T 1 X is naturally Remark 3.16. (1) If X = M f. If the potential identified with the usual Riemannian unit tangent bundle of M 1f e f, it is F : T M → R is H¨ older-continuous for Sasaki’s Riemannian metricRon T 1 M y e a potential as in [Rue1] and [PauPS]. Furthermore, the definition of x F coincides with the one in these references. (2) The quotient function F is Hölder-continuous when Fe is Hölder-continuous. Let Fe, Fe∗ : T 1 X → R be potentials for Γ. We say that Fe∗ is cohomologous to Fe (see, for instance, [Livˇs]) if there exists a continuous Γ-invariant function e : T 1 X → R such that for every ` ∈ G X, the map t 7→ G(v e gt ` ) is differentiable G and d e gt ` ) . (3.16) Fe∗ (v` ) − Fe(v` ) = G(v dt t=0 When one is working with Hölder-continuous potentials, the regularity ree to also be Hölder-continuous. Note that the right-hand side of quirement is for G Equation (3.16) does not depend on the choice of the representative ` of its germ v` . In particular, PerF (γ) = PerF ∗ (γ) for every loxodromic isometry γ if Fe and Fe∗ are cohomologous potentials. A potential Fe is said to be reversible if Fe and Fe ◦ ι are cohomologous.

70

3.3


Poincaré series and critical exponents

Let us fix a potential Fe : T 1 X → R for Γ, and x, y ∈ X. The critical exponent of (Γ, F ) is the element δ = δΓ, F of the extended real line [−∞, +∞] defined by δ = lim sup n→+∞

1 ln n

X

e

R γy x

e F

.

γ∈Γ, n−1 δΓ, F ; (4) δΓ, F +κ = δΓ, F + κ for all κ ∈ R, and (Γ, F ) is of divergence type if and only if (Γ, F + κ) is of divergence type; (5) if Γ0 is a nonelementary subgroup of Γ, denoting by F 0 : Γ0 \T 1 X → R the map induced by Fe, then δΓ0 , F 0 6 δΓ, F ; (6) if δΓ < +∞, then δΓ +

inf

π −1 (C ΛΓ)

Fe 6 δΓ, F 6 δΓ +

sup π −1 (C ΛΓ)

Fe;

(7) δΓ, F > −∞; (8) the map Fe 7→ δΓ, F is convex, subadditive, and 1-Lipschitz for the uniform norm on the vector space of real continuous maps on π −1 (C ΛΓ);7 7 That

f∗ : T 1 X → R are potentials for Γ satisfying the HC property, inducing F, F ∗ : is, if Fe, F

3.3. Poincaré series and critical exponents

71

(9) if Γ00 is a discrete cocompact group of isometries of X such that Fe is Γ00 invariant, denoting by F 00 : Γ00 \T 1 X → R the map induced by Fe, then δΓ, F 6 δΓ00 , F 00 ; (10) if Γ is infinite cyclic, generated by a loxodromic isometry γ of X, then (Γ, F ) is of divergence type and PerF (γ) PerF ◦ι (γ) δΓ, F = max , . λ(γ) λ(γ) Proof. We give details of the proofs of the statements (1)–(7) and (10), which are the ones used in this book, only for the sake of completeness. The proofs from [PauPS, Lem. 3.3] generalise to the current setting, replacing the use of [PauPS, Lem. 3.2] by the following consequence of the HC property: there exist κ1 > 0 and κ2 ∈ ]0, 1] such that for every N ∈ N − {0}, for all x, y, x0 , y 0 ∈ X with d(x, x0 ), d(y, y 0 ) 6 N , we have Z

y

Z Fe −

x

y0

x0

Fe 6 2N κ1 + 2

max 0

π −1 (B(x, N )∪B(x , N )∪B(y, N )∪B(y 0 , N ))

|Fe|

.

(3.17) This is obtained from Equation (HC) by subdividing the segments [x, x0 ] and [y, y 0 ] into N subintervals of equal lengths at most 1 and using the triangle inequality. (1) For x0 , y 0 ∈ X, if N = max{dd(x, x0 )e, dd(y, y 0 )e}, by Equation (3.17) and by the Γ-invariance of Fe, we have, for every γ ∈ Γ, Z γy0 Z γy e| , Fe − Fe 6 c = 2 N κ1 + 2 −1 max | F 0 0 x

π

x0

(B(x, N )∪B(x ,N )∪B(y, N )∪B(y ,N ))

which is finite, since the continuous map Fe is bounded on compact subsets of T 1 X. Hence by the triangle inequality, we have, for every s ∈ R, 0

0

e−c−s d(x, x )−s d(y, y ) QΓ, F, x, y (s) 6 QΓ, F, x0 , y0 (s) 0

0

6 ec+s d(x, x )+s d(y, y ) QΓ, F, x, y (s) . The first claim of Assertion (1) follows. Let Fe∗ : T 1 X → R be a potential for Γ that is cohomologous to Fe. Let e G : T 1 X → R be a continuous Γ-invariant function satisfying Equation (3.16). Γ\T 1 X → R, and if δΓ, F , δΓ, F ∗ < +∞, then δΓ, tF +(1−t)F ∗ 6 t δΓ, F + (1 − t) δΓ, F ∗ for every t ∈ [0, 1], δΓ, F +F ∗ 6 δΓ, F + δΓ, F ∗ , | δΓ, F ∗ − δΓ, F | 6

sup v∈π −1 (C ΛΓ)

f∗ (v) − Fe(v) |. |F

72


e which is finite by continuity. By Γ-invariance, for every Let κx = maxπ−1 (x) |G|, γ ∈ Γ, we have κγy = κy . For every γ ∈ Γ, with ` ∈ G X any geodesic line such that `(0) = x and `(d(x, γy)) = γy, we have Z

γy

Fe∗ −

x

Z

γy

x

Z d(x,γy) d Z d(x,γy) d e gt gs ` ) ds = e gs ` ) ds Fe = G(v G(v dt t=0 ds 0 0 e e = G(v` ) − G(vgd(x,γy) ` ) 6 κx + κy .

Hence by the triangle inequality, we have, for every s ∈ R, e−κx −κy QΓ, F, x, y (s) 6 QΓ, F ∗ , x, y (s) 6 eκx +κy QΓ, F, x, y (s). The second claim of Assertion (1) follows. (2) This assertion follows from Equations (3.11) and (3.10), by the change of variable γ 7→ γ −1 in the summation of the Poincaré series. (3) This assertion is a standard argument of Poincaré series. For every s 6= δΓ, F , let = 12 |δΓ, F − s| > 0. First assume that s > δΓ, F . By the definition of the critical exponent δΓ, F , there exists N ∈ N such that for every integer n > N , X

R γy

e

x

e F

6 en(δΓ, F +) .

γ∈Γ, n−1 0 such that X X Q(s) 6 c + en(δΓ, F +)−s n+|s| = c + e|s| e−n < +∞ . n∈N

n∈N

Now assume that s < δΓ, F . By the definition of δΓ, F , there exists an increasing sequence (nk )k∈N in N such that for every k ∈ N, we have X

e

R γy x

e F

> enk (δΓ, F −) .

γ∈Γ, nk −1

X

enk (δΓ, F −)−s nk −|s|) = e−|s|

k∈N

X

e nk = +∞.

k∈N

This proves Assertion (3). Assertions (4) and (5) are immediate by Assertion (3), since with κ and Γ0 , F 0 as in these assertions, for every s ∈ R, we have QΓ, F +κ, x, y (s) = QΓ, F, x, y (s − κ) and QΓ0 , F 0 , x, y (s) 6 QΓ, F, x, y (s).

3.3. Poincaré series and critical exponents

73

(6) If x is a point in the convex hull C ΛΓ of the limit set of Γ, then for every γ ∈ Γ, the geodesic segment between x and γx is contained in C ΛΓ. Hence Z γx e−s 6 e − s) 6 d(x, γx) e−s . d(x, γx) inf F ( F sup F −1 π

(C ΛΓ)

π −1 (C ΛΓ)

x

This proves Assertion (6) by taking the exponential, summing over γ ∈ Γ with n − 1 < d(x, γy) 6 n, taking the logarithm, dividing by n, and taking the upper limit as n tends to +∞. (7) Let Γ0 be a nonelementary convex-cocompact subgroup of Γ (for instance, a Schottky subgroup of Γ, which exists since Γ is nonelementary). Denote by F 0 : Γ0 \T 1 X → R the map induced by Fe. Since |Fe | is Γ-invariant and bounded on compact subsets of T 1 X, by Assertion (6), we have δΓ0 , F 0 > −∞ as δΓ > 0. Assertion (7) then follows from Assertion (5). (10) For every s ∈ R, if x belongs to the translation axis of γ, we have, by Equation (3.13), X

e

R αx x

e−s) (F

=

α∈Γ

X n∈N

=

X n∈N

e

R γn x x

e−s) (F

+

X

e

R γ −n x x

e−s) (F

n∈N−{0}

en(PerF (γ)−s `(γ)) +

X

en(PerF ◦ι (γ)−s `(γ)) .

n∈N−{0}

Hence the series QΓ, F, x, x (s) converges if and only if PerF (γ) − s `(γ) < 0 and F (γ) PerF ◦ι (γ) PerF ◦ι (γ) − s `(γ) < 0. Let δ = max Per . By Assertion (3), letting λ(γ) , λ(γ) s tend to δΓ, F on the right gives that δΓ, F > δ, and letting s tend to δ on the left gives the reverse inequality. The above computation also gives that QΓ, F, x, x ( δ ) diverges, which proves Assertion (10) and concludes the proof of Lemma 3.17. Examples 3.18. (1) If δΓ is finite and Fe is bounded, then the critical exponent δ is finite by Lemma 3.17 (6). (2) If X is a Riemannian manifold with pinched negative curvature or if X has a compact quotient, then δΓ is finite. See, for instance, [Bou]. (3) There are examples of (X, Γ) with δΓ = +∞ (and hence δ = +∞ if Fe is constant), for instance, when X is the complete ideal hyperbolic triangle complex with three ideal triangles along each edge, see [GaP], and Γ its isometry group. Hence the finiteness assumption of the critical exponent is nonempty in general. For the type of results treated in this book, it is however natural and essential. We may replace upper limits by limits in the definition of the critical exponents, as follows.

74


Theorem 3.19. Assume that (X, Γ, Fe) satisfies the HC property. If c > 0 is large enough, then R γy X 1 e δ = lim ln ex F . n→+∞ n γ∈Γ, n−c 0, then δ = lim

n→+∞

1 ln n

R γy

X

e

x

e F

.

γ∈Γ, d(x,γy)6n

Proof. The proofs of [PauPS, Thm. 4.2 and Thm. 4.3], either using the original arguments of [Rob1] valid when Fe is constant or the supermultiplicativity arguments of [DaPS], extend, using the HC property (see Definition 3.13) instead of [PauPS, Lem. 3.2]. In what follows, we fix a potential Fe for Γ such that (X, Γ, Fe) satisfies the HC property. We define Fe+ = Fe and Fe− = Fe ◦ ι, we denote by F ± : Γ\T 1 X → R their induced maps, and we assume that δ = δΓ, F + = δΓ, F − is finite.

3.4

Gibbs cocycles

The (normalised) Gibbs cocycle associated with the group Γ and the potential Fe± ± is the map C ± = CΓ,F ± : ∂∞ X × X × X → R defined by (ξ, x, y) 7→

Cξ± (x, y)

Z = lim

t→+∞

ξt

(Fe± − δ) −

y

Z

ξt

(Fe± − δ) ,

x

where t 7→ ξt is any geodesic ray with endpoint ξ ∈ ∂∞ X. We will prove in Proposition 3.20 below that this map is well defined, that is, the above limits exist for all (ξ, x, y) ∈ ∂∞ X × X × X and they are independent of the choice of the geodesic rays t 7→ ξt . If Fe± = 0, then C − = C + = δΓ β, where β is the Busemann cocycle. If X is an R-tree, then Z p Z p Cξ± (x, y) = (Fe± − δ) − (Fe± − δ) , (3.18) y

x

where p ∈ X is the point for which [ p, ξ[ = [x, ξ[ ∩ [y, ξ[ ; in particular, the map ξ 7→ Cξ± (x, y) is locally constant on the totally discontinuous space ∂∞ X. The Gibbs cocycles satisfy the following equivariance and cocycle properties: for all ξ ∈ ∂∞ X and x, y, z ∈ X, and for every isometry γ of X, we have ± Cγξ (γx, γy) = Cξ± (x, y) and Cξ± (x, z) + Cξ± (z, y) = Cξ± (x, y) .

(3.19)

3.4. Gibbs cocycles

75

For every ` ∈ G X, for all x and y on the image of the geodesic line `, if `− , x, y, `+ are in this order on `, we have Z y C`−− (x, y) = C`++ (y, x) = −C`++ (x, y) = (Fe+ − δ) . (3.20) x

Proposition 3.20. Assume that (X, Γ, Fe) satisfies the HC property and that δ < +∞. (1) The maps C ± : ∂∞ X × X × X → R are well defined. (2) With the constants κ1 , κ2 of the HC property, for all x, y ∈ X and ξ ∈ ∂∞ X, if we assume that d(x, y) 6 1, then | Cξ± (x, y) | 6 κ1 + 2 |δ| + 2 −1 max |Fe| d(x, y)κ2 , π

(B(x, 1)∪B(y, 1))

and in general, if N = dd(x, y)e, then ± | Cξ (x, y) | 6 N κ1 + 2 |δ| + 2

max

π −1 (B(x, N )∪B(y, N ))

e |F | .

If X is an R-tree, then for all x, y ∈ X and ξ ∈ ∂∞ X, we have | Cξ± (x, y) | 6 d(x, y)

max

π −1 ([x, y])

|Fe± − δ| .

(3) The maps C ± : ∂∞ X × X × X → R are locally H¨ older-continuous (and locally Lipschitz when X is an R-tree). In particular, they are continuous. (4) For all r > 0, x, y ∈ X, and ξ ∈ ∂∞ X, if ξ belongs to the shadow Ox B(y, r) of the ball B(y, r) seen from x, then with the constants κ1 , κ2 of the HC property, if r 6 1, we have Z y ± e± − δ) 6 2 κ1 + 2 |δ| + 2 max |Fe| rκ2 , C (x, y) + ( F ξ −1 π

x

and in general, Z y ± ± e C (x, y) + (F − δ) 6 2 dre κ1 + 2 |δ| + 2 ξ x

If X is an R-tree, then Z y ± e± − δ) 6 2r C (x, y) + ( F ξ x

max

(B(y, 2))

max

π −1 (B(y, 2 dre))

π −1 (B(y, r))

e |F | .

|Fe± − δ| .

Proof. (1) The fact that Cξ± (x, y) is well defined when X is an R-tree follows from Equation (3.18).

76


When X is not an R-tree, let ρ : t 7→ ξt be any geodesic ray with endpoint ξ ∈ ∂∞ X, let t 7→ xt (respectively t 7→ yt ) be the geodesic ray from x (respectively y) to ξ. Let tx = βξ (x, ξ0 ) and ty = βξ (y, ξ0 ), so that the quantity β = ty − tx is equal to βξ (y, x) (which is independent of ρ), and for every t big enough, we have βξ (ξt , xt+tx ) = βξ (ξt , yt+ty ) = 0.

tx

x

ξ0

ξt t

xt+tx ξ yt+ty

ty y

Since X is CAT(−1), if t is large enough, then the distances d(ξt , xt+tx ) and d(ξt , yt+ty ) are at most one, and converge, in a nonincreasing way, exponentially Ry Rx fast to 0 as t → +∞. For s > 0, let as = y s (Fe± − δ) − x s−β (Fe± − δ) (which is independent of ρ). We have, using Equation (HC) as well as the fact that B(x0 , 1) ∪ B(y 0 , 1) ⊂ B(x0 , 2) if d(x0 , y 0 ) 6 1, Z

ξt

(Fe± − δ) −

y

Z

ξt

x

± e (F − δ) − at+ty

Z ξ t Z xt+tx Z yt+ty Z ξt = (Fe± − δ) − (Fe± − δ) + (Fe± − δ) − (Fe± − δ) y y x x 6 2 κ1 + 2 −1 max |Fe − δ| max{d(ξt , xt+tx ), d(ξt , yt+ty )}κ2 , π

(B(ξt , 2))

which converges to 0, since Fe has subexponential growth by the assumptions of the HC property. Hence in order to prove Assertion (1), we have only to prove that lims→+∞ as exists. For all s > t > |β|, we have, by the additivity of the integral along geodesics (see Equation (3.12)) and using again Equation (HC), Z ys Z xs−β ± ± e e |at − as | = (F − δ) − (F − δ) yt

xt−β

6

κ1 + 2 −1 max |Fe − δ| d(yt , xt−β )κ2 π (B(xt−β , 2)) + κ1 + 2 −1 max |Fe − δ| d(ys , xs+β )κ2 . π

(B(ys , 2))

3.4. Gibbs cocycles

77

Again by the subexponential growth of Fe, the above expression converges (exponentially fast, for future use) to 0 as t → +∞ uniformly in s; hence lims→+∞ as exists by a Cauchy-type argument. (2) Let (ξ, x, y) ∈ ∂∞ X × X × X. Assertion (2) of Proposition 3.20 follows from Equation (3.18) when X is an R-tree, since [x, y] = [x, p] ∪ [p, y], where p is the closest point to y on [x, ξ[ . When X is not an R-tree, the first claim of Assertion (2) follows immediately from the HC property of (X, Γ, Fe± − δ), and the second claim from this HC property and the same subdivision argument of the geodesic segment [x, y] into dd(x, y)e subintervals of equal lengths (at most 1), as in the proof of Equation (3.17). (3) Let (ξ, x, y), (ξ 0 , x0 , y 0 ) ∈ ∂∞ X × X × X. By the cocycle property (3.19), we have |Cξ± (x, y) − Cξ±0 (x0 , y 0 )| 6 |Cξ± (x, y) − Cξ±0 (x, y)| + |Cξ±0 (x, x0 )| + |Cξ±0 (y, y 0 )| . (3.21) First assume that X is an R-tree. Let K be a compact subset of X, and let K = inf e−d(x, x0 )−d(x, y) > 0 . x,y∈K

ξ0

y x

q

p

ξ

Let p, q be the points in X such that [x, ξ[ ∩ [y, ξ[ = [p, ξ[ and [x, ξ[ ∩ [x, ξ 0 [ = [x, q]. If dx0 (ξ, ξ 0 ) 6 K , then by the definition of the visual distance (see Equation (2.3)) and by Equation (2.4), we have e−d(x, q) = dx (ξ, ξ 0 ) 6 ed(x, x0 ) dx0 (ξ, ξ 0 ) 6 e−d(x, y) 6 e−d(x, p) . In particular q ∈ [p, ξ[ , so that [x, ξ 0 [ ∩ [y, ξ 0 [ = [p, ξ 0 [ . Thus by Equation (3.18), we have | Cξ± (x, y) − Cξ±0 (x, y) | = 0 . Therefore, by Equation (3.21) and by the R-tree case of Assertion (2), if dx0 (ξ, ξ 0 ) 6 K , if x, y ∈ K and d(x, x0 ), d(y, y 0 ) 6 1, then |Cξ± (x, y) − Cξ±0 (x0 , y 0 )| 6 d(x, x0 )

max

π −1 ([x, x0 ])

|Fe± − δ| + d(y, y 0 )

max

π −1 ([y, y 0 ])

|Fe± − δ| .

Since Fe is bounded on compact subsets of T 1 X, this proves that C ± is locally Lipschitz. Let us now consider the case that X is general. For all distinct ξ, ξ 0 ∈ ∂∞ X, let t 7→ ξt and t 7→ ξt0 be the geodesic rays from x0 to ξ and ξ 0 respectively. By the

78


end of the proof of Assertion (1), for every compact subset K of X, there exist a1 , a2 > 0 such that for every x, y ∈ K, we have for all η ∈ {ξ, ξ 0 } and t > 0, Z ηt Z ηt ± ± ± e e Cη (x, y) − (F − δ) − (F − δ) 6 a1 e−a2 t . y

x

Let T = − 21 ln dx0 (ξ, ξ 0 ). If T > 0, by the properties of CAT(−1) spaces (see Equation (3.6) for the second inequality), there exist two constants a3 , a4 >0 such 0 that d(ξ2T , ξ2T ) 6 a3 and d(ξT , ξT0 ) 6 a4 e−T . Hence if dx0 (ξ, ξ 0 ) 6 min a12 , 1 4 (so that T > 0 and d(ξT , ξT0 ) 6 1), we have, using Equation (HC) for the last inequality, | Cξ± (x, y) − Cξ±0 (x, y) | Z ξT Z ξT Z ξT0 Z ξT0 6 (Fe± − δ) − (Fe± − δ) + (Fe± − δ) − (Fe± − δ) + 2 a1 e−a2 T y y x x 6 2 κ1 + 2 max |Fe± − δ| d(ξT , ξT0 )κ2 + 2 a1 e−a2 T . π −1 (B(ξT , 2))

By the subexponential growth of Fe, there exists a5 > 0 such that | Cξ± (x, y) − Cξ±0 (x, y) | 6 a5 e−

κ2 2

T

+ 2 a1 e−a2 T 6 (a5 + 2 a1 ) dx0 (ξ, ξ 0 )min{

κ2 4

,

a2 2

}

.

We now conclude from Equation (3.21) and Assertion (2) as in the end of the above tree case that C ± is locally Hölder-continuous. (4) Let r > 0, x, y ∈ X, and ξ ∈ ∂∞ X be such that ξ ∈ Ox B(y, r). Let p be the closest point to y on [x, ξ[ , so that d(p, y) 6 r. By Equations (3.20) and (3.19), we have Z y Z p Z y ± ± ± ± ± ± e e e C (x, y) + (F − δ) = Cξ (x, y) − Cξ (x, p) − (F − δ) + (F − δ) ξ x x x Z p Z y ± ± ± e e 6 | Cξ (p, y)| + (F − δ) − (F − δ) . (3.22) x

x

First assume that X is an R-tree. Then by Assertion (2) and by Lemma 3.12, we deduce from Equation (3.22) that Z y ± (Fe± − δ) 6 2 d(y, p) Cξ (x, y) + x

max

π −1 ([y,p])

|Fe± − δ| 6 2r

max

π −1 (B(y, r))

|Fe± − δ| .

In the general case, the result then follows similarly from Equation (3.22) using Assertion (2) and the HC property if r 6 1 or Equation (3.17) in general.

3.5. Systems of conductances on trees and generalised electrical networks

3.5

79

Systems of conductances on trees and generalised electrical networks

Let (X, λ) be a locally finite metric tree without terminal vertices, let X = |X|λ be its geometric realisation, and let Γ be a nonelementary discrete subgroup of Isom(X, λ). Let e c : EX → R be a Γ-invariant function, called a system of (logarithmic) conductances for Γ. We denote by c : Γ\EX → R the function induced by e c, which we also call a system of conductances on Γ\X. Classically, an electric network8 (without sources or reactive elements) is a pair (G, ec ), where G is a graph and c : EG → R a function, such that c is reversible: c(e) = c(e) for all e ∈ EG; see, for example, [NaW], [Zem]. In this text, we do not assume our system of conductances e c to be reversible. In Chapter 6, we will even sometimes assume that the system of conductances is antireversible, that is, satisfying c(e) = − c(e) for every e ∈ EX. Two systems of conductances e c, ce0 : EX → R are said to be cohomologous if there exists a Γ-invariant function f : V X → R such that ce0 − e c = df , where for all e ∈ EX, we have df (e) =

f (t(e)) − f (o(e)) . λ(e)

Proposition 3.21. Let e c : EX → R be a system of conductances for Γ. There exists a potential Fe on T 1 X for Γ such that for all x, y ∈ V X, if (e1 , . . . , en ) is the edge path in X without backtracking such that x = o(e1 ) and y = t(en ), then Z

y

Fe = x

n X

e c(ei ) λ(ei ) .

i=1

Proof. Any germ v ∈ T 1 X determines a unique edge ev of the tree X, the first one into which it enters: if ` is any geodesic line whose class in T 1 X is v, the edge ev is the unique edge of X containing π(v) whose terminal vertex is the first vertex of X encountered at a positive time by `. The function Fe : T 1 X → R defined by 4e c(ev ) Fe(v) = min d(π(v), o(ev )), d(π(v), t(ev )) λ(ev )

(3.23)

is a (indeed Γ-invariant) potential on the R-tree X, with Fe(v) = 0 if π(v) ∈ V X. 8A

potential in this work is not the analogue of a potential in an electric network; we follow the dynamical systems terminology as in, for example, [PauPS].

80


Let us now compute

Ry x

Fe, for all x, y ∈ X. For every λ > 0, let ψλ : [0, λ] → R 2

2

2

be the continuous map defined by ψλ (t) = t2 if t ∈ [0, λ2 ] and ψλ (t) = λ4 − (λ−t) 2 if t ∈ [ λ2 , λ]. Let (e0 , e1 , . . . , en ) be the edge path in X without backtracking such that x ∈ e0 − {t(e0 )} and y ∈ en − {o(en )}. An easy computation shows that Z

y

Fe = x

n−1 X i=0

e c(ei ) λ(ei ) +

4e 4e c(en ) c(e0 ) ψλ(en ) d(y, o(en )) − ψλ(e0 ) d(x, o(e0 )) . λ(en ) λ(e0 )

If x and y are vertices, the expression simplifies to the sum of the lengths of the edges weighted by the conductances. We denote by Fec the potential defined by Equation (3.23) in the above proof, and by Fc : Γ\T 1 X → R the induced potential. Note that Fc is bounded if c is bounded. We call Fec and Fc the potentials associated with the system of conductances e c and c. This is by no means the unique potential with the property required in Proposition 3.21. The following result proves that the choice is unimportant. Given a potential Fe : T 1 X → R for Γ, let us define a map e cF : EX → R by 1 e cF : e 7→ e cF (e) = λ(e)

Z

t(e)

Fe .

(3.24)

o(e)

Note that e cF is Γ-invariant; hence it is a system of conductances for Γ. We denote by cF : Γ\EX → R the function induced by e cF : EX → R. Note that e cF +κ = e cF +κ for every constant κ ∈ R, that e cF is bounded if Fe is bounded, and that cFc = c by the above proposition. Proposition 3.22. (1) Every potential (resp. bounded potential) for Γ is cohomologous to a potential (resp. bounded potential) associated with a system of conductances for Γ. (2) If two systems of conductances ce0 and e c are cohomologous, then their associated potentials Fec0 and Fec are cohomologous. (3) If X has no vertex of degree 2, if two potentials Fe∗ and Fe for Γ are cohomologous, then the systems of conductances e cF ∗ and e cF for Γ are cohomologous. Hence if X has no vertex of degree 2, the map [F ] 7→ [cF ] from the set of cohomology classes of potentials for Γ to the set of cohomology classes of systems of conductances for Γ is bijective, with inverse [c] 7→ [Fc ]. Proof. (1) Let Fe be a potential for Γ, and let Fe∗ = FecF be the potential associated with the system of conductances e cF . Note that if Fe is bounded, then so is e cF by Equation (3.24); hence Fe∗ is bounded by Equation (3.23). For all e ∈ EX and t ∈ ]0, λ(e)[ , let ve, t ∈ T 1 X be the germ of any geodesic line passing at time 0 e : T 1 X → R be the map through the point of e at distance t from o(e). Let G

3.5. Systems of conductances on trees and generalised electrical networks

81

e defined by G(v) = 0 if π(v) ∈ V X and such that for all e ∈ EX and t ∈ ]0, λ(e)[ , Z t e e, t ) = G(v (Fe∗ (ve, s ) − Fe(ve, s )) ds . 0

R t(e) Fe = λ(e) e cF (e) by the construction of e cF and λ(e) e cF (e) = o(e) Fe∗ by e : T 1 X → R is continuous. Let ` be a geodesic line. Proposition 3.21, the map G e The map t 7→ G(vgt ` ) is obviously differentiable at time t = 0 if π(`) ∈ / V X, with ∗ e e derivative F (v` ) − F (v` ). By considering right and left derivatives of this map at R t(e) t = 0 and using the fact that o(e) (Fe − Fe∗ ) = 0 for every e ∈ EX, then by the continuity of Fe and Fe∗ at such a point, this is still true if π(`) ∈ V X. Hence Fe∗ and Fe are cohomologous, and this proves the first claim.

Since

R t(e) o(e)

(2) Assume that ce0 and e c are cohomologous systems of conductances for Γ, and let e f : V X → R be a Γ-invariant function such that ce0 −e c = df . Define G(v) = f (π(v)) if π(v) ∈ V X. For all e ∈ EX and t ∈ ]0, λ(e)[ , define Z t e G(ve, t ) = (Fec0 (ve, s ) − Fec (ve, s )) ds + f (o(e)) , 0

which is Γ-invariant. Its limit as t → 0 is f (o(e)) (independent of the edge e with given initial vertex), and its limit as t → λ(e) is, by the construction of Fec and Fec0 , λ(e) ce0 (e) − e c(e) + f (o(e)) = λ(e) df (e) + f (o(e)) = f (t(e)) e is (independent of the edge e with given terminal vertex). This proves that G e e 0 continuous. One checks as in Assertion (1) that Fc and Fc are cohomologous. (3) In order to prove the third claim, assume that Fe∗ and Fe are two cohomologous e : T 1 X → R be as in the definition of cohomologous potentials for Γ, and let G e for all elements v and potentials: see Equation (3.16). By the continuity of G, 0 1 0 e e 0 ), since (by the v in T X such that π(v) = π(v ) ∈ V X, we have G(v) = G(v assumption on the degrees of vertices) the two edges (possibly equal) into which v and v 0 enter can be extended to geodesic lines with a common negative subray. e x ) for every vx ∈ T 1 X such that Hence for every x ∈ V X, the value f (x) = G(v π(vx ) = x does not depend on the choice of vx . The map f : V X → R thus defined is Γ-invariant. With the above notation and by Equation (3.24), we hence have, for every e ∈ EX, Z λ(e) Z λ(e) 1 1 d e ∗ e e ∗ e cF (e) − e cF (e) = F (ve, t ) − F (ve, t ) dt = G(ve, t ) dt λ(e) 0 λ(e) 0 dt 1 e t(e) ) − G(v e o(e) ) = f (t(e)) − f (o(e)) = df (e) . = G(v λ(e) λ(e) Hence e cF ∗ and e cF are cohomologous.

82


Given a metric tree (X, λ), we define the critical exponent of a Γ-invariant system of conductances e c : EX → R (or of the induced system of conductances c : Γ\EX → R) as the critical exponent of (Γ, Fec ), where Fec is the potential for Γ associated with e c: δc = δΓ, Fc . By Proposition 3.22 (2) and Lemma 3.17 (1), this does not depend on the choice of a potential Fec satisfying Proposition 3.21.

Chapter 4

Patterson–Sullivan and Bowen–Margulis Measures with Potential on CAT(−1) Spaces Let X, x0 , Γ be as in the beginning of Section 2.1,1 and let Fe be a potential for Γ. From now on, we assume that the triple (X, Γ, Fe) satisfies the HC property of Definition 3.13 and that the critical exponent δ = δΓ, F ± is finite. In this chapter, we discuss geometrically and dynamically relevant measures on the boundary at infinity of X and on the space of geodesic lines G X. We extend the theory of Gibbs measures from the case of manifolds with pinched negative sectional curvature treated in [PauPS] 2 to CAT(−1) spaces with the HC property.

4.1

Patterson densities

A family (µ± x )x∈X of finite nonzero (positive Borel) measures on ∂∞ X whose support is ΛΓ is a (normalised) Patterson density for the pair (Γ, Fe± ) if ± γ∗ µ± x = µγx

(4.1)

for all γ ∈ Γ and x ∈ X, and if the following Radon–Nikodym derivatives exist for all x, y ∈ X and satisfy, for (almost) all ξ ∈ ∂∞ X, dµ± −Cξ± (x, y) x . ± (ξ) = e dµy

(4.2)

In particular, the measures µ± x are in the same measure class for all x ∈ X, and by Proposition 3.20, they depend continuously on x for the weak-star convergence 1 That

is, X is a geodesically complete proper CAT(−1) space, x0 ∈ X is a basepoint, and Γ is a nonelementary discrete group of isometries of X. 2 See also the previous works [Led, Ham2, Cou, Moh].


83

84

Chapter 4. Patterson–Sullivan and Bowen–Margulis Measures

of measures. Note that a Patterson density for (Γ, F ± ) is also a Patterson density for (Γ, F ± + s) for every s ∈ R, since the definition involves only the normalised potential Fe± − δ. If F = 0, we get the usual notion of a Patterson–Sullivan density (of dimension δΓ ) for the group Γ; see, for instance, [Pat2, Sul2, Nic, Coo, Bou, Rob2]. Proposition 4.1. There exists at least one Patterson density for the pair (Γ, Fe± ). Proof. The Patterson construction (see [Pat1], [Coo]) modified as in [Led] (with a multiplicative rather than additive parameter s), [Moh], and [PauPS, Section 3.6] (all in the Riemannian manifold case) gives the result, and we give a proof only for the sake of completeness. We start with an independent lemma, generalising [Pat2, Lem. 3.1] with a similar proof. Lemma 4.2. Let δ 0 ∈ R. Let (an )n∈N and (bn )n∈N be sequences of positive P real numbers such that limn→+∞ an = +∞ and the generalised Dirichlet series n∈N bn a−s n converges if s > δ 0 and diverges if s < δ 0 . Then there exists a positive nondecreasing map h on ]0, +∞[ such that

• for every > 0, there exists r0 > 0 such that h(t0 r0 ) 6 t0 h(r0 ) for all t0 > 1 and r0 > r0 ; P 0 • the series n∈N bn a−s n h(an ) converges if and only if s > δ . Proof. Let t0 = 0, t1 = 1, and h1 : ]0, 1] → [1, +∞[ the constant map 1. Let us define by induction on n ∈ N − {0} a positive real number tn and a continuous map hn : ]tn−1 , tn ] → [1, +∞[ . If tn and hn are constructed, let tn+1 ∈ R be such that X hn (tn ) −δ 0 +1/n bk ak >1, 1/n tn k∈N : tn 0, let n = d 1 e. Since ln h(t) is continuous and piecewise affine in ln t with slopes at most n1 on [tn , +∞[ , we have ln(h(t0 r0 )) − ln h(r0 ) 6 n1 ln t0 6 ln t0 if t0 > 1 and r0 > tn , which proves the first claim on h. We have 1/n X X X X ak −δ 0 −δ 0 bn an h(an ) = bk ak h(tn ) > 1 = +∞ . tn n∈N

n∈N

0

k∈N tn 0. Since by construction h(t) = O(t ) as t → +∞, If s > δ 0 , let = s−δ 2 there exists a constant C > 0 such that h(an ) 6 C an for every n ∈ N, and the

4.1. Patterson densities

85

P convergence of the series n∈N bn a−s n h(an ) follows from the convergence of the P generalised Dirichlet series n∈N bn a−δ− , thus proving the second claim on h. n Now, for every z ∈ X, recall that ∆z denotes the unit Dirac mass at z. Let h± : [0, +∞[ → ]0, +∞[ be a nondecreasing map such that • for every > 0, there exists r > 0 such that h± (t + r) 6 e t h± (r) for all t > 0 and r > r ; R γx P 0 e± • if Qx (s) = γ∈Γ e x (F −s) h± (d(x, γx0 )), then Qx (s) diverges if and only if the inequality s 6 δ holds. If (Γ, F ± ) is of divergence type, we may take h± = 1 constant. Otherwise, the existence of h± follows from Lemma 4.2.3 For all s > δ and x ∈ X, define the measure X R γx0 e± 1 e x (F −s) h± (d(x, γx0 )) ∆γx0 µ± x, s = Qx0 (s) γ∈Γ on X. By compactness for the weak-star topology of the space of probability measures on the compact space X∪∂∞ X, there exists a sequence (sk )k∈N in ]δ, +∞[ converging to δ such that the sequence of probability measures (µ± x0 , sk )k∈N weakstar converges to a probability measure µ± on X ∪ ∂ X. Since Q ∞ x0 (δ) = +∞ and x0 ± since the support of µx0 , s in X ∪ ∂∞ X is equal to Γx0 = Γx0 ∪ ΛΓ, the support of µ± x0 is contained in ΛΓ, hence equal to ΛΓ by minimality. The Radon–Nikodym derivative

dµ± x, s

dµ± x0 , s

is the map with support Γx0 defined by

dµ± x, s

(γx0 ) dµ± x0 , s

=e

e± −s)− 0 (F

e± −s) 0 (F

R γx

R γx

x

x0

h± (d(x, γx0 )) . h± (d(x0 , γx0 ))

(4.3)

For every k ∈ N, if d(x0 , γx0 ) is large enough, then h± (d(x, γx0 )) 6 h± (d(x, x0 ) + d(x0 , γx0 )) 6 e(sk −δ)d(x, x0 ) h± (d(x0 , γx0 )) . By the HC property, as k → +∞, the right-hand side of Equation (4.3) with s = sk ± converges to e−Cξ (x, x0 ) uniformly in ξ ∈ ΛΓ as γx0 tends to ξ. Therefore, as k → ± +∞, the measures µ± x, sk converge to a (finite nonzero) measure µx with support ΛΓ such that

dµ± x dµ± x0

±

(ξ) = e−Cξ

(x, x0 )

. Let γ ∈ Γ. Since γ∗ ∆z = ∆γ z for every z ∈ X,

± ± a change of variable in the summation defining µ± x, s gives that γ∗ µx, s = µγx, s . By ± ± the continuity of pushforwards of measures, we have γ∗ µx = µγx . By the cocycle properties of the Radon–Nikodym derivatives and the Gibbs cocycles, the family e± (µ± x )x∈X is a (normalised) Patterson density for (Γ, F ). 3 Let

Rγ x n 0 F e± x

(γn )n∈N be an enumeration of the elements of Γ, take an = ed(x,γn x0 ) and bn = e and then take h± = h ◦ exp for h the map given by Lemma 4.2.

,

86


We refer to Theorem 4.6 for the uniqueness up to scalar multiple of the Patterson density when (Γ, F ± ) is of divergence type and to [DaSU, Cor. 17.1.8] for a characterisation of the uniqueness when F = 0. The Patterson densities satisfy the following extension of the classical Sullivan shadow lemma (which gives the claim when Fe is constant, see [Rob2]), and its corollaries. If µ is a positive Borel measure on a metric space (X, d), the triple (X, d, µ) is called a metric measure space. A metric measure space (X, d, µ) is doubling if there exists c > 1 such that for all x ∈ X and r > 0, µ(B(x, 2 r)) 6 c µ(B(x, r)) . Note that up to changing c, the number 2 may be replaced by any constant larger than 1. See, for instance, [Hei] for more details on doubling metric measure spaces. We refer, for instance, to [DaSU, Ex. 17.4.12] for examples of nondoubling Patterson(–Sullivan) measures, and to [DaSU, Prop. 17.4.4] for a characterisation of the doubling property of the Patterson measures when Γ is geometrically finite and F = 0. A family ((X, µi , di ))i∈I of positive Borel measures µi and distances di on a common set X is called uniformly doubling if there exists c > 1 such that for all i ∈ I, x ∈ X, and r > 0, µi (Bdi (x, 2 r)) 6 c µi (Bdi (x, r)) . ± Lemma 4.3. Let (µ± x )x∈X be a Patterson density for the pair (Γ, F ), and let K be a compact subset of X.

(1) [Mohsen’s shadow lemma] If R is large enough, there exists C > 0 such that for all γ ∈ Γ and x, y ∈ K, R γy ± 1 R γy (Fe± −δ) e −δ) (F x ex 6 µ± . x Ox B(γy, R) 6 C e C

(2) For all x, y ∈ X, there exists c > 0 such that for every n ∈ N R γy ± X e e x (F −δ) 6 c . γ∈Γ : n−1 0 large enough, there exists C = C(R) > 0 such that for all γ ∈ Γ and all x, y ∈ K, ± µ± x (Ox B(γy, 5R)) 6 C µx (Ox B(γy, R)) .

(4) If Γ is convex-cocompact, then the metric measure space (ΛΓ, dx , µ± x ) is doubling for every x ∈ X, and the family ((ΛΓ, dx , µ± x ))x∈CΛΓ of metric measure spaces is uniformly doubling.

4.2. Gibbs measures

87

Proof. For the first assertion, the proof of [PauPS, Lem. 3.10] (see also [Cou, Lem. 4] with the multiplicative rather than additive convention, as well as [Moh]) extends, using Proposition 3.20 (2), (4) instead of [PauPS, Lem. 3.4 (i), (ii)]. The second assertion is similar to that of [PauPS, Lem. 3.11 (i)], and the proof of the last two assertions is similar to that of [PauPS, Prop. 3.12], using Lemma 2.2 instead of [HeP4, Lem. 2.1]. The uniformity in the last assertion follows from the compactness of Γ\CΛΓ and the invariance and continuity properties of the Patterson densities.

4.2

Gibbs measures

± We fix from now on two Patterson densities (µ± x )x∈X for the pairs (Γ, F ). The Gibbs measure m e F on G X (associated with this ordered pair of Patterson densities) is the measure m e F on G X given by the density

dm e F (`) = e

C`− (x0 , `(0)) + C`+ (x0 , `(0)) −

+

+ dµ− x0 (`− ) dµx0 (`+ ) dt

(4.4)

in Hopf’s parametrisation with respect to the basepoint x0 . The Gibbs measure m e F is independent of x0 by Equations (4.2) and (3.19). Hence it is invariant under the action of Γ by Equations (4.1) and (3.19). It is invariant under the geodesic flow by construction (and the invariance of Lebesgue’s measure under translation). Thus,4 it defines a measure mF on Γ\G X that is invariant under the quotient geodesic flow, called the Gibbs measure on Γ\G X (associated with the above ordered pair of Patterson densities). If F = 0 and the Patterson densities − (µ+ x )x∈X and (µx )x∈X coincide, then the Gibbs measure mF coincides with the Bowen–Margulis measure mBM on Γ\G X (associated with this Patterson density); see, for instance, [Rob2]. Remark 4.4. (i) The (positive Borel) measure given by the density −

dλ(ξ, η) = eCξ

(x0 , p) + Cη+ (x0 , p)

+ dµ− x0 (ξ) dµx0 (η)

(4.5)

2 on ∂∞ X is (by the same arguments as above) independent of p ∈ ]ξ, η[ , locally 2 finite, and invariant under the diagonal action of Γ on ∂∞ X. It is a geodesic current for the action of Γ on the Gromov-hyperbolic proper metric space X in the sense of Ruelle–Sullivan–Bonahon; see, for instance, [Bon] and references therein.

(ii) Another parametrisation of G X, also depending on the choice of the basepoint 2 x0 in X, is the map from G X to ∂∞ X × R sending ` to (`− , `+ , s), where now s = β`− (π(`), x0 ) (one may also use the different time parameter s = β`+ (x0 , π(`))). 2 For every (η, ξ) in ∂∞ X, let pη, ξ be the closest point to x0 on the geodesic line between η and ξ. 4 See,

for instance, [PauPS, §2.6] for the precautions in order to push locally forward an invariant measure by an orbifold covering, since the group Γ does not necessarily act freely on G X.

88


x0 s `−

p`− , `+

`(0)

`+

t

For every ` ∈ G X, with (`− , `+ , t) the original Hopf parametrisation, since s − t = β`− (`(0), x0 ) − β`− (`(0), p`− ,`+ ) = β`− (p`− ,`+ , x0 ) depends only on `− and `+ , the measures + − + dµ− x0 (`− ) dµx0 (`+ ) dt and dµx0 (`− ) dµx0 (`+ ) ds

are equal. Hence using the above variant of Hopf’s parametrisation does not change the Gibbs measures m e F and mF . (iii) Since the time-reversal map ι is (`− , `+ , t) 7→ (`+ , `− , −t) in Hopf’s coordinates, the measure ι∗ m e F is the Gibbs measure on G X associated with the switched − pair of Patterson densities (µ+ ) , (µ ) x x∈X x x∈X (and similarly on Γ\G X). The Gibbs property of Gibbs measures. Let us now indicate why the terminology of Gibbs measures is indeed appropriate. This explanation will be the aim of Proposition 4.5, but we need to give some definitions first. For all ` ∈ G X and r > 0, T, T 0 > 0, the dynamical (or Bowen) ball around ` is n o B(`; T, T 0 , r) = `0 ∈ G X : sup d(`(t), `0 (t)) < r . t∈[−T 0 ,T ]

Bowen balls have the following invariance properties: for all s ∈ [−T 0 , T ] and γ ∈ Γ, gs B(`; T, T 0 , r) = B(gs `; T − s, T 0 + s, r) and γB(`; T, T 0 , r) = B(γ`; T, T 0 , r) . The following inclusion properties of the dynamical balls are immediate: If r 6 s, T > S, T 0 > S 0 , then B(`; T, T 0 , r) is contained in B(`; S, S 0 , s). The dynamical balls are almost independent of r: For all r0 > r > 0, there exists Tr, r0 > 0 such that for all ` ∈ G X and T, T 0 > 0, the dynamical ball B(`; T + Tr, r0 , T 0 + Tr, r0 , r0 ) is contained in B(`; T, T 0 , r). This follows from the properties of long geodesic segments with endpoints at bounded distance in a CAT(−1) space. For every ` ∈ Γ\G X, let us define B(`; T, T 0 , r0 ) as the image by the canonical e T, T 0 , r0 ), for any preimage è of ` in G X. projection G X → Γ\G X of B(`; A (gt )t∈R -invariant measure m0 on Γ\G X satisfies the Gibbs property for the potential F with constant c(F ) ∈ R if for every compact subset K of Γ\G X,

4.2. Gibbs measures

89

there exist r > 0 and cK, r > 1 such that for all large enough T, T 0 > 0, for every 0 ` ∈ Γ\G X with g−T `, gT ` ∈ K, we have 1 cK, r

6

m0 (B(`; T, T 0 , r)) e

RT

−T 0

(F (vgt ` )−c(F )) dt

6 cK, r .

We refer to [PauPS, Sect. 3.8] for equivalent variations on the definition of the Gibbs property. The following result shows that the Gibbs measures indeed satisfy the Gibbs property on the dynamical balls of the geodesic flow, thereby justifying the name. We refer, for instance, to [PauPS, Sect. 3.8] for explanations of the connection with symbolic dynamics mentioned in the introduction. See also Proposition 4.14 for a discussion of the case in which X is a simplicial tree; here the correspondence with symbolic dynamics is particularly clear. Proposition 4.5. Let mF be the Gibbs measure on Γ\G X associated with a pair e± of Patterson densities (µ± x )x∈X for (Γ, F ). Then mF satisfies the Gibbs property for the potential F , with constant c(F ) = δ. Proof. The proof is similar to that of [PauPS, Prop. 3.16] (in which the key Lemma 3.17 uses only CAT(−1) arguments), up to replacing [PauPS, Lem. 3.4 (1)] by Proposition 3.20 (2). The Hopf–Tsuji–Sullivan–Roblin theorem. The basic ergodic properties of the Gibbs measures are summarised in the following result. The case in which Fe is constant is due to [Rob2]; see also [BuM, §6]. Theorem 4.6 (Hopf–Tsuji–Sullivan–Roblin). The following conditions are equivalent: (i) The pair (Γ, F ) is of divergence type. (ii)− The conical limit set of Γ has positive measure with respect to µ− x for some (equivalently every) x ∈ X. (ii)+ The conical limit set of Γ has positive measure with respect to µ+ x for some (equivalently every) x ∈ X. 2 + 2 X , Γ) is ergodic for some (equival(iii) The dynamical system (∂∞ X, (µ− x ⊗ µx )|∂∞ ently every) x ∈ X. 2 + 2 X , Γ) is conservative for some (iv) The dynamical system (∂∞ X, (µ− x ⊗ µx )|∂∞ (equivalently every) x ∈ X.

(v) The dynamical system (Γ\G X, mF , (gt )t∈R ) is ergodic. (vi) The dynamical system (Γ\G X, mF , (gt )t∈R ) is conservative. If one of the above conditions is satisfied, then (1) the measures µ± x have no atoms for any x ∈ X, + (2) the diagonal of ∂∞ X × ∂∞ X has measure 0 for µ− x ⊗ µx ,

(3) the Patterson densities (µ± x )x∈X are unique up to a scalar multiple, and

90


(4) for all x, y ∈ X, as n → +∞, max

e

R γy x

e± F

= o(eδ n ) .

γ∈Γ, n−1 t0 , we have U ∩ ht V 6= ∅. We have the following result, due to [Bab1, Thm. 1] in the manifold case, with developments by [Rob2] when Fe = 0, and by [PauPS, Thm. 8.1] for manifolds with pinched negative curvature. Theorem 4.9. If the Gibbs measure is finite, then the following assertions are equivalent: (1) the geodesic flow of Γ\X is mixing for the Gibbs measure, 7 See also [Pau4] for the case of simplicial trees and [DaSU, Thm. 12.4.5] for greater generality on X.

92


(2) the geodesic flow of Γ\X is topologically mixing on its nonwandering set, which is the quotient under Γ of the space of geodesic lines in X both of whose endpoints belong to ΛΓ. (3) the length spectrum of Γ on X is not contained in a discrete subgroup of R.

In the manifold case, the third assertion of Theorem 4.9 is satisfied, for example, if Γ has a parabolic element, if ΛΓ is not totally disconnected (hence if Γ\X is compact), or if X is a surface or a (rank-one) symmetric space; see, for instance, [Dal1, Dal2]. Error terms for the mixing property will be described in Chapter 9. The above result holds for the continuous-time geodesic flow when X is a metric tree. See Theorem 4.17 for a version of this theorem for the discrete-time geodesic flow on simplicial trees. At least when X is an R-tree and Γ is a uniform lattice (so that Γ\X is a finite metric graph), we have a stronger result under additional regularity assumptions; see Section 9.3. Bowen–Margulis measure computations in locally symmetric spaces. Assume in this subsection that the potential Fe is zero. The next result, Proposition 4.10, gathers computations done in [PaP16, PaP17a] of the Bowen–Margulis measures mBM when X is a real or complex hyperbolic space, and Γ is a lattice. We refer to [PaP20] for the case of quaternionic hyperbolic spaces. We begin by giving the notation necessary in order to state Proposition 4.10. When X is a complete simply connected Riemannian manifold with dimension m > 2 and sectional curvature at most −1, we endow the usual unit tangent bundle T 1 X with Sasaki’s Riemannian metric.8 Its Riemannian measure d volT 1 X , called Liouville’s measure, disintegrates (equivariantly with respect to Γ) under the fibration π : T 1 X → X over the Riemannian measure d volX of X, as Z d volT 1 X = d volTx1 X d volX (x) , x∈X

where d volTx1 X is the spherical measure on the fiber Tx1 X of π above x ∈ X. In particular, Vol(T 1 (Γ\X)) = Vol(Sm−1 ) Vol(Γ\X) . Assume furthermore that X is a symmetric space, and that Γ is a lattice in Isom(X). In particular, Γ is geometrically finite and the critical exponent of the stabiliser in Γ of every bounded parabolic fixed point of Γ is strictly less than the critical exponent of Γ; see, for instance, [Dal1, Dal2]. Then the Patterson density is independent of Γ and uniquely defined up to a multiplicative constant by Theorem + 4.8 (2) and Lemma 4.7 (2). We take µ− x = µx for every x ∈ X and we will denote this measure simply by µx . For homogeneity reasons, the Bowen–Margulis measure 8 See

Section 3.1.

4.2. Gibbs measures

93

of Γ on X is proportional to the Liouville measure volT 1 X , and the main point of Proposition 4.10 is to compute the proportionality constant. Let n > 2. We endow Rn−1 with its usual Euclidean norm k · k and its usual Lebesgue measure λn−1 . We use the upper half-space model for the real hyperbolic space HnR of dimension n, that is, HnR = {(x1 , . . . , xn ) ∈ Rn : xn > 0} endowed with the Riemannian metric 1 ds2HnR = 2 (dx21 + · · · + dx2n ) . xn We identify Rn−1 with Rn−1 × {0} in Rn , and again denote by k · k the usual Euclidean norm on Rn . The boundary at infinity of HnR is ∂∞ HnR = Rn−1 ∪ {∞}. Assuming that Γ is a lattice in Isom(HnR ), we normalise its Patterson density so that in the ball model of HnR with center 0, the measure µ0 is the spherical measure on the space at infinity Sn−1 . Let n > 2. We refer to [Gol] and [PaP17a, §3] for background on the complex Pn−1 0 hyperbolic n-space HnC . We denote by ζ · ζ 0 = i=1 ζi ζi the standard Hermitian product and by dζ the standard Lebesgue measure on Cn−1 . We denote by Heis2n−1 the Heisenberg group of dimension 2n − 1, which is the real Lie group structure on Cn−1 × R with law (ζ, u)(ζ 0 , u0 ) = (ζ + ζ 0 , u + u0 + 2 Im ζ · ζ 0 ) . We endow Heis2n−1 with the usual left-invariant Haar measure dλ2n−1 (ζ, u) = dζ du and with the Cygan distance9 dCyg , which is the unique left-invariant distance 1 on Heis2n−1 with dCyg ((ζ, u), (0, 0)) = (|ζ|4 + u2 ) 4 . We use the Siegel domain model of the complex hyperbolic space HnC of dimension n, normalised to have maximum sectional curvature −1, hence to be CAT(−1). This is the manifold Heis2n−1 × ]0, +∞[ endowed with the Riemannian metric given, in the horospherical coordinates (ζ, u, t) ∈ Cn−1 × R× ]0, +∞[ , by 2 1 ds2HnC = 2 dt2 + du + 2 Im dζ · ζ + 4 t dζ · dζ , 4t so that its volume form is 1 d volHnC (ζ, u, t) = n+1 dζ du dt . 4t Note that the action of Heis2n−1 on HnC = Heis2n−1 × ]0, +∞[ by left translations on the first factor, preserving the second one, is isometric. We identify Heis2n−1 with Heis2n−1 ×{0}, and we endow Heis2n−1 ×[0, +∞[ with the distance dCyg extending the Cygan distance on Heis2n−1 , defined by 1/2 dCyg ((ζ, u, t), (ζ 0 , u0 , t0 )) = |ζ − ζ 0 |2 + |t − t0 | + i(u − u0 + 2 Im ζ · ζ 0 ) . 9 See

[Gol, page 160]. It is called the Kor´ anyi distance by many people working in sub-Riemannian geometry, though Kor´ anyi [Kor] attributes it to Cygan [Cyg].

94


The space at infinity ∂∞ HnC of HnC is the Alexandrov compactification Heis2n−1 ∪{∞}, so that the extension at infinity of the isometric action of Heis2n−1 on HnC fixes ∞ and is the left translation on Heis2n−1 . We denote by H∞ = Heis2n−1 ×[1, +∞[ the horoball of HnC centered at ∞ whose boundary contains the point (0, 0, 1). Assuming that Γ is a lattice in Isom(HnC ), we normalise its Patterson density (µx )x∈HnC as follows. If µH∞ is the measure defined in Proposition 7.2 associated with the horoball H∞ , then dµH∞ (ζ, u) = dλ2n−1 (ζ, u) = dζ du . This is possible, since µH∞ is invariant under the isometric action of Heis2n−1 on HnC , which preserves H∞ , hence is a left-invariant Haar measure on ∂∞ HnC −{∞} = Heis2n−1 . Since the arguments of the following result are purely computational and rather long, we do not copy them in this book, but we refer respectively to the proofs of [PaP16, Eq. (19), Eq. (21), Prop. 10] and [PaP17a, Lem. 12 (i), (ii), (iii)]. Analogous computations can be done when X is the quaternionic hyperbolic n-space HnH ; see [PaP20, § 7]. Proposition 4.10. (1) Let Γ be a lattice in Isom(HnR ), with Patterson density (µx )x∈HnR normalised as above. For all x = (x0 , . . . , xn ) in HnR , ξ ∈ ∂∞ HnR − {∞} and v ∈ T 1 HnR such that v± 6= ∞, we have n−1 2xn (i) dµx (ξ) = kx−ξk dλn−1 (ξ), 2 (ii) using a Hopf parametrisation v 7→ (v− , v+ , s), dm e BM (v) =

22(n−1) dλn−1 (v− ) dλn−1 (v+ ) dt , kv+ − v− k2(n−1)

m e BM = 2n−1 VolT 1 HnR ,

(iii) and in particular,

kmBM k = 2n−1 Vol(Sn−1 ) Vol(Γ\HnR ) . (2) Let Γ be a lattice in Isom(HnC ), with Patterson density (µx )x∈HnC normalised as above. For all x = (ζ, u, t) in HnC , (ξ, r) ∈ ∂∞ HnC − {∞} and v ∈ T 1 HnC such that v± 6= ∞, we have tn (i) dµx (ξ, r) = dξ dr ; dCyg (x, (ξ, r))4n (ii) using a Hopf parametrisation v 7→ (v− , v+ , s), dm e BM (v) =

dλ2n−1 (v− ) dλ2n−1 (v+ ) ds ; dCyg (v− , v+ )4n

4.2. Gibbs measures

(iii)

95

m e BM =

1 22n−2

volT 1 HnC ,

and in particular kmBM k =

πn Vol(Γ\HnC ) . 22n−3 (n − 1)!

On the cohomological invariance of Gibbs measures. We end this section with an elementary remark on the independence of Gibbs measures upon replacement of the potential F by a cohomologous one. Remark 4.11. Let Fe∗ : T 1 X → R be a potential for Γ cohomologous to Fe and satisfying the HC property. As usual, let Fe∗+ = Fe∗ and Fe∗− = Fe∗ ◦ ι, and e : T 1 X → R be a continuous Γlet F ∗ : Γ\T 1 X → R be the induced map. Let G e gt ` ) is differentiable invariant function such that for every ` ∈ G X, the map t 7→ G(v d ∗ e e e and F (v` ) − F (v` ) = dt t=0 G(vgt ` ). Furthermore, assume that X is an R-tree or that G is uniformly continuous (for instance, Hölder-continuous). For all x ∈ X and ξ ∈ ∂∞ X, let `x, ξ be any geodesic line with footpoint `x, ξ (0) = x and positive endpoint (`x, ξ )+ = ξ, and let `ξ, x be any geodesic e ` ) is line with `ξ, x (0) = x and origin (`ξ, x )− = ξ. Note that the value G(v x, ξ e e ` ). independent of the choice of `x, ξ , by the continuity of G, and similarly for G(v ξ, x e we have In particular, for all γ ∈ Γ, by the Γ-invariance of G, e ` −1 ) = G(v e ` e ◦ ι (v` ) = G(v e ` ). G(v ) and G γx, ξ x, ξ ξ, x x, γ ξ

(4.6)

e ∗ = − G◦ι, e Note that Fe∗− = Fe∗ ◦ι and Fe− = Fe ◦ι are cohomologous, since if G then for every ` ∈ G X, we have d e gt ι` ) Fe∗ ◦ ι(v` ) − Fe ◦ ι(v` ) = Fe∗ (vι` ) − Fe(vι` ) = G(v dt t=0 d e g−t ` ) = d e ∗ (vgt ` ) . = G(ιv G dt t=0 dt t=0 As already seen in Lemma 3.17 (1) and (2), the critical exponent δΓ, F ∗± is equal to the critical exponent δΓ, F ± , and independent of the choice of ±, and we denote by δ the common value in the definition of the Gibbs cocycle. ± Let us prove that if C ∗± = CΓ, F ∗± is the Gibbs cocycle associated with ∗± ∗± ± (Γ, F ), then C and C are cohomologous: e ` ) − G(v e ` ), Cξ∗+ (x, y) − Cξ+ (x, y) = G(v x, ξ y, ξ

(4.7)

e ∗ (v` ) − G e ∗ (v` ) . Cξ∗− (x, y) − Cξ− (x, y) = G x, ξ y, ξ

(4.8)

and similarly

e ∗ is We prove only the first equality; the second one follows similarly, noting that G e uniformly continuous if G is, since ι is isometric. For all x, y in X and ξ ∈ ∂∞ X, let

96


t 7→ ξt be a geodesic ray with point at infinity ξ, let at = d(x, ξt ), let bt = d(y, ξt ), and for z = x, y, let `z, ξt be any geodesic line with footpoint z passing through ξt . Then Z

ξt

(Fe∗+ − δ) −

y

Z

ξt

x

Z

ξt

= y

ξt

Z (Fe∗+ − δ) −

(Fe∗+ − Fe+ ) −

(Fe+ − δ) −

y

Z

ξt

Z

ξt

(Fe+ − δ)

x

(Fe∗+ − Fe+ )

x bt

Z at d e d e G(vgs `y, ξt ) ds − G(vgs `x, ξt ) ds dt dt 0 0 e ` e ` e gbt ` e gat ` = G(v ) − G(v ) + G(v ) − G(v ) . x, ξt y, ξt x, ξt y, ξt Z

=

As t goes to +∞, the first term of this series of equalities converges to Cξ∗+ (x, y) − Cξ+ (x, y) by the definition of the Gibbs cocycle (see Section 3.4). By continuity, e ` e ` e ` ) and G(v e ` ) respectively. If X is an G(v ) and G(v ) converge to G(v y, ξt x, ξt y, ξ x, ξ R-tree, then if t is large enough, we have vgbt `y, ξt = vgat `x, ξt ; hence Equation (4.7) e since vgbt ` follows. Otherwise, by the uniform continuity of G, and vgat ` are y, ξt

x, ξt

uniformly arbitrarily close as t tends to 0 by the CAT(−1) property, the result also follows. ± Let (µ± x )x∈X be a Patterson density for (Γ, F ). In order to simplify the e+ = G e and G e− = G e ∗ . The family of measures (µ∗± notation, let G x )x∈X defined by setting, for all x ∈ X and ξ ∈ ∂∞ X, e± (v`

−G dµ∗± x (ξ) = e

x, ξ

)

dµ± x (ξ) ,

(4.9)

is also a Patterson density for (Γ, F ∗± ). Indeed, the equivariance property (4.1) ± for (µ∗± x )x∈X follows from the one for (µx )x∈X and from Equation (4.6). The ab± solutely continuous property (4.2) for (µ∗± x )x∈X follows from the one for (µx )x∈X and Equations (4.7) and (4.8). Assume that the Patterson density for (Γ, F ∗± ) defined by Equation (4.9) is chosen in order to construct the Gibbs measure m e F ∗ for (Γ, F ∗ ) on G X. Then using • Hopf’s parametrisation with respect to the basepoint x0 and Equation (4.4) with F replaced by F ∗ for the first equality, • Equations (4.7), (4.8), (4.9) and cancellations for the second equality, e∗ = − G e ◦ ι and again Equation (4.4) for the third equality, • the definition of G • Equation (4.6) and the fact that we may choose ``− , `(0) = ` and ``(0), `+ = ` for the last equality,

4.3. Patterson densities for simplicial trees

97

we have dm e F ∗ (`) = e =e

C`∗− (x0 , `(0)) + C`∗+ (x0 , `(0)) −

+

e∗ (v` C`− (x0 , `(0)) −G `(0), `− −

=e

e e G◦ι(v ``(0), ` ) −G(v``(0), ` )

=e

−

e ` )−G(v e `) G(v

+

∗+ dµ∗− x0 (`− ) dµx0 (`+ ) dt

e ` ) + C`+ (x0 , `(0)) −G(v `(0), `+ +

)

+ dµ− x0 (`− ) dµx0 (`+ ) dt

dm e F (`)

dm e F (`) ,

whence m e F∗ = m eF. In particular, since the Gibbs measure, when finite, is independent up to a multiplicative constant on the choice of the Patterson densities by Corollary 4.7, we have that mF is finite if and only if mF ∗ is finite, and then mF ∗ mF = . kmF ∗ k kmF k

4.3

(4.10)

Patterson densities for simplicial trees

In this section and the following one, we specialise and modify the general framework of the previous sections to treat simplicial trees. Recall10 that a simplicial tree X is a metric tree whose edge length mapbis constant equal to 1. The time 1 map of the geodesic flow (gt )t∈R on the space G X of all generalised geodesic lines of the geometric realisation X = |X|1 of X preserves, for instance, its subset of generalised geodesic lines whose footpoints are at distance at most 1/4 from bvertices. Since both thisbsubset and its complement have nonempty interior in G X, the geodesic flow on G X has no mixing or ergodic measure with bfull support. This is why we considered the discrete-time geodesic flow (gt )t∈Z on G X in Section 2.6. Let X be a locally finite simplicial tree without terminal vertices, and let X = |X|1 be its geometric realisation. Let Γ be a nonelementary discrete subgroup of Aut(X). Let Fe : T 1 X → R be a potential for Γ, and let Fe+ = Fe, Fe− = Fe ◦ ι. Let δ = δΓ,F ± be the critical exponent of (Γ, F ± ), assumed to be finite. Let C ± : ∂∞ X × X × X → R be the associated (normalised) Gibbs cocycles. Let ± (µ± x )x∈X be two Patterson densities on ∂∞ X for the pairs (Γ, F ). Note that only the restrictions of the cocycles C ± to ∂∞ X × V X × V X are useful and that it is often convenient and always sufficient to replace the cocycles by finite sums involving a system of conductances (as defined in Section 3.5); see below. Furthermore, only the restriction (µ± x )x∈V X of the family of Patterson densities to the set of vertices of X is useful. 10 See

Section 2.6.

98


Example 4.12. Let X be a simplicial tree with geometric realisation X and let e c : EX → R be a system of conductances on X. For all x, y in V X and ξ ∈ ∂∞ X, with the usual convention on the empty sums, let c+ ξ (x, y) =

m X

e c(ei ) −

i=1

n X

e c(fj )

j=1

and c− ξ (x, y) =

m X

e c(ei ) −

i=1

n X

e c(fj ) ,

j=1

where if p ∈ V X is such that [p, ξ[ = [x, ξ[ ∩ [y, ξ[ , then (e1 , e2 , . . . , em ) is the geodesic edge path in X from x = o(e1 ) to p = t(em ) and (f1 , f2 , . . . , fn ) is the geodesic edge path in X from v = o(f1 ) to p = t(fn ). x

e1 em

p

ξ

fn y

f1

f2

With δc as defined at the end of Section 3.5 and with C ± the Gibbs cocycles for (Γ, Fec ), by Equation (3.18) and Proposition 3.21, we have, for all ξ ∈ ∂∞ X and x, y ∈ V X, Cξ± (x, y) = − c± ξ (x, y) + δc βξ (x, y) , and Equation (4.2) gives ±

cξ (x, y)−δc βξ (x, y) dµ± dµ± x (ξ) = e y (ξ) .

Using the particular structure of trees, we can prove a version of the shadow lemma (Lemma 4.3), where one can take the radius R to be 0. When F = 0, this result is due to Coornaert [Coo]. Lemma 4.13 (Mohsen’s shadow lemma for trees). Let K be a finite subset of V X. There exists C > 0 such that for all γ ∈ Γ and x, y ∈ K with y ∈ C ΛΓ, we have R γy ± 1 R γy (Fe± −δ) e −δ) (F x ex 6 µ± . x Ox {γy} 6 C e C

Proof. The structure of the proof is the same as for Lemma 4.3 (1) (that is, that of [PauPS, Lem. 3.10]) with differences towards the end of the argument. Let us prove that there exists C = CK > 0 such that for all γ ∈ Γ and x, y ∈ K with y ∈ C ΛΓ, we have 1 6 µ± (4.11) γy Ox {γy} 6 C . C

4.3. Patterson densities for simplicial trees

99

Assuming this, let us prove Lemma 4.13. By Equation (4.2), we have Z ± ± µx Ox {γy} = e−Cξ (x, γy) dµ± γy (ξ) . ξ∈Ox {γy}

Note that Cξ± (x, γy) + Equation (3.20). Hence

R γy x

(Fe± − δ) = 0 if ξ ∈ Ox {γy} (that is, if γy ∈ [x, ξ[ ), by

µ± x (Ox {γy}) = e

R γy x

e± −δ) (F

µ± γy (Ox {γy}) ,

and Lemma 4.13 follows from Equation (4.11). Let us now prove the upper bound in Equation (4.11). Fix z0 ∈ K, and let C0 =

sup x, y∈K, ξ∈∂∞ X

|Cξ± (x, y)| ,

which is finite by Proposition 3.20 (2), since K is compact and Fe± continuous. Then, using Equation (4.1) for the equality and Equation (4.2) for the last inequality, we have 0

± ± C µ± kµ± γy (Ox {γy}) 6 kµγy k = kµy k 6 e z0 k , 0

and the upper bound holds if C > eC kµ± z0 k. Finally, in order to prove the lower bound in Equation (4.11), we assume for a contradiction that there exist sequences (xi )i∈N , (yi )i∈N in K with yi ∈ C ΛΓ and (γi )i∈N in Γ such that µ± γi yi (Oxi {γi yi }) converges to 0 as i → +∞. Up to extracting a subsequence, since K is finite, we may assume that the sequences (xi )i∈N and (yi )i∈N are constant, say with values x and y respectively. Since y ∈ C ΛΓ and every point in C ΛΓ belongs to at least one geodesic line between two limit points of Γ, the geodesic segment from x to γi y may be extended to a geodesic ray from x to a limit point of Γ. Since the support of µ± z is equal to ΛΓ for all z ∈ X, we have µ± (O {γ y}) > 0 for all i ∈ N. Thus, up to taking a subsequence, we can x i γi y −1 assume that γi x converges to ξ ∈ ΛΓ (otherwise by discreteness, we may extract a subsequence such that (γi )i∈N is constant, and µ± γi y (Ox {γi y}) cannot converge to 0). Since X is a tree, there exists a positive integer N such that Oγ −1 x {y} = Oγ −1 x {y} = Oξ {y} i

N

for all i > N . As above, Oξ {y} meets ΛΓ since y ∈ C ΛΓ, thus µ± y (Oξ {y}) > 0. But for every i > N , −1 ± ± µ± y Oξ {y} = (γi )∗ µγi y Oγ −1 x {y} = µγi y Ox {γi y} i

tends to 0 as i → +∞, a contradiction.

100


Let φeµ± : V X → [0, +∞[ be the total mass functions of the Patterson densities: φeµ± (x) = kµ± xk for every x ∈ V X. These maps are Γ-invariant by Equation (4.1); hence they induce maps φµ± : Γ\V X → [0, +∞[ . In the case of real hyperbolic manifolds and vanishing potentials, the total mass functions have important links to the spectrum of the hyperbolic Laplacian (see [Sul1]). See also [CoP3, CoP5] for the case of simplicial trees and the discrete Laplacian, Section 6.1 for a generalisation of the result of Coornaert and Papadopoulos, and, for instance, [BerK] for developments in the field of quantum graphs.

4.4

Gibbs measures for metric and simplicial trees

Let (X, λ) be a locally finite metric tree without terminal vertices, let X = |X|λ be its geometric realisation, and let x0 ∈ V X be a basepoint. Let Γ be a nonelementary discrete subgroup of Aut(X, λ). Let Fe : T 1 X → R be a potential for Γ, and let Fe+ = Fe, Fe− = Fe ◦ ι. Let δ = δΓ,F ± be the critical exponent of (Γ, F ± ), assumed to be finite. Let (µ± x )x∈V X be two (normalised) Patterson densities on ∂∞ X for the pairs (Γ, F ± ). The Gibbs measure m e F on the space of discrete geodesic linesbG X of X, invariant under Γ and under the discrete-time geodesic flow (gt )t∈Z of G X, is defined analogously to the continuous-time case, using the discrete Hopf parametrisation for any basepoint x0 ∈ V X, by dm e F (`) = e

C`− (x0 , `(0)) + C`+ (x0 , `(0)) −

+

+ dµ− x0 (`− ) dµx0 (`+ ) dt ,

(4.12)

where now dt is the counting measure on Z. We again denote by mF the measure that m e F induces on Γ\G X, called the Gibbs measure on Γ\G X. In this section, we prove that the Gibbs measures in the case of trees satisfy a Gibbs property even closer to the one in symbolic dynamics, we give an analytic finiteness criterion of the Gibbs measures for metric trees, and we recall the ergodic properties of tree lattices. As recalled in the introduction, Gibbs measures were first introduced in statistical mechanics and consequently in symbolic dynamics; see, for example, [Bowe3], [ParP], [PauPS]. In order to motivate the terminology used in this book, we recall the definition of a Gibbs measure for the full two-sided shift on a finite alphabet:11 Let Σn = {1, 2, . . . , n}Z be the product space of sequences x = (xk )k∈Z indexed by Z in the finite discrete set {1, 2, . . . , n}, and let σ : Σn → Σn be the shift map defined by σ((xk )k∈Z ) = (xk+1 )k∈Z . A shift-invariant probability measure µ 11 See

Section 5.1 for the appropriate definition when the alphabet is countable.

4.4. Gibbs measures for metric and simplicial trees

101

on Σn satisfies the Gibbs property for an energy function φ : Σn → R if µ([a−m− , a−m− +1 , . . . , am+ −1 , am+ ]) 1 6 6C Pm+ −P (m− +m+ +1)+ k=−m φ(σ k x) C − e for some constants C > 1 and P ∈ R (called the pressure) and for all m± in N with m− 6 m+ and x in the cylinder [a−m− , a−m− +1 , . . . , am+ −1 , am+ ] that consists of those x ∈ Σn for which xk = ak for all k ∈ [−m− , m+ ]. Let x− , x+ ∈ V X and let x0 ∈ V X ∩ [x− , x+ ]. Let us define the tree cylinder of the triple (x− , x0 , x+ ) by [x− , x0 , x+ ] = {` ∈ G X : `± ∈ Ox0 {x± }, `(0) = x0 } . These cylinders are close to the dynamical balls that were introduced in Section 4.2, and the parallel with the symbolic case is obvious, since this cylinder is the set of geodesic lines that coincides on [−m− , m+ ], where m± = d(x0 , x± ), with a given geodesic line passing through x± and through x0 at time t = 0. The Gibbs measure m e F on the space of discrete geodesic lines G X satisfies a variant of the Gibbs property that is even closer to the one in symbolic dynamics than the general case described in Proposition 4.5. Proposition 4.14 (Gibbs property). Let K be a finite subset of V X ∩ C ΛΓ. There exists C 0 > 1 such that for all x± ∈ ΓK and x0 ∈ V X ∩ [x− , x+ ], m e F ([x− , x0 , x+ ]) 1 6 6 C0 . R x+ 0 e −δ C e d(x− , x+ )+ x− F Proof. The result is immediate if d(x− , x+ ) is bounded, since the above denominator and numerator take only finitely many values by the finiteness of K and by invariance under Γ, and the numerator is nonzero, since x± ∈ C ΛΓ so that the tree cylinder [x− , x0 , x+ ] meets the support of m e F . We may hence assume that d(x− , x+ ) > 2. Using the invariance of m e F under the discrete-time geodesic flow, we may thus assume that x0 6= x− , x+ . By Γ-invariance, we may assume that x− varies in the finite set K, and that x0 is at distance 1 from x− , hence also varies in a finite set. Using the discrete Hopf parametrisation with respect to the vertex x0 , we have, by Lemma 4.13, for some C > 0, + m e F ([x− , x0 , x+ ]) = µ− x0 (Ox0 {x− }) µx0 (Ox0 {x+ })

6 C2 e

R x− x0

e◦ι−δ) (F

e

R x+ x0

e−δ) (F

= C 2e

R x+ x−

e−δ) (F

.

This proves the upper bound in Proposition 4.14 with C 0 = C 2 , and the lower bound follows similarly. Next, we give a finiteness criterion of the Gibbs measure for metric trees in terms of the total mass functions of the Patterson densities, extending the case in which Γ is torsion free and Fe = 0, due to [CoP4, Thm. 1.1].

102


Proposition 4.15. Let (X, λ, Γ, Fe) be as in the beginning of this section. (1) If (X, λ) is simplicial and k · k2 is the Hilbert norm of L2 (Γ\V X, volΓ\\X ), we have12 kmF k 6 kφµ+ k2 kφµ− k2 . (2) In general, with k · k2 the Hilbert norm of L2 (Γ\EX, Tvol(Γ\\X,λ) ), we have13 kmF k 6 kφµ+ ◦ ok2 kφµ− ◦ ok2 . Proof. (1) The simplicial assumption on (X, λ) means that all edges have length 1. The space Γ\G X is the disjoint union of the subsets {` ∈ Γ\G X : π(`) = `(0) = [x]} as the orbit [x] = Γx of x ∈ V X ranges over Γ\V X. By Equation (4.12), using the discrete Hopf decomposition with respect to the basepoint x, we have + d(m e F ) {`∈G X : `(0)=x} (`) = dµ− x (`− ) dµx (`+ ) . Let ∆[x] be the unit Dirac mass at [x]. By ramified covering arguments, we hence have the following equality of measures on the discrete set Γ\V X: π∗ mF =

X [x]∈Γ\V X

1 + 2 µ− {(`− , `+ ) ∈ ∂∞ X : x ∈ ]`− , `+ [} ∆[x] . (4.13) x × µx |Γx |

Thus, using the Cauchy–Schwarz inequality and by the definition of the measure volΓ\\X , kmF k = kπ∗ mF k 6

X [x]∈Γ\V X

1 + kµ− x × µx k = hφµ− , φµ+ i2 |Γx |

6 kφµ− k2 kφµ+ k2 . This proves Assertion (1) of Proposition 4.15. (2) The argument is similar to the previous one. Since the singletons in R have zero Lebesgue measure, the space Γ\G X is, up to a measure-zero subset for mF , the disjoint union for [e] ∈ Γ\EX of the sets A[e] consisting of the elements ` ∈ Γ\G X such that `(0) belongs to the interior of the edge [e] and the orientations of ` and e coincide on e. We fix a representative e of each [e] ∈ Γ\EX. For every t ∈ [0, λ(e)], let et be the point of e at distance t from o(e). By Equation (4.4), using Hopf’s 12 The

maps φµ± are defined at the end of Section 4.3. that o : Γ\EX → Γ\V X is the initial vertex map; see Section 2.6.

13 Recall


103

decomposition with respect to the basepoint o(e) in A[e] , we have14 as above kmF k =

X [e]∈Γ\EX

1 |Γe |

Z

Z

`− ∈∂e X

e

Z

`+ ∈∂e X

λ(e)

0

C`− (o(e), et )+C`+ (o(e), et ) −

+

+ dµ− o(e) (`− ) dµo(e) (`+ ) dt .

Since `− , o(e), et , `+ are in this order on the geodesic line ` with `− ∈ ∂e X and `+ ∈ ∂e X, we have C`−− (o(e), et ) + C`++ (o(e), et ) = 0 by Equation (3.20). Hence, by the definition of the measure Tvol(Γ\\X,λ) ,15 kmF k =

X [e]∈Γ\EX

6

X [e]∈Γ\EX

λ(e) − µ (∂e X) µ+ o(e) (∂e X) |Γe | o(e) λ(e) + − + kµ− o(e) k kµo(e) k = hφµ ◦ o, φµ ◦ oi2 |Γe |

6 kφµ− ◦ ok2 kφµ+ ◦ ok2 , which finishes the proof.

Let us give some corollaries of this proposition in the case of simplicial trees. It follows from Assertion (1) of Proposition 4.15 that if the L2 -norms of the total mass of the Patterson densities are finite, then the Gibbs measure mF is finite. − Taking Fe = 0 and (µ+ x )x∈V X = (µx )x∈V X , so that the Gibbs measure mF is the Bowen–Margulis measure mBM , it follows from this proposition that 2

2 kmBM k 6 kφµ± k2 6 Vol(Γ\\X) sup kµ± xk .

(4.14)

x∈V X

In particular, if Γ is a tree lattice16 of X and if the total mass of the Patterson density is bounded (which is the case if X is a uniform tree by the following proposition), then the Bowen–Margulis measure mBM is finite. The following statement summarises the basic ergodic properties of the lattices of (X, λ) when F = 0. Proposition 4.16. Let (X, λ) be a metric or simplicial tree, with geometric realisation X. Assume that (X, λ) is uniform and that Γ is a lattice in Aut(X, λ). Then (1) Γ is of divergence type, and its critical exponent δΓ is the Hausdorff dimension of any visual distance dx on ∂∞ X = ΛΓ; 14 See

Section 2.6 for the definition of ∂e X. Section 2.6. 16 See Section 2.6. 15 See

104


(2) the Patterson density (µx )x∈X coincides, up to a scalar multiple, with the family of Hausdorff measures (µHaus )x∈X of dimension δΓ of the visual disx tances (∂∞ X, dx ); in particular, it is Aut(X, λ)-equivariant: for all γ ∈ Aut(X, λ) and x ∈ X, we have γ∗ µx = µγx ; (3) the Bowen–Margulis measure m e BM of Γ on G X is Aut(X, λ)-invariant, and the Bowen–Margulis measure mBM of Γ on Γ\G X is finite. Proof. Let Γ0 be any uniform lattice of (X, λ), which exists since the metric tree (X, λ) is uniform. It is well known (see, for instance, [Bou]) that the critical exponent δΓ0 of Γ0 is finite and equal to the Hausdorff dimension of any visual distance (∂∞ X, dx ), and that the family (µHaus )x∈V X of Hausdorff measures of the visual x distances (∂∞ X, dx ) is a Patterson density for any discrete nonelementary subgroup of Aut(X, λ) with critical exponent equal to δΓ0 . By [BuM, Cor. 6.5(2)], the lattice Γ in Aut(X, λ) is of divergence type and δΓ = δΓ0 . By the uniqueness property of the Patterson densities when Γ is of divergence type (see Theorem 4.6 (3)), the family (µx )x∈V X coincides, up to a scalar multiple, with (µHaus )x∈V X . x Since the graph Γ0 \X is compact, the total mass function of the Hausdorff measures of the visual distances is bounded, hence so is (kµx k)x∈V X . By Proposition 4.15, since Γ is a tree lattice of (X, λ), hence of X, this implies that the Bowen–Margulis measure mBM of Γ is finite. Note that as in [DaOP], when (X, λ) (or its minimal nonempty Γ-invariant subtree) is not assumed to be uniform, there are examples of Γ that are tree lattices (or are geometrically finite) whose Bowen–Margulis measure mBM is infinite; see Section 15.5 for more details. Assume till the end of this section that (X, λ) is simplicial, that is, that λ ≡ 1. Let us now discuss the mixing properties of the discrete-time geodesic flow on Γ\G X for the Gibbs measure mF . Let LΓ be the length spectrum of Γ, which is, in the present simplicial case, the subgroup of Z generated by the translation lengths in X of the elements of Γ. Recall that x0 ∈ V X is a fixed basepoint. Let Veven X = {x ∈ V X : d(x, x0 ) = 0

mod 2}

be the set of vertices of X at an even distance bfrom the basepoint x0 , and let Vodd X = V bX − Veven X. Let Geven X (respectively Geven X) be the subset of G X (respectively G X) that consists of the geodesic lines (respectively generalised geodesic lines) in X whose origin is in Veven X. Recall17 that a discrete-time one-parameter group (hn )n∈Z of homeomorphisms of a topological space Z is topologically mixing if for all nonempty open subsets U, V of Z, there exists n0 ∈ N such that for all n > n0 , we have U ∩hn (V ) 6= ∅. 17 See

above Theorem 4.9 for the continuous-time version.


105

Recall that given a measured space (Z, m), with m nonzero and finite, endowed with a discrete-time one-parameter group (hn )n∈Z of measure-preserving transformations, the measure m is mixing under (hn )n∈Z if for all f, g ∈ L2 (Z, m), we have Z Z Z 1 lim f g ◦ hn dm = f dm g dm , n→+∞ Z kmk Z Z or equivalently if for every g ∈ L2 (Z, m), the functions g R◦ hn weakly converge in 1 the Hilbert space L2 (Z, m) to the constant function kmk g dm as n → +∞. Z Theorem 4.17. Assume that the smallest nonempty Γ-invariant simplicial subtree of X is uniform, without vertices of degree 2, and that mF is finite. Then the following assertions are equivalent: • the length spectrum of Γ satisfies LΓ = Z; • the discrete-time geodesic flow on Γ\G X is topologically mixing on its nonwandering set; • the quotient graph Γ\X is not bipartite; • the Gibbs measure mF is mixing under the discrete-time geodesic flow (gt )t∈Z on Γ\G X. Otherwise LΓ = 2Z, and the square of the discrete-time geodesic flow (g2t )t∈Z is topologically mixing on the nonwandering subset of Γ\Geven X and the restriction of the Gibbs measure mF to Γ\Geven X is mixing under (g2t )t∈Z . Proof. The nonwandering set of (gt )t∈Z on Γ\G X is Γ\{` ∈ G X : `± ∈ ΛΓ}, and the nonwandering set of (g2t )t∈Z on Γ\Geven X is Ωeven = Γ\{` ∈ Geven X : `± ∈ ΛΓ} . Since the translation axis of any loxodromic element of Γ is contained in the convex hull of the limit set of Γ, we may hence assume that the geometric realisation of X is equal to C ΛΓ. Lemma 4.18. If X is a locally finite tree without vertices of degree 2, if Γ is a nonelementary discrete sugbroup of Aut(X) such that X is tree-minimal (that is, does not contain a Γ-invariant proper nonempty subtree), then the length spectrum LΓ of Γ is equal either to Z or to 2Z, and equal to 2Z if and only if the quotient graph Γ\X is bipartite. Proof. This lemma is essentially due to [GaL]. By, for instance, [Pau1, Lem. 4.3], since X is tree-minimal, every geodesic segment (and in particular any two consecutive edges) is contained in the translation axis of a loxodromic element of Γ. Hence if x and y are the two endpoints of any edge e of X, since they have degree at least 3, there exist at least two loxodromic elements α and β of Γ such that the translation axes Axα and Axβ contain x and y respectively, but do not meet the interior of the edge e, so that d(x, y) = d(Axα , Axβ ). In particular, Axα and Axβ

106


are disjoint, which implies by, for instance, [Pau1, Prop. 1.6] that the translation lengths of α, β, and αβ satisfy λ(αβ) = λ(α) + λ(β) + 2 d(Axα , Axβ ) . Hence 2 = 2 d(x, y) = λ(αβ) − λ(α) − λ(β) ∈ LΓ . Therefore 2Z ⊂ LΓ ⊂ Z, and either LΓ = Z or LΓ = 2Z. Note that for all vertices x, y, z in a simplicial tree, if d(x, y) and d(y, z) are both even or both odd, then d(x, z) = d(x, y) + d(y, z) − 2 d(y, [x, z]) = 0

mod 2 .

(4.15)

Note that for all x ∈ V X and γ ∈ Γ, we have d(x, γx) = λ(γ)

mod 2 .

(4.16)

Indeed, if γ is loxodromic, then d(x, γx) = λ(γ) + 2 d(x, Axγ ), and otherwise, d(x, γx) = 2 d(x, Fix(γ)), where Fix(γ) is the set of fixed points of the elliptic element γ. For future use, this proves that the following assertions are equivalent: (1) LΓ ⊂ 2Z; (2) ∀ x ∈ X, ∀ γ ∈ Γ,

d(x, γx) ∈ 2Z .

(4.17)

Assume that LΓ = 2Z. Then Veven X (hence Vodd X) is Γ-invariant, since for all x ∈ Veven X, the distance d(x, γx) is even by Equation (4.16), and d(γx, γx0 ) = d(x, x0 ) is even, so that d(γx, x0 ) is even by Equation (4.15). Since no edge of X has both endpoints in Veven X, this proves that Γ\X is bipartite, with partition of its set of vertices (Γ\Veven X) t (Γ\Vodd X). Assume conversely that Γ\X is bipartite. The set Veven X, which is the lift of one of the two elements of the partition of its vertices by the canonical projection V X → Γ\V X, is Γ-invariant. By Equation (4.16), this proves that LΓ ⊂ 2Z, hence that LΓ = 2Z. The equivalence of the first, second, and fourth claims in the statement of Theorem 4.17 follows from a discrete-time version with potential of [Rob2, Thm. 3.1] or a discrete-time version of [Bab1, Thm. 1] (which can be extended to CAT(−1) spaces by the remark in [Bab1, page 70]). It can also be recovered from the following arguments when LΓ = 2Z, and we prefer to concentrate on this case, since it requires many modifications and is stated with almost no proof and only when Fe = 0 in [BrP2, Prop. 3.3]. Assume from now on that LΓ = 2Z. Since Veven X is Γ-invariant as seen above, and since d(`(0), g2s `(0)) = 2|s| is even for all ` ∈ G X and s ∈ Z, it follows from the definition of Geven X = {` ∈ G X : π(`) ∈ Veven X} and from Equation (4.15) that Geven X is invariant under the even discrete-time geodesic flow (g2s )s∈Z and


107

under Γ. Note that the discrete Hopf parametrisation of G X with respect to the basepoint x0 gives a homeomorphism from Geven X to 2 (ξ, η, t) ∈ ∂∞ X × Z : t = d(x0 , ]ξ, η[ ) mod 2 . The restriction of the Gibbs measure m e F to Geven X, which we will again denote 2 2 by m e F , disintegrates by the projection on the first factor ∂∞ X × Z → ∂∞ X over 2 the geodesic current m b F , where for every (ξ, η) ∈ ∂∞ X and (any) x ∈ ]ξ, η[ , −

dm b F (ξ, η) = eCξ

(x0 , x) + Cη+ (x0 , x)

+ dµ− x0 (ξ) dµx0 (η) ,

(4.18)

with conditional measure on the fiber over (ξ, η) the counting measure on the discrete set {t ∈ Z : t = d(x0 , ]ξ, η[ ) mod 2}. Since mF is finite and invariant under the discrete-time geodesic flow, it is conservative by Poincaré’s recurrence theorem. Hence the measure quasipreserving action of Γ on the measured space 2 (∂∞ X, m b F ) is ergodic by (the discrete-time version of) Theorem 4.6. Since the distance between two points in a horosphere of a simplicial tree is even, it follows again by Equation (4.15), that every horosphere of X is either entirely contained in Veven X or entirely contained in Vodd X. For every ` ∈ Geven X, its strong stable/unstable leaf W ± (`) = `0 ∈ G X : lim d(`(t), `0 (t)) = 0 t→±∞

is contained in Geven X, since the image by the footpoint projection of a strong stable/unstable leaf is a horosphere (see Equation (2.14)). Thus Geven X is saturated by the partition into strong stable/unstable leaves of G X. We now follow rather closely the arguments of [Bab1, Thm. 1] in order to prove the last claims of Theorem 4.17, the main point being that the geodesic current m b F is a quasi-product measure. The following lemmas are particular cases of respectively Lemma 1 and Fact page 64 of [Bab1], valid for general finite measure-preserving dynamical systems, applied, with the notation of [Bab1], to (Tt )t∈A = (g2t )t∈Z . R Lemma 4.19. Let f ∈ L2 (Γ\Geven X, mF ) be such that f dmF = 0. If there exists an increasing sequence (tn )n∈N in Z such that f ◦ g2tn does not converge to 0 for the weak topology on L2 (Γ\Geven X, mF ), then there exist an increasing sequence (sn )n∈N in Z and a nonconstant18 element f ∗ ∈ L2 (Γ\Geven X, mF ) such that the functions f ◦ g2sn and f ◦ g−2sn both converge to f ∗ for the weak topology on L2 (Γ\Geven X, mF ). Lemma 4.20. If (fn )n∈N is a sequence in L2 (Γ\Geven X, mF ) weakly converging to f ∗ in L2 (Γ\Geven X, mF ), then there exists a subsequence (fnk )k∈N such that the PN 2 −1 Ces` aro averages N12 k=0 fnk converge pointwise almost everywhere to f ∗ as N → +∞. 18 Recall

that an element of L2 (Z, m) is nonconstant if no representative function is almost everywhere constant.

108


Recall that the support of the restriction of mF to Γ\Geven X is the nonwandering set Ωeven . Assume for a contradiction that the restriction of mF to Γ\Geven X is not mixing under the even discrete-time geodesic flow. Then Rthere exists a continuous function f with compact support on Ωeven such that f dmF = 0 and (f ◦ g2n )n∈N does not weakly converge to 0 in L2 (Γ\Geven X, mF ). By Lemmas 4.19 and 4.20, there exist a nonconstant element f ∗ ∈ L2 (Γ\Geven X, mF ) and inPN 2 −1 1 ±2n± k pointwise almost creasing sequences (n± k=0 f ◦ g k )k∈N in N such that N 2 everywhere converges to f ∗ as N → +∞. Let fe∗ = f ∗ ◦ peven , where peven : Geven X → Γ\Geven X is the canonical projection, be the lift of f ∗ to Geven X. Since the conditional measures for the disintegration of m e F over m b F are counting measures on countable sets, there 2 exists a full m b F -measure subset E0 of ∂∞ X such that for every ` ∈ Geven X with (`− , `+ ) ∈ E0 , the above convergences hold after lifting to Geven X at the points g2n ` for all n ∈ Z. For every ` ∈ Geven X, the subgroup A` of 2Z given by the periods of the map 2n 7→ fe∗ (g2n `) depends only on (`− , `+ ). Thus, we have a measurable map from E0 into the (discrete) set of subgroups of 2Z, which is Γ-invariant, hence is constant m b F -almost everywhere by the ergodicity of m b F under Γ. Assume for a contradiction that this almost everywhere constant subgroup is 2Z, that is, that the values of fe∗ almost everywhere do not depend on the time parameter in the discrete Hopf parametrisation of Geven X. Then fe∗ defines a 2 Γ-invariant measurable function on ∂∞ X. Again by ergodicity, this function is almost everywhere constant, contradicting the fact that f ∗ is not almost everywhere constant. Hence there exist a full m b F -measure subset E1 of E0 and κ ∈ N − {0, 1} such that A` = 2 κ Z for every ` ∈ Geven X with (`− , `+ ) ∈ E1 . Let us finally prove that LΓ is contained in 2 κ Z, which contradicts the original assumption that LΓ = 2Z. PN 2 ± 1 f ◦ g±2nk ◦ peven , so that the set Let fe± = lim sup 2 N →+∞ N

k=0

E = (ξ, η) ∈ E1 : ∀ ` ∈ Geven X, if `− = ξ and `+ = η, then fe+ (`) = fe− (`) = fe∗ (`)

has full m b F -measure. By the hyperbolicity of the geodesic flow (see Equation (2.10)) and the uniform continuity of f , the map fe+ is constant along any strong stable leaf of Geven X and fe− is constant along any strong unstable leaf. Let 0 E − = {ξ ∈ ΛΓ : (ξ, η 0 ) ∈ E for µ+ x0 -almost every η ∈ ΛΓ}

and 0 E + = {η ∈ ΛΓ : (ξ 0 , η) ∈ E for µ− x0 -almost every ξ ∈ ΛΓ} . + Since m b F is in the same measure class as the product measure µ− x0 ⊗ µx0 (see − c − Equation (4.18)), it follows by Fubini’s theorem that we have µx0 ( E ) = 0 and c + − + µ+ has full m b F -measure. x0 ( E ) = 0, and the set E × E


109

Let γ be a loxodromic element of Γ, and let x be any vertex of X on the translation axis of γ. Since d(x, γx) is even (see Equation (4.17)), the midpoint y of the geodesic segment [x, γx] is a vertex of X. Since x and y have degree at least 3, there exist ξx and ξy in ∂∞ X whose closest points on the translation axis of γ are respectively x and y. Note that ξx , ξy ∈ ΛΓ, since X = C ΛΓ. Since y and x are the closest points to γ+ and γ− on the geodesic line ]ξx , ξy [ , we have19 [[ξx , γ+ , ξy , γ− ]] = d(x, y). γ+

ξ0

γx

ξy

`2 (0) y

η0

3

` (0) 1

` (0) x ξ

γ−

0

` (0)

`4 (0)

η ξx

Since E ∩(E − ×E + ) has full m b F -measure, there exists (ξ, η) ∈ E ∩(E − ×E + ) arbitrarily close to (γ− , ξx ). Since the set 2 {(ξ 0 , η 0 ) ∈ ∂∞ Γ : (ξ, η 0 ), (ξ 0 , η), (ξ 0 , η 0 ) ∈ E}

has full m b F -measure, there exists such a (ξ 0 , η 0 ) arbitrarily close to (γ+ , ξy ). Let 0 ` ∈ Geven X be such that `0− = ξ and `0+ = η. Let `1 ∈ W + (`0 ) be such that `1− = ξ 0 . Let `2 ∈ W − (`1 ) be such that `2+ = η 0 . Let `3 ∈ W + (`2 ) be such that `3− = ξ. Finally, let `4 ∈ W − (`3 ) be such that `4+ = η. Then `4 = g2s `0 for some s ∈ Z with 2|s| = d(`0 (0), `4 (0)). By the definition of E and since fe+ (resp. fe− ) is constant along the strong stable (resp. unstable) leaves, we have fe∗ (`0 ) = fe+ (`0 ) = fe+ (`1 ) = fe− (`1 ) = fe− (`2 ) = fe+ (`2 ) = fe+ (`3 ) = fe− (`3 ) = fe− (`4 ) = fe∗ (`4 ) = fe∗ (g2 s `0 ) . Hence 2 |s| is a period in A`0 , thus is contained in 2 κ Z. If t > 0 is large enough, we have `0 (t) = `1 (t),

`1 (−t) = `2 (−t),

`2 (t) = `3 (t),

`3 (−t) = `4 (−t) ,

which respectively tend to η, ξ 0 , η 0 , ξ as t → +∞. Since the cross-ratio is locally constant, we have by its properties, in particular its definition in Equation (2.21), 19 See

Section 2.6 for the definition of the cross-ratio of an ordered quadruple of distinct points in ∂∞ X.

110


that `(γ) = d(x, γx) = 2 d(x, y) = 2 [[ξx , γ+ , ξy , γ− ]] = 2 [[η, ξ 0 , η 0 , ξ]] = lim

t→+∞

= lim

t→+∞ 0

d(`0 (t), `3 (−t)) − d(`3 (−t), `2 (t)) + d(`2 (t), `1 (−t)) − d(`1 (−t), `0 (t)) d(`0 (t), `3 (−t)) − d(`1 (−t), `1 (t)) = lim d(`0 (t), `4 (−t)) − 2 t t→+∞

4

= d(` (0), ` (0)) = 2 |s| ∈ 2 κ Z . Thus LΓ ⊂ 2 κ Z, which contradicts the fact that LΓ = 2Z.

By Proposition 4.16, the general assumptions of Theorem 4.17 are satisfied if X is uniform without vertices of degree 2, Γ is a tree lattice of X, and Fe = 0. Thus, if we assume furthermore that Γ\X is not bipartite, then the Bowen–Margulis measure mBM of Γ is mixing under the discrete-time geodesic flow on Γ\G X.

Chapter 5

Symbolic Dynamics of Geodesic Flows on Trees In this chapter, we give a coding of the discrete-time geodesic flow on the nonwandering sets of quotients of locally finite simplicial trees X without terminal vertices by nonelementary discrete subgroups of Aut(X) by a subshift of finite type on a countable alphabet. Similarly we give a coding of the continuous-time geodesic flow on the nonwandering sets of quotients of locally finite metric trees (X, λ) without terminal vertices by nonelementary discrete subgroups of Aut(X, λ) by suspensions of such subshifts. These codings are used in Section 5.4 to prove the variational principle in both contexts, and in Sections 9.2 and 9.3 to obtain rates of mixing of the flows.

5.1

Two-sided topological Markov shifts

In this short and independent section, which will be used in Sections 5.2, 5.3, 5.4, 9.2, and 9.3, we recall some definitions concerning symbolic dynamics on countable alphabets.1 A (two-sided) topological Markov shift2 is a topological dynamical system (Σ, σ) constructed from a countable discrete alphabet A and a transition matrix A = (Ai, j )i, j∈A ∈ {0, 1}A ×A , where Σ is the closed subset of the topological product space A Z defined by Σ = x = (xn )n∈Z ∈ A Z : ∀ n ∈ Z,

Axn ,xn+1 = 1} ,

1 See

for instance [Kit, Sar2]. that the terminology could be misleading: a topological Markov shift comes a priori without a measure, and many probability measures invariant under the shift do not satisfy the Markov chain property that the probability to pass from one state to another depends only on the previous state, not on all past states. 2 Note


111

112

Chapter 5. Symbolic Dynamics of Geodesic Flows on Trees

and σ : Σ → Σ is the (two-sided) shift defined by (σ(x))n = xn+1 for all x ∈ Σ and n ∈ Z. Note that to be given (A , A) is equivalent to be given an oriented graph with countable set of vertices A , whose set of oriented edges is a subset of A × A , and with incidence matrix A such that Ai, j = 1 if there is an oriented edge from the vertex i to the vertex j and Ai, j = 0 otherwise. For all p 6 q in Z, a finite sequence (an )p6n6q ∈ A {p,...,q} is admissible (or A-admissible when we need to make A precise) if Aan , an+1 = 1 for all n ∈ {p, . . . , q − 1}. A topological Markov shift is transitive if for all x, y ∈ A , there exists an admissible finite sequence (an )p6n6q with ap = x and aq = y. This is equivalent to requiring the dynamical system (Σ, σ) to be topologically transitive: for all nonempty open subsets U, V in Σ, there exists n ∈ Z such that U ∩ σ n (V ) 6= ∅. Note that the product space A Z is not locally compact when A is infinite. When the matrix A has only finitely many nonzero entries in each row and each column, then (Σ, σ) is also called a subshift of finite type (on a countable alphabet). The topological space Σ is then locally compact: by diagonal extraction, for all p 6 q in Z and ap , ap+1 , . . . , aq−1 , aq in A , every cylinder [ap , ap+1 , . . . , aq−1 , aq ] = (xn )n∈Z ∈ Σ : ∀ n ∈ {p, . . . , q}, xn = an is a compact open subset of Σ. Given a continuous map Fsymb : Σ → R and a constant cFsymb ∈ R, we say that a measure P on Σ, invariant under the shift σ, satisfies the Gibbs property3 with Gibbs constant cFsymb for the potential Fsymb if for every finite subset E of the alphabet A , there exists CE > 1 such that for all p 6 q in Z and for every x = (xn )n∈Z ∈ Σ such that xp , xq ∈ E, we have 1 P([xp , xp+1 , . . . , xq−1 , xq ]) Pq 6 CE . (5.1) 6 −c n CE e Fsymb (q−p+1)+ n=p Fsymb (σ x) 0 Two continuous maps Fsymb , Fsymb : Σ → R are cohomologous if there exists a continuous map G : Σ → R such that 0 Fsymb − Fsymb = G ◦ σ − G .

5.2

Coding discrete-time geodesic flows on simplicial trees

Let X be a locally finite simplicial tree without terminal vertices, with X = |X|1 its geometric realisation. Let Γ be a nonelementary discrete subgroup of Aut(X), and let Fe : T 1 X → R be a potential for Γ. 3 Note

that some references have a stronger notion of Gibbs measure (see, for instance, [Sar1]), with the constant C independent of E.

5.2. Coding discrete-time geodesic flows on simplicial trees

113

In this section, we give a coding of the discrete-time geodesic flow (gt )t∈Z on the nonwandering subset of Γ\G X by a locally compact transitive (two-sided) topological Markov shift. This explicit construction will be useful later on in studying the variational principle (see Section 5.4) and rates of mixing (see Section 9.2). The main technical aspect of this construction, building on [BrP2, §6], is to allow the case in which Γ has torsion. When Γ is torsion free and Γ/X is finite, the construction is well known, we refer, for instance, to [CoP6] for a more general setting when the potential is 0. In order to consider, for instance, nonuniform tree lattices, it is important to allow torsion in Γ. Our direct approach also avoids the assumption that the discrete subgroup Γ is full, that is, equal to the subgroup consisting of the elements g ∈ Aut(X) such that p ◦ g = p, where p : X → Γ\X is the canonical projection, as in [Kwo] (building on [BuM, 7.3]). Let X0 be the minimal nonempty Γ-invariant simplicial subtree of X whose geometric realisation is C ΛΓ. Since we are interested only in the support of the Gibbs measures, we will code the geodesic flow only on the nonwandering subset Γ\X0 of Γ\G X. The same construction works with the full space Γ\G X, but the resulting Markov shift is then not necessarily transitive. Let (Y, G∗ ) = Γ\\X0 be the quotient graph of groups of X0 by Γ (see, for instance, Example 2.10), and let p : X0 → Y = Γ\X0 be the canonical projection. We denote by [1] = H the trivial double coset in any double coset set H\G/H of a group G by a subgroup H. We consider the alphabet A consisting of the triples (e− , h, e+ ), where • e± ∈ EY satisfy t(e− ) = o(e+ ) and • h ∈ ρe− (Ge− )\Go(e+ ) /ρ e+ (Ge+ ) satisfy h 6= [1] if e+ = e− . This set is countable (and finite if and only if the quotient graph Γ\X0 is finite); we endow it with the discrete topology. We consider the (two-sided) topological Markov shift with alphabet A and transition matrix A(e− , h, e+ ), (e0 − , h0 , e0 + ) = 1 if − e+ = e0 and 0 otherwise. Note that this matrix A = (Ai,j )i,j∈A has only finitely many nonzero entries in each row and each column, since X0 is locally finite and Γ has finite vertex stabilisers in X0 . We consider the subspace + − Z Σ = (e− : ∀ i ∈ Z, e+ i , hi , ei )i∈Z ∈ A i−1 = ei of the product space A Z , and the shift σ : Σ → Σ defined by (σ(x))i = xi+1 for all (xi )i∈Z in Σ and i in Z. As seen above, Σ is locally compact. Let us now construct a natural coding map Θ from Γ\G X0 to Σ, by slightly modifying the construction of [BrP2, §6].

114


fi+1

fi

γi+1

γi eei gei in X0

` p

ei

hi+1 (`) ∈ Gt(ei ) ei+1

eg i+1 g ei+1

in (Y, G∗ ) = Γ\\X0

gi ) t(e

For every discrete geodesic line ` ∈ G X0 , for every i ∈ Z, let fi = fi (`) be the edge of X0 whose geometric realisation is `([i, i + 1]) with initial vertex f (i) and terminal vertex f (i + 1), and let ei = p(fi ), which is an edge in Y. Let us use the notation of Example 2.10: we fix lifts ee and ve of every edge e and vertex g = t(e v of Y in X0 such that ee = e e, and elements ge ∈ Γ such that ge t(e) e). Since p(eei ) = ei = p(fi ), there exists γi = γi (`) ∈ Γ, well defined up to multiplication on the left by an element of Gei = Γeei , such that γi fi = eei for all i ∈ Z. + We define e− i+1 (`) = ei , ei+1 (`) = ei+1 , and hi+1 (`) = ge−−1 (`) γi (`) γi+1 (`)−1 g e+

i+1 (`)

i+1

.

(5.2)

Recall that for every edge e of Y the structural monomorphism ρe from Ge = Γee −1 into Gt(e) = Γt(e) g is the map g 7→ ge gge . Hence the double coset of hi (`) in ρe− (`) (Ge− (`) )\Go(e+ (`)) ρ e+ (`) (Ge+ (`) ) i

i

i

i

i

does not depend on the choice of the γi ’s, and we again denote it by hi (`). The next result shows that, assuming only that Γ is discrete and nonelementary, the time-one discrete geodesic flow g1 on its nonwandering subset of Γ\G X is topologically conjugate to a locally compact transitive (two-sided) topological Markov shift. Theorem 5.1. With X0 = C ΛΓ, the map Θ : Γ\G X0 → Σ defined by + Γ` 7→ (e− i (`), hi (`), ei (`))i∈Z

is a homeomorphism that conjugates the time-one discrete geodesic flow g1 and the shift σ; that is, the following diagram commutes g1

Γ\G X0 −−−−→ Γ\G X0     Θy yΘ Σ

σ

−−−−→

Σ.

The topological Markov shift (Σ, σ) is locally compact and transitive.


115

Furthermore, if we endow Γ\G X0 with the quotient distance of 0 d(`, `0 ) = e− sup n∈N : `|[−n,n] = ` |[−n,n] on G X0 and Σ with the distance d(x, x0 ) = e− sup

n∈N : ∀ i ∈ {−n,...,n}, xi = x0i

,

then Θ is a bilipschitz homeomorphism. Finally, if X0 is a uniform tree without vertices of degree at most 2, if the Gibbs measure mF of Γ is finite, and if the length spectrum LΓ of Γ is equal to Z, then the topological Markov shift (Σ, σ) is topologically mixing. Note that when Y is finite (or equivalently when Γ is cocompact), the alphabet A is finite (hence (Σ, σ) is a standard subshift of finite type). When furthermore the vertex groups of (Y, G∗ ) are trivial (or equivalently when Γ acts freely, and in particular is a finitely generated free group), this result is well known, but it is new if the vertex groups are not trivial. Compare with the construction of [CoP6], whose techniques may be applied because Γ is word-hyperbolic if Y is finite, up to replacing Gromov’s (continuous-time) geodesic flow of Γ by the (discrete-time) geodesic flow on G X0 , thus avoiding the suspension part (see also the end of [CoP6] when Γ is a free group). Proof. For all ` ∈ G X0 and γ ∈ Γ, we can take γi (γ`) = γi (`)γ −1 , and since ± p(γfi ) = p(fi ), we have e± i (γ`) = ei (`) and hi (γ`) = hi (`); hence the map Θ is well defined. By construction, the map Θ is equivariant for the actions of g1 on Γ\G X0 and of σ on Σ. With the distances indicated in the statement of Theorem 5.1, if `, `0 ∈ G X0 ± 0 satisfy that `|[−n, n] = `0 |[−n, n] for some n ∈ N, then we have e± i (`) = ei (` ) for −n 6 i 6 n − 1, and we may take γi (`) = γi (`0 ) for −n 6 i 6 n − 1, so that hi (`) = hi (`0 ) for −n 6 i 6 n − 1. Therefore, we have d( Θ(Γ`), Θ(Γ`0 ) ) 6 e d(Γ`, Γ`0 ) , and Θ is Lipschitz (hence continuous). Let us construct an inverse Ψ : Σ → Γ\G X0 of Θ, by a more general construction that will be useful later on. Let I be a nonempty interval of consecutive integers in Z, either finite or equal to Z (the definition of the inverse of Θ requires only the second case I = Z). For all e− , e+ ∈ EY such that t(e− ) = o(e+ ), we fix once and for all a representative of every double coset in ρe− (Ge− )\Go(e+ ) /ρ e+ (Ge+ ), and we will denote this double coset by its representative. + Let w = (e− i , hi , ei )i∈I be a sequence indexed by I in the alphabet A such − that for all i ∈ I such that i−1 ∈ I, we have e+ i−1 = ei (when I is finite, this means that w is an A-admissible sequence in A , and when I = Z, this means that w ∈ Σ).


116

is the chosen representative of its In particular, the element hi ∈ Go(e+ ) = Γ ^ + o(ei )

i

double coset ρe− (Ge− ) hi ρ e+ (Ge+ ). i

i

i

i

For every i ∈ I, note that + −1 f o hi g + ei = o g ei

But hi g

−1 e+ i

−1

+ ef i

e+ i

+ − −1 f − ^ ] = o(e . i ) = t(ei ) = t ge− ei i

+ −1 f − ef i is not the opposite edge of the edge ge− ei , since the double i

− −1 f − coset of hi is not the trivial one [1] when e+ i = ei ; hence hi does not fix ge− ei . i

Therefore the length-2 edge path (see the picture below) − −1 f + e ge−−1 ef , h g i i i + ei

i

is geodesic. + e− i = ei−1

Y p X

− e+ i = ei+1

e+ i+1

+ ^ o(e i )

0 − ge−−1 ef i

g

i

−1 e+ i

αi−1

hi g

ge+

+ ge+−1 ef i i

fi−1

−1

e+ i

i

+ ef i

hi+1

αi

fi

fi+1

Let us construct by induction a geodesic segment w e in X0 (which will be a discrete geodesic line if I = Z), well defined up to the action of Γ, as follows. We fix i0 ∈ I (for instance, i0 = 0 if I = Z or i0 = min I if I is finite), and αi0 ∈ Γ. Let us define + f = f (w) = α g −1 ef . i0

i0

i0

i0

e+ i

0

Let us then define αi0 −1 = αi0 −1 (w) = αi0 ge+−1 g i0

e+ i

hi−1 0

0

and − fi0 −1 = fi0 −1 (w) = αi0 −1 ge− −1 ef i0 . i0


We have αi0 −1 hi0 g

−1 e+ i

117

+ ef i0 = fi0 , and (fi0 −1 , fi0 ) is a geodesic edge path of length

0

2 (as the image by αi0 −1 of such a path). Let i − 1, i0 ∈ I be such that i0 6 i0 6 i − 1. Assume by increasing induction on i and decreasing induction on i0 that we have constructed a geodesic edge path (fi0 −1 = fi0 −1 (w), . . . , fi−1 = fi−1 (w)) in the tree X0 and a finite sequence (αi0 −1 = αi0 −1 (w), . . . , αi−1 = αi−1 (w)) in Γ such that + fj = αj ge+−1 ef and αj = αj−1 hj g j j

−1 e+ j

ge+ j

for every j ∈ N such that i0 − 1 6 j 6 i − 1, with j > i0 for the equality on the right. If i does not belong to I, we stop the construction on the right-hand side at i − 1. If on the contrary i ∈ I, let us define (see the above picture) αi = αi−1 hi g

−1 e+ i

Then

i

(fi−1 , fi ) =

+ ge+ and fi = fi (w) = αi ge+−1 ef i . i

− αi−1 ge−−1 ef i , αi−1 hi g i

−1 e+ i

+ ef i

is a geodesic edge path of length 2 (as the image by αi−1 of such a path). Since an edge path is geodesic if and only if it has no back-and-forth, (fi0 , . . . , fi ) is a geodesic edge path in X0 . Thus the construction holds at rank i on the right. If i0 − 1 does not belong to I, we stop the construction on the left side at 0 i . Otherwise we proceed as for the construction of αi0 −1 and fi0 −1 in order to construct αi0 −2 and fi0 −2 with the required properties. If I = [p, q] ∩ Z with p 6 q in Z, let I 0 = [p − 1, q] ∩ Z. If I = Z, let I 0 = Z. We have thus constructed a geodesic edge path (fi )i∈I 0 = (fi (w))i∈I 0

(5.3)

in X0 . We denote by w e its parametrisation by R if I = Z and by [p − 1, q + 1] if I = [p, q] ∩ Z, in such a way that w(i) e = o(fi ) for all i ∈ I. In particular, we have fi = w([i, e i + 1]) for all i ∈ I 0 . When I = [p, q] ∩ Z, we consider w e a generalised discrete geodesic line, by extending it to a constant on ] − ∞, p − 1] and on [q + 1, +∞[ . The orbit Γw e of w e does not depend on the choice of αi0 , since replacing αi0 by αi0 0 replaces fi by αi0 0 αi−1 fi for all i ∈ I 0 , hence replaces w e by αi0 0 αi−1 w. e This 0 0 also implies that Γw e does not depend on the choice of i0 ∈ I. Assume from now on that I = Z, and define Ψ : Σ → Γ\G X0 by Ψ(w) = Γw e. −

+

±

+ ± 0 0 0 0 0 Let w = (e− and i , hi , ei )i∈Z and w = (ei , h i , ei )i∈I in Σ satisfy ei = ei 0 hi = h i for all i ∈ {−n, . . . , n} for some n ∈ N. Then we may take the same

118


f0 . We thus have αi (w) = αi (w0 ) i0 = 0 and αi0 in the construction of w e and w 0 and fi (w) = fi (w ) for −n 6 i 6 n. Therefore, with the distances indicated in the statement of Theorem 5.1, we have d( Ψ(w), Ψ(w0 ) ) 6 d(w, w0 ) , and Ψ is Lipschitz. Let us prove that Ψ is indeed the inverse of Θ. As in the construction of Θ, − + for all ` ∈ G X0 and i ∈ Z, we define fi = `([i, i + 1]), e+ i = p(fi ), and ei = ei−1 . + We denote by γ 0 ∈ Γ an element sending f to g −1 ef for all i ∈ Z (see the i

i

i

e+ i

picture below): with the notation above the statement of Theorem 5.1 (page 114), we have γi0 = ge+−1 γi (`). i

X

0

fi−1

fi + ge+−1 ef i i

0 γi−1

p −1

ge− i

g

− ef i

−1 e+ i

+ ^ o(e i )

ge+ i

g

−1

e+ i

h0i+1

+ ef i

h0i

0 γi−1 fi

+ e− i = ei−1

Y

fi+1

γi0

− e+ i = ei+1

e+ i+1

Then γi0 is well defined up to multiplication on the left by an element of the group + to γ 0 f . Γ = ρ + (G + ). Let h0 be an element in G + sending g −1 ef g

−1 + e i

f e+ i

ei

i

ei

o(ei )

e+ i

i

i−1 i

+ ^ It exists, since these two edges have the same origin o(e i ) and the same image under p: f + + 0 p(γi−1 fi ) = p(fi ) = e+ = p g +−1 ef . i = p ei i ei

Furthermore, it is well defined up to multiplication on the right by an element of Γ −1 f+ = ρe+ (Ge+ ), and we have (see the above picture) g

+ e i

ei

i

i

0 γi−1 γi0

−1

ge+−1 g e+ ∈ h0i ρe+ (Ge+ ). i

i

i

i

Using γj (`) = ge+ γj0 for j = i, i − 1 in Equation (5.2) gives j

0 hi (`) = γi−1 γi0

−1

ge−1 + g +. e i

i


119

Hence by the construction of Θ (with γi0 = ge+−1 γi (`) for all i ∈ Z), we have i

+ 0 Θ(`) = (e− i , ρe− (Ge− ) hi ρe+ (Ge+ ), ei )i∈Z . i

i

i

i

Let hi be the chosen representative of the double coset ρe− (Ge− ) h0i ρe+ (Ge+ ): i

i

i

i

there exist α ∈ ρe− (Ge− ) = ρe+ (Ge+ ) and β ∈ ρe+ (Ge+ ) such that hi = αh0i β. i

i

i−1

i−1

i

i

0 0 Up to replacing γi−1 by α−1 γi−1 and h0i by h0i β, we then may have h0i = hi . By −1 0 −1 taking αi0 = γi0 , we have αi = γi0 for all i ∈ Z, and an inspection of the above two constructions gives that Θ ◦ Ψ = id and Ψ ◦ Θ = id. Since the discrete-time geodesic flow is topologically transitive on its nonwandering subset, it follows by conjugation that the topological Markov shift (Σ, σ) is topologically transitive. If X0 is a uniform tree without vertices of degree at most 2, if the length spectrum of Γ is equal to Z and if the Gibbs measure mF for Γ is finite, then by Theorem 4.17, the discrete-time geodesic flow on Γ\G X0 is topologically mixing; hence by conjugation by Θ, the topological Markov shift (Σ, σ) is topologically mixing. This concludes the proof of Theorem 5.1.

When the length spectrum LΓ of Γ is different from Z, the topological Markov shift (Σ, σ) constructed above is not always topologically mixing. We now modify the above construction in order to take care of this problem. Recall that X0 = C ΛΓ and that Geven X0 is the space of geodesic lines ` ∈ G X0 whose origin `(0) is at even distance from the basepoint x0 (we assume as we may that x0 ∈ X0 ), which is invariant under the time-two discrete geodesic flow g2 and, when LΓ = 2Z, under Γ, as seen in the proof of Theorem 4.17. Consider Aeven the alphabet consisting of the quintuples (f − , h− , f 0 , h+ , f + ), where the triples (f − , h− , f 0 ) and (f 0 , h+ , f + ) belong to A and o(f 0 ) is at even distance from the image in Y = Γ\X0 of the basepoint x0 . Consider the transition matrix Aeven = (Aeven, i,j )i,j∈Aeven with row and column indices in Aeven such that 0 + + for all i = (f − , h− , f 0 , h+ , f + ) and j = (f∗− , h− ∗ , f∗ , h∗ , f∗ ), we have Aeven, i,j = 1 + − if and only if f = f∗ . We denote by (Σeven , σeven ) the associated topological Markov shift. We endow Σeven with the slightly modified distance deven (x, x0 ) = e−2 sup

n∈N : ∀ k ∈ {−n, ..., n}, xk = x0k

,

where x = (xk )k∈Z and x0 = (x0k )k∈Z are in Σeven . We have a canonical injection inj : Σeven → Σ sending the infinite sequence 0 + + − + (fn− , h− n , fn , hn , fn )n∈Z to (en , hn , en )n∈Z with, for every n ∈ Z, + − + − − 0 0 + + e− 2n = fn , h2n = hn , e2n = fn , e2n+1 = fn , h2n+1 = hn , e2n+1 = fn .

120


By construction, inj is clearly a homeomorphism onto its image, and Θ(Γ\Geven X0 ) = inj(Σeven ) . If two sequences in Σeven coincide between −n and n, then their images by inj coincide between −2n and 2n. Conversely, if the images by inj of two sequences in Σeven coincide between −2n − 1 and 2n + 1, then these sequences coincide between −n and n. Hence inj is bilipschitz, for the above distances. Let us define Θeven = inj−1 ◦ Θ|Γ\Geven X0 : Γ\Geven X0 → Σeven . The following diagram hence commutes Θ

even Γ\Geven X0 −−− −→ Σeven    inj y y

Γ\G X0

Θ

−−−−→

Σ,

where the vertical map on the left-hand side is the inclusion map. Theorem 5.2. Assume that X0 = C ΛΓ is a uniform tree without vertices of degree at most 2, that the Gibbs measure mF of Γ is finite, and that the length spectrum LΓ of Γ is equal to 2Z. Then the map Θeven : Γ\Geven X0 → Σeven is a bilipschitz homeomorphism that conjugates the time-two discrete geodesic flow g2 and the shift σeven , and the topological Markov shift (Σeven , σeven ) is locally compact and topologically mixing. Proof. The only claim that remains to be proven is the last one, which follows from Theorem 4.17, by conjugation. Let us now study the properties of the image by the coding map Θ of finite Gibbs measures on Γ\G X. Let δ = δΓ,F ± be the critical exponent of (Γ, F ± ) where as previously we define Fe+ = Fe, Fe− = Fe ◦ ι. Let (µ± x )x∈V X be two (normalised) Patterson densities on ∂∞ X for the pairs (Γ, F ± ). Assume that the associated Gibbs measure mF on Γ\G X (using the convention for discrete time of Section 4.3) is finite. Let us define 1 P= Θ∗ mF (5.4) kmF k as the image of the Gibbs measure mF (whose support is Γ\G X0 ) by the homeomorphism Θ, normalised to be a probability measure. It is a probability measure on Σ, invariant under the shift σ. Let (Zn )n∈Z be the random process classically associated with the full shift σ on Σ: it is the random process on the Borel space Σ indexed by Z with values in the discrete alphabet A , where Zn : Σ → A is the (continuous hence measurable) nth projection (xk )k∈N 7→ xn for all n ∈ Z.


121

Proposition 5.5 below summarises the properties of the probability measure P. We start by recalling and giving some notation used in this proposition. For every admissible finite sequence w = (ap , . . . , aq ) in A , with p 6 q in Z, we denote • by [w] = [ap , . . . , aq ] = {(xn )n∈Z ∈ Σ : ∀ n ∈ {p, . . . , q}, xn = an the associated cylinder in Σ, • by w b the associated geodesic edge path in X0 with length q − p + 2 constructed in the proof of Theorem 5.1 (see Equation (5.3)), with origin w b− and endpoint w b+ . For every geodesic edge path α = (fp−1 , . . . , fq ) in X0 , we define (see Section 2.6 for the notation, and the picture below) ∂α+ X0 = ∂fq X0 and ∂α− X0 = ∂ fp−1 X0 , and Gα X = {` ∈ G X : `(p − 1) = o(fp−1 ) and `(q + 1) = t(fq )} . α ∂α− X0

fp−1

fq

∂α+ X0

We define a map Fsymb : Σ → R by Z Fsymb (x) =

t(e+ 0 )

F

(5.5)

o(e+ 0 )

+ if x = (xi )i∈Z with x0 = (e− 0 , h0 , e0 ). Note that for all (xn )n∈Z , (yn )n∈Z ∈ Σ, if x0 = y0 , then Fsymb (x) = Fsymb (y), so that Fsymb is locally constant (constant on each cylinder of length 1 at time 0), hence continuous. For instance, if F = Fc is the potential associated with a system of conductances c : Γ\EX0 → R (see Section 3.5), then

Fsymb (x) = c(e+ 0). Note that if c, c0 : Γ\EX0 → R are cohomologous systems of conductances on 0 Γ\EX0 , then the corresponding maps Fsymb , Fsymb : Σ → R are cohomologous. 0 Indeed, if f : Γ\V X → R is a map such that c0 (e) − c(e) = f (t(e)) − f (o(e)) for every e ∈ Γ\EX0 , with G : Σ → R the map defined by G(x) = f (o(e+ 0 )) if + x = (xi )i∈Z with x0 = (e− , h , e ), then G is locally constant, hence continuous, 0 0 0 + and since t(e+ 0 ) = o(e1 ), we have, for every x ∈ Σ, 0 Fsymb (x) − Fsymb (x) = G(σx) − G(x) .

122


Definition 5.3. Let X00 be a locally finite simplicial tree. A nonelementary discrete subgroup Γ0 of Aut(X00 ) is Markov-good if for every n ∈ N − {0} and every geodesic edge path (e0 , . . . , en+1 ) in C ΛΓ0 , we have |Γ0e0 ∩ · · · ∩ Γ0en | |Γ0en−1 ∩ Γ0en ∩ Γ0en+1 | = |Γ0e0 ∩ · · · ∩ Γ0en+1 | |Γ0en−1 ∩ Γ0en | . (5.6) Remark 5.4. (1) Note that Equation (5.6) is automatically satisfied if n = 1 and that Γ0 is Markov-good if Γ0 acts freely on X00 . (2) A group action on a simplicial tree is 2-acylindrical4 if the stabiliser of every geodesic edge path of length 2 is trivial. If Γ0 is 2-acylindrical on X, then Γ0 is Markov-good, since all groups appearing in Equation (5.6) are trivial. (3) If X00 has degrees at least 3 and if Γ0 is a noncocompact geometrically finite tree lattice of X00 with Abelian edge stabilisers, then Γ0 is not Markov-good. Proof. (3) Since the quotient graph Γ0 \X00 is infinite, the graph of groups Γ0 \\X00 contains at least one cuspidal ray. Consider a geodesic ray in X00 with consecutive edges (fn )n∈N mapping injectively onto this cuspidal ray, pointing towards its end. The stabilisers of the edges fn in Γ0 are hence nondecreasing in n: we have Γ0fn ⊂ Γ0fn+1 for all n ∈ N. By the finiteness of the volume, there exists n > 3 such that Γ0fn−2 is strictly contained in Γ0fn−1 . Since X00 has degrees at least 3, there exists γ ∈ Γ0 fixing the vertex t(fn−1 ) but not fixing the edge fn−1 . Let e0 = f0 , . . . , en−1 = fn−1 , en = γ fn−1 , and en+1 = γ fn−2 . Since the stabilisers of fn−1 and en are conjugated by γ within the Abelian stabiliser of fn , they are equal. Then (e0 , . . . , en+1 ) is a geodesic edge path in the simplicial tree X00 (whose geometric realisation is equal to C ΛΓ0 , since Γ0 is a tree lattice). Since Γ0e0 ∩ · · · ∩ Γ0en = Γ0f0 , Γ0en−1 ∩ Γ0en ∩ Γ0en+1 = Γ0fn−2 , Γ0e0 ∩ · · · ∩ Γ0en+1 = Γ0f0 , Γ0en−1 ∩ Γ0en = Γ0fn−1 , and |Γ0fn−2 | = 6 |Γ0fn−1 |, the subgroup Γ0 is not Markovgood. Recall that a random process (Zn0 )n∈Z on (Σ, P) is a Markov chain if and only if for all p 6 q in Z and ap , . . . , aq , aq+1 in A , we have, when defined, 0 0 P(Zq+1 = aq+1 | Zq0 = aq , . . . , Zp0 = ap ) = P(Zq+1 = aq+1 | Zq0 = aq ) .

(5.7)

Proposition 5.5. (1) For every admissible finite sequence w in A , we have P([w]) =

− 0 + + 0 µ− w b− (∂w b X ) µw b+ (∂w bX ) e

|Γwb | kmF k

Rw b+ w b−

e−δ) (F

.

(2) The random process (Zn )n∈Z on (Σ, P) is a Markov chain if and only if Γ is Markov-good. (3) The measure P on the topological Markov shift Σ satisfies the Gibbs property with Gibbs constant δ for the potential Fsymb . 4 See,

for instance, [Sel, GuL], which require other minor hypotheses that are not relevant here.


123

It follows from the above Assertion (2) and from Remark 5.4 that when X has degrees at least 3 and Γ is a noncocompact geometrically finite tree lattice of X with Abelian edge stabilisers (and more generally, this is not a necessary assumption), then (Zn )n∈Z is not a Markov chain. The fact that codings of discrete-time geodesic flows on trees might not satisfy the Markov chain property had been noticed by Burger and Mozes around the time the paper [BuM] was published.5 When proving the variational principle in Section 5.4 and the exponential decay of correlations in Section 9.2, we will therefore have to use tools that do not involve the Markov chain property. Proof. (1) Let w = (ap , . . . , aq ), with p 6 q in Z, be an admissible finite sequence in A . Recall that [w] = {x ∈ Σ : ∀ i ∈ {p, . . . , q}, xi = ai }. By the construction of Θ, the preimage Θ−1 ([w]) is equal to the image ΓGwb X0 of Gwb X0 in Γ\G X0 . Hence, since Γwb is the stabiliser of Gwb X0 in Γ, P([w]) =

1 1 mF (ΓGwb X0 ) = m e F (Gwb X0 ) . kmF k |Γwb | kmF k

In the expression of m e F given by Equation (4.12), let us use as basepoint the origin w b− of the edge path w, b and note that all elements of Gwb X0 pass through w b− at time t = p − 1, so that by the invariance of m e F under the discrete-time geodesic flow, we have Z Z Z + 0 1−p m e F (Gwb X ) = dm e F (g `) = dµ− w b− (`− )dµw b− (`+ ) 0 `∈Gw bX

− 0 `− ∈∂w X b

+ 0 `+ ∈∂w X b

Rw b+

− 0 + + 0 − − 0 + + 0 = µ− w b− (∂w b X ) µw b− (∂w b X ) = µw b− (∂w b X ) µw b+ (∂w bX ) e

w b−

e−δ) (F

,

where this last equality follows by Equations (4.2) and (3.20) with x = w b− and + 0 y=w b+ , since for every `+ ∈ ∂w X , we have w b ∈ [ w b , ` [ . + − + b (2) Let us fix p 6 q in Z and ap , . . . , aq , aq+1 in A , and let us try to verify Equation (5.7) for the random process (Zn0 )n∈N = (Zn )n∈N . Let α∗ = (ap , . . . , aq ), which is an admissible sequence, since we assumed the conditional probability P(Zq+1 = aq+1 | Zq = aq , . . . , Zp = ap ) to be well defined. We may assume that α = (ap , . . . , aq , aq+1 ) is an admissible sequence; otherwise, both sides of Equation (5.7) are 0. Let us consider Qα =

P([ap , . . . , aq+1 ]) P([aq ]) P(Zq+1 = aq+1 | Zq = aq , . . . , Zp = ap ) = . P(Zq+1 = aq+1 | Zq = aq ) P([ap , . . . , aq ]) P([aq , aq+1 ])

Let us replace each of the four terms in this ratio by its value given by + 0 + + 0 0 0 Assertion (1). Since ∂αb− X0 = ∂α− X0 , ∂α+ X , as well b X = ∂aq\ c∗ X , ∂α c∗ X = ∂a c ,aq+1 q − 0 as ∂ac X = ∂a−q\ X0 , all Patterson measure terms cancel. Denoting by y1 the ,aq+1 q 5 Personal

communication.

124


common origin of α b and α c∗ , by y2 the common origin of b aq and aq\ , aq+1 , by y3 the common terminal point of b aq and α c∗ , and by y4 the common terminal point of aq\ , aq+1 and α b, we thus have, by Assertion (1), Ry

Ry

(F −δ) (F −δ) |Γαc∗ | |Γaq\ e y2 ,aq+1 | e y1 Qα = . Ry Ry e−δ) e−δ) 3 (F 4 (F |Γαb | |Γacq | e y1 e y2 4

3

e

e

Since y1 , y2 , y3 , y4 are in this order on [y1 , y4 ], we have Qα =

|Γαc∗ | |Γaq\ ,aq+1 | |Γαb | |Γacq |

.

Since every geodesic edge path of length n + 1 at least 3 in X0 defines an admissible sequence of length n at least 2 in A , by Equation (5.6), we have Qα = 1 for every admissible sequence α in A if and only if Γ is Markov-good. (3) Let E be a finite subset of the alphabet A , and let w = (ap , . . . , aq ) with p 6 q in Z be an admissible sequence in A such that ap , aq ∈ E. By Assertion (1), we have Rw b P([w]) =

− 0 + + 0 µ− w b− (∂w b X ) µw b+ (∂w bX ) e

+ e (F −δ) w b−

|Γwb | kmF k

.

Since ap , aq are varying in the finite subset E of A , the first and last edges of w b vary among the images under elements of Γ of finitely many edges of X0 . Since w ± 0 is admissible, the sets ∂w b X are nonempty open subsets of ΛΓ; hence they have ± 0 positive Patterson measures. Furthermore, the quantities µ± w b± (∂w b X ) are invariant under the action of Γ on the first/last edge of w. b Hence there exists c1 > 1 ± 0 depending only on E such that 1 6 |Γwb | 6 |Γwb− | 6 c1 and c11 6 µ± w b± (∂w b X ) 6 c1 . Note that the length of w b is equal to q − p + 2. Therefore c31

Rw b+ e e−δ e−δ c21 −δ(q−p+1)+Rwbwb−+ Fe −δ(q−p+1)+ w F b− e 6 P([w]) 6 e . kmF k kmF k

If w b = (fp−1 , fp , . . . , fq ) and x ∈ [w], we have, by the definition of Fsymb , Z

w b+

w b−

Fe =

Z q X i=p−1

t(fi )

o(fi )

Z

t(fp−1 )

Fe =

Fe + o(fp−1 )

q X

Fsymb (σ i (x)) .

i=p

Since Fe is continuous and Γ-invariant, and since o(fp−1 ) remains in the image under Γ of a finite subset of V X0 , there exists c2 > 0 depending only on E such that |Fe(v)| 6 c2 for every v ∈ T 1 X with π(v) ∈ [o(fp−1 ), t(fp−1 )]. Hence we have R t(fp−1 ) | o(fp−1 Fe| 6 c2 , and Assertion (3) of Proposition 5.5 follows (see Equation (5.1) ) for the definition of the Gibbs property).

5.3. Coding continuous-time geodesic flows on metric trees

125

Again in order to consider the case in which the length spectrum LΓ of Γ is 2Z, we define Peven =

1 k(mF )|Γ\Geven X0 k

(Θeven )∗ (mF )|Γ\Geven X0 ,

and (Zeven, n )n∈Z the random process associated with the full shift σeven on Σeven , with Zeven, n : Σeven → Aeven the nth projection for every n ∈ Z. By a proof similar to that of Proposition 5.5, we have the following result. We define a map Fsymb, even : Σeven → R by Z Fsymb,

even (x)

t(f0+ )

=

F

(5.8)

o(f00 ) + + 0 if x = (xi )i∈Z with x0 = (f0− , h− 0 , f0 , h0 , f0 ). As previously, Fsymb, constant, hence continuous.

even

is locally

Proposition 5.6. The measure Peven on the topological Markov shift Σeven satisfies the Gibbs property with Gibbs constant δ for the potential Fsymb, even . Again, if Γ is a noncocompact geometrically finite tree lattice of X with Abelian edge stabilisers and X0 has degrees at least 3, then (Zeven, n )n∈Z is not a Markov chain.

5.3

Coding continuous-time geodesic flows on metric trees

Let (X, λ) be a locally finite metric tree without terminal vertices, with X = |X|λ its geometric realisation. Let Γ be a nonelementary discrete subgroup of Aut(X, λ), and let Fe : T 1 X → R be a potential for Γ. Let X 0 = C ΛΓ, which is the geometric realisation |X0 |λ of a metric subtree (X0 , λ). Let δ = δΓ,F ± be the critical exponent of (Γ, F ± ), assumed to be finite. Let (µ± x )x∈V X be the (normalised) Patterson densities on ∂∞ X for the pairs (Γ, F ± ), and assume that the associated Gibbs measure mF is finite. We also assume in this section that the lengths of the edges of (X0 , λ) have a finite upper bound (which is in particular the case if (X0 , λ) is uniform). They have a positive lower bound by definition (see Section 2.6). In this section, we prove that the continuous-time geodesic flow on Γ\G X 0 is isomorphic to a suspension of a transitive (two-sided) topological Markov shift on a countable alphabet, by an explicit construction that will be useful later on in studying the variational principle (see Section 5.4) and rates of mixing (see Section 9.3). Since we are interested only in the support of the Gibbs measures, we will give such a description only for the geodesic flow on the nonwandering subset Γ\G X 0 of Γ\G X. The same construction works with the full space Γ\G X, but the resulting Markov shift is then not necessarily transitive.

126


We begin by recalling (see, for instance, [BrinS, §1.11]) the definitions of the suspension of an invertible discrete-time dynamical system and the first return map on a cross-section of a continuous-time dynamical system, which allow us to pass from transformations to flows and back, respectively. Let (Z, T, µ) be a metric space Z endowed with a homeomorphism T and a T -invariant (positive Borel) measure µ. Let r : Z → ]0, +∞[ be a continuous map such that for all z ∈ Z, the subset {r(T n z) : n ∈ N} ∪ {− r(T −(n+1) z) : n ∈ N} is discrete in R. Then the suspension (or also special flow) over (Z, T, µ) with roof function r is the following continuous-time dynamical system (Zr , (Trt )t∈R , µr ): • The space Zr is the quotient topological space (Z × R)/ ∼, where ∼ is the equivalence relation on Z × R generated by (z, s + r(z)) ∼ (T z, s) for all (z, s) ∈ Z × R. We denote by [z, s] the equivalence class of (z, s). Note that F = {(z, s) : z ∈ Z, 0 6 s < r(z)} is a measurable strict fundamental domain for this equivalence relation. We endow Zr with the Bowen–Walters distance;, see [BowW] and particularly the appendix in [BarS]. • For every t ∈ R, the map Trt : Zr → Zr is the map [z, s] 7→ [z, s + t]. Equivalently, when (z, s) ∈ F and t > 0, then Trt ([z, s]) = [T n z, s0 ], where n ∈ N and s0 ∈ R are such that t+s=

n−1 X

r(T i z) + s0 and 0 6 s0 < r(T n z) .

i=0

• With ds denoting the Lebesgue measure on R, the measure µr is the pushforward of the restriction to F of the product measure dµ ds by the restriction to F of the canonical projection (Z × R) → Zr . Note that (Trt )t∈R is indeed a continuous one-parameter group of homeomorphisms of Zr , preserving the measure µr . The measure µr is finite if and only if R r dµ is finite, since Z Z Z kµr k = dµds = r dµ . F

Z

We will denote by (Z,T,µ)r the continuous-time dynamical system (Zr ,(Trt )t∈R ,µr ) thus constructed. Conversely, let (Z, (φt )t∈R , µ) be a metric space Z endowed with a continuous one-parameter group of homeomorphisms (φt )t∈R , preserving a (positive Borel) measure µ. Let Y be a cross-section of (φt )t∈R , which is a closed subspace of Z such that for every z ∈ Z, the set {t ∈ R : φt (z) ∈ Y } is infinite and discrete. Let τ : Y → ]0, +∞[ be the (continuous) first return time on the cross-section Y : for every y ∈ Y , τ (y) = min{t > 0 : φt (y) ∈ Y } .


127

Let φY : Y → Y be the (homeomorphic) first return map to (or Poincaré map of) the cross-section Y , defined by φY : y 7→ φτ (y) (y) . By the invariance of µ under the flow (φt )t∈R , the restriction of µ to {φt (y) : y ∈ Y, 0 6 t < τ (y)} disintegrates6 by the (well-defined) map φt (y) 7→ y over a measure µY on Y , which is invariant under the first return map φY : dµ(φt (y)) = dt dµY (y) . Note that if τ has a positive lower bound and if µ is finite, then µY is finite, since kµk > kµY k inf τ , and (Y, µY , φY ) is a discrete-time dynamical system. Recall that an isomorphism from a continuous-time dynamical system (Z, (φt )t∈R , µ) to another one (Z 0 , (φ0t )t∈R , µ0 ) is a homeomorphism between the underlying spaces preserving the underlying measures and commuting with the underlying flows. Example 5.7. If (Z, T, µ) and (Z 0 , T 0 , µ0 ) are (invertible) discrete-time dynamical systems endowed with roof functions r : Z → ]0, +∞[ and r0 : Z 0 → ]0, +∞[ respectively, if θ : Z → Z 0 is a measure-preserving homeomorphism commuting with the transformations T and T 0 (that is, θ∗ µ = µ0 , θ ◦ T = T 0 ◦ θ) and such that r0 ◦ θ = r , b then the map θ : Zr → Zr0 0 defined by [z, s] 7→ [θ(z), s] is an isomorphism between the suspensions (Z, T, µ)r and (Z 0 , T 0 , µ0 )r0 . It is well known (see, for instance, [BrinS, §1.11]) that the above two constructions are inverses one to another, up to isomorphism. In particular, we have the following result. Proposition 5.8. With the general notation above Example 5.7, the suspension (Y, φY , µY )τ over (Y, φY , µY ) with roof function τ is isomorphic to (Z, (φt )t∈R , µ) by the map fY : [y, s] 7→ φs y. In order to describe the continuous-time dynamical system mF Γ\G X 0 , (gt )t∈R , kmF k 6 The

conditional measure on the fiber {φt (y) : 0 6 t < τ (y)} over y ∈ Y is the image of the Lebesgue measure on [0, τ (y)[ by t 7→ φt (y).

128


as a suspension over a topological Markov shift, we will start by describing it as a suspension of the discrete-time geodesic flow on Γ\G X0 . Note that the Patterson densities and Gibbs measures depend not only on the potential, but also on the lengths of the edges.7 We hence need to relate precisely the continuous-time and discrete-time situations, and we will use in this section the left exponent ] to indicate a discrete-time object whenever needed. For instance, we set8 ] X = |X|1 , ] X 0 = |X0 |1 and we denote by (] gt )t∈Z the discrete-time geodesic flow on Γ\G X and Γ\G X0 . Note that X 0 and ] X 0 are equal as topological spaces (but not as metric spaces). The boundaries at infinity of X 0 and ] X 0 , which coincide with their spaces of ends as topological spaces (by the assumption on the lengths of the edges), are hence equal and both denoted by ∂∞ X0 . We may assume by Section 3.5 that the potential Fe : T 1 X → R is the potene tial Fc associated with a system of conductances e c on the metric tree (X, λ) for Γ. Let δc = δFc . We denote by ] e c : EX → R the Γ-invariant system of conductances ]

e c : e 7→ (e c(e) − δc )λ(e)

(5.9)

on the simplicial tree X for Γ, by Fe] c : T 1 ( ] X) → R its associated potential, and by ] c : Γ\EX → R and F] c : Γ\T 1 ( ] X) → R their quotient maps. Note that the inclusion morphism Aut(X, λ) → Aut(X) is a homeomorphism onto its image (for the compact-open topologies), by the assumption of a positive lower bound on the lengths of the edges, hence that Γ is also a nonelementary discrete subgroup of Aut(X). Now let (Σ, σ, P) be the (two-sided) topological Markov shift conjugated to mF the discrete-time geodesic flow Γ\G X0 , ] g1 , kmF] c k by the bilipschitz homeomor]c

phism Θ : Γ\G X0 → Σ of Theorem 5.1 (where the potential F is replaced by F] c ). Let r : Σ → ]0, +∞[ be the map r : x 7→ λ(e+ 0)

(5.10)

+ if x = (xn )n∈Z ∈ Σ and x0 = (e− 0 , h0 , e0 ) ∈ A . This map is locally constant, hence continuous on Σ, and has a positive lower bound, since the lengths of the edges of (X0 , λ) have a positive lower bound. 7 The

fact that the Patterson densities could be singular one with respect to another when the metric varies is a well-known phenomenon, even when the potential vanishes. For instance, let Σ = Γ\H2R and Σ0 = Γ0 \H2R be two closed connected hyperbolic surfaces, uniformised by the real hyperbolic plane (H2R , ds2hyp ) endowed with torsion-free cocompact Fuchsian groups Γ and e : H2 → H2 . Then Γ is a discrete group of Γ0 . Let φ : Σ → Σ0 be a diffeomorphism, with lift φ R R 2 e∗ ds2 . Kuusalo’s theorem [Kuu] says that isometries for the two CAT(−1) metrics ds and φ hyp

hyp

the corresponding two Patterson densities of Γ are absolutely continuous one with respect to the other if and only if φ is isotopic to the identity. See an extension of this result in [HeP1]. See also the result of [KaN] that parametrises the Culler–Vogtmann space using Patterson densities for cocompact and free actions of free groups on metric trees. 8 See Section 2.6 for the definition of the geometric realisation |Y| of a simplicial tree Y. 1


129

Theorem 5.9. Assume that the lengths of the edges of (X0 , λ) have a finite upper bound, and that the Gibbs measure mF is finite. Then there exists a > 0 such mF that the continuous-time dynamical system Γ\G X 0 , (gt )t∈R , km is isomorphic Fk to the suspension (Σ, σ, a P)r over (Σ, σ, a P) with roof function r, by a bilipschitz homeomorphism Θr : Γ\G X 0 → Σr . Proof. Let Y = {` ∈ Γ\G X 0 : `(0) ∈ Γ\V X0 } . Then the (closed) subset Y of Γ\G X 0 is a cross-section of the continuoustime geodesic flow (gt )t∈R , since every orbit meets Y infinitely many times at a discrete set of times and since the lengths of the edges of (X0 , λ) have a positive lower bound. Let τ : Y → ]0, +∞[ be the first return time, let µY be the measure mF ), and let g Y : Y → Y be the first return on Y (obtained by disintegrating km Fk map associated with this cross-section Y . We have a natural reparametrisation map R : Y → Γ\G X0 , defined by ` 7→ ] `, where ] `(n) = (g nY `)(0) is the nth passage of ` in V X, for every n ∈ Z. Since there exist m, M > 0 such that λ(EX) ⊂ [m, M ], the map R is a bilipschitz homeomorphism. It conjugates the first return map g Y and the discrete-time geodesic flow on Γ\G X0 : R ◦ g Y = ] g1 ◦ R . The main point of this proof is the following result relating the measures µY and mF] c . Lemma 5.10. (1) The family (µ± x )x∈V X is a Patterson density for (Γ, F] c ) on the boundary at infinity of the simplicial tree X0 , and the critical exponent δ] c of ] c is equal to 0. (2) We have R∗ kµµYY k =

mF]

kmF] k . c c

Proof. (1) By the definition of the potential associated with a system of conductances,9 for every x, y ∈ V X0 , if (e1 , . . . , en ) is the geodesic edge path in X with o(e1 ) = x and t(en ) = y, then (noting that the integrals along paths depend on the lengths of the edges, the first one below being in X 0 , the second one in ] X 0 ) Z

y

fc − δc ) = (F x

Z n X (e c(ei )λ(ei ) − δc λ(ei )) =

y

x

i=1

Let us denote10 by ]

Q(s) = QΓ, F] c , x, y (s) =

X γ∈Γ

9 See

Section 3.5. Section 3.3.

10 See

e

R γy x

e] −s) (F c

Ff ]c .

(5.11)

130


and Q(s) = QΓ, Fc −δc , x, y (s) =

X

e

R γy x

ec −δc −s) (F

γ∈Γ

the Poincaré series for the simplicial tree with potential Fe] c and for the metric tree with normalised potential Fec − δc , respectively. With M an upper bound on the lengths of the edges of (X, λ), the distances d] X 0 on ] X 0 and dX 0 on X 0 satisfy 1 s d] X 0 > M dX 0 on the pairs of vertices of X0 . We hence have ] Q(s) 6 Q( M ) < +∞ if s s > 0 and ] Q(s) > Q( M ) = +∞ if s < 0. Thus the critical exponent δ] c of (Γ, F] c ) for the simplicial tree X0 is equal to 0; hence F] c is a normalised potential. By the definition11 of the Gibbs cocycles (which uses the normalised potential), Equation (5.11) also implies that the Gibbs cocycles C ± and ] C ± for (Γ, Fc ) and (Γ, F] c ) respectively coincide on ∂∞ X × V X × V X. Thus by Equations (4.1) and (4.2), the family (µ± x )x∈V X is indeed a Patterson density for (Γ, F] c ): for all γ ∈ Γ and x, y ∈ V X, and for (almost) all ξ ∈ ∂∞ X, ± γ∗ µ± x = µγx and

dµ± − ] Cξ± (x, y) x . ± (ξ) = e dµy

(2) We may hence choose these families (µ± x )x∈V X in order to define the Gibbs measure mF] c associated with the potential F] c on Γ\G X. Note that since we will prove that mF] c is finite, the normalised measure

mF]

c

kmF] k

is independent of this

c

choice (see Corollary 4.7). Let Ye = {` ∈ G X 0 : `(0) ∈ V X0 } be the (Γ-invariant) lift of the cross-section e : Ye → G X0 be the lift of R mapping a geodesic line ` ∈ Ye to a Y to G X 0 , let R discrete geodesic line ] ` with same footpoint obtained by reparametrisation, and e e let µf Y be the measure on Y whose induced measure on Y = Γ\Y is µY . We have a partition of Ye into the closed-open subsets Yex = {` ∈ G X 0 : `(0) = x} as x varies in V X0 . m Let us fix x ∈ V X0 . By the definition of µY as a disintegration of kmFFc k with c 0 respect to the continuous time, by lifting to G X , using Hopf’s parametrisation with respect to x and Equation (4.4) with x0 = x, we have for every ` ∈ Yex , dµf Y (`) =

1 + dµ− x (`− ) dµx (`+ ) . kmFc k

Note that `(0) = ] `(0), `− = ] `− , `+ = ] `+ , since the reparametrisation does not change the origin or the two points at infinity. Hence by Assertion (1), we have e∗ (µf R Y)= 11 See

Section 3.4.

1 m e F] c . kmFc k


131

Since µY is a finite measure, since τ has a positive lower bound, this implies that mF] c is finite. By renormalising as probability measures, this proves Assertion (2). Let a = kµY k > 0, so that by Lemma 5.10 (2) we have R∗ µY = b r : Γ\G X0 → ]0, +∞] be the map b r : Γ` 7→ λ `([0, 1])

a mF]

kmF] k . c

Let

c

b given by the length bfor λ of the first edge followed by a discrete geodesic line ` ∈ G X0 . Note that r is locally constant, hence continuous, and that r is a roof function for the discrete-time dynamical system (Γ\G X0 , ] g1 ). Also note that b b r ◦ R = τ and r ◦ Θ = r by the definitions of τ and r. Let us finally define Θr : Γ\G X 0 → Σr as the compositions of the three maps −1 fY mF Γ\G X 0 , (gt )t∈R , −→ (Y, gY , µY )τ (5.12) kmF k ! b b a mF] c R Θ 0 ] 1 −→ Γ\G X , g , −→ (Σ, σ, a P)r , kmF] c k b r

b Proposib where the first one is the inverse of the tautological isomorphism given by tion 5.8, and the last two, given by Example 5.7, are the isomorphisms R and Θ of continuous-time dynamical systems obtained by suspensions of the isomorphisms R and Θ of discrete-time dynamical systems. It is easy to check that Θr is a bilipschitz homeomorphism, using the following description of the Bowen–Walters distance given, for instance, in [BarS, Appendix]. Proposition 5.11. Let (Z, T, µ)r be the suspension over an invertible dynamical system such that T is a bilipschitz homeomorphism, with roof function r having a positive lower bound and a finite upper bound. Let dBW : Zr × Zr → R be the map12 defined (using the canonical representatives) by dBW ([x, s], [x0 , s0 ]) = min{d(x, x0 ) + |s − s0 |, d(T x, x0 ) + r(x) − s + s0 , d(x, T x0 ) + r(x0 ) + s − s0 } Then there exists a constant CBW > 0 such that the Bowen–Walters distance d on Σr satisfies 1 dBW 6 d 6 CBW dBW . CBW This concludes the proof of Theorem 5.9. 12 The

map dBW is actually not a distance, but this proposition says that it may replace the Bowen–Walters true distance when one is working up to multiplicative constants or bilipschitz homeomorphisms.

132

5.4


The variational principle for metric and simplicial trees

In this section, we assume that X is the geometric realisation of a locally finite metric tree without terminal vertices (X, λ) (respectively of a locally finite simplicial tree X without terminal vertices). Let Γ be a nonelementary discrete subgroup of Aut(X, λ) (respectively Aut(X)). We relate in this section the Gibbs measures13 to the equilibrium states14 for the continuous-time geodesic flow on Γ\G X (respectively for the discrete-time geodesic flow on Γ\G X). When X is a Riemannian manifold with pinched negative curvature such that the derivatives of the sectional curvature are uniformly bounded, and when the potential is H¨ older-continuous, the analogues of the results of this section are due to [PauPS, Thm. 6.1]. Their proofs generalise the proofs of Theorems 1 and 2 of [OtaP], with ideas and techniques going back to [LedS]. When Y is a compact locally CAT(−1) space, a complete statement about existence, uniqueness and Gibbs property of equilibrium states for any Hölder-continuous potential is given in [ConLT]. The proof of the metric tree case will rely strongly (via the suspension process described in Section 5.3) upon the proof of the simplicial tree case; hence we start with the latter. The simplicial tree case. Let X be a locally finite simplicial tree without terminal vertices, with geometric realisation X = |X|1 . Let Γ be a nonelementary discrete subgroup of Aut(X). Let e c : EX → R be a system of conductances for Γ on X, and c : Γ\EX → R its quotient map. Let Fec : T 1 X → R be its associated potential, with quotient map Fc : Γ\T 1 X → R, and let δc be the critical exponent of c. fc : G X → R by We define a map F Z t(f0 (`)) f Fc (`) = e c (f0 (`)) = Fec o(f0 (`))

for all ` ∈ G X, where f0 (`) is the edge of X in which ` enters at time t = 0. This map is locally constant, hence continuous, and it is Γ-invariant; hence it induces a continuous map Fc : Γ\G X → R, which is also called a potential.15 The following result proves that the Gibbs measure of (Γ, Fc ) for the discretetime geodesic flow on Γ\G X is an equilibrium state for the potential Fc . We begin by recalling the definition of an equilibrium state;16 see also [Bowe3, Rue3]. 13 See

the definition in Sections 4.2 and 4.3. the definitions below. 15 See after the proof of Theorem 5.12 for a comment on cohomology classes. 16 This definition is given for transformations and not for flows, and for possibly unbounded potentials, in contrast to the definition in the introduction. 14 See

5.4. The variational principle for metric and simplicial trees

133

Let Z be a locally compact topological space, let T : Z → Z be a homeomorphism, and let φ : Z → R be a continuous map. Let Mφ be the set of Borel probability measures m on Z, invariant under the transformation T , such that the negative part φ− = max{0, −φ} of φ is m-integrable. Let hm (T ) be the (metric) entropy of the transformation T with respect to m ∈ Mφ (see, for instance, [BrinS]). The metric pressure for the potential φ of a measure m in Mφ is Z Pφ (m) = hm (T ) + φ dm . Z

The fact that the negative part of φ is m-integrable, which is in particular satisfied if φ is bounded, implies that Pφ (m) is well defined in R ∪ {+∞}. The pressure of the potential φ is the element of R ∪ {+∞} defined by Pφ = sup Pφ (m) . m∈Mφ

A measure m0 in Mφ is an equilibrium state for the potential φ if Pφ (m0 ) = Pφ . Theorem 5.12 (The variational principle for simplicial trees). Let X, Γ, e c be as above. Assume that δc < +∞ and that there exists a finite Gibbs measure mc for mc Fc such that the negative part of the potential Fc is mc -integrable. Then km is ck the unique equilibrium state for the potential Fc under the discrete-time geodesic flow on Γ\G X, and the pressure of Fc coincides with the critical exponent δc of c : PFc = δc . In order to prove this result using the coding of the discrete-time geodesic flow given in Section 5.2, the main tool is the following result of J. Buzzi in symbolic dynamics, building on works of Sarig and Buzzi–Sarig, whose proof is given in the Appendix. Let σ : Σ → Σ be a two-sided topological Markov shift17 with (countable) alphabet A and transition matrix A, and let φ : Σ → R be a continuous map. For every n ∈ N, we denote18 by varn φ =

sup

|φ(x) − φ(y)|

x, y ∈ Σ ∀ i ∈ {−n, ..., n}, xi =yi

the n-variation of φ. For instance, if φ(x) depends only on x0 , where x = (xi )i∈Z , P then varn φ = 0 for every n ∈ N (and hence n∈N (n + 1) varn φ = 0 converges). A weak Gibbs measure for φ with Gibbs constant C(m) ∈ R is a σ-invariant (positive Borel) measure m on Σ such that for every a ∈ A , there exists ca > 1 such that for all n ∈ N − {0} and x ∈ [a] such that σ n (x) = x, we have m([x0 , . . . , xn−1 ]) 1 6 6 ca . Pn−1 −C(m) n e i=0 φ(σ i x) ca e 17 See

Section 5.1 for definitions. is a shift of indices compared with the notation of the Appendix.

18 There

(5.13)

134


Theorem 5.13 (J. Buzzi; see Corollary A.5). Let (Σ, σ) be a two-sided transitive topological Markov P shift on a countable alphabet and let φ : Σ → R be a continuous map such that n∈N (n + 1) varn φ converges.RLet m be a weak Gibbs measure for φ on Σ with Gibbs constant C(m) such that φ− dm < +∞. Then the pressure of φ is finite, equal to C(m), and m is the unique equilibrium state. Proof of Theorem 5.12. In Section 5.2, we have constructed a transitive topological Markov shift (Σ, σ) on a countable alphabet A and a homeomorphism Θ : Γ\G X0 → Σ that conjugates the time-one discrete geodesic flow g1 on the nonwandering subset Γ\G X0 of Γ\G X and the shift σ on Σ (see Theorem 5.1). Let us define a potential Fc, symb : Σ → R by Fc, symb (x) = c(e+ 0)

(5.14)

+ if x = (xi )i∈Z ∈ Σ with x0 = (e− 0 , h0 , e0 ). Note that this potential is the one denoted by Fsymb in Equation (5.5), when the potential F on T 1 X is replaced by Fc . By the construction of Θ and the definition of Fc , we have

Fc, symb ◦ Θ = Fc .

(5.15)

Note that all probability measures on Γ\G X invariant under the discrete-time geodesic flow are supported on the nonwandering set Γ\G X0 . The pushforward of measures Θ∗ hence gives a bijection from the space MFc of g1 -invariant probability measures on Γ\G X for which the negative part of Fc is integrable to the space MFc, symb of σ-invariant probability measures on Σ for which the negative part of Fc, symb is integrable. This bijection induces a bijection between the subsets of equilibrium states. Note that Fc, symb (x) Pdepends only on x0 for every x = (xi )i∈Z ∈ Σ. Hence as seen above, the series n∈N (n + 1) varn Fc, symb converges. mc By definition,19 the measure P is the pushforward of km by Θ. Hence, P is a ck σ-invariant probability measure on Σ for which the negative part of Fc, symb is integrable, by the assumption of Theorem 5.12. By Proposition 5.5 (3), the measure P on Σ satisfies the Gibbs property with Gibbs constant δc for the potential Fc, symb , hence20 satisfies the weak Gibbs property with Gibbs constant δc . Theorem 5.12 then follows from Theorem 5.13. Remark. It follows from Equation (5.15), from the remark above Definition 5.3 and from the fact that Θ ◦ g1 = σ ◦ Θ, that if c, c0 : Γ\EX0 → R are cohomologous systems of conductances on Γ\EX0 , then the corresponding maps Fc , Fc0 : Γ\G X0 → R are cohomologous: there exists a continuous map G : Γ\G X0 → R such that for every ` ∈ Γ\G X0 , Fc0 (`) − Fc (`) = G(g1 `) − G(`) . 19 See

Equation (5.4). every a ∈ A , for the constant ca required by the definition of the weak Gibbs property in Equation (5.13), take the constant CE given by the definition (see Equation (5.1)) of the Gibbs property with E = {a}. 20 For


135

Note that given two bounded cohomologous continuous potentials on a topological dynamical system (Z, T ) as above, one has finite pressure (resp. admits an equilibrium state) if and only if the other one does, and they have the same pressure and same set of equilibrium states. The metric tree case. Let (X, λ) be a locally finite metric tree without terminal vertices with geometric realisation X = |X|λ , let Γ be a nonelementary discrete subgroup of Aut(X, λ), and let e c : EX → R be a system of conductances for Γ on X. Let Fec : T 1 X → R be its associated potential (see Section 3.5), and let δc = δFc be the critical exponent of c. Recall21 that we have a canonical projection G X → T 1 X that associates e\ : G X → R be the to a geodesic line ` its germ v` at its footpoint `(0). Let F c Γ-invariant map obtained by precomposing the potential Fec : T 1 X → R with this canonical projection: e\ : ` 7→ Fec (v` ) . F c Let F\c : Γ\G X → R be its quotient map, which is continuous as a composition of continuous maps. The following result proves that the Gibbs measure of (Γ, Fc ) for the continuous-time geodesic flow on Γ\G X, once renormalised to be a probability measure, is an equilibrium state for the potential F\c . We begin by recalling the definition of an equilibrium state for a possibly unbounded potential under a flow.22 Given (Z, (φt )t∈R ) a topological space endowed with a continuous one-parameter group of homeomorphisms and ψ : Z → R a continuous map (called a potential), let Mψ be the set of Borel probability measures m on Z invariant under the flow (φt )t∈R such that the negative part ψ − = max{0, −ψ} of the potential ψ is m-integrable. Let hm (φ1 ) be the (metric) entropy of the geodesic flow with respect to m ∈ Mψ (see, for instance, [BrinS]). The metric pressure for ψ of a measure m ∈ Mψ is Z Pψ (m) = hm (φ1 ) + ψ dm . Z

The fact that the negative part of ψ is m-integrable, which is in particular satisfied if ψ is bounded, implies that Pψ (m) is well defined in R ∪ {+∞}. The pressure of the potential ψ is the element of R ∪ {+∞} defined by Pψ = sup

Pψ (m) .

m∈Mψ

Note that Pψ+c = Pψ + c for every constant c ∈ R. An element m ∈ Mψ is an equilibrium state for ψ if the least upper bound defining Pψ is attained on m. 21 See

Section 2.3. requires only minor modifications to the definition given in the introduction for bounded potentials. 22 This

136


Note that if ψ 0 is another potential cohomologous to ψ, that is, if there exists a continuous map G : Z → R, differentiable along every orbit of the flow, such that d ψ 0 (x) − ψ(x) = dt G(φt (x)), if ψ and ψ 0 are bounded (so that Mψ0 = Mψ ), t=0 then for every m ∈ Mψ , we have Pψ0 (m) = Pψ (m), Pψ0 = Pψ , and the equilibrium states for ψ 0 are exactly the equilibrium states for ψ. Theorem 5.14 (The variational principle for metric trees). Let (X, λ), Γ, e c be as above. Assume that the lengths of the edges of (X, λ) have a finite upper bound.23 Assume that δc < +∞ and that there exists a finite Gibbs measure mc for Fc such mc is the unique that the negative part of the potential F\c is mc -integrable. Then km ck \ equilibrium state for the potential Fc under the continuous-time geodesic flow on Γ\G X, and the pressure of F\c coincides with the critical exponent δc of c : PF\c = δc . Using the description of the continuous-time dynamical system Γ\G X 0 , (gt )t∈R ,

mc kmc k

as a suspension over a topological Markov shift (see Theorem 5.9), this statement reduces to well-known techniques in the thermodynamic formalism of suspension flows; see, for instance, [IJT], as well as [BarI, Kemp, IJ, JKL]. Our situation is greatly simplified by the fact that our roof function has a positive lower bound and a finite upper bound, and that our symbolic potential is constant on the 1-cylinders {x = (xi )i∈Z ∈ Σ : x0 = a} for a in the alphabet. Proof. Since finite measures invariant under the geodesic flow on Γ\G X are supported on its nonwandering set, up to replacing X by X 0 = C ΛΓ, we assume that X = X 0. Since equilibrium states are unchanged up to adding a constant to the potenmc tial, under the assumptions of Theorem 5.14, let us prove that km is the unique ck equilibrium state for the potential F\c − δc under the continuous-time geodesic flow on Γ\G X, and that the pressure of F\c − δc vanishes. The last claim of Theorem 5.14 follows, since PF\c − δc = PF\c −δc . We refer to the paragraphs before the statement of Theorem 5.9 for the definitions of • the system of conductances ] e c for Γ on the simplicial tree X, • the (two-sided) topological Markov shift (Σ, σ, P) on the alphabet A , conju mF gated to the discrete-time geodesic flow Γ\G X, ] g1 , kmF] c k by the homeo]c

morphism Θ : Γ\G X → Σ, 23 They

have a positive lower bound by definition; see Section 2.6.


137

• the roof function r : Σ → ]0, +∞[ , and • the suspension (Σ, σ, a P)r = (Σr , (σrt )t∈R , a Pr ) over (Σ, σ, a P) with roof function r, conjugated to the continuous-time geodesic flow mc Γ\G X, (gt )t∈R , kmc k by the homeomorphism Θr : Γ\G X → Σr defined at the end of the proof of Theorem 5.9. We will always (uniquely) represent the elements of Σr as [x, s] with x ∈ Σ and 0 6 s < r(x). We denote by F\c, symb : Σr → R the potential defined by F\c, symb = F\c ◦ Θ−1 r ,

(5.16)

which is continuous as a composition of continuous maps. The key technical observation in this proof is the following. R r(x) Lemma 5.15. For every x ∈ Σ, we have F] c, symb (x) = 0 (F\c, symb −δc )([x, s]) ds. For every x ∈ Σ, the sign of F\c, symb ([x, s]) is constant on s ∈ [0, r(x)[ . + Proof. Let x = (xn )n∈Z ∈ Σ and x0 = (e− 0 , h0 , e0 ) ∈ A . By the definition of the first return time r in Equation (5.10), we have in particular

r(x) = λ(e+ 0). By Equation (5.14) and by the definition of ] e c in Equation (5.9), we have + + F] c, symb (x) = ] c(e+ 0 ) = (c(e0 ) − δc )λ(e0 ) .

Using in the following sequence of equalities respectively • the two definitions of the potential F\c, symb in Equation (5.16) and of the suspension flow (σrt )t∈R , • the fact that the suspension flow (σrt )t∈R is conjugated to the continuous-time geodesic flow by Θr , • the definition of Θr using the reparametrisation map24 R of continuous-time geodesic lines with origin on vertices to discrete-time geodesic lines,25 • the definition of the potential F\c , • the fact that e+ 0 is the first edge followed by the discrete-time geodesic line Θ−1 x, hence by the continuous-time geodesic line R−1 Θ−1 x, and the relation between c and the potential Fc associated with c (see Proposition 3.21), 24 See 25 See

its definition above Lemma 5.10. Equation (5.12), with the notation of Example (5.7) and Proposition 5.8.

138


we have Z r(x) Z F\c, symb ([x, s]) ds =

r(x)

Z

r(x)

0

s F\c Θ−1 r σr [x, 0] ds =

r(x)

Z

0

F\c gs Θ−1 r [x, 0] ds

0

F\c

=

s

g R

−1

−1

Θ

Z

λ(e+ 0 )

x ds =

0

Fc vgs R−1 Θ−1 x ds

0

+ = c(e+ 0 )λ(e0 ) .

R r(x) Since 0 δc ds = δc λ(e+ 0 ), the first claim of Lemma 5.15 follows. The second claim follows by the definition of the potential Fc associated with c; see Equation (3.23). By Equation (5.16), the pushforwards of measures by the homeomorphism Θr , which conjugates the flows (gt )t∈R and (σrt )t∈R , is a bijection from MF\c to MF\ such that c, symb

PF\

c, symb

((Θr )∗ m) = PF\c (m)

mc for every m ∈ MF\c . In particular, we have only to prove that (Θr )∗ km = a Pr is ck

the unique equilibrium state for the potential F\c, symb − δc under the suspension flow (σrt )t∈R , and that the pressure of F\c, symb − δc vanishes. The uniqueness follows, for instance, from [IJT, Thm. 3.5],26 since the roof function r is locally constant and the potential g = F\c, symb is such that the R r(x) map27 from Σ to R defined by x 7→ 0 g([x, s]) ds is locally Hölder-continuous by Lemma 5.15 and since F] c, symb is locally constant. Let us now relate the σ-invariant measures on Σ to the (σrt )t∈R -invariant measures on Σr . Recall that we denote the Lebesgue measure on R by ds and the points in Σr by [x, s] with x ∈ Σ and 0 6 s < r(x). Lemma 5.16. The map S : MF] c, symb → MF\ , which associates to any measure c, symb m in MF] c, symb on Σ the measure d S(m)([x, s]) = R Σ

1 dµ(x) ds r dm

on Σr is a bijection such that, for every m ∈ MF] c, symb , PF\

(S(m)) c, symb −δc

26 Note

=

PF] c, symb (m) R . r dm Σ

that a topological Markov shift, which is (incorrectly) called topologically mixing in [IJT, page 551], is actually (topologically) transitive with the definition in this book, Section 5.1. 27 This map is denoted by ∆ in loc. cit. g


139

R Proof. Note that Σ r dm is the total mass of the measure dµr ([x, s]) = dµ(x) ds on Σr . In particular, S(m) is indeed a probability measure on Σr . Since r has a positive lower bound and a finite upper bound, it is well known since [AmK], see also [IJT, §2.4], that the map S defined above28 is a bijection from the set of σ-invariant probability measures m on Σ to the set of (σrt )t∈R -invariant probability measures on Σr . Furthermore, for every σ-invariant probability measure m on Σ, we have the following Kac formula, by the definition of the probability measure S(m) and by Lemma 5.15, Z Z F\c, symb d S(m) − δc = (F\c, symb − δc ) d S(m) Σr

Σr

=R Σ

=R Σ

Z Z r(x) 1 (F\c, symb − δc )([x, s]) dm(x) ds r dm x∈Σ 0 Z 1 F] dm . (5.17) r dm Σ c, symb

By the comment on the signs at the end of Lemma 5.15, this computation also proves that the negative part of F\c, symb is integrable for S(m) if and only if the negative part of F] c, symb is integrable for m. Hence S is indeed a bijection from MF] c, symb to MF\ . c, symb

By Abramov’s formula [Abr], see also [IJT, Prop. 2.14], we have hS(m) (σr1 ) = R

hm (σ) . r dm Σ

(5.18)

The last claim of Lemma 5.16 follows by summation from Equations (5.17) and (5.18). By the proof of Theorem 5.12 (replacing the potential c by ] c), the pressure of the potential F] c, symb is equal to the critical exponent δ] c of the potential ] c, and by Lemma 5.10 (1), we have δ] c = 0. Hence for every m ∈ MF] c, symb , we have PF\

c, symb −δc

(S(m)) =

PF] PF] c, symb (m) δ] c R 6 R c, symb = R =0. r dm r dm r dm Σ Σ Σ

In particular, the pressure of the potential F\c, symb − δc is at most 0, since S is a bijection. By the proof of Theorem 5.12 (replacing the potential c by ] c), we know that P is an equilibrium state for the potential F] c, symb . Hence PF\

(S(P)) c, symb −δc

28 This

map is denoted by R in [IJT, §2.4].

=

PF] c, symb (P) R =0. r dP Σ

140


Therefore, S(P) is an equilibrium state of the potential F\c, symb − δc , with pressure 0. But aPr , which is equal to kPPrr k since aPr is a probability measure, is by construction equal to S(P). The result follows. With slightly different notation, this result implies Theorem 1.1 in the introduction. Proof of Theorem 1.1. Every bounded potential Fe for Γ on T 1 X is cohomologous to a bounded potential Fec associated with a system of conductances (see Proposition 3.22). If two potentials Fe and Fe0 for Γ on T 1 X are cohomologous,29 then the potentials ` 7→ Fe(v` ) and ` 7→ Fe0 (v` ) for Γ on G X are cohomologous for the definition given before the statement of Theorem 5.14. Since the existence and uniqueness of an equilibrium state depends only on the cohomology class of the bounded30 potentials on G X, the result follows.

29 See 30 A

the definition at the end of Section 3.2. bounded potential has integrable negative part for any probability measure.

Chapter 6

Random Walks on Weighted Graphs of Groups Let X be a locally finite simplicial tree without terminal vertices, and let X = |X|1 be its geometric realisation. Let Γ be a nonelementary discrete subgroup of Aut(X). In Section 6.1, given a (logarithmic) system of conductances c : Γ\EX → R, we define an operator ∆c on the functions defined on the set of vertices of the quotient graph of groups Γ\\X. This operator is the infinitesimal generator1 of the random walk on Γ\\X associated with the (normalised) exponential of this system of conductances. When Γ is torsion free and the system of conductances vanishes, the construction recovers the standard Laplace operator on the graph Γ\X. Under appropriate antireversibility assumptions on the system of conductances c, using techniques of Sullivan and Coornaert–Papadopoulos, we prove that the total mass of the Patterson densities is a positive eigenvector for the operator ∆c associated with c. In Section 6.2, we study the asymmetric nearest neighbour random walks on V X associated with antireversible systems of conductivities, and we show that the Patterson densities are the harmonic measures of these random walks.

6.1

Laplacian operators on weighted graphs of groups

Let X be a locally finite simplicial tree without terminal vertices, and let X = |X|1 be its geometric realisation. Let Γ be a nonelementary discrete subgroup of Aut(X). Let e c : EX → R be a (Γ-invariant) system of conductances for Γ. 1 More

precisely, it is the infinitesimal generator of the continuous-time random process on the graph Γ\X whose so-called discrete skeleton or jump chain is the aforementioned random walk. The process waits an exponentially distributed time with parameter 1 at a vertex x, then instantaneously jumps along an edge e starting from x with probability i(e)ec(e) / degc (x). See, for instance, [AlF, §2.1.2].


141

142

Chapter 6. Random Walks on Weighted Graphs of Groups

We define e c+ = e c and e c − : e 7→ e c(e), which is another system of conductances for Γ. Recall (see Section 3.5) that e c is reversible (respectively antireversible) if e c− = e c + (respectively e c− = −e c + ). The quotient graph of groups Γ\\X is endowed with the quotient maps c± : Γ\EX → R of e c ± . For every x ∈ V X, we define the ± weighted degree of x in (X, e c ) by X

degec ± (x) =

eec

±

(e)

.

e∈EX, o(e)=x

Note that the quantity degec ± (x) is constant on the Γ-orbit of x. Hence, it defines a map degc± : Γ\V X → ]0, +∞[ . On the vector space CV X of maps from V X to C, we consider the operator ∆ec ± , called the (weighted) Laplace operator of (X, c± ),2 defined by setting, for all f ∈ CV X and x ∈ V X, ∆ec ± f (x) =

1

X

degec ± (x)

e∈EX, o(e)=x

eec

±

(e)

f (x) − f (t(e)) .

(6.1)

This is the standard Laplace operator3 of a weighted graph for the weight function ± e 7→ eec (e) , except that usually one requires that e c(e) = e c(¯ e). Note that the c e ± (e)

function p± on EX defined by p± (e) = dege ± (o(e)) is a Markov transition kernel c e on the tree X; see Section 6.2 below. The weighted Laplace operator ∆ec ± is invariant under Γ: for all f ∈ CV X and γ ∈ Γ, we have ∆ec ± (f ◦ γ) = (∆ec ± f ) ◦ γ . In particular, this operator induces an operator on functions defined on the quotient graph Γ\X, as follows. Let (Y, G∗ ) be a graph of finite groups. We denote4 by L2 (V Y, vol(Y,G∗ ) ) the Hilbert space of maps f : V Y → C with finite norm kf kvol for the following scalar product: X 1 hf, givol = f (x) g(x) . |Gx | x∈V Y

5

2

We denote by L (EY, Tvol(Y,G∗ ) ) the Hilbert space of maps φ : EY → C with finite norm kφkTvol for the following scalar product: hφ, ψiTvol =

1 X 1 φ(e) ψ(e) . 2 |Ge | e∈EY

is also called the Laplace operator on X associated with the system of conductances e c ±. for example, [Car] with the opposite choice of sign, or [ChGY]. 4 See Section 2.6 for the definition of the measure vol (Y,G∗ ) on the discrete set V Y. 5 See Section 2.6 for the definition of the measure Tvol (Y,G∗ ) on the discrete set EY. 2 It

3 See,

6.1. Laplacian operators on weighted graphs of groups

143

Let i : EY → N−{0} be the index map i(e) = [Go(e) : Ge ]. For every function c : EY → R, let degc : V Y → R be the positive function defined by X degc (x) = i(e) ec(e) . e∈EY, o(e)=x 6

The Laplace operator of (Y, G∗ , c) is the operator ∆c = ∆Y,G∗ ,c on L2 (V Y, volY,G∗ ) defined by ∆c f : x 7→

1 degc (x)

X

i(e) ec(e) f (x) − f (t(e)) .

e∈EY, o(e)=x

Remark 6.1. (1) Let (Y, G∗ ) = Γ\\X be a quotient graph of finite groups with p0 : V X → V Y = Γ\V X the canonical projection. Let e c : EX → R be a potential for Γ and let c : EY = Γ\EX → R be the map induced by e c. An easy computation shows that for all f ∈ CV Y and x ∈ V Y, we have ∆c f (x) = ∆c fe(e x) if fe = f ◦ p0 : V X → C and x e ∈ V X satisfies p0 (e x) = x. (2) For every x ∈ V Y, let X

i(x) =

i(e) .

e∈EY, o(e)=x

Then i(x) is the degree of any vertex of any universal cover of (Y, G∗ ) above x. In particular, the map i : V Y → R is bounded if and only if the universal cover of (Y, G∗ ) has uniformly bounded degrees. When c = 0, we denote the Laplace operator by ∆ = ∆Y, G∗ and for every x ∈ V Y, we have X 1 ∆f (x) = i(e) f (x) − f (t(e)) . i(x) e∈EY, o(e)=x

We thus recover the Laplace operator of [Mor] on the edge-indexed graph (Y, i). Proposition 6.2. Let (Y, G∗ ) be a graph of finite groups, whose map i : V Y → R is bounded. Let c : EY → R be a system of conductances on Y, and let p(e) =

ec(e) degc (o(e))

for every e ∈ EY. The following properties hold: (1) The Laplace operator ∆c : L2 (V Y, vol(Y,G∗ ) ) → L2 (V Y, vol(Y,G∗ ) ) is linear and bounded. 6 See,

for instance, [Mor] when c = 0.


144

(2) The map dc : L2 (V Y, vol(Y,G∗ ) ) → L2 (EY, Tvol(Y,G∗ ) ) defined by p dc (f ) : e 7→ p(e) f (t(e)) − f (o(e)) is linear and bounded, and its dual operator d∗c : L2 (EY, Tvol(Y,G∗ ) ) → L2 (V Y, vol(Y,G∗ ) ) is given by d∗c (φ) : x 7→

X e∈EY, o(e)=x

p i(e) p p(e) φ(e) − p(e) φ(e) . 2

(3) Assume that c is reversible and that the map degc : V Y → R is constant. Then ∆c = d∗c dc . In particular, ∆c is self-adjoint and nonnegative. Proof. By the assumptions, there exists M ∈ N such that i(x) 6 M for every |G | x ∈ V Y, and hence i(e) 6 M for every e ∈ EY. Note that i(e) = |Go(e) , that e| Ge = Ge , and that p(e) 6 1. (1) For every f ∈ L2 (V Y, vol(Y,G∗ ) ), using in the following computations • the Cauchy–Schwarz inequality for the first inequality, • the fact that for every x ∈ V Y, we have X X X i(e)2 p(e)2 6 M 2 p(e) 6 M 2 e∈EY, o(e)=x

e∈EY, o(e)=x

i(e)p(e) = M 2

e∈EY, o(e)=x

for the second inequality, • the fact that |Go(e) | > |Ge | for every e ∈ EY for the third inequality, and P • the change of variable e 7→ e in e∈EY |G1e | |f (t(e))|2 (since Ge = Ge ) for the first equality on the fifth line of the computations, we have X X 2 1 i(e) p(e) f (x) − f (t(e)) k∆c f k2vol = |Gx | x∈V Y e∈EY, o(e)=x X X X 1 f (x) − f (t(e)) 2 6 i(e)2 p(e)2 |Gx | x∈V Y e∈EY, o(e)=x e∈EY, o(e)=x X X 1 2 f (x) + f (t(e)) 2 6 2 M2 |Gx | x∈V Y e∈EY, o(e)=x X 1 f (o(e)) 2 + f (t(e)) 2 6 2 M2 |Ge | e∈EY

6.1. Laplacian operators on weighted graphs of groups

= 4 M2

X e∈EY

= 4 M2

X 2 1 1 f (o(e)) = 4 M 2 |Ge | |Gx | x∈V Y

145

X

i(e) |f (x)|2

e∈EY, o(e)=x

X i(x) X 1 |f (x)|2 6 4 M 3 |f (x)|2 = 4 M 3 kf k2vol . |Gx | |Gx |

x∈V Y

x∈V Y

Hence the linear operator ∆c is bounded. (2) For every f ∈ L2 (V Y, vol(Y,G∗ ) ), we have 2 1 X p(e) f (t(e)) − f (o(e)) 2 |Ge | e∈EY X X 2 2 2 1 1 f (t(e)) + f (o(e)) = 2 f (o(e)) 6 |Ge | |Ge | e∈EY e∈EY X i(e) X i(x) f (o(e)) 2 = 2 =2 |f (x)|2 6 2 M kf k2vol . |Go(e) | |Gx |

kdc f k2Tvol =

e∈EY

x∈V Y

Hence the linear operator dc is bounded. For all f ∈ L2 (V Y, vol(Y,G∗ ) ) and φ ∈ L2 (EY, Tvol(Y,G∗ ) ), using again the change of variable e 7→ e, we have 1 X 1 p hφ, dc f iTvol = p(e) φ(e) f (t(e)) − f (o(e) 2 |Ge | e∈EY p X p X p(e) p(e) 1 = φ(e) f (o(e)) − φ(e) f (o(e)) 2 |Ge | |Ge | e∈EY e∈EY X X p i(e) p 1 p(e) φ(e) − p(e) φ(e) f (x) . = |Gx | 2 x∈V Y

This gives the formula for

e∈EY, o(e)=x

d∗c .

(3) Let f, g ∈ L2 (V Y, vol(Y,G∗ ) ). Note that p(e) = p(e)7 by the reversibility of c and the fact that degc is constant. Hence, by developing the products in the first line and by making the change of variable e 7→ e in half the values, we have 1 1 X p(e) f (t(e)) − f (o(e)) g(t(e)) − g(o(e)) hdc f, dc giTvol = 2 |Ge | e∈EY X i(e) = p(e) f (o(e)) g(o(e)) − f (t(e)) g(o(e)) |Go(e) | e∈EY X X 1 = i(e) p(e) f (x) − f (t(e)) g(x) |Gx | x∈V Y

e∈EY, o(e)=x

= h∆c f, givol . This proves the last claim in Proposition 6.2. 7 This

is the usual reversibility requirement for the corresponding Markov chain.

146


The following result is an extension to antireversible systems of conductances of [CoP2, Prop. 3.3] (which treated the case of zero conductances), which is a discrete version of Sullivan’s analogous result for hyperbolic manifolds (see [Sul1]). Let Fec : T 1 X → R be the potential for Γ associated with e c,8 so that (Fec )± = Fec± , ± and let δc be their common critical exponent. Let C : ∂∞ X × V X × V X → R be the associated Gibbs cocycles. Let (µ± x )x∈V X be two Patterson densities on ∂∞ X for the pairs (Γ, Fc± ). Proposition 6.3. Assume that X is (q + 1)-regular, that the system of conductances e c is antireversible and that the map degec ± : V X → R is constant with value κ± . Then the total mass φµ± : x 7→ kµ± x k of the Patterson density is a positive eigenvector associated with the eigenvalue 1−

eδc + qe−δc κ±

for the Laplace operator ∆ec ± on CV X . ±

Proof. The function e c ± : EX → R is bounded, since eec (e) 6 degec ± (o(e)) = κ± for every e ∈ EX. Hence (Fec )± = Fec± is bounded by its definition in Section 3.5. Since the tree X is (q + 1)-regular, the critical exponent δΓ is finite, and hence the critical exponent δc = δΓ, Fc± is finite by Lemma 3.17 (6). Since Z φµ± (x) =

dµ± x =

∂∞ X

Z

±

e−Cξ

(x, x0 )

∂∞ X

dµ± x0 ,

by Equation (4.2) and by linearity, we have only to prove that for every fixed ξ ∈ ∂∞ X the map ± f : x 7→ e−Cξ (x, x0 ) δc

−δc

is an eigenvector with eigenvalue 1 − e +qe for ∆ec ± . κ± For every e ∈ EX, recall9 that ∂e X is the set of points at infinity of the geodesic rays in X whose initial edge is e. By Equation (3.20) and by the definition of the potential associated with a system of conductances,10 for all e ∈ EX and η ∈ ∂e X, since t(e) ∈ [o(e), η[ (independently of the choice of sign ±), we have Cη± (t(e), o(e)) =

Z

t(e)

o(e)

(Fec± − δc ) = e c ± (e) − δc .

Thus if ξ ∈ ∂e X, we have ±

f (t(e)) = e−Cξ 8 See

(t(e), o(e))−Cξ± (o(e), x0 )

= e−ec

±

(e)+δc

Section 3.5. Section 2.6. 10 See Proposition 3.21 with the edge length map λ constant equal to 1. 9 See

f (o(e)) ,

6.2. Patterson densities as harmonic measures for simplicial trees

147

and otherwise ±

f (t(e)) = eCξ

(t(e), o(e))−Cξ± (o(e), x0 )

= eec

±

(e)−δc

f (o(e)) .

For every x ∈ V X, let eξ be the unique edge of X with origin x such that ξ ∈ ∂eξ X. Then, ∆ec ± f (x) = f (x) − = f (x) −

1

X

degec ± (x)

o(e)=x

eec

±

(e)

f (t(e))

1 ec ± (eξ ) 1 e f (t(eξ )) − ± ± κ κ

X

eec

±

(e)

f (t(e))

e6=eξ , o(e)=x

eδc q e−δc = 1− ± − f (x) . κ κ± This proves the result.

Note that the antireversibility of the potential is used in an essential way in order to get the last equation in the proof of Proposition 6.3.

6.2

Patterson densities as harmonic measures for simplicial trees

In this section, we define and study a Markov chain on the set of vertices of a simplicial tree endowed with a discrete group of automorphisms and with an appropriate system of conductances such that the associated (asymmetric, nearest neighbour) random walk converges almost surely to points in the boundary of the tree, and we prove that the Patterson densities, once normalised, are the corresponding harmonic measures. We thereby generalise the zero potential case treated in [CoP2], which is also a special case of [CoM] when X is a tree under the additional restriction that the discrete group is cocompact. For other connections between harmonic measures and Patterson measures, we refer, for instance, to [CoM, BlHM, Tan, GouMM] and their references. Let X be a (q+1)-regular simplicial tree, with q > 2. Let Γ be a nonelementary discrete subgroup of Aut(X). Let e c : EX → R be an antireversible system of conductances for Γ such that the associated map degec : V X → R on the vertices of X is constant. Let (µx )x∈V X be a Patterson density for (Γ, Fc ), where Fc is the potential associated with c. We denote by φµ : x 7→ kµx k the associated total mass function on V X. We begin this section by recalling a few facts about discrete Markov chains, for which we refer, for instance, to [Rev, Woe1]. A state space is a discrete and countable set I. A transition kernel on I is a map p : I × I → [0, 1] (considered as

148


a square matrix with coefficients in [0, 1], and with row and column indices in I) such that for every x ∈ I, X p(x, y) = 1 . y∈I

Let λ be a probability measure on I. A (discrete) Markov chain on a state space I with initial distribution λ and transition kernel p is a sequence (Zn )n∈N of random variables with values in I such that for all n ∈ N and x0 , . . . , xn+1 ∈ I, the probability of events P satisfies (1) P(Z0 = x0 ) = λ({x0 }), (2) P(Zn+1 = xn+1 | Z0 = x0 , Z1 = x1 , . . . , Zn = xn ) = P(Zn+1 = xn+1 | Zn = xn ) = p(xn , xn+1 ). The associated random walk consists in choosing a point x0 in I with law λ, and by induction, once xn is constructed, in choosing xn+1 in I with probability p(xn , xn+1 ). Note that P(Z0 = x0 , Z1 = x1 , . . . , Zn = xn ) = λ({x0 }) p(x0 , x1 ) · · · p(xn−1 , xn ) . When the initial distribution λ is the unit Dirac mass ∆x at x ∈ I, the Markov chain is then uniquely determined by its transition kernel p and by x, and is denoted by (Znx )n∈N . For every n ∈ N, we denote by p(n) the iterated matrix product of the transition kernel p: we define p(0) (x, y) to be the Kronecker symbol δx,y for all x, y ∈ I, and by induction p(n+1) = p p(n) , that is, for all x, z ∈ I, X p(n+1) (x, z) = p(x, y)p(n) (y, z) . y∈I

Note that p(n) (x, y) = P(Znx = y) is the probability for the random walk starting at time 0 from x of being at time n at the point y. The Green kernel of p is the map Gp from I × I to [0, +∞] defined by X (x, y) 7→ Gp (x, y) = p(n) (x, y) , n∈N

and its Green function is the following power series in the complex variable z : X Gp (x, y | z) = p(n) (x, y) z n . n∈N

Recall that if Gp (x, y) 6= 0 for all x, y ∈ I, then the random walk is recurrent11 if Gp (x, y) = ∞ for any (hence all) (x, y) ∈ I × I, and transient otherwise. Note 11 This

x = y} = ∞ for every y ∈ I (or equivalently, there exists means that Card{n ∈ N : Zn x = y} = ∞). y ∈ I such that Card{n ∈ N : Zn


149

that, using again matrix products of I × I matrices, Gp = Id +p Gp .

(6.2)

We will from now on consider as state space the set V X of vertices of X. If a Markov chain (Znx )n∈N starting at time 0 from x converges almost surely in x x V X ∪ ∂∞ X to a random variable Z∞ with values in ∂∞ X, the law of Z∞ is called the harmonic measure (or hitting measure on the boundary) associated with this Markov chain, and is denoted by x νx = (Z∞ )∗ (P) .

Note that νx is a probability measure on ∂∞ X. For instance, the transition kernel of the simple nearest neighbour random walk on X is defined by taking as transition kernel the map p where p(x, y) =

1 A(x, y) q+1

for all x, y ∈ V X, with A : V X × V X → {0, 1} the adjacency matrix of the tree X, defined by A(x, y) = 1 for any two vertices x, y of X that are joined by an edge in X and A(x, y) = 0 otherwise. We denote by X G(x, y | z) = p(n) (x, y) z n n∈N √ and which diverges the Green function of p, whose radius of convergence is r = 2q+1 q at z = r; see, for example, [Woe1], [Woe2, Ex. 9.82], [LyP2, §6.3].

The antireversible system of conductances e c : EX → R defines a cocycle on the set of vertices of X, as follows. For all u, v ∈ V X, let c(u, v) = 0 if u = v and otherwise let n X c(x, y) = e c(ei ) , i=1

where (e1 , e2 , . . . , en ) is the geodesic edge path in X from u = o(e1 ) to v = t(en ). Lemma 6.4. (1) For every edge path (e01 , e02 , . . . , e0n0 ) from u to v, we have 0

c(u, v) =

n X

e c(e0i ) .

i=1

(2) The map c : V X × V X → R has the following cocycle property: for all vertices u, v, w ∈ V X, c(u, v) + c(v, w) = c(u, w) and hence c(v, u) = −c(u, v) . Rv (3) We have c(u, v) = u Fec .

150


(4) For all ξ ∈ ∂∞ X and u, v ∈ V X, if C·c (·, ·) is the Gibbs cocycle associated with Fec , we have Cξc (u, v) = c(v, u) + δc βξ (u, v) . Proof. (1) Since X is a simplicial tree, every nongeodesic edge path from u to v has a back-and-forth on some edge, which contributes to 0 in the sum defining c(x, y) by the antireversibility assumption on the system of conductances. Therefore, by induction, the sum in Assertion (1) indeed does not depend on the choice of the edge path from u to v. Assertion (2) is immediate from Assertion (1). Assertion (3) follows from the definition of c(·, ·) by Proposition 3.21. (4) For every ξ ∈ ∂∞ X, if p ∈ V X is such that [u, ξ[ ∩ [v, ξ[ = [p, ξ[ , then using Equation (3.18) and Assertions (3) and (2), we have Cξc (u, v)

Z

p

Z

p

(Fec − δc ) −

= v

(Fec − δc ) = c(v, p) − c(u, p) + δc βξ (u, v) u

= c(v, u) + δc βξ (u, v) .

Let κc =

q+1 , eδc + q e−δc

√ ] , with κc = which belongs to ]0, 2q+1 q

q+1 √ 2 q

if and only if eδc =

this constant κc is less than the radius of convergence r =

√ q. Note that

q+1 √ 2 q

of the Green

1 2

function G(x, y | z) if and only if δc 6= ln q. The computation (due to Kesten) of the Green function of p is well known, and gives the following formula; see, for instance, [CoP2, Prop. 3.1]: if δc 6= 12 ln q, then there exists α > 0 such that12 for all x, y ∈ V X, G(x, y | κc ) = α e−δc d(x, y) . (6.3) We now define the transition kernel pc associated with13 the (logarithmic) system of conductances c by, for all x, y ∈ V X, pc (x, y) = κc

φµ (y) c(x, y) e p(x, y) . φµ (x)

From now on, we denote by (Znx )n∈N the Markov chain with initial distribution ∆x and transition kernel pc . 12 We

q+1 actually have α = eδc +(q−1) . e−δc transition kernel also depends on the choice of the Patterson density if Γ is not of divergence

13 The

type.


151

Lemma 6.5. (1) The map pc is a transition kernel on V X. (2) The Green kernel Gc = Gpc of pc is Gc (x, y) = ec(x, y)

φµ (y) G(x, y | κc ) . φµ (x) 1 2

In particular, the Green kernel of pc is finite if δc 6= (3) Assume that δc 6=

1 2

(6.4)

ln q.

ln q. For all x, y, z ∈ V X, we have

φµ (y) Gc (y, z) = ec(y, x)+δc (d(x, z)−d(y, z)) . φµ (x) Gc (x, z) If furthermore z ∈ / [x, y[ , then for every ξ ∈ Ox (z),14 c φµ (y) Gc (y, z) = eCξ (x, y) . φµ (x) Gc (x, z)

Proof. (1) By the proof of Proposition 6.3, the positive function φµ is an eigenvector with eigenvalue eδc + q e−δc for the operator X f 7→ {x 7→ eec(e) f (t(e))} . e∈EX, o(e)=x

Since p(o(e), t(e)) = X pc (x, y) = y∈V X

1 q+1

for every e ∈ EX, we hence have X pc (x, t(e))

e∈EX, o(e)=x

=

(eδc

1+q + q e−δc ) φµ (x)

X

eec(e) φµ (t(e)) p(x, t(e)) = 1 .

e∈EX, o(e)=x

(2) Let us first prove that for all x, y ∈ V X and n ∈ N, we have n p(n) c (x, y) = (κc )

φµ (y) c(x, y) (n) e p (x, y) . φµ (x)

(6.5)

Indeed, by the cocycle property of c(·, ·) and by a telescopic cancellation argument, we have X p(n) pc (x, x1 ) pc (x1 , x2 ) · · · pc (xn−2 , xn−1 ) pc (xn−1 , y) c (x, y) = x1 ,..., xn−1 ∈V X

= (κc )n

= (κc )n

φµ (y) c(x, y) e φµ (x)

X x1 ,..., xn−1 ∈V X

p(x, x1 ) p(x1 , x2 ) · · · · · · p(xn−2 , xn−1 ) p(xn−1 , y)

φµ (y) c(x, y) (n) e p (x, y) . φµ (x)

that given x, z ∈ V X, the shadow Ox (z) of z seen from x is the set of points at infinity of the geodesic rays from x through z. 14 Recall

152


Equation (6.4) follows from Equation (6.5) by summation on n. As we have already seen, κc < r if and only if δc 6= 12 ln q. The last claim of Assertion (2) follows. (3) Let x, y, z ∈ V X. Using (twice) Assertion (2), the cocycle property of c and (twice) Equation (6.3), we have ec(y, z) φµ (z) φµ (x) G(y, z | κc ) Gc (y, z) φµ (x) α e−δc d(y, z) = c(x, z) = ec(y, x) Gc (x, z) φµ (y) α e−δc d(x, z) e φµ (y) φµ (z) G(x, z | κc ) φµ (x) c(y, x)+δc (d(x, z)−d(y, z)) e . = φµ (y) This proves the first claim of Assertion (3). Under the additional assumptions on x, y, z, ξ, we have βξ (x, y) = d(x, z) − d(y, z) . The last claim of Assertion (3) hence follows from Lemma 6.4 (4).

Using the criterion that the random walk starting from a given vertex of X with transition probabilities pc is transient if and only if the Green kernel Gc (x, y) of pc is finite (for any, hence for all, x, y ∈ V X), Lemma 6.5 (2) implies that if δc 6= 12 ln q, then (Znx )n∈N almost surely leaves every finite subset of V X. The following result strengthens this remark. Proposition 6.6. If δc 6= 12 ln q, then for every x ∈ V X, the Markov chain (Znx )n∈N (with initial distribution ∆x and transition kernel pc ) converges almost surely in V X ∪ ∂∞ X to a random variable with values in ∂∞ X. In particular, the harmonic measure νx of (Znx )n∈N is well defined if δc 6= 12 ln q. Proof. Since X is a tree, if (xn )n∈N is a sequence in V X such that d(xn , xn+1 ) = 1 for every n ∈ N and that does not converge to a point in ∂∞ X, then there exists a point y such that this sequence passes infinitely often through y, that is, the set {n ∈ N : xn = y} is infinite. The result then follows from the fact that the Markov chain (Znx )n∈N is transient, since δc 6= 12 ln q. The following result, generalising [CoP2, Thm. 4.5] when e c = 0, says that the Patterson measures associated with the system of conductances e c, once renormalised to probability measures, are exactly the harmonic measures for the random walk with transition probabilities pc . Theorem 6.7. Let (X, Γ, e c, (µx )x∈V X ) be as in the beginning of Section 6.2. If δc 6= 12 ln q, then for every x ∈ V X, the harmonic measure of the Markov chain (Znx )n∈N is µx νx = . kµx k


153

Proof. We fix x ∈ V X. For every n ∈ N, we denote by S(x, n) and B(x, n) the sphere and (closed) ball of center x and radius n in V X, and we define two maps f1 , f2 : V X → R with finite support by f1 (z) =

µx (Ox (z)) νx (Ox (z)) and f2 (z) = kµx k Gc (x, z) Gc (x, z)

if z ∈ S(x, n), and f1 (z) = f2 (z) = 0 otherwise. Let us prove that f1 = f2 for every n ∈ N. Since {Ox (z) : z ∈ V X} generates the Borel σ-algebra of ∂∞ X, this proves that the Borel measures νx and kµµxx k coincide. We will use the following criterion. For all maps G : V X × V X → R and f : V X → R such that f has finite support, let us again denote by G f : V X → R the matrix product of the square matrix G and the column matrix f , defined by, for every y ∈ V X, X G f (y) = G(y, z)f (z) . z∈V X 0

Lemma 6.8. For all f, f : V X → R with finite support, if Gc f = Gc f 0 , then f = f 0. Proof. By Equation (6.2), we have f 0 = Gc f 0 − pc Gc f 0 = Gc f − pc Gc f = f .

Let us hence fix n ∈ N and prove that Gc f1 = Gc f2 . Theorem 6.7 then follows. Step 1: For every y ∈ B(x, n), since {Ox (z) : z ∈ S(x, n)} is a Borel partition of ∂∞ X, by Equation (4.2), since z ∈ / [x, y[ if z ∈ S(x, n) and y ∈ B(x, n), it follows by the second claim of Lemma 6.5 (3) that we have Z X Z c 1 1 1= dµy = e−Cξ (y, x) dµx (ξ) φµ (y) ∂∞ X φµ (y) z∈S(x,n) Ox (z) Z X 1 φµ (y) Gc (y, z) = dµx (ξ) φµ (y) Ox (z) φµ (x) Gc (x, z) z∈S(x,n)

=

X z∈S(x,n)

Gc (y, z)

µx (Ox (z)) = (Gc f1 )(y) . kµx k Gc (x, z)

(6.6)

Step 2: For all y, z ∈ V X such that z ∈ / [x, y[ , any random walk starting at time 0 from y and converging to a point in Ox (z) goes through z. Let us denote by Cx (z) the set of vertices different from z on the geodesic rays from z to the points in Ox (z). Partitioning by the last time the random walk passes through z, using the Markov property saying that what happens before the random walk arrives at z and after it leaves z are independent, we have y νy (Ox (z)) = P(Z∞ ∈ Ox (z)) = Gc (y, z) P(∀ n > 0, Znz ∈ Cx (z)) ,

154


so that

νy (Ox (z)) Gc (y, z) = . νx (Ox (z)) Gc (x, z)

(6.7)

Step 3: For every y ∈ B(x, n), again since {Ox (z) : z ∈ S(x, n)} is a Borel partition of ∂∞ X, by Equation (6.7), we have 1 = kνy k =

X

X

νy (Ox (z)) =

z∈S(x,n)

Gc (y, z)

z∈S(x,n)

νx (Ox (z)) = (Gc f2 )(y) . (6.8) Gc (x, z)

Step 4: By Steps 1 and 3, we have (Gc f1 )(y) = (Gc f2 )(y) for every y ∈ B(x, n). Let now y ∈ V X − B(x, n). Define y 0 ∈ S(x, n) as the point at distance exactly n from x on the geodesic segment [x, y]. For every z ∈ S(x, n), note that we have d(y 0 , z) − d(y, z) = −d(y, y 0 ), which is independent of z. z

x

y

y0

n Since y 0 ∈ B(x, n), we have, as just said, (Gc f1 )(y 0 ) = (Gc f2 )(y 0 ). Hence by the first claim of Lemma 6.5 (3), we have X (Gc f1 )(y) = Gc (y, z) f1 (z) z∈S(x,n)

=

0

X

0

ec(y, y )+δc (d(y , z)−d(y, z))

z∈S(x,n) 0

0

0

0

= ec(y, y )−δc d(y, y ) = ec(y, y )−δc d(y, y )

φµ (y 0 ) Gc (y 0 , z) f1 (z) φµ (y)

φµ (y 0 ) (Gc f1 )(y 0 ) φµ (y) φµ (y 0 ) (Gc f2 )(y 0 ) = (Gc f2 )(y) . φµ (y)

This proves that Gc f1 = Gc f2 , thereby concluding the proof of Theorem 6.7.

Chapter 7

Skinning Measures with Potential on CAT(−1) Spaces In this chapter, we introduce skinning measures as weighted pushforwards of the Patterson–Sullivan densities associated with a potential to the unit normal bundles of convex subsets of a CAT(−1) space. The development follows [PaP14a] with modifications to fit the present context. Let X, x0 , Γ, Fe be as at the beginning of Chapter 4, and Fe± , F ± , δ = δΓ, F ± the associated notation with δ < +∞. Let (µ± x )x∈X be (normalised) Patterson densities on ∂∞ X for the pairs (Γ, F ± ).1

7.1

Skinning measures

Let D be a nonempty proper closed convex subset of X. The outer skinning mea+ 1 sure σ eD on the outer normal bundle ∂+ D of D and the inner skinning measure − 1 σ eD on the inner normal bundle ∂− D of D associated with the Patterson densities ± ± 1 e± (µ± eD =σ eD, x )x∈X for (Γ, F ) are the measures σ F ± on ∂± D defined by ± de σD (ρ) = e

Cρ± (x0 , ρ(0)) ±

dµ± x0 (ρ± ) ,

(7.1)

1 1 where ρ ∈ ∂± D, using the endpoint homeomorphisms ρ 7→ ρ± from ∂± D to ∂∞ X − ∂∞ D, and we note that ρ(0) = PD (ρ± ) depends continuously on ρ± . When Fe = 0, the skinning measure has been defined by Oh and Shah [OhS2] for the outer unit normal bundles of spheres, horospheres, and totally geodesic subspaces in real hyperbolic spaces. The definition was generalised in [PaP14a] to the outer unit normal bundles of nonempty proper closed convex sets in Riemannian manifolds with variable negative curvature. Note that the Gibbs measure is defined on the space G X of geodesic lines, the potential is defined on the space T 1 X of germs at time t = 0 of geodesic lines, 1 See

Section 4.1.


155

Chapter 7. Skinning Measures with Potential on CAT(−1) Spaces

156

1 and since ∂± D is contained in G±, 0 X (see Section 2.4), the skinning measures are defined on the spaces G±, 0 X of (generalised) geodesic rays. In the manifold case, all the above spaces are canonically identified with the standard unit tangent bundle, but in general, the natural restriction maps G X → T 1 X and G X → G±, 0 X have infinite (though compact) fibers.

Remark 7.1. (1) If D = {x} is a singleton, then ± de σD (ρ) = dµ± x (ρ± ) ,

(7.2)

where ρ is a geodesic ray starting (at time t = 0) from x. (2) When the potential Fe is equal to Fe ◦ ι (in particular when F = 0), we have + C − = C + , and we may take µ− eF = m e F and x = µx for all x ∈ X, hence ι∗ m − + σ eD = ι∗ σ eD .

e : T 1 X → R be a continuous Γ-invariant More generally, if Fe is reversible, let G e gt ` ) is differentiable and function such that for every ` ∈ G X, the map t 7→ G(v d ∗ e e e F (v` ) − F (v` ) = dt t=0 G(vgt ` ). Furthermore, assume that X is an R-tree or that G is uniformly continuous (for instance, Hölder-continuous). Then we have, for 1 every ρ ∈ ∂+ D, denoting by ρb ∈ G X any extension of ρ to a geodesic line in X, − + d ι∗ σ eD (ρ) = e−G(vρb) d σ eD (ρ) . e

Indeed, for all x, y ∈ X and ξ ∈ ∂∞ X, let `x, ξ be any geodesic line with footpoint `x, ξ (0) = x and positive endpoint (`x, ξ )+ = ξ. Then by Remark 4.11, we have e ` ) − G(v e ` ), Cξ− (x, y) − Cξ+ (x, y) = G(v x, ξ y, ξ and we may take −G(v`x, ξ ) dµ− dµ+ x (ξ) = e x (ξ) . e

1 Hence for every ρ ∈ ∂+ D, we have − dσ eD (ιρ) = e

=e

− C(ιρ) (x0 , (ιρ)(0)) −

Cρ− (x0 , ρ(0))

dµ− x0 ((ιρ)− ) = e

e ` Cρ+ (x0 , ρ(0))−G(v ) ρ(0), ρ +

+

+

dµ+ x0 (ρ+ ) = e

dµ− x0 (ρ+ )

e ` −G(v ) ρ(0), ρ +

+ dσ eD (ρ) .

(3) The (normalised) Gibbs cocycle being unchanged when the potential F is replaced by the potential F + σ for any constant σ, we may take the Patterson densities, hence the Gibbs measure and the skinning measures, to be unchanged by such a replacement. When D is a horoball in X, let us now relate the skinning measures of D to previously known measures on ∂∞ X, constructed using techniques due to Hamenst¨ adt.

7.1. Skinning measures

157

Let H be a horoball centered at a given point ξ ∈ ∂∞ X. Recall that the map PH : ∂∞ X − {ξ} → ∂H is the closest-point map on H , mapping η 6= ξ to the intersection with the boundary of H of the geodesic line from η to ξ. The following result is proved in [HeP3, §2.3] when F = 0. Proposition 7.2. Let ρ : [0, +∞[ → X be the geodesic ray starting from any point of the boundary of H and converging to ξ. The weak-star limit dµ± H (η) = lim

t→+∞

e

−

R PH (η) ρ(t)

e± −δ) (F

dµ± ρ(t) (η)

of measures on ∂∞ X − {ξ} exists, and it does not depend on the choice of ρ. The measure µ± H is invariant under the elements of Γ preserving H , and it satisfies, for every x ∈ X and (almost) every η ∈ ∂∞ X − {ξ}, ± dµ± H (η) = e−Cη (PH (η), x) . dµ± x

Proof. We prove all three assertions simultaneously. Let us fix x ∈ X. For all t > 0 and η ∈ ∂∞ X − {ξ}, let zt be the closest point to PH (η) on the geodesic ray from ρ(t) to η. ρ(0) ρ(t)

zt

ξ

η PH (η) H

Using Equation (4.2) with x replaced by ρ(t) and y by the present x, by the cocycle equation (3.19) and by Equation (3.20), as zt ∈ [ρ(t), η[ , we have e

−

R PH (η) ρ(t)

e± −δ) (F

dµ± ρ(t) (η) = e

−

R PH (η)

e± −δ) −C ± (ρ(t), x) (F η

=e

−

R PH (η)

e± −δ) −C ± (ρ(t), z ) −C ± (z , x) (F t t η η

=e

−

R PH (η)

R e± −δ)+ zt (F e± −δ) −C ± (z , x) (F t ρ(t) η

ρ(t)

ρ(t)

ρ(t)

e

dµ± x (η)

e

e

e

dµ± x (η)

dµ± x (η) .

As t → +∞, note that zt converges to PH (η) and that by the HC property (and since Fe is bounded on every compact neighbourhood of PH (η)), we have Z PH (η) Z zt ± ± e e (F − δ) − (F − δ) → 0 . ρ(t)

ρ(t)

The result then follows by the continuity of the Gibbs cocycle (see Proposition 3.20 (3)).


158

Using this proposition and the cocycle property of C ± in the definition (4.4) of the Gibbs measure, we obtain, for every ` ∈ G X such that `± 6= ξ, dm e F (`) = e

C`− (PH (`− ), `(0)) + C`+ (PH (`+ ), `(0)) −

+

+ dµ− H (`− ) dµH (`+ ) dt .

(7.3)

1 Note that it is easy to see that for every ρ ∈ ∂± H , we have ± de σH (ρ) = dµ± H (ρ± ) .

(7.4)

When F = 0, we obtain Hamenst¨ adt’s measure µH = lim eδΓ t µρ(t) t→+∞

(7.5)

on ∂∞ X − {ξ} associated with the horoball H , which is independent of the choice of the geodesic ray ρ starting from a point of the horosphere ∂H and converging to ξ. Note that for every t > 0, if H [t] is the horoball contained in H whose boundary is at distance t from the boundary of H , we then have µH [t] = e−δΓ t µH .

(7.6)

Assume till the end of Proposition 7.3 that the potential Fe is zero. The next result gathers computations done in [PaP16, PaP17a] of the skinning measures of horoballs and some totally geodesic subspaces when X is a real or complex hyperbolic space and Γ is a lattice. We consider the notation HnR , HnC , H∞ , Heis2n−1 , λ2n−1 introduced in Section 4.2, and we again endow T 1 HnR and T 1 HnC with their Sasaki’s Riemannian metric. Recall that a complex hyperbolic line in HnC is a totally geodesic plane with constant sectional curvature −4. Since the arguments of the following result are purely computational and rather long, we do not copy them in this book, but we refer respectively to the proofs of [PaP16, Prop. 11 (1), (2)] and [PaP17a, Lem. 12 (iv), (v), (vi)]. Analogous computations can be done when X is the quaternionic hyperbolic n-space HnH , see [PaP20, § 7]. Proposition 7.3. (1) Let Γ be a lattice in Isom(HnR ), with Patterson density (µx )x∈HnR normalised as in Section 4.2. (i) If D is a horoball in HnR , if vol∂±1 D is the Riemannian measure of the sub1 manifold ∂± D in T 1 HnR , then ± σ eD = 2n−1 vol∂±1 D .

If the point at infinity of D is a parabolic fixed point of Γ, with stabiliser ΓD in Γ, then2 ± 1 kσD k = 2n−1 Vol(ΓD \∂± D) = 2n−1 Vol(ΓD \∂D) = 2n−1 (n − 1) Vol(ΓD \D) . 2 See,

for instance, [Hers, p. 473] for the last equality.


159

(ii) If D is a totally geodesic hyperbolic subspace of dimension k ∈ {1, . . . , n − 1} 1 in HnR , if vol∂±1 D is the Riemannian measure of the submanifold ∂± D in 1 n T HR , then ± σ eD = vol∂±1 D . With ΓD the stabiliser of D in Γ and m the order of the pointwise stabiliser of D in Γ, if ΓD \D has finite volume, then ± kσD k=

Vol(Sn−k−1 ) Vol(ΓD \D) . m

(2) Let Γ be a lattice in Isom(HnC ), with Patterson density (µx )x∈HnC normalised as in Section 4.2. 1 (i) Using the homeomorphism v 7→ v± from ∂± H∞ to ∂∞ HnC −{∞} = Heis2n−1 , we have ± de σH (v) = dλ2n−1 (v± ) . ∞

For every horoball D in HnC , if vol∂D is the Riemannian measure of the hypersurface ∂D in HnC , then ± π∗ σ eD = 2 vol∂D .

If the point at infinity of D is a parabolic fixed point of Γ, with stabiliser ΓD in Γ, then ± kσD k = 4n Vol(ΓD \D) . (ii) For every geodesic line D in HnC , if vol∂±1 D is the Riemannian measure of the 1 submanifold ∂± D in T 1 HnC , we have ± dπ∗ σ eD =

n dπ∗ vol∂±1 D . 4n−1 (2n − 1)

With ΓD the stabiliser of D in Γ and m the order of the pointwise stabiliser of D in Γ, if ΓD \D has finite length, then ± kσD k=

2 π n−1 n! Vol(ΓD \D) . m (2n − 1)!

(iii) For every complex geodesic line D in HnC , if vol∂+1 D is the Riemannian mea1 sure of the submanifold ∂+ D in T 1 HnC , we have + dπ∗ σ eD =

1 dπ∗ vol∂+1 D . 22n−3

With ΓD the stabiliser of D in Γ and m the order of the pointwise stabiliser of D in Γ, if ΓD \D has finite area, then + kσD k=

π n−1 Vol(ΓD \D) . (n − 2)!

m 4n−2

160


The following results give the basic properties of the skinning measures analogous to those in [PaP14a, Sect. 3] when the potential is zero. Proposition 7.4. Let D be a nonempty proper closed convex subset of X, and let ± 1 σ eD be the skinning measures on ∂± D for the potential Fe. ± (i) The skinning measures σ eD are independent of x0 .

± ± ± (ii) For all γ ∈ Γ, we have γ∗ σ eD = σ eγD . In particular, the measures σ eD are invariant under the stabiliser of D in Γ. 1 (iii) For all s > 0 and w ∈ ∂± D, we have3 C ± (π(w), π(g±s w)) ± ± dσ eN ((g±s w) ±[0,+∞[ ) = e w± de σD (w) sD −

=e

R π(g±s w) π(w)

e± −δ) (F

± de σD (w) .

± (iv) The support of σ eD is ± 1 {v ∈ ∂± D : v± ∈ ΛΓ} = PD (ΛΓ − (ΛΓ ∩ ∂∞ D)) . ± In particular, σ eD is the zero measure if and only if ΛΓ is contained in ∂∞ D.

For future use, the version4 of Assertion (iii) when F = 0 is ± d(g±s )∗ σ eD (g±s w) = e−δΓ s , ± dσ eN sD

(7.7)

1 1 1 where w ∈ ∂± D and we again denote by g±s the map from ∂± D to ∂± N1 D defined ±s by w 7→ (g w) ±[0,+∞[ .

As another particular case of Assertion (iii) for future use, consider the case in which X = |X|λ is the geometric realisation of a metric tree (X, λ) and Fe = Fec is the potential associated with a system of conductances e c on X for a subgroup 1 Γ of Aut(X) (see Equation (3.23) and Proposition 3.22). Then for all w ∈ ∂+ D 1 (respectively w ∈ ∂− D), if ew is the first (respectively the last) edge followed by w, with length λ(ew ), then Z π(g±λ(ew ) w) Fe± = e c(ew ) λ(ew ) π(w)

by Proposition 3.21, so that ± ± dσ eN ((g±λ(ew ) w) ±[0,+∞[ ) = e−(ec(ew )−δ)λ(ew ) de σD (w) . sD

(7.8)

Proof. The proofs of the claims are straightforward modifications of those for zero + potential in [PaP14a, Prop. 4]. We give details of the proofs for the measure σ eD , − the case of σ eD being similar. denote by (g±s w) ±[0,+∞[ the element of G±, 0 that coincides with g±s w on ±[0, +∞[ . 4 It is contained in [PaP14a, Prop. 4]. 3 We


161

(i) The claim follows from Equation (4.2) and the cocycle property (3.19). (ii) The claim follows from Equation (4.1), the first part of Equation (3.19) and Assertion (i). 1 (iii) Since (gs w) [0,+∞[ + = w+ and since a geodesic ray w belongs to ∂+ D if and s 1 only if (g w) [0,+∞[ ∈ ∂+ Ns D, we have, using the definition (7.1) of the skinning measure and the cocycle property (3.19), for all s > 0, + C + (x , π(gs w)) Cw (π(w), π(gs w)) + + + dσ eN ((gs w) [0,+∞[ ) = e w+ 0 dµ+ dσ eD (w) . x0 (w+ ) = e sD

+ This proves Assertion (iii) for σ eD , using Equation (3.20).

(iv) The assertions follow from the fact that the support of any Patterson measure is ΛΓ; see Section 4.1. Given two nonempty closed convex subsets D and D0 of X, let AD,D0 = ∂∞ X − (∂∞ D ∪ ∂∞ D0 ) ± ± ± ± −1 and let h± D,D 0 : PD (AD,D 0 ) → PD 0 (AD,D 0 ) be the restriction of PD 0 ◦ (PD ) ± 1 1 to PD (AD,D0 ). It is a homeomorphism between open subsets of ∂± D and ∂± D0 , 0 associating to the element w in the domain the unique element w in the range 0 with w± = w± .

Proposition 7.5. Let D and D0 be nonempty closed convex subsets of X and let ± ± ± ± h± = h± eD and σ eD 0 on PD 0 (AD,D 0 ) are absolutely conD,D 0 . The measures (h )∗ σ tinuous one with respect to the other, with ± d(h± )∗ σ eD −C ± (π(w), π(w0 )) (w0 ) = e w± , ± de σD 0 ± for (almost) all w ∈ PD (AD,D0 ) and w0 = h± (w). 0 Proof. Since w± = w± , we have ± 0 de σD 0 (w ) = e

=e

± 0 Cw 0 (x0 , π(w )) ±

± Cw (π(w), π(w0 )) ±

± Cw 0 (x0 , π(w))

0 dµ± x0 (w± ) = e

±

e

± 0 Cw 0 (π(w), π(w )) ±

0 dµ± x0 (w± )

± de σD (w)

using the definition (7.1) of the skinning measure and the cocycle property (3.19). 1 Let w ∈ G± X. With Nw± : W ± (w) → ∂∓ HB± (w) the canonical homeomorphism defined in Section 2.4, we define the skinning measures µW ± (w) on the strong stable or strong unstable leaves W ± (w) by ∓ µW ± (w) = ((Nw± )−1 )∗ σ eHB , ± (w)


162

so that dµW ± (w) (`) = e

C`∓ (x0 , `(0)) ∓

dµ∓ x0 (`∓ )

±

for every ` ∈ W (w). By Proposition 7.4 (ii) and the naturality of γ ∈ Γ, we have γ∗ µW ± (w) = µW ± (γw) .

(7.9) Nw± ,

for every (7.10)

By Proposition 7.4 (iv), the support of µW ± (w) is {` ∈ W ± (w) : `∓ ∈ ΛΓ}. For all t ∈ R and ` ∈ W ± (w), we have, using Equations (7.9), (3.19), and (3.20), and since `± = w± , d(g−t )∗ µW ± (w) t C ∓ (`(t), `(0)) C ± (`(0), `(t)) (g `) = e `∓ = e w± . d µW ± (gt w)

(7.11)

Let w ∈ G± X. The homeomorphisms W ± (w) × R → W 0± (w) defined by (`, s) 7→ `0 = gs ` conjugate the actions of R by translation on the second factor of the domain and by the geodesic flow on the range, and the actions of Γ (trivial on the second ∓ factor of the domain). Let us consider the measures νw on W 0± (w) given, using the above homeomorphism, by ± Cw (w(0), `(0))

∓ 0 dνw (` ) = e

±

dµW ± (w) (`) ds .

(7.12)

± ± They satisfy (gt )∗ νw = νw for all t ∈ R (since if `0 = gs `, then g−t `0 = gs−t `, and by invariance under translations of the Lebesgue measure on R). Furthermore, ± ± γ∗ νw = νγw for all γ ∈ Γ. In general, they depend on w, not only on W ± (w). ± Furthermore, the support of νw is {`0 ∈ W 0± (w) : `0∓ ∈ ΛΓ}. These properties follow easily from the properties of the skinning measures on the strong stable or strong unstable leaves.

Lemma 7.6. (i) For every nonempty proper closed convex subset D0 in X, there 1 exists R0 > 0 such that for all R > R0 , η > 0, and w ∈ ∂± D0 , we have ± ∓ 5 νw (Vw, η, R ) > 0. ∓ (ii) For all w ∈ G± X and t ∈ R, the measures νg∓t w and νw are proportional:

νg∓t w = e

± Cw (w(t), w(0)) ±

∓ νw .

Proof. (i) Let us show, as in [PaP14a, Lem. 7], that there exists R0 > 0 (depending 1 only on D0 and on the Patterson densities) such that for all R > R0 , w ∈ ∂+ D0 , 0 1 0 + − 0 and w ∈ ∂− D , we have µW + (w) (B (w, R)) > 0 and µW − (w0 ) (B (w , R)) > 0. ± ∓ The result follows from this by the definitions of νw and Vw, η, R . 5 See

± Section 2.4 for the definition of Vw, η, R .


163

We give the proof of the claim on B + (w, R), the proof of the claim on B − (w, R) is similar. For all w in 1 ∂+ D0 and ξ 0 in D0 ∪ ∂∞ D0 , by a standard comparison and convexity argument applied to the geodesic triangle with vertices π(w), w+ , ξ 0 , the point π(w) is at distance √ at most 2 ln( 1+2 5 ) from the intersection between the stable horosphere H+ (w) and the geodesic ray or line between ξ 0 and w+ .

W + (w) D

0

w+ = `0+

π(w)

π(`0 ) `0 ξ 0 = `0−

The triangle inequality and the definition of Hamenstädt distances imply that for all `, `0 ∈ W + (w), 1

0

dW + (w) (`, `0 ) 6 e 2 d(π(`), π(` )) .

(7.13)

Hence, for every ξ 0 ∈ ∂∞ D0 , for every extension w b ∈ G X of w, if `0 is the element + 0 0 of W (w) such that `− = ξ , we have √ 1+ 5 0 dW + (w) (w, b `)6 . 2 √

Thus, if ∂∞ D0 ∩ ΛΓ 6= ∅, then we may take R0 = 2 > 1+2 5 , since by Proposition 7.4 (iv), the support of µW + (w) is {` ∈ W + (w) : `− ∈ ΛΓ}. Assume now that ∂∞ D0 ∩ ΛΓ = ∅. For a contradiction, assume that for all 1 n ∈ N, there exists wn ∈ ∂+ D0 such that µW + (wn ) (B + (wn , n)) = 0. Assume 1 first that (wn )n∈N has a convergent subsequence with limit w ∈ ∂+ D0 . Since the measure µW + (w0 ) depends continuously on w0 ∈ G+ X, for every compact subset K of W + (w), we have µW + (w) (K) = 0. By Proposition 7.4 (iv) and Equation (7.9), this implies that the support of the Patterson measure µ− x0 , which is the limit set of Γ, is contained in {w+ }. This is impossible, since Γ is nonelementary. In the remaining case, the points π(wn ) in D0 converge, up to extracting a + 1 0 subsequence, to a point ξ in ∂∞ D0 . By definition of the map PD 0 and of ∂+ D , the points at infinity (wn )+ converge to ξ. For every η in ∂∞ X different from ξ, the geodesic lines from η to (wn )+ converge to the geodesic line from η to ξ. By convexity, if n is large enough, the geodesic (w ) n + line ]η, (wn )+ [ meets N1 D0 , hence passes at distance η at most 2 from π(wn ). This implies by Equation w wn (7.13) that if n is large enough, then there exists ` ∈ B + (wn , n) such that η = `− . ξ Since we assumed that µW + (wn ) (B + (wn , n)) = 0 for all n ∈ N, Proposition 7.4 (iv) implies that we have η ∈ / ΛΓ. Hence ΛΓ is contained in {ξ}, a contradiction, since Γ is nonelementary. (ii) For all w ∈ G± X, s, t ∈ R, and ` ∈ W ± (w), we have by Equations (7.12) and (7.11), and by the cocycle property (3.19) of C ± , dνg∓t w (gs `) = dνg∓t w (gs−t gt `) = e

± t t C(g t w) (g w(0), g `(0)) ±

dµW ± (gt w) (gt `) d(s − t)


164

=e

± Cw (w(t), `(t))

=e

±

e

± Cw (w(t), `(0)) ±

=e

± Cw (w(t), w(0)) ±

± −Cw (`(0), `(t))

e

±

dµW ± (w) (`) ds

± −Cw (w(0), `(0)) ±

∓ s dνw (g `)

∓ s dνw (g `) .

The following disintegration result of the Gibbs measure over the skinning measures of any closed convex subset is a crucial tool for our equidistribution and counting results. Recall the definition in Equation (2.15) of the flow-invariant open ± 1 sets UD± and the definition of the fibrations fD : UD± → ∂± D from Section 2.4. Proposition 7.7. Let D be a nonempty proper closed convex subset of X. The restriction to the open set UD± of the Gibbs measure m e F disintegrates by the fibration ± ± 1 fD : UD± → ∂± D over the skinning measure σ eD of D, with conditional measure ± −1 1 νρ∓ on the fiber (fD ) (ρ) = W 0± (ρ) of ρ ∈ ∂± D: when ` ranges over UD± , we have Z ± dm e F U ± (`) = dνρ∓ (`) de σD (ρ) . D

1 D ρ∈∂±

+ Proof. In order to prove the claim for the fibration fD , let φ ∈ Cc (UD+ ). Using in the various steps below:

• Hopf’s parametrisation with time parameter t and the definitions of m e F (see Equation (4.4)) and UD+ (see Equation (2.15)), 1 • the positive endpoint homeomorphism w 7→ w+ from ∂+ D to ∂∞ X − ∂∞ D, 0 0 and the negative endpoint homeomorphism ` 7→ `− from the strong stable leaf W + (w) to ∂∞ X − {w+ }, with s ∈ R the real parameter such that we have `0 = g−s ` ∈ W + (w), where ` ∈ W 0+ (w), noting that t − s depends only on `+ = w+ and `− = `0− , + • the definitions in Equations (7.9) and (7.1) of the measures µW + (w) and σ eD , ± and the cocycle property (3.19) of C ,

• Equation (3.20) and the cocycle property (3.19) of C + , we have Z + `∈UD

φ(`)dm e F (`)

Z

Z

Z

=

φ(`) e

C`− (x0 ,π(`))+C`+ (x0 ,π(`)) −

+

`+ ∈∂∞ X−∂∞ D `− ∈∂∞ X−{`+ } t∈R

Z

Z

Z

= 1D w∈∂+

`0 ∈W + (w)

s∈R

φ(gs `0 ) e

+ dtdµ− x0 (`− )dµx0 (`+ )

+ C`−0 (x0 ,π(gs `0 ))+Cw (x0 ,π(gs `0 )) −

+

0 + dsdµ− x0 (`− )dµx0 (w+ )

7.2. Equivariant families of convex subsets and their skinning measures

Z

Z

Z

φ(gs `0 ) e

= 1D w∈∂+

Z

`0 ∈W + (w)

Z

+ C`−0 (π(`0 ),π(gs `0 ))+Cw (π(w),π(gs `0 )) +

−

s∈R

Z

+ dsdµW + (w) (`0 )de σD (w)

φ(gs `0 ) e

=

165

+ Cw (π(w),π(`0 )) +

1 D `0 ∈W + (w) s∈R w∈∂+

+ dsdµW + (w) (`0 )de σD (w),

+ which implies the claim for the fibration fD , by the definition (7.12) of the measure − − νw . The proof for the fibration fD is similar. 1 For every u ∈ G− X, if D = HB− (u), we have ∂+ D = Nu− (W − (u)) and

UD+ = G X − W 0+ (ιu) =

[

W 0+ (w) .

w∈W − (u)

Applying the above proposition and a change of variable, the restriction to the set G X − W 0+ (ιu) of the Gibbs measure m e F disintegrates over the strong unstable + measure µW − (u) = ((Nu− )−1 )∗ σ eD , with conditional measure on the fiber W 0+ (w) − − of w ∈ W − (u) the measure νw = νN : for every φ ∈ Cc (G X − W 0+ (ιu)), we − u (w) have Z φ(`) dm e F (`) `∈G X−W 0+ (ιu) Z Z Z C + (π(w), π(`0 )) = φ(gs `0 ) e w+ ds dµW + (w) (`0 ) dµW − (u) (w) . w∈W − (u)

`0 ∈W + (w)

s∈R

(7.14) Note that if the Patterson densities have no atoms, then the stable and unstable leaves have measure zero for the associated Gibbs measure. This happens, for instance, if the Gibbs measure mF is finite, see Corollary 4.7 and Theorem 4.6.

7.2

Equivariant families of convex subsets and their skinning measures

b endowed with a left action of Γ. A family D = (Di )i∈I of Let I be an index set subsets of X or of G X indexed by I is Γ-equivariant if γDi = Dγi for all γ ∈ Γ and i ∈ I. We will denote by ∼ = ∼D the equivalence relation on I defined by i ∼ j if Di = Dj and there exists γ ∈ Γ such that j = γi. This equivalence relation is Γ-equivariant: for all i, j ∈ I and

166


γ ∈ Γ, we have γi ∼ γj if and only ifb i ∼ j. We say that D is locally finite if for every compact subset K in X or in G X, the quotient set {i ∈ I : Di ∩ K 6= ∅}/∼ is finite. Examples. (1) Fixing a nonempty proper closed convex subset D of X, taking I = Γ with the left action by translations (γ, i) 7→ γi, and setting Di = iD for every i ∈ Γ gives a Γ-equivariant family D = (Di )i∈I . In this case, we have i ∼ j if and only if i−1 j belongs to the stabiliser ΓD of D in Γ, and I/∼ = Γ/ΓD . Note that γD depends only on the class [γ] of γ in Γ/ΓD . We could also take I 0 = Γ/ΓD with the left action by translations (γ, [γ 0 ]) 7→ [γγ 0 ], and D 0 = (γD)[γ]∈I 0 , so that for all i, j ∈ I 0 , we have i ∼D 0 j if and only if i = j, and besides, D 0 is locally finite if and only if D is locally finite. The following choices of D yield equivariant families with different characteristics: (a) Let γ0 ∈ Γ be a loxodromic element with translation axis D = Axγ0 . The family (γD)γ∈Γ is locally finite and Γ-equivariant. Indeed, by Lemma 2.1, only finitely many elements of the family (γD)γ∈Γ/ΓD meet any given bounded subset of X. (b) Let ` ∈ G X be a geodesic line whose image under the canonical projection map G X → Γ\G X has a dense orbit in Γ\G X under the geodesic flow, and let D = `(R) be its image. Then the Γ-equivariant family (γD)γ∈Γ is not locally finite. (c) More generally, let D be a convex subset such that ΓD \D is compact. Then the family (γD)γ∈Γ is a locally finite Γ-equivariant family. (d) Let ξ ∈ ∂∞ X be a bounded parabolic limit point of Γ, and let H be any horoball in X centered at ξ. Then the family (γH )γ∈Γ is a locally finite Γ-equivariant family. (2) More generally, let (Dα )α∈A be a finite family of nonempty proper closed convex subsets S of X, and for every α ∈ A, let Fα be a finite set. Define an index set I = α∈A Γ × {α} × Fα with the action of Γ by left translation on the first S factor, and for every i = (γ, α, x) ∈ I, let Di = γDα . Then we have I/∼ = α∈A Γ/ΓDα × {α} × Fα , and the Γ-equivariant family D = (Di )i∈I is locally finite if and only if the family (γDα )γ∈Γ is locally finite for every α ∈ A. The cardinalities of Fα for α ∈ A contribute to the multiplicities (see Section 12.2). Let D = (Di )i∈I be a locally finite Γ-equivariant family of nonempty proper b convex subsets of X. Let Ω = (Ωi )i∈I be a Γ-equivariant family of subsets closed 1 Di for all i ∈ I (the sign ± being of G X, where Ωi is a measurable subset of ∂± constant), such that Ωi = Ωj if i ∼D j. Then ± σ eΩ =

X i∈I/∼

± σ eD | i Ωi

7.2. Equivariant families of convex subsets and their skinning measures

167

b

is a well-defined Γ-invariant locally finite measure on G X, whose support is con± b in G±, 0 X. Hence, the measure σ tained eΩ induces6 a locally finite measure on ± ± 1 1 Γ\ G X, denoted by σΩ . When Ω = ∂± D = (∂± Di )i∈I , the measure σ eΩ is denoted by X ± ± σ eD = σ eD . i i∈I/∼ + σ eD

− σ eD

band b skinning The measures are respectively called the outer and inner + − b σD and σD on Γ\ G X are the measures of D on G X, and their induced measures outer and inner skinning measures of D on Γ\ G X. Example. Consider the Γ-equivariant family D = (γD)γ∈Γ/Γx with D = {x} a ± −1 1 singleton in X. With π ± = (PD ) : ∂± D → ∂∞ X the homeomorphism ρ 7→ ρ± , ± ± ± we have (π )∗ σ eD = µx by Remark 7.1 (1), and7 ± kσD k =

6 See,

kµ± xk . |Γx |

(7.15)

b for instance, [PauPS, §2.6] and the beginning of Chapter 12 for details on the definition of the induced measure when Γ may have torsion, hence does not necessarily act freely on G X. 7 See also the beginning of Chapter 12.

Chapter 8

Explicit Measure Computations for Simplicial Trees and Graphs of Groups In this chapter, we compute skinning measures and Bowen–Margulis measures for some highly symmetric simplicial trees X endowed with a nonelementary discrete subgroup Γ of Aut(X). These computations are parallel to those given in Section 4.2 when X is a rank-one symmetric space. The potentials F are supposed to be 0 in this chapter, and we assume that the Patterson densities (µ+ x )x∈V X and (µ− ) of Γ are equal, denoted by (µ ) . Since the study of geometrically x∈V X x x∈V X x finite discrete subgroups of Aut(X) mostly reduces to the study of particular tree lattices (see Remark 2.12), we will assume that Γ is a tree lattice in this chapter. The results of these computations will be useful when we state special cases of the equidistribution and counting results in regular and biregular trees and, in particular, in the arithmetic applications in Part III. The reader interested in only the continuous-time case may skip directly to Chapter 9.

x0

p0 + 1

p1

p2

p3

...

A rooted simplicial tree (X, x0 ) is spherically symmetric if X is not reduced to x0 and has no terminal vertex, and if the stabiliser of x0 in Aut(X) acts tran© Springer Nature Switzerland AG 2019 A. Broise-Alamichel et al., Equidistribution and Counting Under Equilibrium States in Negative Curvature and Trees, Progress in Mathematics 329, https://doi.org/10.1007/978-3-030-18315-8_8

169

170

Chapter 8. Explicit Measure Computations for Simplicial Trees

sitively on each sphere of center x0 . The set of isomorphism classes of spherically symmetric rooted simplicial trees (X, x0 ) is in bijection with the set of sequences (pn )n∈N in N − {0}, where pn + 1 is the degree of any vertex of X at distance n from x0 . If (X, x0 ) is spherically symmetric, it is easy to check that the simplicial tree X is uniform1 if and only if the sequence (pn )n∈N is periodic with palindromic period in the sense that there exists N ∈ N − {0} such that pn+N = pn for every n ∈ N and pN −n = pn for every n ∈ N such that n 6 N . If N = 1, then X = Xp0 is the regular tree of degree p0 + 1, and if N = 2, then X = Xp0 , p1 is the biregular tree of degrees p0 + 1 and p1 + 1. We denote by X = |X|1 the geometric realisation of X. The Hausdorff dimension hX of ∂∞ X for any visual distance is then hX =

1 ln(p0 · · · pN −1 ) ; N

(8.1)

see, for example, [Lyo, p. 935].

8.1

Computations of Bowen–Margulis measures for simplicial trees

The next result gives examples of computations of the total mass of Bowen– Margulis measures for lattices of simplicial trees having some regularity properties. Analogous computations can be performed for Riemannian manifolds having appropriate regularity properties. We refer, for instance, to [PaP16, Prop. 10] and [PaP17b, Prop. 20 (1)] for computations of Bowen–Margulis measures for lattices in the isometry group of real hyperbolic spaces, and to [PaP17a, Lem. 12 (iii)] for the computation in the complex hyperbolic case; see also Section 4.2 and [PaP20]. In both cases, the main point is the computation of the proportionality constant between the Bowen–Margulis measure and Sasaki’s Riemannian volume of the unit tangent bundle. In dealing now with simplicial trees, similar consequences of homogeneity properties will appear below. We refer to Section 2.6 for the definitions of vol, Vol, T π, Tvol, TVol appearing in the following result. Proposition 8.1. Let (X, x0 ) be a spherically symmetric rooted simplicial tree, with associated sequence (pn )n∈N , such that X is uniform, and let Γ be a tree lattice of X. (1) For every x ∈ V X, let rx = d(x, Aut(X)x0 ), and let cx = 1 See

(prx − 1)e2 rx hX 2p0 + 2 2 2 (p0 + 1) p1 . . . prx −1 prx (p0 + 1)2

Section 2.6 for the terminology concerning simplicial trees.

8.1. Computations of Bowen–Margulis measures for simplicial trees

if rx 6= 0 and cx =

p0 p0 +1

171

if rx = 0. Then

X 1 kµx k2 − µx (∂e X)2 |Γx | [x]∈Γ\V X e∈EX : o(e)=x X cx 2 = kµx0 k . |Γx |

kmBM k =

X

(8.2)

[x]∈Γ\V X

(2) If X = Xp, q is the biregular tree of degrees p+1 and q+1, with V X = Vp X t Vq X the corresponding partition of the set of vertices of X, if the Patterson density √ (µx )x∈V X of Γ is normalised so that kµx k = p+1 p for all x ∈ Vp X, then (T π)∗ mBM = TvolΓ\\X and X

kmBM k = TVol(Γ\\X) =

[x]∈Γ\Vp X

p+1 + |Γx |

X [x]∈Γ\Vq X

q+1 . |Γx |

(8.3)

(3) If X = Xq is the regular tree of degree q + 1, if the Patterson density (µx )x∈V X of Γ is normalised to be a family of probability measures, then π∗ mBM =

q volΓ\\X , q+1

kmBM k =

q Vol(Γ\\X) . q+1

and in particular, (8.4)

Proof. We begin by proving the first equality of Assertion (1). For every x ∈ V X, we may partition the set of geodesic lines ` ∈ G X with `(0) = x according to the two edges starting from x contained in the image of `. The only restriction for the edges is that they are required to be distinct. For every e ∈ EX, recall from Section 2.6 that ∂e X is the set of points at infinity of the geodesic rays whose initial edge is e. For all e ∈ EX and x ∈ V X, say that e points away from x if o(e) ∈ [x, t(e)], and that e points towards x otherwise. In particular, all edges with origin x point away from x. Hence by Equation (4.13), + and since µx = µ− x = µx , we have π∗ mBM =

X [x]∈Γ\V X

=

X [x]∈Γ\V X

1 |Γx |

X 0

+ µ− x (∂e X) µx (∂e0 X) ∆[x]

(8.5)

0

e, e ∈EX : o(e)=o(e )=x, e6=e0

 1  |Γx |

X



2 µx (∂e X)

e∈EX : o(e)=x

−

X

µx (∂e X)2  ∆[x] .

e∈EX : o(e)=x

(8.6) This gives the first equality of Assertion (1).

172


Let us prove the second equality of Assertion (1). By homogeneity, we assume that kµx0 k = 1, and we will prove that X cx kmBM k = . |Γx | [x]∈Γ\V X

Let N ∈ N − {0} be such that pn+N = pn for every n ∈ N and pN −n = pn for every n ∈ {0, . . . , N }, which exists, since X is assumed to be uniform. Then the automorphism group Aut(X) of the simplicial tree X acts transitively on the set of vertices at distance a multiple of N from x0 . Hence for every x ∈ V X, the distance rx = d(x, Aut(X)x0 ) belongs to 0, 1, . . . , b N2 c , and there exist γx , γx0 ∈ Aut(X) such that d(x, γx x0 ) = rx ,

x ∈ [γx x0 , γx0 x0 ], and d(γx x0 , γx0 x0 ) = N .

The map x 7→ rx is constant on the orbits of Γ in V X,2 hence so is the map x 7→ cx , and thus the right-hand side of Equation (8.2) is well defined. Since the family (µHaus )x∈V X of Hausdorff measures of the visual distances x (∂∞ X, dx ) is invariant under any element of Aut(X), since Γ is a tree lattice, and by Proposition 4.16, we have δΓ = hX and γ∗ µx = µγx for all x ∈ V X and γ ∈ Aut(X). Since (X, x0 ) is spherically symmetric, and since µx0 is a probability measure, we have by induction, for every e ∈ EX pointing away from x0 with d(x0 , o(e)) = n, µx0 (∂e X) =

1 (p0 + 1) p1 · · · pn

(8.7)

if n 6= 0, and µx0 (∂e X) = p01+1 otherwise. For every fixed x ∈ V X, let us now compute µx (∂e X) for every edge e of X with origin x. Let γ = γx , γ 0 = γx0 ∈ Aut(X) be as above. By spherical transitivity, we may assume that e or e belongs to the edge path from γx0 to γ 0 x0 . rx e γx0

x N − rx

γ 0 x0

There are two cases to consider. Case 1: Assume first that e points away from γx0 . There are p0 + 1 such edges starting from x if rx = 0, and prx otherwise. By Equation (8.7) and by invariance under Aut(X) of (µHaus )x∈V X , we have x µγx0 (∂e X) = 2 In

1 , (p0 + 1) p1 · · · prx

fact, it is constant on the orbits of Aut(X).

8.1. Computations of Bowen–Margulis measures for simplicial trees

173

with the convention that the denominator is p0 + 1 if rx = 0. Since the map ξ 7→ βξ (x, γx0 ) is constant with value −rx on ∂e X, and by the quasi-invariance property of the Patterson density (see Equation (4.2)), we have µx (∂e X) = e−δΓ (−rx ) µγx0 (∂e X) =

erx hX , (p0 + 1) p1 · · · prx

with the same convention as above. Case 2: Assume now that e points towards γx0 . This implies that rx > 1, and there is one and only one such edge starting from x. Then as above we have µγ 0 x0 (∂e X) =

1 , (pN + 1) pN −1 · · · prx

and µx (∂e X) = e−δΓ (−(N −rx )) µγ 0 x0 (∂e X) = For every x ∈ V X, let X Cx =

µx (∂e X)

e∈EX : o(e)=x

2

e(N −rx )hX . (pN + 1) pN −1 · · · prx

X

−

µx (∂e X)2 .

(8.8)

e∈EX : o(e)=x

If rx 6= 0, since the stabiliser of γx0 in Aut(X) acts transitively on the prx edges with origin x pointing away from γx0 , since eN hX = p0 p1 · · · pN −1 and pN = p0 , we have 2 erx hX e(N −rx )hX Cx = prx + (p0 + 1) p1 · · · prx (pN + 1) pN −1 · · · prx 2 2 rx hX e e(N −rx )hX − prx + (p0 + 1) p1 · · · prx (pN + 1) pN −1 · · · prx 2

(prx − prx ) e2 rx hX 2 prx eN hX + 2 2 2 (p0 + 1) p1 · · · prx (p0 + 1) p1 · · · prx prx · · · pN −1 (pN + 1) 2 rx hX 2 p0 (prx − 1) e + = cx . = (p0 + 1)2 p1 2 · · · prx −1 2 prx (p0 + 1)2 =

If rx = 0, we have Cx = (p0 + 1)

1 2 1 2 p0 − (p0 + 1) = = cx . p0 + 1 p0 + 1 p0 + 1

Assertion (1) now follows from Equation (8.6). Let us prove Assertion (2). Note that X = Xp, q is spherically symmetric with respect to any vertex of X, and that by Equation (8.1), hX =

1 ln(pq) . 2

174


Let e be an edge of X, with x = o(e) ∈ Vp X and y = t(e) ∈ Vq X. For every z ∈ V X, we define Cz as in Equation (8.8). Note that by homogeneity, we have Cz = Cx and kµz k = kµx k for all z ∈ Vp X, as well as Cz = Cy and kµz k = kµy k for all z ∈ Vq X. Hence the normalisation of the Patterson density as in the statement of Assertion (2) is possible. By the spherical symmetry at x and the normalisation of the measure, we have µx (∂e X) = √1p and √ µx (∂e X) = p. Therefore kµy k = µy (∂e X) + µy (∂e X) = ehX µx (∂e X) + e−hX µx (∂e X) 1 q+1 1 √ √ p= √ . = pq √ + √ p pq q This symmetry in the values of kµy k and kµx k explains the choice of our normalisation of the Patterson density. We have kµ k 2 p x = kµx k2 = p + 1 Cx = kµx k2 − (p + 1) p+1 p+1 q and similarly Cy = q+1 kµy k2 = q + 1. This proves the second equality in Equation (8.3) by the first equation of Assertion (1). In order to prove that (T π)∗ mBM = TvolΓ\\X , we now partition Γ\G X as [ Γ\ ` ∈ G X : `(0) = π(o(e)), `(1) = π(t(e)) . [e]∈Γ\EX

Using on every element of this partition Hopf’s decomposition with respect to the basepoint o(e), we have, by a ramified covering argument already used in the proof of the second part of Proposition 4.15, X 1 (T π)∗ mBM = µo(e) (∂∞ X − ∂e X) µo(e) (∂e X) ∆[e] . |Γe | [e]∈Γ\EX

Since µo(e) (∂e X) = e−hX µt(e) (∂e X) and by homogeneity, we have 1 deg o(e) − 1 deg t(e) − 1 kµo(e) k kµt(e) k e−hX ∆[e] |Γe | deg o(e) deg t(e) [e]∈Γ\EX √ X 1 kµo(e) k kµt(e) k pq ∆[e] = TvolΓ\\X . = |Γe | (p + 1)(q + 1)

(T π)∗ mBM =

X

[e]∈Γ\EX

The first equality of Equation (8.3) follows, since pushforwards of measures preserve the total mass. Finally, the last claim of Assertion (3) of Proposition 8.1 follows from Equaq tion (8.2), since cx = q+1 for every x ∈ V Xq (or by taking q = p in Equation (8.3) and by renormalising). The first claim of Assertion (3) follows from the first claim of Assertion (2), using Equation (2.23) and renormalising.

8.2. Computations of skinning measures for simplicial trees

175

Remark 8.2. (1) In particular, if X = Xq is regular, if the Patterson density is normalised to be a family of probability measures, and if Γ is torsion free, then q π∗ mBM is q+1 times the counting measure on Γ\V X. In this case, Equation (8.4) is given by [CoP4, Rem. 2]. (2) If X = Xp, q is biregular with p 6= q, then π∗ mBM is not proportional to volΓ\\X . In particular, if Γ is torsion free and if the Patterson density is normalised to be a p family of probability measures, then π∗ mBM is the sum of p+1 times the counting q measure on Γ\Vp X and q+1 times the counting measure on Γ\Vq X. This statement is consistent with the well-known fact that in pinched but variable curvature, the Bowen–Margulis measure is generally not absolutely continuous with respect to Sasaki’s Riemannian measure on the unit tangent bundle (they would then be proportional by ergodicity of the geodesic flow in the lattice case).

8.2

Computations of skinning measures for simplicial trees

We now give examples of computations of the total mass of skinning measures (for zero potentials), after introducing some notation. Let X be a locally finite simplicial tree without terminal vertices, and let Γ be a discrete subgroup of Aut(X). For every simplicial subtree D of X, we define the boundary ∂V D of V D in X as ∂V D = {x ∈ V D : ∃ e ∈ EX, o(e) = x, t(e) ∈ / V D} . The boundary ∂D of D is the maximal subgraph (which might not be connected) of X with set of vertices ∂V D. It is contained in D. The stabiliser ΓD of D in Γ acts discretely on ∂D. For every x ∈ V X, we define the codegree of x in D as codegD (x) = 0 if x ∈ /D and otherwise codegD (x) = degX (x) − degD (x) . Note that codegD (x) = 0 if x ∈ / ∂V D, and that the codegree codegN1 D (x) of x ∈ V X in the 1-neighbourhood N1 D of D is equal to 0 unless x lies in the boundary of N1 D, in which case it is equal to degX (x) − 1. Let D = (Di )i∈I be a locally finite Γ-equivariant family of simplicial subtrees of X, and let x ∈ V X. We define the multiplicity3 of x in (the boundary of) D as (see Section 7.2 for the definition of ∼D ) mD (x) = 3 See

Card{i ∈ I/∼D : x ∈ ∂V Di } . |Γx |

Section 12.2 for explanations on the terminology.

176


The numerator and the denominator are finite by the local finiteness of the family D and the discreteness of Γ, and they depend only on the orbit of x under Γ. Note that if D is a simplicial subtree of X that is precisely invariant under Γ (that is, whenever γ ∈ Γ is such that D ∩ γD is nonempty, then γ belongs to the stabiliser ΓD of D in Γ), if D = (γD)γ∈Γ/ΓD , and if x ∈ ∂V D, then mD (x) =

1 . |Γx |

In particular, if furthermore Γ is torsion free, then mD (x) = 1 if x ∈ ∂V D, and mD (x) = 0 otherwise. Example 8.3. Let G be a connected graph without vertices of degree 6 2 and let X be its universal cover, with covering group Γ. If C is a cycle in G = Γ\X and if D is the family of geodesic lines in X lifting C, then mD (x) = 1 for all x ∈ V X whose image in G = Γ\X belongs to C if C is a simple cycle (that is, if C passes through no vertex twice). We define the codegree of x in D as X codegD (x) = codegDi (x) , i∈I/∼D

which is well defined, since codegDi (x) depends only on the class of i ∈ I modulo ∼D . Note that codegD (x) = (degX x − k) |Γx | mD (x) (8.9) if degDi (x) = k for every x ∈ ∂V Di and i ∈ I. If every vertex of X has degree at least 3, this is in particular the case with k = 2 if Di is a line for all i ∈ I and with k = 1 if Di is a horoball for all i ∈ I. We will say that a simplicial subtree D of X, with stabiliser ΓD in Γ, is almost precisely invariant if there exists N ∈ N such that for every x ∈ V D, the number of γ ∈ Γ/ΓD such that x ∈ γV D is at most N . It follows from this property that if D = (γD)γ∈Γ , then D is locally finite and codegD (x) 6 N codegD (x) for every x ∈ V X. Proposition 8.4. Assume that X is a regular or biregular simplicial tree with degrees at least 3, and that Γ is a tree lattice of X. (1) For every simplicial subtree D of X, we have π∗ σ eD± =

X x∈V X

kµx k codegD (x) ∆x . degX (x)

(2) If D = (Di )i∈I is a locally finite Γ-equivariant family of simplicial subtrees of X, then X kµx k codegD (x) ± π∗ σ D = ∆[x] . |Γx | degX (x) [x]∈Γ\V X


177

(3) Let k ∈ N and let D be a simplicial subtree of X such that degD (x) = k for every x ∈ ∂V D and the Γ-equivariant family D = (γD)Γ/ΓD is locally finite. Then X

± π∗ σ D =

ΓD y ∈ ΓD \∂V D

kµy k (degX (y) − k) ∆Γy . |(ΓD )y | degX (y)

(4) If D is a simplicial subtree of X such that the Γ-equivariant family D = (γD)γ∈Γ ± is locally finite, then the skinning measure σD is finite if and only if the graph of groups ΓD \\∂D has finite volume. Before proving Proposition 8.4, let us give some immediate consequences of its Assertion (3). If X = Xp, q is biregular of degrees p + 1 and q + 1, let V X = Vp X t Vq X be the corresponding partition of the set of vertices of X, and for r ∈ {p, q}, let ∂r D be the edgeless graph with set of vertices ∂V D ∩ Vr X. Corollary 8.5. Assume that (X, Γ) is as in Proposition 8.4. Let D be a simplicial subtree of X such that the Γ-equivariant family D = (γD)γ∈Γ/ΓD is locally finite. (1) If X = Xp, q is biregular of degrees p + 1 and q + 1 and if the Patterson density (µx )x∈V X of Γ is normalised so that kµx k = √ degX (x) for all x ∈ V X, then degX (x)−1

• if D is a horoball, ± kσD k=

√

p Vol(ΓD \\∂p D) +

√ q Vol(ΓD \\∂q D) ,

• if D is a line, p−1 q−1 ± kσD k= √ Vol(ΓD \\∂p D) + √ Vol(ΓD \\∂q D) . p q

(8.10)

(2) If X = Xq is the regular tree of degree q + 1 and if the Patterson measures (µx )x∈V X are normalised to be probability measures, then • if D is a horoball, ± kσD k=

q Vol(ΓD \\∂D); q+1

± kσD k=

q−1 Vol(ΓD \\D) . q+1

(8.11)

• if D is a line, (8.12)

Proof of Proposition 8.4. (1) We may partition the outer/inner unit normal bundle 1 1 ∂± D of D according to the first/last edge of the elements in ∂± D. On each of the elements of this partition, for the computation of the skinning measures using its definition and its independence of the basepoint (see Section 7.1), we take as basepoint the initial/terminal point of the corresponding edge. Since D is a

178


simplicial tree, note that for every e ∈ EX such that o(e) ∈ V D, we have e ∈ ED if and only if t(e) ∈ V D. Thus, we have X

π∗ σ eD+ =

µo(e) (∂e X) ∆o(e)

e∈EX : o(e)∈V D, t(e)∈V / D

X

=

µx (∂e X) ∆x ,

X e∈EX : o(e)=x, t(e)∈V / D

x∈∂V D

and similarly, X

π∗ σ eD− =

µt(e) (∂e X) ∆t(e)

e∈EX : t(e)∈V D, o(e)∈V / D

X

=

x∈∂V D

µx (∂e X) ∆x .

X e∈EX : o(e)=x, t(e)∈V / D

As in the proof of Proposition 8.1 (2), since X is spherically homogeneous around each point and since Γ is a tree lattice (so that the Patterson density is actually kµx k Aut(X)-equivariant, see Proposition 4.16), we have µx (∂e X) = deg for all x ∈ X (x) V X and e ∈ EX with o(e) = x. Assertion (1) follows, since X

1 = codegD (x)

e∈EX : o(e)=x, t(e)∈V / D

if x ∈ ∂V D and codegD (x) = 0 otherwise. 4 (2) By the definitionP of the skinning measures associated with Γ-equivariant fam± ilies, we have σ eD = i∈I/∼ σ eD±i , where ∼ = ∼D . Hence by Assertion (1), ± π∗ σ eD =

X

π e∗ σD±i =

X X x∈V X

X

i∈I/∼ x∈V X

i∈I/∼

=

X

codegDi (x)

kµx k codegDi (x) ∆x degX (x) X kµx k codeg (x) kµx k D ∆x = ∆x . degX (x) degX (x) x∈V X

i∈I/∼

By the definition of the measure induced in Γ\V X when Γ may have torsion (see, for instance, [PauPS, §2.6] and the beginning of Chapter 12), Assertion (2) follows. (3) It follows from Assertion (2) and from Equation (8.9) that ± π∗ σ D =

X [x]∈Γ\V X

4 See

Section 7.2.

degX (x) − k kµx k mD (x) ∆[x] . degX (x)


179

For every x ∈ V X, by the definition of mD (x), we have, by partitioning ∂V D into its orbits under ΓD , 1 Card{γ ∈ ΓD \Γ : γx ∈ ∂V D} |Γx | X 1 Card{γ ∈ ΓD \Γ : ΓD γx = ΓD y} = |Γx |

mD (x) =

ΓD y ∈ ΓD \∂V D

1 = |Γx | =

1 |Γx |

X

Card{γ ∈ ΓD \Γ : ΓD γy = ΓD y}

ΓD y ∈ ΓD \∂V D, Γx=Γy

X

[Γy : (ΓD )y ] =

ΓD y ∈ ΓD \∂V D, Γx=Γy

This proves Assertion (3), since

P

X ΓD y ∈ ΓD \∂V D, Γx=Γy

[x]∈Γ\V X, Γx=Γy

1 . |(ΓD )y |

∆[x] = ∆Γy .

(4) It follows from Assertion (2) that ± kσD k=

X [x]∈Γ\V X

kµx k codegD (x) . |Γx | degX (x)

Note that for every x ∈ ∂V D, we have |Γx | mD (x) 6 codegD (x) 6 degX (x) |Γx | mD (x) . Let m = minx∈V X kµx k and M = maxx∈V X kµx k, which are positive and finite, since the total mass of the Patterson measures takes at most two values, as Γ is a tree lattice and X is biregular. By arguments similar to those in the proof of Assertion (3), we hence have m ± Vol(ΓD \\∂D) 6 kσD k 6 M Vol(ΓD \\∂D). minx∈V X degX (x) The result follows.

We now give a formula for the skinning measure (with zero potential) of a geodesic line in the simplicial tree X, using5 Hamenstädt’s distance dH and measure µH associated with a fixed horoball H in X. This expression for the skinning measure will be useful in Part III. Lemma 8.6. Let H be a horoball in X centered at a point ξ ∈ ∂∞ X. Let L be a 1 geodesic line in X with endpoints L± ∈ ∂∞ X − {ξ}. Then for all ρ ∈ ∂+ L such that ρ+ 6= ξ, + de σL (ρ) = 5 See

dH (L+ , L− )δΓ dµH (ρ+ ) . dH (ρ+ , L− )δΓ dH (ρ+ , L+ )δΓ

the definitions of Hamenst¨ adt’s distance and measure in Sections 2.3 and 7.1 respectively.

180


Proof. By Equations (2.13) and (7.6), the power dH δΓ of the distance and the measure µH scale by the same factor when the horoball is replaced by another one centered at the same point. Thus, we can assume in the proof that L does not intersect the interior of H . 1 Fix ρ ∈ ∂+ L such that ρ+ 6= ξ. Let y be the closest point to ξ on L, let x0 be the closest point to L on H , and let z be the closest point to ξ on ρ([0, +∞[). Let t 7→ xt be the geodesic ray starting from x0 at time t = 0 and converging to ξ. When t is large enough, the points ρ+ , z, xt , and ξ are in this order on the geodesic line ]ρ+ , ξ[ . We have, by the definition in Equation (7.1) of the skinning measure, the cocycle property of the Busemann function, Equation (3.20), and the definition of z, + de σL (ρ) = eδΓ βρ+ (xt , ρ(0)) dµxt (ρ+ ) = eδΓ βρ+ (xt , z)+δΓ βρ+ (z, ρ(0)) dµxt (ρ+ )

= e−δΓ βξ (xt , z)−δΓ d(z, ρ(0)) dµxt (ρ+ ) = eδΓ t−δΓ βξ (x0 , z)−δΓ d(z, ρ(0)) dµxt (ρ+ ) , and, by the definition of Hamenstädt’s measure µH (see Equation (7.5)), dµH (ρ+ ) = eδΓ t dµxt (ρ+ ) . ξ

ξ xt

∂H

x0 L−

y

xt

∂H

ρ+

x0 = z

z = ρ(0)

L+

L−

y = ρ(0)

ρ+ L+

Case 1: Assume first that ρ(0) 6= y. We may assume that ρ(0) ∈ [y, L+ [ . Then z = ρ(0) and z is the closest point to H on the geodesic line ]L+ , ρ+ [ . Thus dH (L− , L+ ) = dH (L− , ρ+ ) and dH (L+ , ρ+ ) = e−d(z, x0 ) = eβξ (x0 , z) , and the claim follows. Case 2: Assume now that y = ρ(0). Then [y, z] = [y, ξ[ ∩ [y, ρ+ [ , and we may assume that x0 = z up to adjusting the horoball H while keeping its point at infinity. Thus dH (L− , L+ ) = e−d(y, x0 ) = e−d(z, ρ(0)) and dH (L− , ρ+ ) = dH (L+ , ρ+ ) = 1, and the claim follows.

Chapter 9

Rate of Mixing for the Geodesic Flow ± e± Let X, x0 , Γ, Fe, (µ± x )x∈X be as of the beginning of Chapter 7, and F , F , δ = δΓ, F ± < ∞, m e F , mF the associated notation. In this chapter, we begin by collecting in Section 9.1 known results on the rate of mixing of the geodesic flow for manifolds. The main part of the chapter then consists in proving analogous bounds for the discrete-time and continuous-time geodesic flow for quotient spaces of simplicial and metric trees respectively. mF when the Gibbs measure mF is finite. Recall that this We define mF = km Fk measure is nonzero, since Γ is nonelementary. Let α ∈ ]0, 1].1 We will say that the (continuous-time) geodesic flow on Γ\G X is exponentially mixing for the α-H¨ older regularity or that it has exponential decay of α-H¨ older correlations for the potential F if there exist C, κ > 0 such that for all φ, ψ ∈ Cbα (Γ\G X) and t ∈ R, we have Z Z Z −t φ ◦ g ψ dmF − φ dmF ψ dmF Γ\G X

Γ\G X

Γ\G X

6 C e−κ|t| kφkα kψkα ,

(9.1)

and that it is polynomially mixing for the α-Hölder regularity or has polynomial decay of α-H¨ older correlations if there exist C > 0 and n ∈ N − {0} such that for all φ, ψ ∈ Cbα (Γ\G X) and t ∈ R, we have Z Z Z −t φ◦g ψ dmF − φ dmF ψ dmF 6 C (1+|t|)−n kφkα kψkα . Γ\G X

9.1

Γ\G X

Γ\G X

Rate of mixing for Riemannian manifolds

f is a complete simply connected Riemannian manifold with pinched When X = M negative sectional curvature with bounded derivatives, then the boundary at infinrefer to Section 3.1 for the definition of the Banach space Cbα (Z) of bounded α-H¨ oldercontinuous functions on a metric space Z. 1 We


181

182

Chapter 9. Rate of Mixing for the Geodesic Flow

f, the strong unstable, unstable, stable, and strong stable foliations of T 1 M f ity of M are H¨ older-smooth and only Hölder-smooth in general.2 Hence the assumption of f is appropriate for these manifolds. H¨ older regularity on functions on T 1 M The geodesic flow is known to have exponential decay of Hölder correlations f when for compact manifolds M = Γ\M • M is two-dimensional and F is any Hölder-continuous potential by [Dol1], • M is 1/9-pinched and F = 0 by [GLP, Cor. 2.7], • mF is the Liouville measure by [Live]; see also [Tsu], [NZ, Cor. 5], who give more precise estimates, • M is locally symmetric and F is any Hölder-continuous potential by [Sto]; see also [MO]. f is a symmetric space, then the boundary at infinity of M f, the strong When M f are smooth. Hence unstable, unstable, stable, and strong stable foliations of T 1 M f makes sense. For every ` ∈ N, talking about leafwise C ` -smooth functions on T 1 M we will denote by Cc` (T 1 M ) the vector space of real-valued C ` -smooth functions f with compact support in T 1 M , and by kψk` the on the orbifold T 1 M = Γ\T 1 M `,2 Sobolev W -norm of any ψ ∈ Cc` (T 1 M ). This space consists of functions induced f. on T 1 M by C ` -smooth Γ-invariant functions with compact support on T 1 M 1 Given ` ∈ N, we will say that the geodesic flow on T M is exponentially mixing for the `-Sobolev regularity (or that it has exponential decay of `-Sobolev correlations) for the potential F if there exist c, κ > 0 such that for all functions φ, ψ ∈ Cc` (T 1 M ) and all t ∈ R, we have Z Z Z −t −κ|t| φ ◦ g ψ dm − φ dm ψ dm kψk` kφk` . F F F 6 ce T 1M

T 1M

T 1M

f (the When F = 0 and Γ is an arithmetic lattice in the isometry group of M Gibbs measure then coincides, up to a multiplicative constant, with the Liouville measure), this property, for some ` ∈ N, follows from [KM1, Theorem 2.4.5], with the help of [Clo, Theorem 3.1] to check its spectral gap property, and of [KM2, Lemma 3.1] to deal with finite cover problems.

9.2

Rate of mixing for simplicial trees

Let X be a locally finite simplicial tree without terminal vertices, with geometric realisation X = |X|1 , and x0 ∈ V X. Let Γ be a nonelementary discrete subgroup of Aut(X) and let e c : EX → R be a system of conductances for Γ on X. Let us consider Fec : T 1 X → R the associated potential of e c, and c : Γ\EX → R, Fc : Γ\T 1 X → R the quotient functions. Let δc = δΓ, Fc be the critical exponent 2 See,

f has a compact quotient (a result first proved by Anosov), and for instance, [Brin] when M [PauPS, Thm. 7.3].

9.2. Rate of mixing for simplicial trees

183

of c, assumed to be finite.3 Let (µ± x )x∈V X be (normalised) Patterson densities on ∂∞ X for the pairs (Γ, Fc± ), and let m ec = m e Fc and mc = mFc be the associated Gibbs measures on G X and Γ\G X. In this section, building on the end of Section 4.4 concerning the mixing properties themselves,4 we now study the rate of mixing of the discrete-time geodesic flow on Γ\G X for the Gibbs measure mc = mFc , when it is mixing. Let (Z, T, m) be a dynamical system with (Z, m) a probability space and T : Z → Z a (not necessarily invertible) measure-preserving map. For all n ∈ N and φ, ψ ∈ L2 (m), the (well-defined) nth correlation coefficient of φ, ψ is Z Z Z n covm, n (φ, ψ) = φ ◦ T ψ dm − φ dm ψ dm . Z

Z

Z

Let α ∈ ]0, 1] and let us assume that Z is a metric space (endowed with its Borel σ-algebra). Similarly as for the case of flows at the beginning of Chapter 9, we will say that the dynamical system (Z, T, m) is exponentially mixing for the α-H¨ older regularity or that it has exponential decay of α-H¨ older correlations if there exist C, κ > 0 such that for all φ, ψ ∈ Cbα (Z) and n ∈ N, we have | covm, n (φ, ψ)| 6 C e−κ n kφkα kψkα . Note that this property is invariant under measure-preserving conjugations of dynamical systems by bilipschitz homeomorphisms. The main result of this section is a simple criterion for the exponential decay of correlations of the discrete-time geodesic flow on Γ\G X. mc We define mc = km when the Gibbs measure mc on Γ\G X is finite, and ck we use the dynamical system (Γ\G X , g1 , mc ) in the definition of the correlation coefficients. Given a finite subset E of Γ\V X, we denote by τE : Γ\G X → N ∪ {+∞} the first return5 time to E of the discrete-time geodesic flow: τE (`) = inf{n ∈ N − {0} : gn `(0) ∈ E} , with the usual convention that inf ∅ = +∞. Theorem 9.1. Let X, Γ, e c be as above, with δc finite. Assume that the Gibbs measure mc is finite and mixing for the discrete-time geodesic flow on Γ\G X. Assume moreover that there exist a nonempty finite subset E of Γ\V X and C 0 , κ0 > 0 such that for all n ∈ N, we have 0 mc {` ∈ Γ\G X : `(0) ∈ E and τE (`) > n} 6 C 0 e−κ n . (9.2) 3 That

is, δc < +∞, since δc > −∞ by Lemma 3.17 (7). Theorem 4.17. 5 Actually, a more precise terminology is “first positive passage time,” but we use the shorter one. If ` ∈ Γ\G X is such that `(0) ∈ E, then “return” is appropriate. 4 See

184


Then the discrete-time geodesic flow on Γ\G X has exponential decay of α-H¨ older correlations for the system of conductances c. A similar statement holds for the square of the discrete-time geodesic flow on Γ\Geven X when mc is finite, C ΛΓ is a uniform simplicial tree with degrees at least 3, and LΓ = 2Z. Note that the crucial Hypothesis (9.2) of Theorem 9.1 is in particular satisfied if Γ\X is finite, by taking E = Γ\V X. In this case, the result is quite well known: when Γ is torsion free, it follows from Bowen’s result [Bowe3, 1.26] that a mixing subshift of finite type is exponentially mixing. Proof. Let X0 = C ΛΓ. Using the coding introduced in Section 5.2, we first reduce this statement to a statement in symbolic dynamics. Step 1: Reduction to two-sided symbolic dynamics. Let (Σ, σ) be the (two-sided) topological Markov shift with alphabet A and transition matrix A constructed in Section 5.2, that we proved to be conjugated to (Γ\G X0 , g1 ) by the homeomormc phism Θ : Γ\G X0 → Σ (see Theorem 5.1). Let P = Θ∗ km , which is a mixing ck σ-invariant probability measure on Σ with full support, since Γ\G X0 is the support of mc . Let E = {(e− , h, e+ ) ∈ A : t(e− ) = o(e+ ) ∈ E} . The set E is finite, since the degrees and the vertex stabilisers of X are finite. For all x ∈ Σ and k ∈ Z, we denote by xk the kth component of x = (xn )n∈Z . Let τE (x) = inf{n ∈ N − {0} : xn ∈ E } be the first return6 time to E of x under iteration of the shift σ. Let π+ : Σ → A N be the natural extension7 (xn )n∈Z 7→ (xn )n∈N . Theorem 9.1 will follow from the following two-sided symbolic dynamics result.8 Theorem 9.2. Let (Σ, σ) be a locally compact transitive two-sided topological Markov shift with alphabet A and transition matrix A, and let P be a mixing σ-invariant probability measure with full support on Σ. Assume that (1) for every A-admissible finite sequence w = (w0 , . . . , wn ) in A , the Jacobian of the map fw from the cylinder {(xk )k∈N ∈ π+ (Σ) : x0 = wn } to the cylinder {(yk )k∈N ∈ π+ (Σ) : y0 = w0 , . . . , yn = wn } defined by (x0 , x1 , x2 , . . . ) 7→ (w0 , . . . , wn , x1 , x2 , . . . ), with respect to the restrictions of the pushforward measure (π+ )∗ P, is constant; 6 See

the previous footnote. authors are not responsible for this questionable terminology, rather standard in symbolic dynamics. 8 Assumption (1) of Theorem 9.2 is far from being optimal, but will be sufficient for our purpose. 7 The


185

(2) there exist a nonempty finite subset E of A and C 0 , κ0 > 0 such that for all n ∈ N, we have 0 P {x ∈ Σ : x0 ∈ E and τE (x) > n} 6 C 0 e−κ n . (9.3) Then (Σ, σ, P) has exponential decay of α-H¨ older correlations. Proof that Theorem 9.2 implies Theorem 9.1. Since m e c is supported on G X0 , up 0 to replacing X by X , we may assume that ∂∞ X = ΛΓ. By the construction of Θ just before the statement of Theorem 5.1, for every + e e e + e e ` = Γè ∈ Γ\G X, we have (Θ`)0 = (e− 0 (` ), h0 (` ), e0 (` )) with e0 (` ) = p(`([0, 1])), + e where p : X → Γ\X is the canonical projection, so that o e0 (` )) = `(0). Since Θ conjugates g1 to σ, for every n ∈ N, we have (Θ`)n = (σ n (Θ`))0 = (Θ(gn `))0 . Thus (Θ`)n ∈ E if and only if gn `(0) ∈ E, and τE (Θ`) = τE (`) . Therefore Theorem 9.1 will follow from Theorem 9.2 by conjugation, since Θ is bilipschitz, once we have proved that Hypothesis (1) of Theorem 9.2 is satisfied for the two-sided topological Markov shift (Σ, σ) conjugated by Θ to (Γ\G X, g1 ), which is the main point in this proof. We hence fix an A-admissible finite sequence w = (w0 , . . . , wn ) in A . We consider the (one-sided) cylinders [wn ] = {(xk )k∈N ∈ π+ (Σ) : x0 = wn } , [w] = {(yk )k∈N ∈ π+ (Σ) : y0 = w0 , . . . , yn = wn }, and the map fw : [wn ] → [w] with (x0 , x1 , x2 , . . . ) 7→ (w0 , . . . , wn , x1 , x2 , . . . ) that appear in Hypothesis (1). We denote by w e and w fn the discrete generalised geodesic lines in X associated with w and wn (see the proof of Theorem 5.1 just after Equation (5.3)). Since w ends with wn , by the construction of Θ, there exists γ ∈ Γ sending the two consecutive edges of w en to the last two consecutive edges of w. We denote by x = w(0) e and y = w fn (0) the footpoints of w e and w fn respectively. ξ

w e x = w(0) e w fn

γy γ

γ −1 ξ

y=w fn (0) γ −1 η

η

186


b

For every discrete generalised geodesic line ω ∈ G X that is isometric exactly on an interval I containing 0 in its interior (as for ω = w, e w fn ), let Gω X = {` ∈ G X : `|I = ω|I } be the space of extensions of ω|I to geodesic lines. With ∂ω± X = {`± : ` ∈ Gω X} its set of points at ±∞, we have a homeomorphism Gω X → (∂ω− X × ∂ω+ X) defined by ` 7→ (`− , `+ ), using Hopf’s parametrisation with respect to the point ω(0), since all the geodesic lines in Gω X are at the point ω(0) at time t = 0. Using as basepoint x0 = ω(0) in the definition of the Gibbs measure (see Equation (4.12)), this homeomorphism sends the restriction to Gω X of the Gibbs measure d m e c (`) + to the product measure dµ− (` ) dµ (` ). Hence the pushforward of m e − + c |Gω X ω(0) ω(0) − + − by the positive endpoint map e+ : ` 7→ `+ is µω(0) (∂ω X) dµω(0) (`+ ), and note that − µ− ω(0) (∂ω X) is a positive constant. Let pe : G X → Γ\G X be the canonical projection. Since π+ : Σ → Σ+ is the + map that forgets about the past, there exist measurable maps uw : ∂w e X → [w] + and uwn : ∂wfn X → [wn ] such that the following diagrams commute: π+ ◦ Θ ◦ p e

Gwe X e+

#

/ [w]
0 be such that for all bounded α-Hölder-continuous functions φ0 , ψ 0 : Σ+ → R and n ∈ N, we have | cov(π+ )∗ µ, n (φ0 , ψ 0 )| 6 C kφ0 kα kψ 0 kα e−κn . Let φ, ψ : Σ → R be bounded α-Hölder-continuous maps and n ∈ N. Denoting by ± t any value in [−t, t] for every t > 0, we have, by the first part of Lemma 9.3 10 See

Section 3.1 for the definition of the H¨ older norm k · kα .


189

and for every N ∈ N,11 Z Z n φ ◦ σ ψ dµ = φ ◦ σ n+N ψ ◦ σ N dµ Σ Σ Z = (φ(N ) ◦ π+ ± kφk0α e−α N ) ◦ σ n (ψ (N ) ◦ π+ ± kψk0α e−α N ) dµ ZΣ n = φ(N ) ◦ σ+ ψ (N ) d(π+ )∗ µ ± kφkα kψkα e−α N . Σ+

A similar estimate holds for the second term in the definition of the correlation coefficients. Hence, by the second part of Lemma 9.3, | covµ, n (φ, ψ)| 6 | cov(π+ )∗ µ, n (φ(N ) , ψ (N ) )| + 2 kφkα kψkα e−α N 6 C (kφk∞ + kφk0α eα N ) (kψk∞ + kψk0α eα N ) e−κ n + 2 kφkα kψkα e−α N 6 kφkα kψkα (C e2α N −κ n + 2 e−α N ) . Taking N = b κ4 αn c, C 0 = C + 2 eα and κ0 = κ4 , we have 0

| covµ, n (φ, ψ)| 6 C 0 kφkα kψkα e−κ n , and the result follows.

In order to conclude Step 2, we now state the one-sided version of Theorem 9.2 and prove how it implies Theorem 9.2.12 For every finite subset E of A , let τE (x) = inf{n ∈ N − {0} : xn ∈ E } be the first return time13 of x ∈ Σ+ under iteration of the one-sided shift. Theorem 9.5. Let (Σ+ , σ+ ) be a locally compact transitive one-sided topological Markov shift with alphabet A and transition matrix A, and let P+ be a mixing σ+ -invariant probability measure with full support on Σ+ . Assume that (1) for every A-admissible finite sequence w = (w0 , . . . , wn ) in A , the Jacobian of the map (wn , x1 , x2 , . . . ) 7→ (w0 , . . . , wn , x1 , x2 , . . . ) from [wn ] to [w] with respect to the restrictions of the measure P+ is constant; (2) there exist a nonempty finite subset E of A and C 0 , κ0 > 0 such that for all n ∈ N, we have 0 P+ {x ∈ Σ+ : x0 ∈ E and τE (x) > n} 6 C 0 e−κ n . (9.4) Then (Σ+ , σ+ , P+ ) has exponential decay of α-H¨ older correlations. Proof that Theorem 9.5 implies Theorem 9.2. Let (Σ, σ, P, E ) be as in the statement of Theorem 9.2. Let P+ = (π+ )∗ P, which is a mixing σ+ -invariant probability measure on Σ+ with full support. Note that Hypothesis (1) in Theorem 9.5 follows 11 We

will choose N appropriately below. (1) of Theorem 9.5 is far from being optimal, but will be sufficient for our purpose. 13 Or rather the first positive passage time. 12 Assumption

190


from Hypothesis (1) of Theorem 9.2. Similarly, Equation (9.4) follows from Equation (9.3). Hence Theorem 9.2 follows from Theorem 9.5 and Proposition 9.4. Let us now consider Theorem 9.5. The scheme of its proof, using inducing and Young tower arguments, was communicated to us by O. Sarig. Step 3: Proof of Theorem 9.5. In this final step, using inducing of the dynamical S system (Σ+ , σ+ ) on the subspace {x ∈ Σ+ : x0 ∈ E } = a∈E [a] (a finite union of 1-cylinders), we present (Σ+ , σ+ ) as a Young tower to which we will apply the results of [You2]. Note that since σ+ is mixing and P+ has full support in Σ+ , there exists a σ+ -invariant measurable subset of full measure ∆ of Σ+ such that the orbit under σ+ of every element of ∆ passes infinitely many times inside the nonempty open S subset a∈E [a]. WeS again denote by τE : ∆ → N − {0} the restriction to ∆ of the first return time in a∈E [a], so that if [ ∆0 = {x ∈ ∆ : x0 ∈ E } = ∆ ∩ [a] , a∈E n σ+ (x)

then τE (x) = min{n ∈ N − {0} : ∈ ∆0 } for all x ∈ ∆. We denote by F : ∆ → ∆0 the first return map to ∆0 under iteration of the one-sided shift, that is, τ (x) F : x 7→ σ+E (x) . Let W be the set of admissible sequences w of length |w| at least 2 such that if w = (w0 , . . . , wn ) with n = |w| − 1, then w0 , wn ∈ E

and w1 , . . . , wn−1 ∈ /E .

We have the following properties: • the sets ∆a = ∆ ∩ [a] for a ∈ E form a finite measurable partition of ∆0 and for every a ∈ E , the sets ∆w = ∆ ∩ [w] for w ∈ W and w0 = a form a countable measurable partition of ∆a ; • for every w ∈ W , the first return time τE is constant (equal to |w| − 1) on ∆w , and if w|w|−1 = b, then the first return map F is a bijection from ∆w to ∆b ; • for all w ∈ W and x, y ∈ ∆w , since x, y have the same |w| first components, we have |w|−1

d(F (x), F (y)) = d(σ+

|w|−1

x, σ+

y) = e|w|−1 d(x, y) > e d(x, y) ;

• for all w ∈ W , n ∈ {0, . . . , |w| − 2} and x, y ∈ ∆w , we have n n d(σ+ x, σ+ y) = en d(x, y) 6 e|w|−2 d(x, y) < d(F (x), F (y)) ;

• for every w ∈ W , the Jacobian of the first return map F : ∆w → ∆w|w|−1 for the restrictions to ∆w and ∆w|w|−1 of P+ is constant.14 14 Actually,

only a much weaker assumption is required, such as a H¨ older continuity property of


191

By an easy adaptation of [You2, Thm. 3] (see also [Mel1, §2.1]), which considers the case in which E is a singleton, we have the following noneffective15 exponential decay of correlation: there exists κ > 0 such that for every φ, ψ ∈ Cbα (Σ+ ), there exists a constant Cφ,ψ > 0 such that | covP+ , n (φ, ψ)| 6 Cφ,ψ e−κn . By an elegant argument using the principle of uniform boundedness, it is proved in [ChCS, Appendix B] that this implies that there exist C, κ > 0 such that for every φ, ψ ∈ Cbα (Σ+ ), we have | covP+ , n (φ, ψ)| 6 C kφkα kψkα e−κ n . This concludes the proof of Theorem 9.5, hence the proof of Theorem 9.1.

The next result gives examples of applications of Theorem 9.1 when Γ\X is infinite. It strengthens [AtGP, Thm. 2.1], which applies only to arithmetic lattices and only for the locally constant regularity (see Section 15.4); see also [BekL] for an approach using spectral gaps. It was claimed in [Kwo], but was retracted by the author. Corollary 9.6. Let X be a locally finite simplicial tree without terminal vertices. Let Γ be a geometrically finite subgroup of Aut(X) such that the smallest nonempty Γ-invariant subtree of X is uniform without vertices of degree 2. Let α ∈ ]0, 1]. (1) If LΓ = Z, then the discrete-time geodesic flow on Γ\G X has exponential decay of α-H¨ older correlations for the zero system of conductances. (2) If LΓ = 2Z, then the square of the discrete-time geodesic flow on Γ\Geven X has exponential decay of α-H¨ older correlations for the zero system of conductances, that is, there exist C, κ > 0 such that for all φ, ψ ∈ Cbα (Γ\Geven X) and n ∈ Z, we have Z φ ◦ g−2n ψ d mBM Γ\Geven X Z Z 1 − φ d mBM ψ d mBM mBM (Γ\Geven X) Γ\Geven X

6 Ce

−κ|n|

Γ\Geven X

kφkα kψkα .

The main point in order to obtain this corollary is to prove the exponential decay of volumes of geodesic lines going high in the cuspidal rays of Γ\X, stated as Assumption (9.2) in Theorem 9.1. There is a long history of similar results, starting from the exponential decay of volumes of small cusp neighbourhoods in noncompact finite-volume hyperbolic manifolds (based on the description of their ends) used by Sullivan to deduce Diophantine approximation results (see [Sul3, this Jacobian; see [You2]. 15 Actually, there is in [You2] (see also [CyS]) a control on the constant in terms of some norms of the test functions, but these norms are not the ones we are interested in.

192


§ 9]).16 These results were extended to the case of locally symmetric Riemannian manifolds by Kleinbock–Margulis [KM2] (based on the description of their ends using Siegel sets). Note that the geometrically finite tree lattice assumption on Γ is here in order to obtain similar descriptions of the ends of Γ\X. Proof. Up to replacing X by C ΛΓ, we assume that X is a uniform simplicial tree with degrees at least 3 and that Γ is a geometrically finite tree lattice of X. We use the zero system of conductances. (1) By [Pau4],17 the graph Γ\X is the union of a nonempty finite graph Y and finitely many cuspidal rays Ri for i ∈ {1, . . . , k}. If (xi, n )n∈N is the sequence of vertices in increasing order along Ri for i = 1, . . . , k, then the vertex group Gxi, n of xi, n in the quotient graph of groups Γ\\X satisfies Gxi, n ⊂ Gxi, n+1 for every n ∈ N, and the edge group of the edge ei,n with initial vertex xi,n and terminal vertex xi, n+1 is equal to Gxi, n .18 Note that since the degrees of the vertices of X are at least 3, we have [Gxi, n+1 : Gxi,n ] > 2 and |Gxi, 0 | > 1, so that for every n ∈ N, |Gxi, n | > 2n .

(9.5)

Let E be the (finite nonempty) set of vertices V Y of Y. Note that for all n ∈ N − {0} and ` ∈ Γ\G X, if `(0) ∈ E and τE (`) > 2n, then ` leaves Y after time 0 and it travels (geodesically) inside some cuspidal ray for a time at least n, so that there exists i ∈ {1, . . . , k} such that `(n) = xi, n . Hence for all n ∈ N, using • the invariance of mBM under the discrete-time geodesic flow in order to get the third term, • Equation (8.5) where x ei, n is a fixed lift of xi, n in V X for the fifth term, and • Equation (9.5), since |Γxei, n | = |Gxi, n |, and the facts that the degrees of the uniform simplicial tree X are uniformly bounded and that the total mass of the Patterson measures of the tree lattice Γ are uniformly bounded (see Proposition 4.16) for the last term, we have mBM {` ∈ Γ\G X : `(0) ∈ E and τE (`) > 2n} Xk 6 mBM {` ∈ Γ\G X : `(n) = xi,n } i=1 Xk Xk = mBM {` ∈ Γ\G X : `(0) = xi,n } = π∗ mBM ({xi,n }) i=1 i=1 Xk X 1 = µxei, n (∂e X) µxei, n (∂e0 X) i=1 |Γx ei, n | 0 0 0 e, e ∈EX : o(e)=o(e )= x ei, n , e6=e

1 6 k n max deg(x)2 max kµx k2 . x∈V X 2 x∈V X 16 And

by probabilists in order to study the statistics of cusp excursions (see, for instance, [EF]). also Section 2.6. 18 We identify the edge group G of an edge e with its image by the structural map G → G e e o(e) . 17 See


193

The result then follows from Theorem 9.1 using the above finite set E, which satisfies Assumption (9.2), as we just proved, and using Proposition 4.16 and Theorem 4.17 in order to check that under the assumption that LΓ = Z, the Bowen–Margulis measure mBM of Γ is finite and mixing under the discrete-time geodesic flow on Γ\G X. (2) The proof of Assertion (2) is similar to that of Assertion (1).

Remark. The techniques introduced in the above proof in order to check the main hypothesis of Theorem 9.1 may be applied to numerous other examples. For instance, let X be a locally finite simplicial tree without terminal vertices. Let Γ be a nonelementary discrete subgroup of Aut(X) such that the smallest nonempty Γ-invariant subtree of X is uniform without vertices of degree 2, and such that LΓ = Z. Let α ∈ ]0, 1]. Assume that Γ\X is the union of a finite graph A and finitely many trees T1 , . . . , Tk meeting A in one and exactly one vertex ∗1 , . . . , ∗n such that for every edge e in Ti pointing away from the root ∗i of Ti , the canonical morphism Ge → Go(e) between edge and vertex groups of the quotient graph of groups Γ\\X = (Γ\X, G∗ ) is an isomorphism. Assume that there exist C, κ > 0 such that for all n ∈ N, X i=1,...,k, x∈V Ti : d(x,∗i )=n

1 6 Ce−κ n . |Gx |

Then the discrete-time geodesic flow on Γ\G X has exponential decay of α-Hölder correlations for the zero system of conductances.

q−1 2

1

q−1 1 1 q−1 q−1

1 q−1 1

1 q−1 1 1

q−1 1

q−1 2

1

q−1

q−1

q−1 1 1 q−1 q−1

q−1 1

1

q−1

q−1

This is in particular the case for every k, q ∈ N such that k > 2, q > 2k + 1 and q − k is odd, when the quotient graph of groups Γ\\X has underlying edgeindexed graph19 a loop-edge with both indices equal to q−k+1 glued to the root of 2 a regular k-ary rooted tree, with indices 1 for the edges pointing towards the root 19 See

the definition in Section 2.6.

194


and q − k + 1 for the edges pointing away from the root (see the above picture where k = 2). Note that X is then the (q + 1)-regular tree, and that the loop edge is here in order to ensure that LΓ = Z. For instance, the vertex group of a point n at distance n from the root may be chosen to be Z/( q−k+1 )Z × Z/(q − k + 1)Z . 2

9.3

Rate of mixing for metric trees

Let (X, λ), X, Γ, Fe, Fe± , δ = δΓ, F ± < +∞, and (µ± x )x∈V X be as at the beginning of Section 4.4. Let m e F and mF be the associated Gibbs measures on G X and Γ\G X. The aim of this section is to study the problem of finding conditions on these data under which the (continuous-time) geodesic flow on Γ\G X is polynomially mixing for the Gibbs measure mF . We will actually prove a stronger property, though it applies only to observables that are smooth enough along the flow. Let us fix α ∈ ]0, 1]. Let (Z, (φt )t∈R , µ) be a topological space Z endowed with a continuous one-parameter group (φt )t∈R of homeomorphisms preserving a (Borel) probability measure µ on Z. For all k ∈ N, let Cbk, α (Z) be the real vector space of maps f : Z → R such that for all z ∈ Z, the map t 7→ f (φt z) is C k -smooth, and such that the maps ∂ti f : Z → R defined di by z 7→ dt f (φt z) for 0 6 i 6 k are bounded and α-Hölder-continuous. It is a i t=0 Banach space when endowed with the norm kf kk, α =

k X

k∂ti f kα ,

i=0

and it is contained in L2 (Z, µ) by the finiteness of µ. We denote by Cck, α (Z) the vector subspace of elements of Cbk, α (Z) with compact support. For all ψ, ψ 0 ∈ L2 (Z, µ) and t ∈ R, let Z Z Z 0 0 covµ, t (ψ, ψ ) = ψ ◦ φt ψ dµ − ψ dµ ψ 0 dµ Z

Z

Z

be the (well-defined) correlation coefficient of the observables ψ, ψ 0 at time t under the flow (φt )t∈R for the measure µ. We say20 that the (continuous-time) dynamical system (Z, (φt )t∈R , µ) has superpolynomial decay of α-H¨ older correlations if for every n ∈ N there exist two constants C = Cn > 0 and k = kn ∈ N such that for all ψ, ψ 0 ∈ Cbk, α (Z) and t ∈ R, we have | covµ, t (ψ, ψ 0 )| 6 C (1 + |t|)−n kψkk, α kψ 0 kk, α . 20 See

[Dol2], and more precisely [Mel1, Def. 2.2], whose definition is slightly different but implies the one given in this paper by the principle of uniform boundedness argument of [ChCS, Appendix B] already used in Section 9.2.

9.3. Rate of mixing for metric trees

195

Following Dolgopyat, we say that the dynamical system (Z, (φt )t∈R , µ) is rapidly mixing if there exists α > 0 such that (Z, (φt )t∈R , µ) has superpolynomial decay of α-H¨ older correlations. We will use the following two assumptions on our data, introduced respectively in [Dol2] and [Mel1]. Recall that the Gibbs measure mF , when finite, is mixing if and only if the length spectrum LΓ is dense in R (see Theorem 4.9). The rapidly mixing property will require stronger assumptions on LΓ . We say that the length spectrum LΓ of Γ is 2-Diophantine if there exists a ratio of two translation lengths of elements of Γ that is Diophantine. Recall that a real number x is Diophantine if there exist α, β > 0 such that x − p > α q −β q for all p, q ∈ Z with q > 0. e be the set of vertices Let E be a finite subset of vertices of Γ\X, and let E of X mapping to E. We denote by TE the set of triples (λ(γ), d(γ), q(γ, p)), where γ ∈ Γ has translation length λ(γ) > 0, has d(γ) vertices on its translation axis e ∩ Ax(γ) under Ax(γ) modulo γ Z , and if the first return time of a vertex p in E the discrete-time geodesic flow along the translation axis has period q(γ, p). We say that the length spectrum LΓ of Γ is 4-Diophantine with respect to E if for all sequences (bk )k∈N in [1 + ∞[ converging to +∞ and (ωk )k∈N , (ϕk )k∈N in [0, 2π[ , there exists N ∈ N such that for all a > N and C, β > 1, there exist k > 1 and (τ, d, q) ∈ TE such that d (bk τ + ωk d)bβ ln bk c + qϕk , 2πZ) > C q b−a k . We define the first return time after time on a finite subset E of vertices of Γ\X as the map τE> : Γ\G X → [0,+∞] defined by τE> (`) = inf{t > : `(t) ∈ E}. Theorem 9.7. Assume that the Gibbs measure mF is finite and mixing for the (continuous-time) geodesic flow, and that the lengths of the edges of (X, λ) have a finite upper bound.21 Furthermore, assume that (a) either Γ\X is compact and the length spectrum of Γ is 2-Diophantine, (b) or there exists a finite subset E of vertices of Γ\X satisfying the following properties: (1) there exist C, κ > 0 and ∈ ]0, min λ[ such that for all t > 0, mF ({` ∈ Γ\G X : d(`(0), E) 6 and τE> (`) > t}) 6 C e−κ t , (2) the length spectrum of Γ is 4-Diophantine with respect to E. Then the (continuous-time) geodesic flow on Γ\G X has superpolynomial decay of mF α-H¨ older correlations for the normalised Gibbs measure km . Fk 21 They

have a positive lower bound by definition; see Section 2.6.

196


Note that the existence of E satisfying the exponentially small tail Hypothesis (1) is in particular satisfied if Γ is geometrically finite with E the set of vertices of a finite subgraph of Γ\X whose complement in Γ\X is the underlying graph of a union of cuspidal rays in Γ\\X. See the proof of Corollary 9.6 and use the hypothesis on the lengths of edges. Note that the exponentially small tail Hypothesis (1) might be weakened to a superpolynomially small tail hypothesis while keeping the same conclusion; see [Mel2]. Since the former is easier to check than the latter, we prefer to state Theorem 9.7 as it is. We will follow a scheme of proof analogous to the one in Section 9.2 for simplicial trees, by reducing the study to a problem of suspensions of Young towers, and then apply results of [Dol2] and [Mel1] for the rapid mixing property of suspensions of hyperbolic and nonuniformly hyperbolic dynamical systems. Proof. Since the Gibbs measure normalised to be a probability measure depends only on the cohomology class of the potential (see Equation (4.10)), we may assume by Proposition 3.22 that F = Fc is the potential on Γ\T 1 X associated with a system of conductances e c : EX → R for Γ. We denote by δc the critical exponent of (Γ, Fc ), and by mc the Gibbs measure mFc . Up to replacing X by its minimal nonempty Γ-invariant subtree, we assume that X = C ΛΓ. Step 1: Reduction to a suspension of a two-sided symbolic dynamics. We refer to the paragraphs before the statement of Theorem 5.9 and at the beginning of Section 5.3 for the definitions of • the system of conductances ] c for Γ on the simplicial tree X, • the (two-sided) topological Markov shift (Σ, σ, P) on the alphabet A , conju m gated to the discrete-time geodesic flow Γ\G X, ] g1 , km]] c k by the homeoc morphism Θ : Γ\G X → Σ, • the roof function r : Σ → ]0, +∞[ , • the suspension (Σ, σ, a P)r = (Σr , (σrt )t∈R , a Pr ) over (Σ, σ, a P) with roof function r, where a = kP1r k . The suspension (Σ, σ, a P)r is conjugated to the geodesic flow mc Γ\G X, (gt )t∈R , kmc k by the bilipschitz homeomorphism Θr : Γ\G X → Σr defined at the end of the proof of Theorem 5.9. We will always (uniquely) represent the elements of Σr as [x, s] with x ∈ Σ and 0 6 s < r(x). t t Note that since the map Θ−1 r conjugates (σr )t∈R and (g )t∈R , we have for all f : Γ\G X → R and x ∈ Σr , when defined,

∂ti (f ◦ Θ−1 r )(x) =

di di t i −1 f ◦ Θ−1 f (gt Θ−1 r (σr x) = r (x)) = (∂t f ) ◦ Θr (x) . i t=0 dt dti t=0


197

Hence if f : Γ\G X → R is C k -smooth along the orbits of (gt )t∈R , then f ◦ Θ−1 r is C k -smooth along the orbits of (σrt )t∈R . Furthermore, since Θr is bilipschitz, the k, α precomposition map by Θ−1 (Γ\G X) r is a continuous linear isomorphism from Cb to Cbk, α (Σr ). mc Note that since Θr conjugates (gt )t∈R and (σrt )t∈R , and sends km to kPPrr k , ck we have, for all ψ, ψ 0 ∈ L2 (Γ\G X) and t ∈ R, 0 mc cov km , t (ψ, ψ ) = cov k c

Pr ,t kPr k

0 −1 (ψ ◦ Θ−1 r , ψ ◦ Θr ) .

Therefore we have only to prove that the suspension (Σr , (σrt )t∈R , kPPrr k ) is rapidly mixing under one of the assumptions (a) and (b). Step 2: Reduction to a suspension of a one-sided symbolic dynamics. In this step, we explain the rather standard reduction concerning mixing rates from suspensions of two-sided topological Markov shifts to suspensions of one-sided topological Markov shifts. We use the obvious modifications of the notation and constructions concerning the suspension of a noninvertible transformation to a semiflow, given for invertible transformations at the beginning of Section 5.3. We consider the one-sided topological Markov shift (Σ+ , σ+ , P+ ) over the alphabet A constructed at the beginning of Step 2 of the proof of Theorem 9.1, where the system of conductances c is now replaced by ] c, so that we have P = Θ∗ (m] c /km] c k). Let π+ : Σ → Σ+ be the natural extension and recall that P+ = (π+ )∗ P and π+ ◦ σ = σ+ ◦ π+ . We are going to construct in Step 2, as the suspension of (Σ+ , σ+ , P+ ) with an appropriate roof function r+ , a semiflow ((Σ+ )r+ , (σ+ )tr+ t>0 , (P+ )r+ ), and prove that the flow Σr , (σrt )t∈R , kPPrr k is rapidly mixing if the semiflow

(Σ+ )r+ , (σ+ )tr+

, t>0

(P+ )r+ k(P+ )r+ k

is rapidly mixing. Let r+ : Σ+ → ]0, +∞[ be the map defined by r+ : x 7→ λ(e+ 0),

(9.6)

− + where if x = (xn )n∈N , the edge e+ 0 is such that x0 = (e0 , h0 , e0 ). Note that this map has a positive lower bound and a finite upper bound, and that it is locally constant (and even constant on the 1-cylinders of Σ+ ). By Equation (5.10), we have r+ ◦ π+ = r . (9.7) t We denote by ((Σ+ )r+ , (σ+ )r+ t>0 , (P+ )r+ ) the suspension semiflow over (Σ+ , σ+ , P+ ) with roof function r+ . We (uniquely) represent the points of the

198


suspension space (Σ+ )r+ as [x, s] for x ∈ Σ+ and 0 6 s < r+ (x). For all t > 0, n we have (σ+ )tr+ ([x, s]) = [σ+ x, s0 ], where n ∈ N and s0 ∈ R are such that we have Pn−1 i n t + s = i=0 r+ (σ+ x) + s0 and 0 6 s0 < r+ (σ+ x). We define the suspended natural extension as the map π+r : Σr → (Σ+ )r+ by π+r : [x, s] 7→ [π+ (x), s] , which is well defined by Equation (9.7). Note that π+r is 1-Lipschitz for the Bowen– Walters distances on Σr and (Σ+ )r+ (see Proposition 5.11).22 For all ψ : Σr → R and T > 0, let us construct ψ (T ) : (Σ+ )r+ → R as follows. For every [x, s] ∈ (Σ+ )r+ , let N ∈ N and s0 > 0 be such that we have N (σ+ )Tr+ [x, s] = [σ+ x, s0 ], with N −1 X

N 0 6 s0 < r+ (σ+ x) and s + T =

i r+ (σ+ x) + s0 .

i=0

Let ψ (T ) ([x, s]) = ψ([y, s0 ]), where y = (yn )n∈Z is such that yi = xi+N if i > −N and yi = (z x0 )i+N otherwise. N Note that y0 = xN , hence r(y) = r+ (σ+ (x)) and 0 6 s0 < r(y), so that the above map is well defined. Finally, for every ψ ∈ Cbk, α (Σr ) or ψ ∈ Cbk, α ((Σ+ )r+ ), let kψkk, ∞ =

k X

k∂ti ψk∞ and kψk0k, α =

i=0

k X

k∂ti ψk0α ,

i=0

so that kψkk, α = kψkk, ∞ + kψk0k, α . Lemma 9.8. Let T > 0 and ψ ∈ Cbk, α (Σr ). (1) For all t > 0, we have (σ+ )tr+ ◦ π+r = π+r ◦ σrt . (2) With α0 = such that |ψ ◦

α sup λ , there σrT − ψ (T ) ◦

exists a constant C1 > 1 (independent of k, T , and ψ) 0

π+r | 6 C1 kψk0α e−α

T

.

(3) We have ψ (T ) ∈ Cbk, α ((Σ+ )r+ ) and kψ (T ) kk, ∞ 6 kψkk, ∞ . With α00 = there exists a constant C2 > 1 (independent of k, T , and ψ) such that 00

kψ (T ) k0k, α 6 C2 eα 22 Note

T

kψk0k, α .

α inf λ ,

(9.8)

that Proposition 5.11 is stated for suspensions of invertible maps, but the roof function is constant on 1-cylinders, and the branches of the inverse of σ+ on these 1-cylinders are uniformely Lipschitz; hence the proof of [BarS, Appendix] extends.


199

Proof. (1) For all t > 0 and [x, s] ∈ Σr , let n ∈ N and s0 > 0 be such that t+s=

n−1 X

i n r+ (σ+ π+ (x)) + s0 and 0 6 s0 < r+ (σ+ π+ (x)) .

i=0 i Since r+ ◦ σ+ ◦ π+ = r ◦ σ i for every i ∈ N, these two conditions are equivalent to

t+s=

n−1 X

r(σ i x) + s0 and 0 6 s0 < r(σ n x) .

i=0

Hence n (σ+ )tr+ ◦ π+r ([x, s]) = (σ+ )tr+ ([π+ (x), s]) = [σ+ π+ (x), s0 ]

and π+r ◦ σrt ([x, s]) = π+r ([σ n x, s0 ]) = [π+ (σ n x), s0 ] . This proves Assertion (1), since π+ ◦ σ = σ+ ◦ π+ . (2) By Proposition 5.11, we may assume that in the formula of the Hölder norms, the Bowen–Walters distance is replaced by the function dBW , since this will only α change C1 by CBW C1 . For every [x, s] ∈ Σr , with [y, s0 ] and N associated with π+r ([x, s]) = [π+ (x), s] as in the definition of ψ (T ) ([π+ (x), s]), we have dBW (σrT [x, s], [y, s0 ]) = dBW ([σ N x, s0 ], [y, s0 ]) 6 d(σ N x, y) 6 e−N . Since the positive roof function r is bounded from above by the least upper bound sup λ of the lengths of the edges, we have N>

N −1 X i=0

r(σ i x) 1 T = (s + T − s0 ) > −1. sup λ sup λ sup λ

Hence |ψ ◦ σrT ([x, s]) − ψ (T ) (π+r (x))| = |ψ(σrT [x, s]) − ψ([y, s0 ])| α

6 kψk0α dBW (σrT [x, s], [y, s0 ])α 6 kψk0α e− sup λ T +α . (3) Let us prove that ψ (T ) is C k along semiflow lines. Fix [x, s] ∈ (Σ+ )r+ . With [y, s0 ] and N as in the construction of ψ (T ) ([x, s]), let us consider > 0 small N enough that < r+ (x) − s and < r+ (σ+ (x)) − s0 = r(y) − s0 . Then ψ (T ) ◦ (σ+ )r+ ([x, s]) = ψ (T ) ([x, s + ]) = ψ([y, s0 + ]) = ψ ◦ σr ([y, s0 ]) . Therefore by taking derivatives with respect to in this formula, ψ (T ) is indeed C k along semiflow lines, and for i = 0, . . . , k, we have (T ) ∂ti ψ (T ) = ∂ti ψ . (9.9)

200


The inequality kψ (T ) k∞ 6 kψk∞ is immediate by construction. Using the above centered equation, we have kψ (T ) kk, ∞ 6 kψkk, ∞ . Let us prove that there exists a constant C2 > 1 (independent of T and ψ) such that 00 kψ (T ) k0α 6 C2 eα T kψk0α . (9.10) Let [x, s] ∈ (Σ+ )r+ , and take [y, s0 ] and N as in the definition of ψ (T ) ([x, s]). Let [x, s ] ∈ (Σ+ )r+ , and take [y, s0 ] and N as in the definition of ψ (T ) ([x, s ]). Up to exchanging [x, s] and [x, s ], we assume that N > N . By Proposition 5.11,23 we may assume that in the Hölder norms formulas, the Bowen–Walters distance is replaced by the function dBW , since this will only 2α change C2 by CBW C2 . Let C3 = min{e−1 , inf λ} . We have (T ) ψ ([x, s]) − ψ (T ) ([x, s ]) = ψ([y, s0 ]) − ψ([y, s0 ]) 6 kψk0α dBW ([y, s0 ], [y, s0 ])α .

(9.11)

Note that the map dBW on Σr × Σr is bounded from above by 1 + sup λ, since the distance on Σ is at most 1 and since the roof function r is bounded from above by T sup λ. If dBW ([x, s], [x, s ]) > C3 e− inf λ , then dBW ([y, s0 ], [y, s0 ]) 6 1 + sup λ 6

1 + sup λ T e inf λ dBW ([x, s], [x, s ]) . C3

Therefore Equation (9.10) follows from Equation (9.11) whenever C2 >

(1+sup λ)α . C3α

T

Conversely, suppose that dBW ([x, s], [x, s ]) < C3 e− inf λ . Assume that dBW ([x, s], [x, s ]) = d(x, x) + |s − s| ; the other possibilities are treated similarly. Since s+T =

N −1 X

i r+ (σ+ x) + s0

i=0

and since the roof function r+ is bounded from below by the constant inf λ, we have T > N inf λ − inf λ, or equivalently N 6 infT λ + 1. Therefore we have T d(x, x) < C3 e− inf λ 6 e−N by the definition of C3 . In particular, the sequences x and x indexed by N have the same first N + 1 coefficients. Since r+ (z) depends 23 See

the previous footnote.


201

i i only on z0 for all z ∈ Σ+ , we thus have r+ (σ+ x) = r+ (σ+ x) for i = 0, . . . , N . Note that we have N −1 X i s+T = r+ (σ+ x) + s0 . i=0

If N = N , then by taking the difference of the last two centered equations, we have s−s = s0 −s0 , and by construction, the sequences y and y indexed by Z satisfy yi = (y)i if i 6 0 and if 0 6 i 6 − ln d(x, x) − N . Therefore d(y, y) 6 eN d(x, x) and dBW ([y, s0 ], [y, s0 ]) 6 d(y, y) + |s0 − s0 | 6 eN d(x, x) + |s − s| T

6 eN dBW ([x, s], [x, s ]) 6 e inf λ +1 dBW ([x, s], [x, s ]) . Therefore Equation (9.10) follows from Equation (9.11) whenever C2 > eα . If N > N , then again by subtraction, N −1

s−s=

X

i N r+ (σ+ x) + r+ (σ+ x) − s0 + s0 .

i=N +1 T

N x) − s0 > 0, and that |s − s| < C3 e− inf λ 6 inf λ by Note that s0 > 0, that r+ (σ+ N the definition of C3 . Hence we have N = N + 1 and s − s = r+ (σ+ x) − s0 + s0 . By construction, the sequences σy and y indexed by Z satisfy (σy)i = (y)i if i 6 0 and if 0 6 i 6 − ln d(x, x) − N − 1. Hence by the definition of dBW and since N r(y) = r+ (σ+ x), we have

dBW ([y, s0 ], [y, s0 ]) 6 d(σy, y) + r(y) − s0 + s0 6 eN +1 d(x, x) + |s − s| T

6 eN +1 dBW ([x, s], [x, s ]) 6 e inf λ +2 dBW ([x, s], [x, s ]) . Therefore Equation (9.10) follows from Equation (9.11) whenever C2 > e2α . This ends the proof of Equation (9.10). Now note that Equations (9.9) and (9.10) imply Equation (9.8) by summation (using the independence of C2 on ψ), thus concluding the proof of Lemma 9.8. Proposition 9.9. Let µ be a (σrt )t∈R -invariant probability measure on Σr . Assume that the dynamical system (Σ+ )r+ , ((σ+ )tr+ )t∈R , (π+r )∗ µ has superpolynomial decay of α-H¨ older correlations. Then (Σr , (σrt )t∈R , µ) has superpolynomial decay of α-H¨ older correlations. λ Proof. We fix n ∈ N. Let N = 1 + 2d sup inf λ e. Let k ∈ N and C4 > 0 (depending on n) be such that for all ψ, ψ 0 ∈ Cbk, α ((Σ+ )r+ ), we have for all t > 1,

cov(π r )

+ ∗ µ, t

(ψ, ψ 0 ) 6 C4 kψkk, α kψ 0 kk, α tN n .

(9.12)

202


Now let ψ, ψ 0 ∈ Cbk, α (Σr ). We again denote by ± a any value in [−a, a] for every a > 0. By invariance of µ under (σrt )t∈R , by Lemma 9.8 (2) and by Lemma 9.8 (1), we have, for every T > 0 (to be chosen appropriately later on), Z

Z ψ ◦ σrt ψ 0 dµ = ψ ◦ σrT +t ψ 0 ◦ σrT dµ Σr Σr Z 0 0 (T ) (T ) r = (ψ ◦ π+ ± C1 kψk0α e−α T ) ◦ σrt (ψ 0 ◦ π+r ± C1 kψ 0 k0α e−α T ) dµ Σ Z r 0 (T ) = ψ (T ) ◦ (σ+ )tr+ ψ 0 d(π+r )∗ µ ± C12 kψkα kψ 0 kα e−α T . (Σ+ )r+

A similar estimate holds for the second term in the definition of the correlation (T ) coefficients. Hence, applying Equation (9.12) to the observables ψ (T ) and ψ 0 , α00 T by Lemma 9.8 (3), we have since C2 e > 1, | covµ, t (ψ, ψ 0 )| 6 | cov(π+r )∗ µ, t (ψ (T ) , ψ 0 00

6 C4 (kψkk, ∞ + C2 kψk0k, α eα 0

+ 2 C12 kψkα kψ 0 kα e−α

T

(T )

0

)| + 2 C12 kψkα kψ 0 kα e−α 00

) (kψ 0 kk, ∞ + C2 kψ 0 k0k, α eα

T

T

) t−N n

T 00

6 kψkk, α kψ 0 kk, α (C4 C22 e2α

T

0

t−N n + 2 C12 e−α 00

Take T = αn0 ln t > 0. Since N = 1 + 2d αα0 e, we have 2α00 with C5 = C4 C22 + 2 C12 , we have for all t > 1,

T

).

n α0

− N n 6 −n. Hence

| covµ, t (ψ, ψ 0 )| 6 C5 kψkk, α kψ 0 kk, α t−n . This concludes the proof of Proposition 9.9.

Step 3: Conclusion of the proof of Theorem 9.7. In this step, we prove that the semiflow (P+ )r+ (Σ+ )r+ , (σ+ )tr+ t>0 , k(P+ )r+ k is rapidly mixing, which concludes the proof of Theorem 9.7, using Proposition 9.9 with µ = kPPrr k . Recall24 that Y = {` ∈ Γ\G X : `(0) ∈ V X} is a cross-section of the geodesic flow on Γ\G X, and that if R : Y → Γ\G X is the reparametrisation map of ` ∈ Y to a discrete geodesic line ] ` ∈ Γ\G X with the same origin, then the measure µY , induced by the Gibbs measure mc on the cross-section Y by disintegration along the flow, maps by R∗ to a constant multiple of m] c = mF] c (see Lemma 5.10 24 See

the proof of Theorem 5.9.


203

(2)). Hence for all n ∈ N − {0} and ∈ ]0, 21 inf λ[ , by Assumption (b) (1) in the statement of Theorem 9.7, we have n o ] `(0) ∈ E m] c ] ` ∈ Γ\G X : ] ∀ k ∈ {1, . . . , n − 1}, `(k) ∈ /E n o km] c k R−1 (] `)(0) ∈ E 6 µY R−1 (] `) ∈ Y : −1 ] ∀ t ∈ ]0, n inf λ[ , R ( `)(t) ∈ /E kµY k n km] c k 0 6 s 6 , d(gs R−1 (] `)(0), E) 6 o mc gs R−1 (] `) ∈ Γ\G X : 6 ∀ t ∈ ], n inf λ − [ , gs R−1 (] `)(t) ∈ /E kµY k 6

km] c k C e−κ (inf λ) n+κ . kµY k

Therefore Equation (9.2) (where c is replaced by ] c) is satisfied, with C0 =

km] c k C eκ kµY k

and

κ0 = κ inf λ.

As seen in the proof of Theorem 9.1, this implies that there exists a finite subset E of the alphabet A such that Equation (9.4) is satisfied. We now apply [Mel1, Thm. 2.3] when the dynamical system (X, T, m0 ) is (Σ+ , σ+ , P+ ) (using the system of conductances ] c) and the roof function h = r+ . This dynamical system is presented as a Young tower in Step 3 of the proof of Theorem 9.1. Equation (9.4) for the first return map τE and the 4-Diophantine hypotheses are exactly the hypothesis needed in order to apply [Mel1, Thm. 2.3]. (P+ )r Thus the semiflow ((Σ+ )r+ , (σ+ )tr+ t>0 , k(P+ )r+ k ) has superpolynomial decay of + α-H¨ older correlations. When Γ\X is compact, the alphabet A is finite and (Σ+ , σ+ , P+ ) is a (onesided) subshift of finite type, hence we do not need the exponentially small tail assumption, but only the 2-Diophantine hypothesis, and we may apply [Dol2]. Corollary 9.10. Assume that the Gibbs measure mF is finite and mixing for the (continuous-time) geodesic flow, that the lengths of the edges of (X, λ) have a finite upper bound, and that Γ is geometrically finite. There exists a full-measure subset A of R4 (for the Lebesgue measure) such that if Γ has a quadruple of translation lengths in A, or if the length spectrum is 4-Diophantine, then the (continuoustime) geodesic flow on Γ\G X has superpolynomial decay of α-H¨ older correlations for the Bowen–Margulis measure mBM . Proof. The exponentially small tail assumption (b) (1) is checked as in the proof of Corollary 9.6. The deduction of Corollary 9.10 from Theorem 9.7 then proceeds, by an argument going back in part to Dolgopyat, as for the deduction of Corollary 2.4 from Theorem 2.3 in [Mel1]. Note that under the general assumptions of Theorem 4.9, the geodesic flow on Γ\G X might not be exponentially mixing; see, for instance, [Pol1, page 162] or [Rue2] for analogous behaviour.

Part II Geometric Equidistribution and Counting

Chapter 10

Equidistribution of Equidistant Level Sets to Gibbs Measures Let X be a geodesically complete proper CAT(−1) space, let Γ be a nonelementary discrete group of isometries of X, let Fe be a continuous Γ-invariant map on T 1 X such that δ = δΓ, F ± is finite and positive and that the triple (X, Γ, Fe) satisfies ± the HC property,1 and let (µ± x )x∈X be Patterson densities for the pairs (Γ, F ). In this chapter, we prove that the skinning measure on (any nontrivial piece of) the outer unit normal bundle of any properly immersed nonempty proper closed convex subset of X, pushed a long time by the geodesic flow, equidistributes towards the Gibbs measure, under finiteness and mixing assumptions. This result gives four important extensions of [PaP14a, Thm. 1], one for general CAT(−1) spaces with constant potentials, one for Riemannian manifolds with pinched negative curvature and Hölder-continuous potentials, one for R-trees with general potentials, and one for simplicial trees.

10.1

A general equidistribution result

Before stating this equidistribution result, we begin with a technical construction that will also be useful in the following chapter. We refer to Section 2.4 for the ∓ notation concerning the dynamical neighbourhoods (including Vw, η 0 , R ) and to ± Chapter 7 for the notation concerning the skinning measures (including νw ). Technical construction of bump functions. Let D± be nonempty proper closed ∓ ± 00 convex subsets of X, and let R > 0 be such that νw (Vw, η 00 , R ) > 0 for all η > 0 1 1 and w ∈ ∂∓ D± . Let η > 0 and let Ω± be measurable subsets of ∂∓ D± . We now construct functions φ± η, R, Ω± : G X → [0, +∞[ 1 See

Definition 3.13.


207

208

Chapter 10. Equidistribution of Equidistant Level Sets to Gibbs Measures

f is whose supports are contained in dynamical neighbourhoods of Ω± . If X = M a Riemannian manifold and Fe = 0, we recover the same bump functions as in [PaP14a] after the standard identifications concerning the unit tangent bundle. For all η 0 > 0, let h± η, η 0 : G∓ X → [0, +∞[ be the Γ-invariant measurable maps defined by 1 h± (10.1) η, η 0 (w) = ± ∓ νw (Vw, η, η 0 ) ∓ ± if νw (Vw, η, η 0 ) > 0 (which is, for instance, satisfied if w± ∈ ΛΓ and for every 1 ± w ∈ ∂∓ D if η 0 = R by the choice of R) and h± η, η 0 (w) = 0 otherwise. ± These functions hη, η0 have the following behaviour under precomposition by the geodesic flow. By Lemma 7.6 (ii), by Equation (2.18), and by the invariance ± of νw under the geodesic flow, we have, for all t ∈ R and w ∈ G± X, ∓t h∓ η, η 0 (g w) = e

± Cw (w(0), w(∓t)) ±

h∓ η, e−t η 0 (w) .

(10.2)

0 Let us also describe the behaviour of h± η, η 0 when η is small. Let w ∈ G± X be such that w is isometric at least on ±[0, +∞[ , which is, for instance, the case if 1 w ∈ ∂± D∓ . For all η 0 > 0 and ` ∈ B ± (w, η 0 ), let w b be an extension of w such that dW ± (w) (`, w) b < η 0 . Then w(0) b = w(0) by the assumption on w, and using Lemma 2.4, we have

d(`(0), w(0)) = d(`(0), w(0)) b 6 dW ± (w) (`, w) b < η0 . Hence, with κ1 and κ2 the constants in Definition 3.13, if η 0 6 1 and c1 = κ1 + 2 δ + 2

sup π −1 (B(w(0), 2))

|Fe| ,

we have, by Proposition 3.20 (2), ± | Cw (w(0), `(0)) | 6 c1 (η 0 )κ2 . ± ∓ Using Equation (7.12) defining νw , for all s ∈ R, η 0 ∈ ]0, 1], and ` ∈ B ± (w, η 0 ), we have 0 κ2 0 κ2 ∓ s e−c1 (η ) ds dµW ± (w) (`) 6 dνw (g `) 6 ec1 (η ) ds dµW ± (w) (`) . 1 It follows that for all η 0 ∈ ]0, 1] and w ∈ ∂± D∓ such that w± ∈ ΛΓ, we have the ∓ following control of hη, η0 (w): 0 κ2

0 κ2

ec1 (η ) e−c1 (η ) ∓ 6 h (w) 6 . 0 η, η 2η µW ± (w) (B ± (w, η 0 )) 2η µW ± (w) (B ± (w, η 0 ))

(10.3)

When X is an R-tree, we may take κ2 = 1 and c1 = supπ−1 (B(w(0),1)) |Fe − δ| in this equation, as asserted in the last claim of Proposition 3.20 (2). Note that c1

10.1. A general equidistribution result

209

is bounded when w ranges over any compact subset of G± X, and is uniformly bounded when Fe is bounded. Recall that 1A denotes the characteristic function of a subset A. We now define the test functions φ∓ η, R, Ω∓ : G X → [0, +∞[ with support in a dynamical ∓ neighbourhood of Ω by ∓ ± φ∓ η, R, Ω∓ = hη, R ◦ fD ∓

1Vη,±R (Ω∓ ) ,

(10.4)

± ± ∓ where Vη,±R (Ω∓ ) and fD ∓ are as in Section 2.4. Note that if ` ∈ Vη, R (Ω ), then ± `± ∈ / ∂∞ D∓ by convexity. Thus, ` belongs to the domain of definition UD±∓ of fD ∓. ∓ ∓ ± ∓ Hence φη, R, Ω∓ (`) = hη, R ◦ fD∓ (`) is well defined. By convention, φη, R, Ω∓ (`) = 0 if ` ∈ / Vη,±R (Ω∓ ).

We now globalise these test functions in order to apply them to equivariant families of supports. Let η > 0. Let D = (Di )i∈I be a locally finite Γ-equivariant family of nonempty proper closed convex subsets of X with Γ\I finite, and ∼ = ∼D . Let ∓ ± 00 1 R > 0 be such that νw (Vw, η 00 , R ) > 0 for all η > 0, i ∈ I, and w ∈ ∂∓ Di . bLet

Ω = (Ωi )i∈I be a locally finite Γ-equivariant family of measurable subsets of G X, 1 with Ωi ⊂ ∂± Di for all i ∈ I and Ωi = Ωj if i ∼ j. We define the global test ∓ e functions Φη : G X → [0, +∞[ by e∓ = Φ η

X

φ∓ η, R, Ωi =

i∈I/∼

X

± h∓ η, R ◦ fDi

1Vη,±R (Ωi ) .

(10.5)

i∈I/∼

b

b A measurable subset ∆Γ of G X is a fundamental domain for the action of Γ if the interiors of its translates are disjoint and every compact subset of G X meets only finitely many translates of ∆Γ : if mF is finite, a fundamental domain with boundary of zero measure exists by [Rob2, p. 13], using the fact that m e F has no atoms according to Corollary 4.7 (1) and Theorem 4.6. The following properties of the bump functions are proved as in [PaP14a, Prop. 18]. Lemma 10.1. (1) For every η > 0, the functions φ∓ η, R, Ω± are measurable, nonnegative, and satisfy Z ± ∓ φ∓ eF = σ eD ∓ (Ω ) . η, R, Ω± dm GX

e ∓ is well defined, measurable, Γ-invariant and (2) For every η > 0, the function Φ η defines, by passing to the quotient, a measurable function Φ∓ η : Γ\G X → [0, +∞[ such that Z ± Φ∓ (10.6) η dmF = kσΩ k . Γ\G X

210


± ± ∓ ∓ Proof. (1) Recall that the fiber of the restriction of fD ∓ to Vη, R (Ω ) over w ∈ Ω ± 0± is the open subset Vw, (w). By the disintegration result of Proposition η, R of W ∓ 7.7, by the definition of hη, R and by the choice of R, we have Z Z ± φ∓ d m e = h∓ e F (`) ∓ F η, R ◦ fD ∓ (`) dm η, R, Ω ± ∓ `∈Vη, R (Ω )

GX

Z = w∈Ω∓

h∓ η, R (w)

Z ± `∈Vw, η, R

± ± ∓ ∓ dνw (`) de σD eD ∓ (w) = σ ∓ (Ω ) .

± ± e∓ (2) The function Φ η is well defined, since Ωi = Ωj and thus Vη, R (Ωi ) = Vη, R (Ωj ) ± ± if i ∼ j, since h∓ η, R ◦ fDi (`) is finite if ` ∈ Vη, R (Ωi ) (by the definition of R), e∓ and since the sum defining Φ η has only finitely many nonzero terms, by the local e∓ finiteness of the family Ω (given ` ∈ G X, the summation over I/∼ giving Φ η (`)

may be replaced by a summation over the finite set {i ∈ I : ` ∈ Vη,±R (Ωi )}/∼). e∓ The function Φ η is Γ-invariant, since

1Vη,±R (Ωi ) ◦ γ = 1γ −1 Vη,±R (Ωi ) = 1Vη,±R (Ωγ−1 i ) and2 ± ∓ ± ∓ ± h∓ η, R ◦ fDi ◦ γ = hη, R ◦ γ ◦ fγ −1 Di = hη, R ◦ fD

γ −1 i

and by a change of index in Equation (10.5). If ∆Γ is a fundamental domain for the action of Γ, we have by Assertion (1), Z Z X Z ∓ e Φ∓ dm = Φ d m e = φ∓ eF F F η η η, R, Ωi ∩∆Γ dm Γ\G X

∆Γ

=

X

i∈I/∼

GX

± ± ± σ eD (∆Γ ∩ Ωi ) = σ eΩ (∆Γ ) = kσΩ k. i

i∈I/∼

We now state and prove the aforementioned equidistribution result. Note that since the elements of the outer unit normal bundles are only geodesic rays on [0, +∞[ , their pushforwards by the geodesic flow at time t are geodesic rays on [−t, +∞[ , and the convergence towards geodesic linesb(defined on ] − ∞, +∞[ ) takes place in the full space of generalised geodesic lines G X. This explains why it is important not to forget to consider the negative times in order for the skinning measures, supported on geodesic rays, when pushed by the geodesic flow, to have a chance to weak-star converge to Gibbs measures, supported on geodesic lines, up to renormalisation. The proof of the following result has similarities to that of [PaP14a, Thm. 1], but the computations do not apply in the present context because the proof in 2 See

Equation (2.16).

10.1. A general equidistribution result

211

loc. cit. does not keep track of the past: here we can no longer reduce our study to the outer unit normal bundle of the t-neighbourhood of the elements of D. Theorem 10.2. Let (X, Γ, Fe) be as at the beginning of Chapter 10. Assume that the Gibbs measure mF on Γ\G X is finite and mixing for the geodesic flow. Let D = (Di )i∈I be a locally finite Γ-equivariant family of nonempty proper closed b Ω = (Ωi )i∈I be a locally finite Γ-equivariant family of convex subsets of X. Let 1 measurable subsets of G X, with Ωi ⊂ ∂± Di for all i ∈ I, such that Ωi = Ωj if ± i ∼D j. Assume that σΩ is finiteband nonzero. Then as t → +∞, for the weak-star convergence of measures on Γ\ G X, 1

∗

± k(g±t )∗ σΩ k

± (g±t )∗ σΩ *

1 mF . kmF k

Proof. We give the proof only when ± = +; the other case is treated similarly. Given three numbers a, b, c (depending on some parameters), we write a = b ± c if |a − b| 6 c. Let η ∈ ]0, 1]. We may assume that Γ\I is finite, since for every > 0, there exists a Γ-invariant partition I = I 0 ∪ I 00 with Γ\I 0 finite such that if Ω0 = (Ωi )i∈I 0 + + + + + −t and Ω00 = (Ωi )i∈I 00 , then σΩ = σΩ )∗ σΩ 0 + σΩ00 with k(g 00 k = kσΩ00 k < . Hence, + − using Lemma 7.6 (i), we may fix R > 0 such that νw (Vw, η, R ) > 0 for all i ∈ I and 1 w ∈ ∂+ Di . b b Fix ψ ∈ Cc (Γ\ G X), a continuous function with compact support on Γ\ G X. Let us prove that Z Z 1 1 + t ψ dmF . lim ψ d(g )∗ σΩ = t→+∞ k(gt )∗ σ + k kmF k Γ\G X Γ\G X Ω b

Let ∆Γ be a fundamental domain for the action of Γ on G X such that the boundary b of ∆Γ has zero measure. By a standard argument of finite partition of unity and e up to modifying ∆Γ , we may assume that there exists a function ψ : G X → R b b whose support has a small neighbourhood contained in ∆Γ such that ψe = ψ ◦ p on this neighbourhood, where p : G X → Γ\ G X is the canonical projection (which is Lipschitz). Fix > 0. Since ψe is uniformly continuous, for every η > 0 small enough and for every t > 0 large enough, for all w ∈ G+ X isometric on [−t, +∞[ + and ` ∈ Vw, η, e−t R , we have e = ψ(w) e ψ(`) ± . 2

(10.7)

If t is large enough and η small enough, we have, using respectively e • the definition of the global test function Φη = Φ− η , since the support of ψ − + is contained in ∆Γ and the support of φη, R, Ωi is contained in UDi , for the second equality, + • the disintegration property of fD in Proposition 7.7 for the third equality, i

212


+ + • the fact that if ` is in the support of νρ− , then fD (g−t `) = fD (`) = ρ and i i t the change of variables by the geodesic flow w = g ρ for the fourth equality, • the fact that the support of νg−−t w is contained in W 0+ (g−t w), and that

W 0+ (g−t w) ∩ gt Vη,+R (Ωi ) = gt W 0+ (g−t w) ∩ Vη,+R (Ωi )) = gt Vg+−t w, η, R for the fifth equality, • Equation (10.7) for the sixth equality, and • the definition of h− , the invariance of the measure νg−−t w and the Gibbs + measure m e F under the geodesic flow, and the definition of the measure σΩ for the last two equalities: Z Z e η ◦ g−t dm ψ Φη ◦ g−t dmF = ψe Φ eF Γ\G X ∆Γ ∩ G X X Z −t e φ− = ψ(`) e F (`) η, R, Ωi (g `) dm i∈I/∼

=

i∈I/∼

=

Z

1D ρ∈∂+ i

+ `∈UD

1D w∈gt ∂+ i

η,R (Ωi )

1D w∈gt ∂+ i

1D w∈gt ∂+ i

+ (g−t `) dνρ− (`) de σD (ρ) i

i

+ `∈gt Vη,R (Ωi )

+ e h− (g−t w)) dν −−t (`) d(gt )∗ σ ψ(`) eD (w) η, R g w i

Z

X Z i∈I/∼

e h− (f + (g−t `))1 + ψ(`) η, R Di V

Z

X Z i∈I/∼

=

i

X Z i∈I/∼

=

+ `∈UD

X Z

`∈gt V + −t g

+ e h− (g−t w) dν −−t (`) d(gt )∗ σ ψ(`) eD (w) η, R g w i w, η, R

− + −t t + t e ψ(w) h− eD (w) η, R (g w) νg−t w (g Vg−t w, η, R ) d(g )∗ σ i

Z e η ◦ g−t dm Φ eF 2 ∆Γ ∩ G X Z X Z + e d(gt )∗ σ b ψ = Φη ◦ g−t dmF eD ± i 2 Γ\G X G X i∈I/∼ Z Z b ψ d(gt )∗ σ + ± = Φη dmF . Ω 2 Γ\G X Γ\ G X ±

(10.8)

We then conclude as at the end of theR proof of [PaP14a, Thm. 19]. By Equation + + (10.6), we have k(gt )∗ σΩ k = kσΩ k = Γ\G X Φη dmF . By the mixing property of the geodesic flow on Γ\G X for the Gibbs measure mF , for t > 0 large enough (while η is small but fixed), we hence have R b R R + ψ d(gt )∗ σΩ Φ ◦ g−t ψ dmF ψ dmF Γ\G X η Γ\G X Γ\ G X R = ± . ± = + t 2 kmF k Φ dmF k(g )∗ σΩ k Γ\G X η This proves the result.

10.2. Rate of equidistribution of equidistant level sets for manifolds

213

Recall that by Proposition 3.14, Theorem 10.2 applies to Riemannian manifolds with pinched negative curvature and for R-trees for which the geodesic flow is mixing and that satisfy the finiteness requirements of the theorem. Since pushforwards of measures are weak-star continuous and preserve total mass, we have, under the assumptions of Theorem 10.2, the following equidistribution result in X of the immersed t-neighbourhood of a properly immersed nonempty proper closed convex subset of X: as t → +∞, 1 1 ∗ + t π∗ m F . + π∗ (g )∗ σΩ * kmF k kσΩ k

10.2

(10.9)

Rate of equidistribution of equidistant level sets for manifolds

f is a Riemannian manifold and if the geodesic flow of Γ\M f is mixing If X = M with exponentially decaying correlations, we get a version of Theorem 10.2 with error bounds. See Section 9.1 for conditions on Γ and Fe that imply the exponential mixing. f be a complete simply connected Riemannian manifold with Theorem 10.3. Let M negative sectional curvature. Let Γ be a nonelementary discrete group of isometries f. Let Fe : T 1 M f → R be a bounded Γ-invariant H¨ of M older-continuous function with critical exponent δ = δΓ, F . Let D = (Di )i∈I be a locally finite Γ-equivariant f, with finite nonzero skinning family of nonempty proper closed convex subsets of M + 1 f measure σD = σD . Let M = Γ\M and let F : T M → R be the potential induced by Fe. (i) If M is compact and if the geodesic flow on T 1 M is mixing with exponential speed for the H¨ older regularity for the potential F , then there exist α ∈ ]0, 1] and κ00 > 0 such that for all ψ ∈ Ccα (T 1 M ), we have, as t → +∞, Z Z 00 1 1 ψ d(gt )∗ σD = ψ dmF + O(e−κ t kψkα ) . kσD k kmF k f is a symmetric space, if Di has smooth boundary for every i ∈ I, if mF (ii) If M is finite and smooth, and if the geodesic flow on T 1 M is mixing with exponential speed for the Sobolev regularity for the potential F , then there exist ` ∈ N and κ00 > 0 such that for all ψ ∈ Cc` (T 1 M ), we have, as t → +∞, Z Z 00 1 1 t ψ d(g )∗ σD = ψ dmF + O(e−κ t kψk` ) . kσD k kmF k f is a symmetric space and M has finite volume, then M is Note that if M geometrically finite, and Theorem 4.8 implies that mF is finite if F is small enough. f, Γ, F, D, and the speeds of mixing. The maps O(·) depend on M

214


Proof. Up to rescaling, we may assume under the assumptions of Claim (i) or (ii) that the sectional curvature is bounded from above by −1. The critical exponent δ and the Gibbs measure mF are finite under the assumptions of the theorem. Let us consider Claim (i). Under its assumptions, there exists α ∈ ]0, 1[ such that the geodesic flow on T 1 M is exponentially mixing for the Hölder regularity f is α-Hölder.3 α and such that the strong stable foliation of T 1 M First assume that Γ\I is finite. Fix R > 0 large enough and for every η > 0, let us consider the test function Φη = Φ− η as in the proof of Theorem 10.2. Up to replacing Di by N1 Di , we may assume that the boundary of Di is C 1,1 -smooth, for every i ∈ I; see, for instance, [Walt]. Fix ψ ∈ Ccα (T 1 M ). We may assume as in the proof of Theorem 10.2 that f → R of ψ whose support is contained in the interior of there exists a lift ψe : T 1 M f. There exist η0 > 0 a given fundamental domain ∆Γ for the action of Γ on T 1 M f and t0 > 0 such that for every η ∈ ]0, η0 ], for every t ∈ [t0 , +∞[ , every w ∈ T 1 M + and v ∈ Vw, η, e−t R , we have e e ψ(v) = ψ(w) + O (η + e−t )α kψkα ,

(10.10)

since d(v, w) = O(η + e−t ) by Equation (2.8) and Lemma 2.4. mF Let mF = km be the normalisation of the Gibbs measure mF to a probaFk bility measure. As in the proof of Theorem 10.2 using Equation (10.10) instead of Equation (10.7), we have R

+ ψ d(gt )∗ σD = + k(gt )∗ σD k

T 1M

R

Φη ψ ◦ gt dmF + O (η + e−t )α kψkα . Φ dmF T 1M η

T 1R M

Since M is compact, the Patterson densities and the Bowen–Margulis measure are doubling measures by Lemma 4.3 (4).4 Using discrete convolution approximation,5 there exist κ0 > 0 and, for every η > 0, a nonnegative function RΦη ∈ Ccα (T 1 M ) such that R R • T 1 M RΦη dmF = T 1 M Φη dmF , R R • T 1 M | RΦη − Φη | dmF = O(η T 1 M Φη dmF ), 0 R • k RΦη kα = O(η −κ T 1 M Φη dmF ). Hence, applying the exponential mixing of the geodesic flow, with κ > 0 as in its definition (9.1), we have, for η ∈ ]0, η0 ] and t ∈ [t0 , +∞[ , R

+ ψ d(gt )∗ σD = + k(gt )∗ σD k

T 1M

3 See

R

RΦη ψ ◦ gt dmF + O η kψk∞ + (η + e−t )α kψkα Φ dmF T 1M η

T 1M R

Section 9.1. also [PauPS, Prop. 3.12]. 5 See, for instance, [Sem, pp. 290–292] or [KinKST]. 4 See

10.3. Equidistribution of equidistant level sets on simplicial graphs

215

R Z RΦη dmF T 1M R = ψ dmF Φ dmF T 1M T 1M η k RΦη kα + O e−κt R kψkα + η kψk∞ + (η + e−t )α kψkα Φ mF T 1M η Z 0 = ψ dmF + O (e−κt η −κ + η + (η + e−t )α )kψkα . T 1M

Taking η = e−tλ for λ small enough (for instance, λ = κ/(2κ0 )), the result follows (for instance, with κ00 = min{κ/2, κ/(2κ0 ), α min{1, κ/(2κ0 )}}), when Γ\I is finite. Since the implied constants do not depend on the family D, the result holds in general. For claim (ii), the required smoothness of mF (that is, the fact that mF is absolutely continuous with respect to the Lebesgue measure with smooth Radon– Nikodym derivative) allows us to use the standard convolution approximation described, for instance, in [Zie, §1.6], instead of the operator R as above, and the proof proceeds similarly.

10.3

Equidistribution of equidistant level sets on simplicial graphs and random walks on graphs of groups

Let X, X, Γ, e c, c, Fec , Fc , δc < +∞, (µ± ec = m e Fc , mc = mFc be as at the x )x∈V X , m beginning of Section 9.2. b In this section, we state an equidistribution result analogous to Theorem 10.2, which now holds in the space of generalised discrete geodesic lines Γ\ G X, but whose proof is completely analogous. Theorem 10.4. Let X, Γ, e c, (µ± x )x∈V X be as above. Assume that the Gibbs measure mc on Γ\G X is finite and mixing for the discrete-time geodesic flow. Let D = (Di )i∈I be a locally finite Γ-equivariant family of nonempty proper simplicial b Ω = (Ωi )i∈I be a locally finite Γ-equivariant subtrees of X, and Di = |Di |1 . Let 1 family of measurable subsets of G X, with Ωi ⊂ ∂+ Di for all i ∈ I and Ωi = Ωj if + b nonzero. Then, i ∼D j. Assume that the skinning measure σΩ of Ω is finite and as n → +∞, for the weak-star convergence of measures on Γ\ G X, 1 1 ∗ + n mF . + (g )∗ σΩ * kmF k k(gn )∗ σΩ k

We leave to the reader the analogue of this result when the restriction to Γ\Geven X of the Gibbs measure is finite and mixing for the square of the discretetime geodesic flow.

216


Using Proposition 4.16 and Theorem 4.17 in order to check that the Bowen– Margulis measure mBM on Γ\G X is finite and mixing, we have the following consequence of Theorem 10.4, using the system of conductances e c = 0. Corollary 10.5. Let X be a uniform simplicial tree. Let Γ be a tree lattice of X such that the graph Γ\X is not bipartite. Let D = (Di )i∈I be a locally finite Γequivariant family of nonempty proper simplicial subtrees of X and Di = |Di |1 . b Ω = (Ωi )i∈I be a locally finite Γ-equivariant family of measurable subsets of Let 1 G X, with Ωi ⊂ ∂+ Di for all i ∈ I and Ωi = Ωj if i ∼D j. Assume that the skinning + b nonzero. Then, as n → +∞, measure σΩ (with vanishing potential) is finite and for the weak-star convergence of measures on Γ\ G X, 1 k(gn )

+ ∗ σΩ k

∗

+ (gn )∗ σΩ *

1 mBM . kmBM k

When furthermore X is regular, we have the following corollary, using Proposition 8.1 (3). Corollary 10.6. Let X be a regular simplicial tree of degree at least 3. Let Γ be a tree lattice of X such that the graph Γ\X is not bipartite. Let D = (Di )i∈I be a locally finite Γ-equivariant family of nonempty proper simplicial subtrees of X and Di = |Di |1 .bLet Ω = (Ωi )i∈I be a locally finite Γ-equivariant family of measurable 1 Di for all i ∈ I and Ωi = Ωj if i ∼D j. Assume that subsets of G X, with Ωi ⊂ ∂+ + the skinning measure σΩ (with vanishing potential) is finite and nonzero. Then, as n → +∞, for the weak-star convergence of measures on Γ\V X, 1 1 ∗ + n volΓ\\X . + π∗ (g )∗ σΩ * Vol(Γ\\X) k(gn )∗ σΩ k

Let us give an application of Corollary 10.6 in terms of random walks on graphs of groups, which might also be deduced from general results on random walks, as indicated to the third author by M. Burger and S. Mozes. Let (Y, G∗ ) be a graph of finite groups with finite volume, and let (Y0 , G0∗ ) be a connected subgraph of subgroups.6 Assume that (Y0 , G0∗ ) also has finite volume. P x| We say that (Y, G∗ ) is locally homogeneous if e∈EY, o(e)=x |G |Ge | is constant at least 3 for all x ∈ V Y. We say that a graph of groups is 2-acylindrical if the action of its fundamental group on its Bass–Serre tree is 2-acylindrical (see Remark 5.4). In particular, this action is faithful if the graph has at least two edges. The nonbacktracking simple random walk on (Y, G∗ ) starting transversally to (Y0 , G0∗ ) is the following Markovian random process (Xn = (fn , γn ))n∈N , where fn ∈ EY and γn is a double coset or right coset of Go(fn ) for all n ∈ N. Choose at random a vertex y0 of Y0 for the probability measure Vol(Y10 ,G0 ) volY0 ,G0∗ (we ∗ will call y0 the origin of the random path). Then choose uniformly at random 6 See

Section 2.6 for definitions and background.


217

X0 = (f0 , γ0 ), where f0 ∈ EY is such that o(f0 ) = y0 and γ0 is a double coset in G0y0 \Gy0 /ρ f0 (Gf0 ) such that if f0 ∈ EY0 , then γ0 ∈ / G0y0 ρ f0 (Gf0 ).7 Assuming Xn = (fn , γn ) constructed, choose uniformly at random Xn+1 = (fn+1 , γn+1 ), where fn+1 ∈ EY is such that o(fn+1 ) = t(fn ) and γn+1 ∈ Go(fn+1 ) /ρ fn+1 (Gfn+1 ) is such that if fn+1 = fn , then γn+1 ∈ / ρ fn+1 (Gfn+1 ). The nth vertex of the random process (Xn = (fn , γn ))n∈N is o(fn ). Corollary 10.7. Let (Y, G∗ ) be a locally homogeneous 2-acylindrical nonbipartite graph of finite groups with finite volume, and let (Y0 , G0∗ ) be a locally homogeneous nonempty proper connected finite subgraph of subgroups. Then the nth vertex of the nonbacktracking simple random walk on (Y, G∗ ) starting transversally to (Y0 , G0∗ ) 1 converges in distribution to Vol(Y,G volY,G∗ as n → +∞. ∗) Proof. Let Γ be the fundamental group of (Y, G∗ ) (with respect to any choice of basepoint in V Y0 ), which is a tree lattice of the Bass–Serre tree X of (Y, G∗ ), since Γ acts faithfully on X and (Y, G∗ ) has finite volume. Note that X is regular, since (Y, G∗ ) is locally homogeneous. Let p : X → Y = Γ\X be the canonical projection. Let Γ0 be the fundamental group of (Y0 , G0∗ ) (with respect to the same choice of basepoint). As seen in Section 2.6, there exists a simplicial subtree X0 whose stabiliser in Γ is Γ0 such that the quotient graph of groups Γ0 \\X0 identifies with (Y0 , G0∗ ) and the map (Γ0 \X0 ) → (Γ\X) is injective. Similarly, X0 is regular, since (Y0 , G0∗ ) is locally homogeneous. Let D = (γX0 )γ∈Γ , which is a locally finite (since Y0 is finite) Γ-equivariant family of nonempty proper simplicial subtrees of X. Using the notation of Example 2.10 for the graph of groups Γ\\X (which identifies with (Y, G∗ )), we fix lifts fe and ye in X by p of every edge f and vertex g) = t(fe). We may y of Y such that fe = fe, and elements gf ∈ Γ such that gf t(f 0 0 0 0 assume that fe ∈ EX if f ∈ EY , that ye ∈ V X if y ∈ V Y , and that gf ∈ Γ0 if f ∈ EY0 , which is possible by Equation (2.22). Let (Ω, P) be the (canonically constructed) probability space of the random walk (Xn = (fn , γn ))n∈N . For all n ∈ N, let yn = o(fn ) be the random variable (with values in the discrete space Y = Γ\X) of the nth vertex of the random process (Xn )n∈N . b b b image contained in Let us define a measurable map Ψ : Ω → Γ\ G X, with 1 0 the image of ∂+ X by the canonical projection G X → Γ\ G X, such that Ψ∗ P is

the normalised skinning measure all n ∈ N:

+ σD + kσD k

b / Γ\ G X

Ψ

Ω yn

{ Y

7 This

and the following diagram commutes for

π◦gn

.

last condition says that γ0 is not the double coset of the trivial element.

(10.11)

218


Assuming that we have such a map, we have + σD 1 + n = + + π∗ (g )∗ σD , kσD k k(gn )∗ σD k

(yn )∗ P = (π∗ ◦ (gn )∗ ◦ Ψ∗ )P = π∗ (gn )∗ so that the convergence of the law of yn to 1 10.6 applied to Ω = (∂+ Di )i∈I . y0

f0

1 Vol(Y,G∗ )

yn

fn

volY,G∗ follows from Corollary yn+1

fn+1

yn+2

Y gf

p

−1

fe0

gfn ff n gfn−1 g fn g f0

X

g f −1 ff n n

yf n g f −1 fe0 0

ye0

−1

n+1

f] n+1 γn+1

y] n+1

γn αn+1

αn

γ0

e0

en+1

en

Let (Xn = (fn , γn ))n∈N be a random path with origin y0 ∈ Y0 , corresponding to ω ∈ Ω. Fix a representative of γn in its right class for every n > 1, and a representative of γ0 in its double class, which we still denote by γn and γ0 respectively. Using ideas introduced for the coding in Section 5.2, let us construct by induction an infinite geodesic edge path (en )n∈N with origin o(e0 ) = ye0 and a sequence (αn )n∈N in Γ such that en = αn γn g f

−1

n

ff n .

(10.12)

] Let α0 = id and e0 = γ0 g f −1 fe0 . Since o(fe0 ) = g f0 o(f 0 ) = g f0 ye0 by the 0 construction of the lifts and since γ0 ∈ Gy0 = Γye0 , we have o(e0 ) = ye0 . Since the stabiliser of g f −1 fe0 is ρ f0 (Gf0 ), the edge e0 does not depend on the choice of 0 the representative γ0 modulo ρ f0 (Gf0 ) on the right, but depends on the choice of the representative γ0 modulo G0y0 = Γ0ye0 on the left. The hypothesis that if f0 ∈ EY0 then γ0 ∈ / G0y0 ρ f0 (Gf0 ) ensures that the 0 edge e0 does not belong to EX . Indeed, assume otherwise that e0 belongs to EX0 . Then f0 = p(e0 ) ∈ EY0 , and by the assumptions on the choice of lifts, the edges g f −1 fe0 and e0 both belong to EX0 . Since they are both mapped to f0 by 0 the map X0 → Y0 = Γ0 \X0 , and by Equation (2.22), they are mapped one to the other by an element of Γ0ye0 = G0y0 . Let γ00 ∈ Γ0ye0 be such that γ00 e0 = g f −1 fe0 . 0 −1 Then γ 0 γ0 belongs to the stabiliser in Γ of the edge g −1 fe0 , which is equal to 0

gf

0

−1

f0

Γfe0 g f0 = ρ f0 (Gf0 ). Therefore γ0 ∈ G0y0 ρ f0 (Gf0 ), a contradiction.


219

Assume by induction that en and αn are constructed. Define αn+1 = αn γn g f

−1

n

gfn

and en+1 = αn+1 γn+1 g f

−1

n+1

f] n+1 ,

so that the induction formula (10.12) at rank n+1 is satisfied. By the construction of the lifts, since yn+1 = t(fn ) = o(fn+1 ), we have −1 f y] n+1 = gfn t(fn ) = g f

−1

n+1

o(f] n+1 ) .

Hence, since γn+1 ∈ Gyn+1 = Γy] fixes y] n+1 , using the induction formula (10.12) n+1 at rank n for the last equality, we obtain o(en+1 ) = αn+1 γn+1 g f

−1

n+1

o(f] ] ] n+1 ) = αn+1 γn+1 y n+1 = αn+1 y n+1

= αn+1 gf−1 t(ff n ) = αn γn g f n

−1

n

t(ff n ) = t(en ) .

In particular, the sequence (en )n∈N is an edge path in X. Since the stabiliser of g −1 f] n+1 is fn+1

gf

−1

n+1

the edge γn+1 g f

−1

n+1

Γf] g = ρ fn+1 (Gfn+1 ) , n+1 fn+1

f] n+1 does not depend on the choice of the representative

−1 ] of the right coset γn+1 . If the length-2 edge path (gf−1 ff fn+1 ) is n , γn+1 g f n n+1 not geodesic, then the two edges of this path are opposite one to another; hence fn+1 = fn using the projection p : X → Y, and therefore g fn+1 = gfn . Thus −1 f γn+1 maps gf−1 ff n to gfn fn , hence belongs to ρfn (Gfn ) = ρ fn+1 (Gfn+1 ), a n contradiction by the assumptions on the random walk. By construction, the element αn+1 of Γ sends the above length-2 geodesic −1 ] edge path (gf−1 ff fn+1 ) to (en , en+1 ). This implies on the one n , γn+1 g f n n+1

hand that the edge path (en , en+1 ) is geodesic, and on the other hand that αn+1 is uniquely defined, since the action of Γ on X is 2-acylindrical. In particular, (en )n∈N is the sequence of edges followed by a (discrete) geodesic ray in X, starting from a point of X0 but not by an edge of X0 , that is, 1 0 an element of ∂+ X . Furthermore, this ray is well defined up to the baction of Γ0ye0 ; hence its image, which we denote by Ψ(ω), is well defined in Γ\ G X. Since p(o(en )) = p(f yn ) = yn for all n ∈ N, the commutativity of the diagram (10.11) is immediate.

220


1 0 1 0 For every x ∈ V X0 , let ∂+ X (x) be the subset of ∂+ X consisting of the elements w with w(0) = x. By construction, the above map from the subset of 1 0 random paths in Ω starting from y0 to the set Γ0ye0 \∂+ X (ye0 ), which associates to 0 (Xn )n∈N the Γye0 -orbit of the geodesic ray with consecutive edges (en )n∈N , is clearly a bijection. This bijection maps the measure P to the normalised skinning measure + σD + , kσD k

1 0 since by homogeneity, the restriction of σ eX+0 to ∂+ X (ye0 ), normalised to be

1 0 a probability measure, is the restriction to ∂+ X (ye0 ) of the Aut(X)x -homogeneous probability measure on the space of geodesic rays with origin x in the regular tree X. This proves the result.

When Y is finite, all the groups Gy for y ∈ V Y are trivial and Y0 is reduced to a vertex,8 the above random walk is the nonbacktracking simple random walk 1 on the nonbipartite regular finite graph Y, and Vol(Y,G volY,G∗ is the uniform ∗) distribution on V Y. Hence this result (as well as Corollary 1.3 in the introduction) is classical. See, for instance, [OW, Thm. 1.2] and [AloBLS], which under further assumptions on the spectral properties of Y give precise rates of convergence, and also the book [LyP2], including its Section 6.3 and its references.

10.4

Rate of equidistribution for metric and simplicial trees

In this section, we give error terms for the equidistribution results stated in Theorem 10.2 for metric trees, and in Theorem 10.4 for simplicial trees, under additional assumptions required in order to get the error terms for the mixing property discussed in Chapter 9. We first consider the simplicial case, for the discrete-time geodesic flow. Let X, X, Γ, e c, c, Fec , Fc , δc < +∞, (µ± e c , mc be as at the beginning of x )x∈V X , m Section 9.2. Theorem 10.8. Assume that δc is finite and that the Gibbs measure mc on Γ\G X is finite. Assume furthermore that + (1) the families (ΛΓ, µ− x , dx )x∈V C ΛΓ and (ΛΓ, µx , dx )x∈V C ΛΓ of metric measure 9 spaces are uniformly doubling, (2) there exists α ∈ ]0, 1] such that the discrete-time geodesic flow on (Γ\G X, mc ) is exponentially mixing for the α-H¨ older regularity.

Let D = (Di )i∈I be a locally finite Γ-equivariant family of nonempty proper simplicial subtrees of X with Γ\I finite.b Let Ω = (Ωi )i∈I be a locally finite Γ-equivariant 1 family of measurable subsets of G X, with Ωi ⊂ ∂± Di for all i ∈ I and Ωi = Ωj if ± i ∼D j. Assume that σΩ is finite and positive. Then there exists κ0 > 0 such that 8 The 9 See

result for general Y0 follows by averaging. Section 4.1 for definitions.

10.4. Rate of equidistribution for metric and simplicial trees

221

b

for all ψ ∈

Ccα (Γ\ G X),

1 ± ±n k(g )∗ σΩ k

Z

we have, as n → +∞,

± ψ d (g±n )∗ σΩ =

1 kmc k

Z

0

ψ d mc + O(kψkα e−κ n ) .

Remarks. (1) If e c = 0, if the simplicial subtree X0 of X satisfying |X0 |1 = C ΛΓ is uniform, if LΓ = Z, and if Γ is a tree lattice of X0 , then we claim that δc = δΓ is finite, mc = mBM is finite and mixing, and (ΛΓ, µx = µ± x , dx )x∈V C ΛΓ is uniformly doubling. Indeed, the above finiteness and mixing properties follow from the results of Section 4.4. Since X0 is uniform, it has a cocompact discrete group of isometries Γ0 whose Patterson density (for the vanishing potential) is uniformly doubling on ΛΓ0 = ΛΓ, by Lemma 4.3 (4). Since e c = 0 and Γ is a tree lattice, the Patterson densities of Γ and Γ0 coincide (up a scalar multiple) by Proposition 4.16 (2). (2) Assume that e c = 0, that the simplicial subtree X0 of X satisfying |X0 |1 = C ΛΓ is uniform without vertices of degree 2, that LΓ = Z, and that Γ is a geometrically finite tree lattice of X. Then all assumptions of Theorem 10.8 are satisfied by the first remark and by Corollary 9.6. Therefore we have an exponentially small error term in the equidistribution of the equidistant levels sets. Proof. We give the proof only when ± = +; the other case is treated similarly. We follow the proof of Theorem 10.2, concentrating on the newbfeatures. We now have Γ\I finite by assumption. Let η ∈ ]0, 1[ and ψ ∈ Ccα (Γ\ G X). We consider the constant R > 0, the test function Φη , the fundamental domain ∆Γ , and the lift ψe of ψ as in the proof of Theorem 10.2. + For all n ∈ N, all w ∈ G+ X isometric on [−n, +∞[ , and all ` ∈ Vw, η, e−n R = + −n 10 B (w, e R), by Lemma 2.7, where we can take η = 0, we have d(`, w) = O(e−n ). Since p is Lipschitz, the map ψe is α-Hölder-continuous with α-Hölder norm at most kψkα . Hence for all n ∈ N, all w ∈ G+ X isometric on [−n, +∞[ , + and all ` ∈ Vw, η, e−n R , we have e = ψ(w) e ψ(`) + O(e−n α kψkα ) .

(10.13)

As in the proof of Theorem 10.2 with t replaced by n, using Equation (10.13) instead of Equation (10.7) in the series of Equations (10.8), since the symbols w that appear in them are indeed generalised geodesic lines isometric on [−n, +∞[ , we have R b R + ψ d(gn )∗ σΩ ψ Φη ◦ g−n dmc Γ\ G X Γ\G X R = + O(e−n α kψkα ) . (10.14) + Φ dm k(gn )∗ σΩ k η c Γ\G X Let us now apply the assumption on the decay of correlations. In order to do that, we need to regularise our test functions Φη . + + stated in Section 2.6, the subsets Vw, η, s and B (w, s) of the space of discrete geodesic lines G X are equal for every s > 0 since X is simplicial and η < 1. 10 As

222


By the definition of the Gibbs measures,11 Lemma 3.3 implies that, for all > 0 small enough and ` ∈ G X, 1 √ 1 √ + µ− B `(−∞), µ B `(+∞), d d `(0) `(0) `(0) `(0) c0 c0 6m e c (Bd (`, )) 6 µ− `(0) Bd`(0) (`(−∞), c0

√ + √ ) µ`(0) Bd`(0) (`(+∞), c0 ) .

Since the Patterson densities are uniformly doubling for basepoints in C ΛΓ, since the footpoints of the geodesic lines in the support of m e c belong to C ΛΓ, the Gibbs mc measure mc is hence doubling on its support. Let mc = km . As in the proof of ck Theorem 10.3, using discrete convolution approximation, there exist κ00 > 0 and, for every η > 0, a nonnegative function RΦη ∈ Ccα (Γ\G X) such that R R (1) Γ\G X RΦη d mc = Γ\G X Φη d mc , R R (2) Γ\G X | RΦη − Φη | d mc = O η Γ\G X Φη d mc , 00 R (3) k RΦη kα = O η −κ Γ\G X Φη d mc . R + By Equation (10.6), the integral Γ\G X Φη dmc = kσΩ k is constant (in particular independent of η). All integrals below besides the first one being over Γ\G X, and using • Equation (10.14) and the above property (2) of the regularised map RΦη for the first equality, • the assumption of exponential decay of correlations for the second one, involving some constant κ > 0, for the second equality, • the above properties (1) and (3) of the regularised map RΦη for the last equality, we hence have R b + ψ d(gn )∗ σΩ Γ\ G X + k(gn )∗ σΩ k

ψ RΦη ◦ g−n d mc R + O(e−n α kψkα + η kψk∞ ) Φη d mc R R RΦη d mc ψ d mc R + O(e−n α kψkα + η kψk∞ = Φη d mc 1 +R e−κ n k RΦη kα kψkα ) Φη d mc Z 00 = ψ d mc + O (e−n α + η + e−κ n η −κ )kψkα . R

=

Taking η = e−λ n with λ = 11 See

κ 2κ00 ,

the result follows with κ0 = min{α, 2κκ00 , κ2 }.

Equation (4.4), using x0 = `(0) as the basepoint for the Hopf parametrisation, and the fact that if > 0 is small enough and `0 ∈ Bd (`, ), then `0 (0) = `(0), as seen in the proof of Lemma 3.3.

10.4. Rate of equidistribution for metric and simplicial trees

223

Let us now consider the metric tree case, for the continuous-time geodesic flow, where the main change is to assume a superpolynomial decay of correlations and hence get a superpolynomial error term, for observables that are smooth enough along the flow lines. Let (X, λ), X, Γ, Fe, δ = δΓ, F ± < ∞, (µ± eF, x )x∈X , m and mF be as at the beginning of Section 9.3. Theorem 10.9. Assume that the Gibbs measure mF on Γ\G X is finite. Assume furthermore that + (1) the families (ΛΓ, µ− x , dx )x∈C ΛΓ and (ΛΓ, µx , dx )x∈C ΛΓ of metric measure 12 e spaces are uniformly doubling, and F is bounded on T 1 C ΛΓ, (2) there exists α ∈ ]0, 1] such that the (continuous-time) geodesic flow on (Γ\G X, mF ) has superpolynomial decay of α-H¨ older correlations.

Let D = (Di )i∈I be a locally finite Γ-equivariant family of nonempty proper b Ω = (Ωi )i∈I be a locally finite closed convex subsets of X with Γ\I finite. Let 1 Γ-equivariant family of measurable subsets of G X, with Ωi ⊂ ∂± Di for all i ∈ I ± b Then for evand Ωi = Ωj if i ∼D j. Assume that σΩ is finite and nonzero. ery n ∈ N, there exists k ∈ N such that for all ψ ∈ Cck, α (Γ\ G X), we have, as t → +∞, Z Z 1 1 ± ±t ψ d (g ) σ = ψ d mF + O(kψkk, α t−n ) . ∗ Ω ± kmF k k(g±t )∗ σΩ k Remarks. (1) If F = 0, if the metric subtree X 0 = C ΛΓ of X is uniform, if the length spectrum of Γ on X is not contained in a discrete subgroup of R, and if Γ is a tree lattice of X 0 , then we claim that δ = δΓ is finite, mF = mBM is finite and mixing, and (ΛΓ, µx = µ± x , dx )x∈X 0 is uniformly doubling. Indeed, the above finiteness and mixing properties follow from Proposition 4.16 and Theorem 4.9. Since X 0 is uniform, it has a cocompact discrete subgroup of isometries Γ0 whose Patterson density (for the vanishing potential) is uniformly doubling on ΛΓ0 = ΛΓ, by Lemma 4.3 (4). Since F = 0 and Γ is a tree lattice, the Patterson densities of Γ and of Γ0 coincide (up to a scalar multiple) by Proposition 4.16. (2) Assume that F = 0, that the metric subtree X 0 = C ΛΓ of X is uniform, that the length spectrum of Γ on X is 4-Diophantine, and that Γ is a geometrically finite tree lattice of X 0 . Then all assumptions of Theorem 10.9 are satisfied by the first remark and by Corollary 9.10. Therefore we have a superpolynomially small error term in the equidistribution of the equidistant level sets. Proof. The proof is similar to that of Theorem 10.8, except for the doubling property of the Gibbs measure on its support and the conclusion of the proof. Let X 0 = C ΛΓ. The modification of Lemma 3.3 used in the previous proof is now the third assertion of Lemma 3.4. 12 See

Section 4.1 for definitions.

224


If the footpoints of `, `0 ∈ G X 0 are at distance bounded by c0 0 , then by Proposition 3.20 (2), since |Fe| is bounded on T 1 X 0 by assumption, the quantities |Cξ± (`(0), `0 (0))| for ξ ∈ ΛΓ are bounded by c00 = c0 0 (maxT 1 X 0 |Fe − δ|). By the definition of the Gibbs measures (see Equation (4.4)), Assertion (3) of Lemma 3.4 hence implies that if 6 0 , then for every ` ∈ G X, 0 1 √ 1 √ + e−2c0 µ− B `(−∞), µ B `(+∞), d`(0) d`(0) `(0) `(0) c0 c0 6m e c (Bd (`, )) 0

6 e2c0 µ− `(0) Bd`(0) (`(−∞), c0

√ + √ ) µ`(0) Bd`(0) (`(+∞), c0 ) .

As in the simplicial case, since the Patterson densities are uniformly doubling for basepoints in X 0 , the Gibbs measure mc is hence doubling on its support. Fix n ∈ N. As at the end of the proof of Theorem 10.8, using the assumption of superpolynomial decay of correlations, involving some degree of regularity k in order to have polynomial decay in t−N n , where N = bdκ00 e + 1, instead of the exponential one, we have, for all t > 1 and ψ ∈ Cck,α (Γ\ G X), R b + Z ψ d(gt )∗ σΩ 00 Γ\ G X = ψ d mc + O (e−t α + η + t−N n η −κ )kψkk, α . + t k(g )∗ σΩ k Taking η = t−n , by the definition of N , we hence have R b + Z ψ d(gt )∗ σΩ Γ\ G X = ψ d mc + O t−n kψkk, α . + t k(g )∗ σΩ k This proves Theorem 10.9.

Chapter 11

Equidistribution of Common Perpendicular Arcs In this chapter, we prove the equidistribution of the initial and terminal vectors of common perpendiculars of convex subsets, at the universal covering space level, for Riemannian manifolds and for metric and simplicial trees. The results generalise [PaP17b, Thm. 8]. From now until the end of Section 11.3, we consider the continuous-time situation in which X is a proper CAT(−1) space that is either an R-tree without terminal point or a complete Riemannian manifold with pinched negative curvature at most −1. In Section 11.4, X will be the geometric realisation of a simplicial tree X, and we will consider the discrete-time geodesic flow. Let Γ be a nonelementary discrete group of isometries of X. Let x0 be any basepoint in X. Let Fe be a continuous Γ-invariant potential on T 1 X, which is H¨ older-continuous if X is a manifold. Assume that δ = δΓ, F ± is finite and positive ± and let (µ± x )x∈X be (normalised) Patterson densities for the pairs (Γ, F ), with − + − + associated Gibbs measure mF . Let D = (Di )i∈I − and D = (Dj )j∈I + be locally finite Γ-equivariant families of nonempty proper closed convex subsets of X. For every (i, j) in I − × I + such that the closures Di− and Dj+ of Di− and Dj+ in X ∪ ∂∞ X have empty intersection, let λi,j = d(Di−b , Dj+ ) be the length of the

− common perpendicular from Di− to Dj+ , and let αi, j ∈ G X be its parametrisation: it is the unique map from R to X such that − − − • αi, j (t) = αi, j (0) ∈ Di if t 6 0, − − + • αi, j (t) = αi, j (λi, j ) ∈ Dj if t > λi, j , and − • αi,j |[0, λi,j ] = αi, j is the shortest geodesic arc starting from a point of Di− and ending at a point of Dj+ . + − Let αi,j = gλi,j αi,j . In particular, we have g

λi,j 2

− αi,j =g

−λi,j 2

+ αi,j .


225

226

Chapter 11. Equidistribution of Common Perpendicular Arcs

We now state our main equidistribution result of common perpendiculars between convex subsets in the continuous-time and upstairs settings. We will give the discrete-time version in Section 11.4, and the downstairs version in Chapter 12. Theorem 11.1. Let X be either a proper R-tree without terminal points or a complete simply connected Riemannian manifold with pinched negative curvature at most −1. Let Γ be a nonelementary discrete group of isometries of X and let Fe be a bounded Γ-invariant potential on X that is H¨ older-continuous if X is a manifold. Let D ± = (Dk± )k∈I ± be locally finite Γ-equivariant families of nonempty proper closed convex subsets of X. Assume that the critical exponent δ is positive,1 and that the Gibbs measure mF is finite and mixing for the geodesic flow on Γ\G X. Then lim

t→+∞

X

δ kmF k e−δ t

R

e

αi,γj

e F

∆α−

i, γj

−

+

⊗ ∆α+

γ −1 i, j

+ − = σ eD eD − ⊗σ +

i∈I /∼ , j∈I /∼ , γ∈Γ − + D i ∩ D γj =∅, λi, γj 6t

b

b

for the weak-star convergence of measures on the locally compact space G X × G X. The proof of Theorem 11.1 follows that of [PaP17b, Thm. 8], which proves this result when X is a manifold and F = 0. The first two and a half steps work for both trees and manifolds and are given in Section 11.1. The differences begin in Step 3T. After this, the steps for trees are called 3T and 4T and are given in Section 11.2 and the corresponding steps for manifolds are 3M and 4M, given in Section 11.3. In the special case of D − = (γx)γ∈Γ and D + = (γy)γ∈Γ for some x, y ∈ X, this statement,2 with Equation (7.2), gives the following version with potentials of Roblin’s double equidistribution theorem [Rob2, Thm. 4.1.1] when F = 0; see [PauPS, Thm. 9.1] for general F when X is a Riemannian manifold with pinched sectional curvature at most −1. Corollary 11.2. Let X, Γ, Fe, δ, mF be as in Theorem 11.1, and let x, y ∈ X. We have R γy X e − lim δ kmF k e−δ t e x F ∆γy ⊗ ∆γ −1 x = µ+ x ⊗ µy t→+∞

γ∈Γ : d(x,γy)6t

for the weak-star convergence of measures on (X ∪ ∂∞ X) × (X ∪ ∂∞ X). Let us give a version of Theorem 11.1 without the assumption δ > 0. 1 It

is finite, since Fe is bounded; see Lemma 3.17 (6). rather the following Equation (11.1).

2 Or


227

Theorem 11.3. Let X, Γ, Fe, δ, mF be as in Theorem 11.1, except that the critical exponent δ is not assumed to be positive. Then for every τ > 0, we have lim

t→+∞

δ kmF k −δ t e 1 − e−τ δ

R

X

e

αi,γj

e F

∆α− ⊗∆α+ i, γj

i∈I − /∼ , j∈I + /∼ , γ∈Γ − + D i ∩ D γj =∅, t−τ 0 and, as t → +∞, X

eκ f (i) g(i) ∼

i∈I, f (i)6t

e(δ+κ)t , δ+κ

then for every c > 0, as t → +∞, X

g(i) ∼

i∈I, t−c 0 be such that δΓ, F +κ = δΓ, F + κ > 0. Since the definition of the Gibbs measure involves only the normalised potential, we have kmF +κ k = kmF k. Thus, the statement of Theorem 11.1 for the potential F + κ is equivalent to the claim that as t → +∞, R X e F − + eκλi, γj e αi,γj ψ(αi, γj , αγ −1 i, j ) i∈I − /∼ , j∈I + /∼ , γ∈Γ − + D i ∩ D γj =∅, λi, γj 6t

e(δ+κ) t ∼ (δ + κ) kmF k

Z

+ − ψ d(e σD eD − ⊗ σ +) b b

for all nonzero nonnegative functions ψ ∈ Cc (G X × G X). Since the function (i, j, γ) 7→ λi,γj is proper by the local finiteness of the families D − and D + , Lemma 11.4 implies that for every τ > 0, as t → +∞, R X e F − + e αi,γj ψ(αi, γj , αγ −1 i, j ) i∈I − /∼ , j∈I + /∼ , γ∈Γ + D γj =∅, t−τ 0, let X Z −t/2 t/2 −1 aη (t) = φ− `) φ+ γ `) dm e F (`) . (11.5) η (g η (g γ∈Γ

`∈G X

As in [PaP17b], the heart of the proof is to give two pairs of upper and lower bounds, when T > 0 is large enough and η ∈ ]0, 1] is small enough, of the (Cesàrotype) quantity Z T iη (T ) = eδ t aη (t) dt . (11.6) 0

11.1. Part I of the proof of Theorem 11.1: The common part

229

By passing to the universal cover, the mixing property of the geodesic flow on Γ\G X for the Gibbs measure mF gives that for every > 0, there exists T = T,η > 0 such that for all t > T , we have Z Z Z Z e− e + − φ− d m e φ d m e 6 a (t) 6 φ d m e φ+ eF . F F η F η η η dm kmF k G X η km k F GX GX GX Hence by Lemma 10.1 (1), for all > 0 and η ∈ ]0, 1], there exists c = c,η > 0 such that for every T > 0, we have e−

eδ T eδ T σ e+ (Ω− ) σ e− (Ω+ ) − c 6 iη (T ) 6 e σ e+ (Ω− ) σ e− (Ω+ ) + c . δ kmF k δ kmF k (11.7)

Step 3: Second upper and lower bounds. Let T > 0 and η ∈ ]0, 1]. By Fubini’s theorem for nonnegative measurable maps, the definition3 of the test functions φ± η ± and the flow-invariance4 of the fibrations fD ∓ , we have iη (T ) =

X Z γ∈Γ

T

eδ t

0

Z `∈G X

+ + − −1 h− `) η, R ◦ fD − (`) hη, R ◦ fD + (γ

× 1gt/2 V +

− η, R (Ω )

(`) 1g−t/2 V −

+ η, R (γΩ )

(11.8) (`) dm e F dt .

We begin the computations by rewriting the product term involving the functions + − h± η, R . For all γ ∈ Γ and ` ∈ UD − ∩ UγD + , define (using Equation (2.16)) + − − −1 w − = fD and w+ = fγD `) ∈ G−, 0 X . − (`) ∈ G+, 0 X + (`) = γfD + (γ

(11.9)

This notation is ambiguous (w− depends on `, and w+ depends on ` and γ), but it makes the computations less heavy. By Equations (10.2) and (3.20), we have, for every t > 0, R w− (t/2)

− − −t/2 t/2 − h− (g w ) = e η, R (w ) = hη, R ◦ g

w− (0)

e−δ) (F

h− (gt/2 w− ) . η, e−t/2 R

Similarly, −1 + h+ w )=e η, R (γ

R w+ (0) w+ (−t/2)

e−δ) (F

h+ (g−t/2 w+ ) . η, e−t/2 R

Hence, + + − −1 h− `) η, R ◦ fD − (`) hη, R ◦ fD + (γ

= e−δ t e 3 See 4 See


R w− (t/2) w− (0)

(11.10)

R + e+ w (0) F + w

(−t/2)

e F

h− (gt/2 w− ) h+ (g−t/2 w+ ) . η, e−t/2 R η, e−t/2 R

230


11.2

Part II of the proof of Theorem 11.1: The metric tree case

In this section, we assume that X is an R-tree, and we will consider the manifold case separately in Section 11.3. Step 3T. Consider the product term in Equation (11.8) involving the characteristic functions. By Lemma 2.8 (applied by replacing D+ by γD+ ), there exists t0 > 2 ln R + 4 such that for all η ∈ ]0, 1], t > t0 , γ ∈ Γ, and ` ∈ G X, if 1gt/2 V + (Ω− ) (`) 1g−t/2 V − (γΩ+ ) (`) 6= 0, or equivalently by Equation (2.20) if η, R

η, R

` ∈ Vη,+e−t/2 R (gt/2 Ω− ) ∩ Vη,−e−t/2 R (γg−t/2 Ω+ ) , then the following facts hold. − (i) By the convexity of D± , we have ` ∈ UD+− ∩ UγD +.

(ii) By the definition5 of w± , we have w− ∈ Ω− and w+ ∈ γΩ+ . The notation (w− , w+ ) here coincides with the notation (w− , w+ ) in Lemma 2.8. (iii) There exists a common perpendicular αγ from D− to γD+ , whose length λγ satisfies | λγ − t | 6 2η , (11.11) whose origin is αγ− (0) = w− (0), whose endpoint is γ αγ+ (0) = w+ (0), such that the points w− ( 2t ) and w+ (− 2t ) are at distance at most η from `(0) ∈ αγ . See the picture following Lemma 2.8 (replacing D+ by γD+ ). Hence, by Lemma 3.12 and since Fe is bounded, e−2η kF k∞ e

R αγ

e F

R w− (t/2)

6e

w− (0)

R + e+ w (0) F + w

(−t/2)

e F

6 e2η kF k∞ e

R αγ

e F

.

(11.12)

For all η ∈ ]0, 1], γ ∈ Γ, and T > t0 , let Aη,γ (T ) = (t, `) ∈ [t0 , T ] × G X : ` ∈ Vη,+e−t/2 R (gt/2 Ω− ) ∩ Vη,−e−t/2 R (γg−t/2 Ω+ ) and ZZ jη, γ (T ) = (t, `)∈Aη,γ (T )

h− (gt/2 w− ) h+ (g−t/2 w+ ) dt dm e F (`) . η, e−t/2 R η, e−t/2 R

By the above, since the integral of a function is equal to the integral on any Borel set containing its support, and since the integral of a nonnegative function 5 See

equation (11.9).

11.2. Part II of the proof of Theorem 11.1: The metric tree case

231

is nondecreasing in the integration domain, there hence exists c4 > 0 such that for all T > 0 and η ∈ ]0, 1], we have R X e F iη (T ) > − c4 + e−2ηkF k∞ e αγ jη, γ (T ) , γ∈Γ : t0 +26λγ 6T −2η − + + α− γ |[0, λγ ] ∈Ω |[0, λγ ] , αγ |[−λγ , 0] ∈Ω |[−λγ , 0]

and similarly, for every T 0 > T (later on, we will take T 0 to be T + 4η), R X e F iη (T ) 6 c4 + e2η kF k∞ e αγ jη, γ (T 0 ) . γ∈Γ : t0 +26λγ 6T +2η − + + α− γ |[0, λγ ] ∈Ω |[0, λγ ] , αγ |[−λγ , 0] ∈Ω |[−λγ , 0]

Step 4T: Conclusion. Let > 0. Let γ ∈ Γ be such that D− and γD+ do not intersect and the length of their common perpendicular satisfies λγ > t0 +2. Let us prove that if η is small enough and λγ is large enough,6 then for every T > λγ +2η, we have 1 − 6 jη, γ (T ) 6 1 + . (11.13) This estimate proves the claim (11.4), as follows. For every > 0, if η > 0 is small enough, we have I − + (T ) IΩ− ,Ω+ (t0 + 2) Ω ,Ω iη (T + 2η) > −c4 + e−2ηkF k∞ (1 − ) − δ kmF ke−δ T δ kmF ke−δ(t0 +2) and by Equation (11.7), iη (T + 2η) 6 c +

e σ e+ (Ω− ) σ e− (Ω+ ) . δ kmF k e−δ(T +2η)

Thus, for η small enough, σ e+ (Ω− ) σ e− (Ω+ ) >

1− I − + (T ) + o(1) e2 Ω ,Ω

as T → +∞, which gives lim sup IΩ− ,Ω+ (T ) 6 σ e+ (Ω− ) σ e− (Ω+ ). T →+∞

A similar estimate for the lower limit proves the claim (11.4). In order to prove the claim (11.13), let η ∈ ]0, 1] and T > λγ + 2η. In order to simplify the notation, let rt = e−t/2 R, wt− = gt/2 w− , and wt+ = g−t/2 w+ . By the definition of jη, γ , using the inequalities (10.3) (with the comment following them), where the constant c1 is uniform, since Fe is bounded, and the fact that 6 Here

the “enough”s are independent of γ.

232


rt = O(e−λγ /2 ) by Equation (11.11), we hence have ZZ − + + jη, γ (T ) = h− e F (`) η, rt (wt ) hη, rt (wt ) dt dm

(11.14)

(t,`)∈Aη,γ (T )

−λγ /2

=

eO(e (2η)2

) ZZ

dt dm e F (`) . rt )) µW − (w+ ) (B − (wt+ , rt ))

− + (t, `)∈Aη,γ (T ) µW + (w− ) (B (wt , t

t

Let xγ be the midpoint of the common perpendicular αγ , so that d(xγ , `(0)) = O(η) for every (t, `) ∈ Aη, γ (T ) by the above claim (iii). Let us use the Hopf parametrisation of G X with basepoint xγ , denoting by s its time parameter. When (t, `) ∈ Aη,γ (T ), by Definition (4.4) of the Gibbs measure m e F , by the R-tree case of Proposition 3.20 (2), and since Fe is bounded, we have dm e F (`) = e

C`− (xγ , `(0)) + C`+ (xγ , `(0)) −

+

+ dµ− xγ (`− ) dµxγ (`+ ) ds

+ = eO(η) dµ− xγ (`− ) dµxγ (`+ ) ds .

(11.15)

Let Pγ be the plane domain of the (t, s) ∈ R2 such that |λγ − t| 6 2η and λ −t there exist s± ∈ ] − η, η[ with s∓ = γ2 ± s. It is easy to see that Pγ is a rhombus centered at (λγ , 0) whose area is (2η)2 . s 2η

111 000 0000 1111 00000 11111 000 111 0000 1111 00000 11111 00000 11111 000 111 0000 1111 00000 11111 000 111 0000 1111 000 111 0000 1111 Pγ

λγ − 2η −2η

λγ + 2η

t

Let ξγ− (resp. ξγ+ ) be the point at infinity of any fixed geodesic ray from xγ through αγ− (0) (resp. γαγ+ (0)). If A is a subset of G X, we denote by A± the subset {`± : ` ∈ A} of ∂∞ X. Lemma 11.5. For every t > t0 such that |λγ − t| 6 2η, we have (B ± (wt∓ , rt ))∓ = Bdxγ (ξγ∓ , R e−

λγ 2

).

Proof. We prove the statement for the negative endpoints; the proof of the claim λ for positive endpoints is similar. Since R > 1 and d(xγ , αγ− (0)) = 2γ , the term on − the right-hand side does not depend on the choice of ξγ . Let us first prove the inclusion of the set on the left-hand side in the set on the right-hand side. Let `0 ∈ B + (wt− , rt ), so that7 there exists a geodesic 7 See

the definition of the strong stable ball B + (wt− , rt ) in Section 2.3.

11.2. Part II of the proof of Theorem 11.1: The metric tree case

233

line w bt− ∈ G X, an extension of the geodesic ray wt− : [− 2t , +∞[ → X, with dW + (w− ) (w bt− , `0 ) < rt . We may assume that (w bt− )− = ξγ− and `0− 6= ξγ− . Let p ∈ X t be such that [`0 (0), ξγ− [ ∩ [`0 (0), `0− [ = [`0 (0), p]. `0− p ξγ−

−

w (0) =

xγ

αγ− (0)

w− ( 2t )

0

= ` (0)

Since t > t0 > 2 ln R, we have rt < 1, hence8 `0 (0) = wt− (0) = w− ( 12 ) and p ∈ ]`0 (0), ξγ− [ . Since 1 t t0 d p, w− = − ln dW + (w− ) (w bt− , `0 ) > − ln rt = − ln R > − ln R > 1 > η t 2 2 2 λ and d(xγ , w− ( 12 )) = 2γ − 2t 6 η, we have p ∈ [xγ , ξγ− [ . Hence dxγ (`0− , ξγ− ) = e−d(p, xγ ) 6 q

n e−d(p, w− ( 2t ))−d(w− ( 12 ), xγ ) − 1 − t e−d(p, w ( 2 ))+d(w ( 2 ), xγ )

if t 6 λγ , otherwise.

In both cases, dxγ (`0− , ξγ− ) = dW + (w− ) (w bt− , `0 ) e−

λγ 2

+ 2t

t

< rt e−

λγ 2

+ 2t

= R e−

λγ 2

.

λγ

Conversely, if ξ ∈ Bdxγ (ξγ− , R e− 2 ), let `0 ∈ G X be such that `0 (0) = w− ( 12 ) and `0− = ξ. We may assume that ξ 6= ξγ− . Let w bt− be the extension of wt− such − that (w bt )− = ξγ− . Let p ∈ X be such that [`0 (0), ξγ− [ ∩ [`0 (0), `0− [ = [`0 (0), p]. Then as above, we have R e−

λγ 2

< 1, and hence 0

dW + (w− ) (w bt− , `0 ) = e−d(p, ` (0)) = dxγ (`0− , ξγ− ) e

λγ 2

− 2t

t

< rt ,

and thus `0 ∈ B + (wt− , rt ).

It follows from this lemma that for all t > t0 , s± ∈ ] − η, η[ and ` ∈ G X, we ∓ have g∓s ` ∈ B ± (wt∓ , rt ) if and only if d(`(0), αγ± (0)) = s± + 2t (or equivalently, by the definition of the time parameter s of ` in Hopf’s parametrisation with −λγ λ basepoint xγ , when s± + 2t = 2γ ± s), and `± ∈ Bdxγ (ξγ± , R e 2 ). Thus, Aη,γ (T ) = Pγ × Bdxγ (ξγ− , R e− 8 See

λγ 2

) × Bdxγ (ξγ+ , R e−

the definition of the Hamenst¨ adt distance dW + (w− ) in Section 2.3. t

λγ 2

).

234


To finish Step 4T and the proof of the theorem for R-trees, note that by the definition of the skinning measure (using again the Hopf parametrisation with basepoint xγ ), by the above Lemma 11.5, by the claim for R-trees of Assertion (2) of Proposition 3.20, and the boundedness of Fe, we have ∓ − µW ± (w∓ ) (B ± (wt∓ , rt )) = eO(η) µ∓ xγ (Bdxγ (ξγ , R e t

λγ 2

)) .

(11.16)

Thus, by the above and by Equations (11.14) and (11.15) (and also noting that O(η) ± O(η) = O(η)),

jη, γ (T ) =

e

=e

λ γ O e− 2 O(η)

4η 2

e

λγ O η+e− 2

(2η)

2

− − µ− xγ (Bdxγ (ξγ , Re − µ− xγ (Bdxγ (ξγ , Re

λγ 2

λ − 2γ

λγ 2

))

λ − 2γ

))

+ − ))µ+ xγ (Bdxγ (ξγ , Re + ))µ+ xγ (Bdxγ (ξγ , Re

,

which gives the inequalities (11.13).

(11.17)

The effective control on jη, γ (T ) given by Equation (11.17) is stronger than what is needed in order to prove Equation (11.13) in Step 4T. We will use it in Section 12.6 in order to obtain error terms.

11.3

Part III of the proof of Theorem 11.1: The manifold case

The proof of Theorem 11.1 for manifolds is the same as that for trees until Equation (11.10). The remaining part of the proof that we give below is more technical than f is for trees, but the structure of the proof is similar. In this section, X = M 1 a Riemannian manifold, and we identify G X and T X with the standard unit f, as explained in Section 2.3. tangent bundle of M Step 3M. Consider the product term in Equation (11.8) involving the characteristic functions. The quantity 1V + (Ω− ) (g−t/2 v) 1V − (Ω+ ) (γ −1 gt/2 v) is different η, R η, R from 0 (hence equal to 1) if and only if v ∈ gt/2 Vη,+R (Ω− ) ∩ γg−t/2 Vη,−R (Ω+ ) = Vη,+e−t/2 R (gt/2 Ω− ) ∩ Vη,−e−t/2 R (γg−t/2 Ω+ ) ; see Section 2.4 and in particular Equation (2.20). By Lemma 2.9 (applied by replacing D+ by γD+ and w by v), there exist t0 , c0 > 0 such that for all η ∈ ]0, 1] f, if 1 + − (g−t/2 v) 1 − + (γ −1 gt/2 v) 6= 0, then and t > t0 , for all v ∈ T 1 M Vη, R (Ω ) Vη, R (Ω ) the following facts hold:

11.3. Part III of the proof of Theorem 11.1: The manifold case

235

− (i) by the convexity of D± , we have v ∈ UD+− ∩ UγD +,

(ii) by the definition9 of w± , we have w− ∈ Ω− and w+ ∈ γΩ+ (the notation (w− , w+ ) here coincides with the notation (w− , w+ ) in Lemma 2.9), (iii) there exists a common perpendicular αγ from D− to γD+ ,10 whose length λγ satisfies | λγ − t | 6 2η + c0 e−t/2 , (11.18) whose origin π(vγ− ) is at distance at most c0 e−t/2 from π(w− ), whose endpoint π(vγ+ ) is at distance at most c0 e−t/2 from π(w+ ), such that both points π(gt/2 w− ) and π(g−t/2 w+ ) are at distance at most η + c0 e−t/2 from π(v), which is at distance at most c0 e−t/2 from some point pv of αγ .

π(v)

vγ−

pv

π(vγ− )

v

vγ0

π(vγ+ )

xγ

αγ

vγ+

D−

γD+

Using (iii) and the HC property,11 which introduces a constant κ2 ∈ ]0, 1], f for which and since Fe is bounded, for all η ∈ ]0, 1], t > t0 , and v ∈ T 1 M 1V + (Ω− ) (g−t/2 v) 1V − (Ω+ ) (γ −1 gt/2 v) 6= 0, we have η, R

η, R

R π(gt/2 w− )

e

π(w− )

+ R e+ π(w ) F −t/2 π(g

w+ )

e F

R pv

− π(vγ )

=e R

=e

αγ

e F

R π(v+ ) e+ p γ F e+O((η+e−t/2 )κ2 ) F v −λγ /2 κ2 ) )

eO((η+e

.

(11.19)

For all η ∈ ]0, 1], γ ∈ Γ, and T > t0 , define f : v ∈ V + −t/2 (gt/2 Ω− )∩V − −t/2 (γg−t/2 Ω+ ) Aη,γ (T ) = (t, v) ∈ [t0 , T ]×T 1 M η, e R η, e R and ZZ jη, γ (T ) = (t, v)∈Aη,γ (T )

h− (gt/2 w− ) h+ (g−t/2 w+ ) dt dm e F (v) . η, e−t/2 R η, e−t/2 R

By the above, since the integral of a function is equal to the integral on any Borel set containing its support, and since the integral of a nonnegative function 9 See

Equation (11.9) then denote as previously by vγ− its tangent vector at its origin, by vγ+ its tangent vector at its terminal point, and by vγ0 its tangent vector at its midpoint. 11 See Definition 3.13. 10 We

236


is nondecreasing in the integration domain, there hence exists c4 > 0 such that for all T > 0 and η ∈ ]0, 1], we have R

X

iη (T ) > − c4 +

e

αγ

−λγ /2 κ2 ) )

e F

jη, γ (T ) e− O((η+e

,

γ∈Γ : t0 +2+c0 6λγ 6T −O(η+e−λγ /2 ) − vγ− ∈ N −λγ /2 Ω , − O(η+e

vγ+ ∈ γN

− O(η+e

)

−λγ /2

)

Ω+

and similarly, for every T 0 > T , R

X

iη (T ) 6 c4 +

e

αγ

−λγ /2

γ∈Γ : t0 +2+c0 6λγ 6T +O(η+e − vγ− ∈ N −λγ /2 Ω , O(η+e

vγ+ ∈ γN

O(η+e

e F

−λγ /2 κ2

jη, γ (T 0 ) eO((η+e

)

)

.

)

)

−λγ /2

)

Ω+

We will take T 0 to be of the form T + O(η + e−λγ /2 ), for a bigger O(·) than the one appearing in the index of the above summation. Step 4M: Conclusion. Let γ ∈ Γ be such that D− and γD+ have a common perpendicular with length λγ > t0 + 2 + c0 . Let us prove that for all > 0, if η is small enough and λγ is large enough, then for every T > λγ + O(η + e−λγ /2 ) (with the “enough”s and O(·) independent of γ), we have 1 − 6 jη, γ (T ) 6 1 + .

(11.20)

Note that σ e± (Nε (Ω∓ )) and σ e± (N−ε (Ω∓ )) tend to σ e± (Ω∓ ) as ε → 0 (since ± ∓ σ e (∂Ω ) = 0 as required in Step 1). Using Steps 2, 3M, and 4M, this will prove Equation (11.4), hence will complete the proof of Theorem 11.1. f, Γ, Fe) has radius-continuous strong stable/unstable ball masWe say that (M f, if ses if for every > 0, if r > 1 is close enough to 1, then for every v ∈ T 1 M B − (v, 1) meets the support of µ+ , then W − (v) − + − µ+ W − (v) (B (v, r)) 6 e µW − (v) (B (v, 1)),

and if B + (v, 1) meets the support of µ− W + (v) , then + − + µ− W + (v) (B (v, r)) 6 e µW + (v) (B (v, 1)) .

f, Γ, Fe) has radius-H¨ We say that (M older-continuous strong stable/unstable ball masses if there exist c ∈ ]0, 1] and c0 > 0 such that for every ∈ ]0, 1], for every f, if B − (v, 1) meets the support of µ+ − , then v ∈ T 1M W (v) 0 c

− c + µ+ µW − (v) (B − (v, 1)), W − (v) (B (v, e )) 6 e


237

and if B + (v, 1) meets the support of µ− W + (v) , then 0 c

+ c − µ− µW + (v) (B + (v, 1)) . W + (v) (B (v, e )) 6 e

Note that when F = 0 and M is locally symmetric with finite volume, the conditional measures on the strong stable/unstable leaves are homogeneous. Hence f, Γ, Fe) has radius-Hölder-continuous strong stable/unstable ball masses. (M f has bounded derivatives and when When the sectional curvature of M f, Γ, Fe) has radius-Hölder-continuous strong stable/unstable ball masses, we (M will prove a stronger statement than Equation (11.20): with a constant c7 > 0 and functions O(·) independent of γ, for all η ∈ ]0, 1] and T > λγ + O(η + e−λγ /2 ), we have e−λγ /2 2 −λγ /2 c7 ) ) jη, γ (T ) = 1 + O eO((η+e . (11.21) 2η This stronger version will be needed for the error term estimate in Section 12.3. In order to obtain Theorem 11.1, only the fact that jη, γ (T ) tends to 1 as firstly λγ tends to +∞, secondly η tends to 0 is needed. A reader not interested in the error term may skip many technical details below. Given a, b > 0 and a point x in a metric space X (with a, b, x depending on parameters), we will denote by B(x, a eO(b) ) any subset Y of X such that there exists a constant c > 0 (independent of the parameters) with B(x, a e−c b ) ⊂ Y ⊂ B(x, a ec b ) .

(11.22)

Let η ∈ ]0, 1] and T > λγ + O(η + e−λγ /2 ). In order to simplify the notation, let rt = e−t/2 R, wt− = gt/2 w− and wt+ = g−t/2 w+ . By the definition of jη, γ , using the inequalities (10.3), where the constant c1 is uniform, since Fe is bounded, and the fact that rt = O(e−λγ /2 ) by Equation (11.18), we hence have ZZ − + + jη, γ (T ) = h− e F (v) (11.23) η, rt (wt ) hη, rt (wt ) dt dm (t,v)∈Aη,γ (T )

=

e

O(e−κ2 λγ /2 )

(2η)2

ZZ

dt dm e F (v) − + + − (t,v)∈Aη,γ (T ) µW + (w− ) (B (wt , rt )) µW − (w+ ) (B (wt , rt )) t

.

t

We begin the proof of Equation (11.20) by defining parameters s+ , s− , s, v 0 , v 00 associated with (t, v) ∈ Aη,γ (T ).

238


W + (g t/2 w− )

−

00

W − (g −t/2 w+ )

W + (v 0 ) W (v ) s− −s+ v s 0 t/2 − v g w

g −t/2 w+

v 00 xγ W − (vγ0 )

vγ0 W + (vγ0 )

We have (t, v) ∈ Aη,γ (T ) if and only if there exist s± ∈ ] − η, η[ such that ∓

g∓s v ∈ B ± (g±t/2 w∓ , e−t/2 R) . In order to define the parameters s, v 0 , v 00 , we use the well-known local product structure of the unit tangent bundle in negative curvature. If v ∈ T 1 M is close + enough to vγ0 (in particular, v− 6= (vγ0 )+ and v+ 6= (vγ0 )− ), then let v 0 = fHB 0 (v) − (v ) γ

− 0 be the unique element of W − (vγ0 ) such that v+ = v+ , let v 00 = fHB 0 (v) be + (v ) γ

00 the unique element of W + (vγ0 ) such that v− = v− , and let s be the unique element of R such that g−s v ∈ W + (v 0 ). The map v 7→ (s, v 0 , v 00 ) is a homeomorf to a neighbourhood of (0, vγ0 , vγ0 ) in phism from a neighbourhood of vγ0 in T 1 M − 0 + 0 R × W (vγ ) × W (vγ ). Note that if v = gr vγ0 for some r ∈ R close to 0, then

w− = vγ− , w+ = vγ+ , s = r, v 0 = v 00 = vγ0 , λγ − t λγ − t s− = + s, s+ = − s. 2 2 Up to increasing t0 (which does not change Step 3M, up to increasing c4 ), we may assume that for every (t, v) ∈ Aη,γ (T ), the vector v belongs to the domain of this f at v 0 . local product structure of T 1 M γ The vectors v, v 0 , v 00 are close to vγ0 if t is large and η small, as the following result shows. We denote (also) by d the Riemannian distance induced by Sasaki’s f. metric on T 1 M Lemma 11.6. For every (t, v) ∈ Aη,γ (T ), we have d(v, vγ0 ), d(v 0 , vγ0 ), d(v 00 , vγ0 ) = O(η + e−t/2 ). f defined by Proof. Consider the distance d0 on T 1 M f, ∀ v1 , v2 ∈ T 1 M

d0 (v1 , v2 ) = max d π(gr v1 ), π(gr v2 ) . r∈[−1,0]


239

As seen in claim (iii) of Step 3M, we have d(π(w± ), π(vγ± )), d(π(v), αγ ) = O(e−t/2 ), λ and furthermore, d(π(g±t/2 w∓ ), π(v)), 2γ − 2t = O(η + e−t/2 ). Hence d(π(v), π(vγ0 )) = O(η + e−t/2 ). By Lemma 2.4, we have t

−

−

t

d(π(g− 2 −s v), π(vγ− )) 6 d(π(g− 2 −s v), π(w− )) + d(π(w− ), π(vγ− )) 6 R + c0 e−t/2 . By an exponential pinching argument, we hence have d0 (v, vγ0 ) = O(η + e−λγ /2 ). Since d and d0 are equivalent by Proposition 3.5,12 we therefore have d(v, vγ0 ) = O(η + e−λγ /2 ). f and V ∈ Tw T 1 M f, we may uniquely write V = V − + V 0 + For all w ∈ T 1 M d + − − 0 V with V ∈ Tw W (w), V ∈ R dt t0 gt w, and V + ∈ Tw W + (w). By [PauPS, Lem. 7.4],13 Sasaki’s metric (with norm k · k) is equivalent to the Riemannian metric with (product) norm p kV k0 = k V − k2 + k V 0 k2 + k V + k2 . f in a neighbourhood of vγ0 and By the dynamical local product structure of T 1 M f at by the definition of v 0 , v 00 , the result follows, since the exponential map of T 1 M 0 vγ is almost isometric close to 0, and the projection to a factor of a product norm is Lipschitz. We now use the local product structure of the Gibbs measure to prove the following result. Lemma 11.7. For every (t, v) ∈ Aη,γ (T ), we have −λγ /2 κ2

dt dm e F (v) = eO((η+e

)

dt ds dµW − (vγ0 ) (v 0 ) dµW + (vγ0 ) (v 00 ) .

)

Proof. By the definition of the measures (see Equations (4.4) and (7.9)), since the above parameter s differs, when v− , v+ are fixed, only up to a constant from the time parameter in Hopf’s parametrisation with respect to the basepoint xγ = π(vγ0 ), we have dm e F (v) = e dµW − (vγ0 ) (v 0 ) = e dµW + (vγ0 ) (v 00 ) = e

Cv− (xγ , π(v)) + Cv+ (xγ , π(v)) −

Cv+0 (xγ , π(v 0 )) +

+

0 dµ+ xγ (v+ ) ,

Cv−00 (xγ , π(v 00 )) −

+ dµ− xγ (v− ) dµxγ (v+ ) ds,

00 dµ− xγ (v− ) .

fact, Proposition 3.5 considers the distance δ2 (v, v 0 ) = supr∈[0,1] d(π(gr v), π(gr v 0 )) instead of d0 , but the argument is similar. 13 This reference builds on [Brin], whose compactness assumption on M and torsion-free assumption on Γ are not necessary for this; the pinched negative curvature assumption is sufficient.

12 In

240


By Proposition 3.20 (2), since F is bounded, we have | Cξ± (z, z 0 ) | = O(d(z, z 0 )κ2 ) f and z, z 0 ∈ M f with d(z, z 0 ) bounded. Since the map π : T 1 M f→M f for all ξ ∈ ∂∞ M 0 00 is Lipschitz, and since v+ = v+ and v− = v− , the result follows from Lemma 11.6. When λγ is large, the submanifold gλγ /2 Ω− has a second-order contact at with W − (vγ0 ), and similarly, g−λγ /2 Ω+ has a second-order contact at vγ0 with + 0 W (vγ ). Let Pγ be the plane domain of (t, s) ∈ R2 such that |λγ −t| 6 2η+c0 e−t/2 λ −t and there exist s± ∈ ] − η, η[ with s∓ = γ2 ± s + O(e−λγ /2 ). Note that its area is (2η + O(e−λγ /2 ))2 . By the above, we have14 vγ0

−λγ /2

Aη,γ (T ) = Pγ × B − (vγ0 , rλγ eO(η+e

)

−λγ /2

) × B + (vγ0 , rλγ eO(η+e

)

).

By Lemma 11.7, we hence have ZZ

−λγ /2 κ2

Aη,γ (T )

dt dm e F (v) = eO((η+e

)

)

(2η + O(e−λγ /2 ))2 ×

−λγ /2

µW − (vγ0 ) (B − (vγ0 , rλγ eO(η+e

)

(11.24) −λγ /2

))µW + (vγ0 ) (B + (vγ0 , rλγ eO(η+e

)

)) .

The last ingredient of the proof of Step 4M is the following continuity property of the masses of balls in the strong stable and strong unstable manifolds as their center varies. This result generalises [PaP17b, Lem. 11]. The precise control for the error term is used in Section 12.3. f, Γ, Fe) has radius-continuous strong stable/unstable Lemma 11.8. Assume that (M ball masses. There exists c5 > 0 such that for every > 0, if η is small enough and λγ large enough, then for every (t, v) ∈ Aη,γ (T ), we have µW − (w+ ) (B − (wt+ , rt )) = eO(

c5

)

µW − (vγ0 ) (B − (vγ0 , rλγ ))

µW + (w− ) (B + (wt− , rt )) = eO(

c5

)

µW + (vγ0 ) (B + (vγ0 , rλγ )) .

t

and t

f has bounded derivaIf we furthermore assume that the sectional curvature of M f e tives and that (M , Γ, F ) has radius-H¨ older-continuous strong stable/unstable ball masses, then we may replace by (η + e−λγ /2 )c6 for some constant c6 > 0. Proof. We prove the (second) claim for W + ; the (first) one for W − follows similarly. The final statement is used only for the error estimates in Section 12.3. 14 With

the obvious meaning of a double inclusion by Equation (11.22).


241

B + (w− , R) w−

t/2

wt−

vγ−

w− vγ−

O(η + e−λγ /2 ) vγ0

O(e−λγ /2 ) `γ /2

−λγ /2

B + (vγ− , R eO(η+e

)

)

Using respectively Equation (2.18), since wt− = gt/2 w− and rt = e−t/2 R, Equation (7.11) where (`, t, w) is replaced by (v, t/2, w− ), and Equation (3.20), we have Z µW + (w− ) (B + (wt− , rt )) = dµW + (gt/2 w− ) (gt/2 v) t v∈B + (w− , R) Z C − (π(v), π(gt/2 v)) = e v− dµW + (w− ) (v) v∈B + (w− , R)

Z =

e

R π(gt/2 v) π(v)

e−δ) (F

dµW + (w− ) (v) .

(11.25)

dµW + (vγ− ) (v) .

(11.26)

v∈B + (w− , R)

Similarly, for every a > 0, we have Z µW + (vγ0 ) (B + (vγ0 , art )) =

R π(gt/2 v)

e

π(v)

e−δ) (F

v∈B + (vγ− , aR)

Let h− : B + (w− , R) → W + (vγ− ) be the map such that (h− (v))− = v− , which is well defined and a homeomorphism onto its image if λγ is large enough (since R is fixed). By Proposition 7.5 applied with D = HB+ (w− ) and D0 = HB+ (vγ− ), we have, for every v ∈ B + (w− , R), −Cv− (π(v), π(h− (v)))

dµW + (w− ) (v) = e

−

dµW + (vγ− ) (h− (v)) .

Let us fix > 0. The strong stable balls of radius R centered at w− and vγ− are very close (see the above picture). More precisely, recall that R is fixed, and that d(π(w− ), π(vγ− )) = O(e−λγ /2 ) and d(π(gt/2 w− ), π(gλγ /2 vγ− )) = O(η + e−λγ /2 ) . Therefore we have d(π(v), π(h− (v))) 6 for every v ∈ B + (w− , R) if η is small enough and λγ is large enough. If we furthermore assume that the sectional curvature has bounded derivatives, then by Anosov’s arguments, the strong stable foliation is H¨ older-continuous; see, for instance, [PauPS, Thm. 7.3]. Hence we have d(π(v), π(h− (v))) = O((η + e−λγ /2 )c6 ) for every v ∈ B + (w− , R), for some constant c6 > 0, under the additional regularity assumption on the curvature. We also have h− (B + (w− , R)) = B + (vγ− , R eO() ) and, under the additional hypothesis −λγ /2 c6 ) )

on the curvature, h− (B + (w− , R)) = B + (vγ− , R eO((η+e

).

242


In what follows, we assume that = (η + e−λγ /2 )c6 under the additional assumption on the curvature. By Proposition 3.20 (2), since Fe is bounded, we hence have, for every v ∈ B + (w− , R), κ2

dµW + (w− ) (v) = eO(

)

dµW + (vγ− ) (h− (v)),

and, using the HC property and the boundedness of Fe, Z

π(gt/2 v)

Z (Fe − δ) −

π(v)

π(gt/2 h− (v))

(Fe − δ) = O(κ2 ) .

π(h− (v))

The result follows, by Equations (11.25) and (11.26) and the continuity properties in the radius of the strong stable/unstable ball masses. Now Lemma 11.8 (with as in its statement, and when its hypotheses are satisfied) implies that ZZ dt dm e F (v) − − + + + − (t,v)∈Aη,γ (T ) µW + (w− ) (B (wt , rt )) µW − (w+ ) (B (wt , rt )) t t c5 RR eO( ) (t,v)∈Aη,γ (T ) dt dm e F (v) = − . + + 0 µW + (v0 ) (B (vγ , rt )) µW − (v0 ) (B − (vγ0 , rt )) γ

γ

By Equations (11.23) and (11.24), we hence have −λγ /2 κ2 ) )

jη, γ (T ) = eO((η+e

eO(

c5

)

(2η + O(e−λγ /2 ))2 (2η)2

under the technical assumptions of Lemma 11.8. The assumption on radius-continuity of strong stable/unstable ball masses can be bypassed using bump functions, as explained in [Rob2, page 81]. This completes the proof of Equation (11.20), hence the proof of Theorem 11.1.

11.4

Equidistribution of common perpendiculars in simplicial trees

In this section, we prove a version of Theorem 11.1 for the discrete-time geodesic flow on simplicial trees (and we leave to the reader the version without the assumption that the critical exponent of the system of conductances is positive). Let X, X, x0 , Γ, e c, c, Fec , Fc , δc < +∞, (µ± e c , mc be as at the x )x∈V X , m + + − and D beginning of Section 9.2. Let D − = (D− ) = (D ) i∈I i j j∈I + be locally finite Γ-equivariant families of nonempty proper simplicial subtrees of X. We denote by ± ± Dk± = |D± k |1 the geometric realisation of Dk for k ∈ I .

11.4. Equidistribution of common perpendiculars in simplicial trees

243

For every edge path α = (e1 , . . . , en ) in X, we set e c(α) =

n X

e c(ei ) .

i=1

Theorem 11.9. Assume that the critical exponent δc of e c is positive and that the Gibbs measure mc is finite and mixing for the discrete-time geodesic flow on Γ\G X. Then lim

t→+∞

eδc − 1 kmc k e−δc t eδc

X

eec(αi,γj ) ∆α−

i, γj

i∈I − /∼ , j∈I + /∼ , γ∈Γ + Di− ∩Dγj =∅, λi, γj 6t

⊗ ∆α+

γ −1 i, j

+ − =σ eD eD − ⊗ σ +

b

b

for the weak-star convergence of measures on the locally compact space G X × G X. Proof. The proof is a modification of the continuous-time proof for metric trees in Sections 11.1 and 11.2. Here, we indicate the changes to adapt the proof to the discrete time. We use the conventions for the discrete-time geodesic flow described in Section 2.6. Note that for all i ∈ I − , j ∈ I + , γ ∈ Γ, the common perpendicular αi,γj is + now an edge path from Di− to Dγj , and that by Proposition 3.21, we have Z αi,γj

Fec = e c(αi,γj ) .

In the definition of the bump functions in Section 10.1, we assume (as we 1 may) that η < 1, so that for all η 0 ∈ ]0, 1[ and w ∈ ∂∓ D± such that w∓ ∈ ΛΓ, we have ± ± 0 Vw,η,η 0 = B (w, η ) , see Equation (2.17) and recall that we are considering only discrete geodesic lines. Since `(0) = w(0) for every ` ∈ B ± (w, η 0 ), since η 0 < 1, and since the time is now discrete, Equations (10.1) and (7.12) give h∓ η, η 0 (w) =

1 . µW ± (w) (B ± (w, η 0 ))

(11.27)

This is a considerable simplification compared with the inequalities of Equation (10.3). In the whole proof, we restrict to t = n ∈ N and T = N ∈ N. We keep the notation of Equation (11.2), as well as the only assumptions on the Borel 1 sets Ω± ⊂ ∂∓ D± to have finite positive skinning measure, with boundary of zero

244


skinning measure. In Steps 1 and 2, we define instead of Equation (11.3), R X e F IΩ− , Ω+ (N ) = (eδc − 1) kmc k e−δc (N +1) e αγ c , γ∈Γ : 0 2) and let Γ be a tree lattice of X such that Γ\X is not bipartite. Assume that the Patterson density for the zero system of conductances is normalised to be a family of probability measures. Let D± be nonempty proper simplicial subtrees of X with stabilisers ΓD± ∓ in Γ such that D ± = (γD± )γ∈Γ/ΓD± is locally finite. Let σ eD ± be their skinning

246


measures for the zero system of conductances. Then lim

t→+∞

q−1 Vol(Γ\\X) q −t q+1

X

∆α−

α, γβ

⊗∆α+

γ −1 α, β

(α, β, γ)∈Γ/ΓD− ×Γ/ΓD+ ×Γ 0 2 and Γ is a tree lattice of X. Theorem 11.11. Assume that the smallest nonempty Γ-invariant simplicial subtree of X is uniform, without vertices of degree 2, and that the length spectrum LΓ of Γ is 2Z. Assume that the critical exponent δc of e c is positive, that the Gibbs measure mc is finite, and that its restriction to Γ\Geven X is mixing for the square of the discrete-time geodesic flow on Γ\Geven X. Then lim

t→+∞

e2 δc − 1 kmc k e−δc t 2 e2 δc

X


i, γj

−

⊗ ∆α+

+

γ −1 i, j

i∈I /∼ , j∈I /∼ , γ∈Γ + Di− ∩Dγj =∅, λi, γj 6t + − = σ eD eD − ⊗ σ +

b

b

for the weak-star convergence of measures on the locally compact space G X × G X. b ± ± ± Proof. We denote by σ ebD of σ eD eD ∓ , even the ∓ to Geven X, and by σ ∓ , odd the b restriction b ± restriction of σ eD ∓ to Godd X = G X− Geven X. We denote by Veven X the subset of V X consisting of the vertices at even distance from x0 , and by Vodd X = V X − Veven X


247

its complement. The subsets Veven X and Vodd X are Γ-invariant if LΓ = 2Z by Equation (4.17). Let us first prove that lim

t→+∞

e2 δc − 1 kmc k e−δc t 2 e2 δc

X


i, γj

−

⊗ ∆α+

γ −1 i, j

+

i∈I /∼ , j∈I /∼ , γ∈Γ + π(α− ) ∈ Veven X i,γj ), π(α −1 γ

i,j

+ Di− ∩Dγj =∅, λi, γj 6t

+ − = σ eD eD (11.28) − , even ⊗ σ + , even b b

for the weak-star convergence of measures on Geven X × Geven X.

The proof of Equation (11.28) is a modification of the proof of Theorem 11.9. We now restrict to t = 2n ∈ N, T = 2N ∈ N, and we replace m e c by (m e c ) Geven X and (gt )t∈Z by (g2t )t∈Z . Note that since m e c is invariant under the time 1 of the geodesic flow, which maps Γ\Geven X to Γ\G X − Γ\Geven X, we have

(mc )

= 1 kmc k . Γ\Geven X 2

(11.29)

− Note that for all i ∈ I − , j ∈ I + , and γ ∈ Γ, if π(αi,γj ) and π(αγ+−1 i,j ) belong to Veven X, then the distance between Di− and γDj+ is even.15 b 1 In Steps 1 and 2, we now consider Ω± two Borel subsets of ∂∓ D± ∩ Geven X, and we define instead of Equation (11.3),

IΩ− , Ω+ (2N ) = (e2 δc − 1)

kmc k −2 δc (N +1) e × 2 X

R

e

αγ

ec F

,

+ γ∈Γ : 0 0, there exists T = T,η > 0 such that for all n > T , we have R R e− Geven X φ− e c Geven X φ+ ec η dm η dm k(mc ) Γ\Geven X k R R e Geven X φ− e c Geven X φ+ ec η dm η dm 6 aη (2n) 6 . k(mc ) k Γ\Geven X

Note that Geven X is saturated by the strong stable and strong unstable leaves, since two points x, y on a given horosphere of center ξ ∈ ∂∞ X are at even distance one from another (equal to 2d(x, p), where [x, ξ[ ∩ [y, ξ[ = [p, ξ[ ). By the disintegration statement in Proposition 7.7, for every nonempty proper subtree D of X and with D = |D|1 , as ` ranges over UD± ∩ Geven X, we have Z ± b dm e c |U ± ∩ Geven X (`) = dνρ∓ (`) de σD (ρ) . D

1 D∩ G ρ∈∂± even X

Hence the proof of Lemma 10.1 extends to give Z ± φ∓ ec = σ eeven (Ω∓ ) , η dm

(11.30)

Geven X

± ± where in order to simplify notation, σ eeven =σ eD ∓ , even .

Therefore, by Equations (11.29) and (11.30), and by a geometric sum argument, the pair of inequalities (11.7) becomes + − 2 e− e2δc (N +1) σ eeven (Ω− ) σ eeven (Ω+ ) − c (e2δc − 1) kmc k

6 iη (2N ) 6

+ − 2 e e2δc (N +1) σ eeven (Ω− ) σ eeven (Ω+ ) + c . (e2δc − 1) kmc k

Up to replacing the summations from n = 0 to N to summations on even numbers between 0 to 2N , and replacing bn/2c by 2bn/2c as well as dn/2e by 2dn/2e, the remaining part of the proof applies and gives the result, noting that in claim (iii) of Step 3T, we furthermore have that the origin and endpoint of the constructed common perpendicular αγ are in Veven X. This concludes the proof of Equation (11.28). b consists b in proving b b The remainder of the proof of Theorem 11.11 versions of b equidistribution b the result Equation (11.28) in Godd X × Godd X, Geven X × Godd X, Godd X × Geven X respectively, and in summing these four contributions.


249

b b By applying Equation (11.28) by replacing x0 by a vertex x00 in Vodd X, which ± ± exchanges Veven X and Vodd X, Geven X and Godd X, as well as σ eD eD ∓ , even and σ ∓ , odd , we have X e2 δc − 1 lim kmc k e−δc t eec (αi,γj ) ∆α− ⊗ ∆α+ 2 δ c i, γj t→+∞ 2 e γ −1 i, j − + i∈I /∼ , j∈I /∼ , γ∈Γ + π(α− ) ∈ Vodd X i,γj ), π(α −1 γ

i,j


+ − = σ eD eD − , odd ⊗ σ + , odd b b

(11.31)

for the weak-star convergence of measures on Godd X × Godd X.

b Let us now apply Equation (11.28) by replacing therein Db− = (Di− )i∈I − by − N1 D − = (N1 Di )i∈I − . Let us consider the map ϕ+ : G X → G X, which maps a generalised geodesic line ` to the generalised geodesic line that coincides with g+1 ` b that this b map is on [0, +∞[ and is constant (with value `(1)) on ] b− ∞, 0[ . Note b continuous and Γ-equivariant, and that it maps Geven X in Godd X and Godd X in Geven X. b Furthermore, by convexity, ϕ+ induces forb every i ∈ I − a homeomorphism − − − − 1 1 1 1 from ∂+ Di to ∂+ N1 Di , which sends ∂+ Di b∩ Godd X to ∂+ N1 Di ∩ Geven X, such

1 that by Equation (7.8), for all w ∈ ∂+ Di− ∩ Godd X, if ew is the first edge followed by w, + + de σD (w) = eec (ew )−δc de σN (ϕ+ (w)) . − , odd D − , even 1

i

i

Note that for all ` > 0, there is a one-to-one correspondence between the set of common perpendiculars of length `, with origin and endpoint both in Veven , between N1 Di− , and γDj+ for all i ∈ I − , j ∈ I + and γ ∈ Γ, and the set of common perpendiculars of length ` + 1, with origin in Vodd and endpoint in Veven , between − Di− and γDj+ for all i ∈ I − , j ∈ I + , and γ ∈ Γ. In particular, ϕ+ (αi,γj ) is the − + common perpendicular between N1 Di and γDj , starting at time t = 0 from N1 Di− . Therefore Equation (11.28) applied by replacing therein D − = (Di− )i∈I − by − N1 D = (N1 Di− )i∈I − gives lim

t→+∞

e2 δc − 1 kmc k e−δc t 2 e2 δc X

e c (eα−

e

i, γj

)+e c (ϕ+ (α− i,γj ))

∆α−

i, γj

−

+

i∈I /∼ , j∈I /∼ , γ∈Γ + π(α− i,γj ) ∈ Vodd X, π(αγ −1 i,j ) ∈ Veven X − + Di ∩Dγj =∅, λi, γj 6t+1 + − = eδc σ eD eD − , odd ⊗ σ + , even

b

b

for the weak-star convergence of measures on Godd X × Geven X.

⊗ ∆α+

γ −1 i, j

250

Chapter 11. Equidistribution of Common Perpendicular Arcs − − Since e c(eα− ) + e c(ϕ+ (αi,γj )) = e c(αi,γj ), replacing t by t − 1 and simplifying i, γj

by eδc , we get lim

t→+∞

e2 δc − 1 kmc k e−δc t 2 e2 δ c X

−


i, γj

⊗ ∆α+

γ −1 i, j

i∈I − /∼ , j∈I + /∼ , γ∈Γ + π(α− ) ∈ Veven X i,γj ) ∈ Vodd X, π(α −1 γ

i,j


+ − = σ eD eD − , odd ⊗ σ + , even

(11.32) b

b

for the weak-star convergence of measures on Godd X × Geven X. Now Theorem 11.11 follows by summing Equation (11.28), Equation (11.31), Equation (11.32), and the formula, proven similarly, obtained from Equation (11.28) by replacing D + = (Dj+ )j∈I + by (N1 Dj+ )j∈I + . The following result for bipartite graphs (of groups) is used in the arithmetic applications in Part III (see Section 15.4). Corollary 11.12. Let X be a (p + 1, q + 1)-biregular simplicial tree (with p, q > 2, possibly with p = q), with corresponding partition V X = Vp X t Vq X. Let Γ be a tree lattice of X such that this partition is Γ-invariant. Assume that the Patterson √ density for the zero system of conductances is normalised so that kµx k = p+1 p for ± every x ∈ Vp X. Let D be nonempty proper simplicial subtrees of X with stabilisers ∓ ΓD± in Γ, such that the families D ± = (γD± )γ∈Γ/ΓD± are locally finite. Let σ eD ± be their skinning measures for the zero system of conductances. Then lim

t→+∞

pq − 1 √ −t−2 TVol(Γ\\X) pq 2

X

∆α−

α, γβ

⊗ ∆α+

γ −1 α, β

(α, β, γ)∈Γ/ΓD− ×Γ/ΓD+ ×Γ 0 0 and two unit tangent vectors v, w ∈ T 1 M , we define the number nt (v, w) of locally geodesic paths having v and w as initial and terminal tangent vectors respectively, weighted by the potential F , with length at most t, by R X nt (v, w) = Card(Γα ) e α F , α

12.1. Multiplicities and counting functions in Riemannian orbifolds

255

where the sum ranges over the locally geodesic paths α : [0, s] → M in the Riemannian orbifold M such that α(0) ˙ = v, α(s) ˙ = w and s ∈ ]0, t], and Γα is the f stabiliser in Γ of any geodesic path α e in M mapping to α by the quotient map f → M . If F = 0 and Γ is torsion free, then nt (v, w) is precisely the number M of locally geodesic paths having v and w as initial and terminal tangent vectors respectively, with length at most t. Let D − = (Di− )i∈I − and D + = (Dj+ )j∈I + be Γ-equivariant families of proper f. Let Ω− = (Ω− )i∈I − and Ω+ = (Ω+ )j∈I + nonempty closed convex subsets of M i

j

f, where Ω∓ is a measurable subset of be Γ-equivariant families of subsets of T 1 M k ∓ ± ± 1 ∓ ∂± Dk for all k ∈ I and Ωk = Ωk0 if k ∼D ± k 0 . We will denote by NΩ− , Ω+ , F : ]0, +∞[ → R the following counting function: for every t > 0, let NΩ− , Ω+ , F (t) be the number of common perpendiculars whose initial vectors belong to the images in T 1 M of the elements of Ω− and terminal vectors to the images in T 1 M of the elements of Ω+ , counted with multiplicities and weighted by the potential F ; that is, X NΩ− , Ω+ , F (t) = mΩ− (v) mΩ+ (w) nt (v, w) . v, w ∈ T 1 M 1 When Ω± = ∂∓ D ± , we denote NΩ− , Ω+ , F by ND − , D + , F .

Remark 12.2. Let Y be a negatively curved complete connected Riemannian manifold and let Ye → Y be its Riemannian universal cover. Let D± be a locally convex3 geodesic metric space endowed with a continuous map f ± : D± → Y such that e ± → D± is a locally isometric universal cover and if fe± : D e ± → Ye is a lift if D e ± an isometric embedding of f ± , then fe± is on each connected component of D whose image is a proper nonempty closed convex subset of Ye , and the family of images under the covering group of Ye → Y of the images by fe± of the connected e ± is locally finite. Then D± (or the pair (D± , f ± )) is a proper components of D nonempty properly immersed closed locally convex subset of Y . If Γ is a discrete subgroup without torsion of isometries of a CAT(−1) Riee ± )γ∈Γ , where D e ± is a nonempty proper closed mannian manifold X, if D ± = (γ D ± convex subset of X such that the family D is locally finite, and if D± is the image e ± by the covering map X → Γ\X, then D± is a proper nonempty properly of D immersed closed convex subset of Γ\X. Under these assumptions, ND − , D + , F is the counting function ND− , D+ , F given in the introduction. Let us continue fixing the notation used in Sections 12.2 and 12.3. For every (i, j) in I − × I + such that Di− and Dj+ have a common perpendicular,4 we denote 1 by αi, j this common perpendicular, by λi, j its length, by vi,−j ∈ ∂+ Di− its initial 3 Not 4 This

necessarily connected. occurs exactly when the closures Di− and Dj+ in X ∪ ∂∞ X have empty intersection.

256

Chapter 12. Equidistribution and Counting of Common Perpendiculars

1 tangent vector, and by vi,+j ∈ ∂− Dj+ its terminal tangent vector. Note that if i0 ∼ i, 0 j ∼ j, and γ ∈ Γ, then

γ αi0 , j 0 = αγi, γj ,

± λi0 , j 0 = λγi, γj , and γ vi±0 , j 0 = vγi, γj .

(12.1)

When Γ has no torsion, we have, for the diagonal action of Γ on I − × I + , R X e F ND − , D + , F (t) = e αi, j . (i, j)∈Γ\((I − /∼ )×(I + /∼ )) : Di− ∩Dj+ =∅, λi, j 6t

When the potential F is zero and Γ has no torsion, ND − , D + , F (t) is the number of common perpendiculars of length at most t, and the counting function t 7→ ND − , D + , 0 (t) has been studied in various special cases of negatively curved manifolds since the 1950s and in a number of recent works; see the introduction. The asymptotics of ND − , D + , 0 (t) as t → +∞ in the case that X is a Riemannian manifold with pinched negative curvature are described in general in [PaP17b, Thm. 1], where it is shown that if the Bowen–Margulis measure mBM is finite and + − mixing, and if the skinning measures σD − and σD + are finite and nonzero, then as s → +∞, − δΓ s kσ +− k kσD +k e ND − , D + , 0 (s) ∼ D . (12.2) kmBM k δΓ

12.2

Common perpendiculars in Riemannian orbifolds

Corollary 12.3 below is the main result of this text on the counting with weights of common perpendiculars and on the equidistribution of their initial and terminal tangent vectors in negatively curved Riemannian manifolds endowed with a H¨ older-continuous potential. We use the notation of Section 12.1. f be a complete simply connected Riemannian manifold with Corollary 12.3. Let M pinched negative sectional curvature at most −1. Let Γ be a nonelementary discrete f. Let Fe : T 1 M f → R be a bounded Γ-invariant H¨ group of isometries of M oldercontinuous function with positive critical exponent δ. Let D − = (Di− )i∈I − and D + = (Dj+ )j∈I + be locally finite Γ-equivariant families of nonempty proper closed f. Assume that the Gibbs measure mF is finite and mixing for convex subsets of M the geodesic flow on T 1 M . Then, X lim δ kmF k e−δ t m∂+1 D − (v) m∂−1 D + (w) nt (v, w) ∆v ⊗ ∆w t→+∞

=

+ σD −

v, w∈T 1 M

⊗

− σD +

(12.3)

for the weak-star convergence of measures on the locally compact product space + − T 1 M × T 1 M . If σD − and σD + are finite, the result also holds for narrow convergence.

12.2. Common perpendiculars in Riemannian orbifolds

257

Furthermore, for all Γ-equivariant families Ω± = (Ω± k )k∈I ± of subsets of ± ± ± 1f 1 ± 0 T M with Ωk a Borel subset of ∂∓ Dk for all k ∈ I and Ω± k = Ωk0 if k ∼D ± k , ∓ ∓ 1 with nonzero finite skinning measure σΩ± and with boundary in ∂± Dk of zero skinning measure, we have, as t → +∞, + − kσΩ − k kσΩ+ k eδ t . δ kmF k

NΩ− , Ω+ , F (t) ∼

Proof. Note that the sum in Equation (12.3) is locally finite, hence it defines a locally finite measure on T 1 M × T 1 M . We are going to rewrite the sum in the statement of Theorem 11.1 in a way that makes it easier to push it down from f × T 1M f to T 1 M × T 1 M . T 1M f, let For every ve ∈ T 1 M 1 m∓ (e v ) = Card {k ∈ I ∓ /∼ : ve ∈ ∂± Dk∓ } , 1 so that for every v ∈ T 1 M , the multiplicity of v with respect to the family ∂± D ∓ is5

m∂±1 D ∓ (v) =

m∓ (e v) , Card(StabΓ ve)

f. for every preimage ve of v in T 1 M f, there exists (i, j) ∈ (I − /∼ ) × (I + /∼ ) such For all γ ∈ Γ and ve, w e ∈ T 1M − + + that ve = vi,γj and w e = vγ −1 i,j = γ −1 vi,γj if and only if γ w e ∈ gR ve, there exists 1 1 i0 ∈ I − /∼ such that ve ∈ ∂+ Di−0 , and there exists j 0 ∈ I + /∼ such that γ w e ∈ ∂− Dj+0 . 0 0 Then the choice of such elements (i, j), as well as i and j , is free. We hence have R X e F e αi, γj ∆v− ⊗ ∆v+ i, γj

i∈I − /∼ , j∈I + /∼ , γ∈Γ − 0 0, by applying Corollary 12.3 with the potential F + σ (see Remark 7.1 (2)) as in the proof of Theorem 11.3. Using the continuity of the pushforwards of measures for the weak-star and f × T 1M f to the narrow topologies, applied to the basepoint maps π × π from T 1 M 1 1 f f M × M , and from T M × T M to M × M , we have the following result of equidistribution of the ordered pairs of endpoints of common perpendiculars between two f or two families of locally convex sets in M . equivariant families of convex sets in M When M has constant curvature and finite volume, F = 0 and D − is the Γ-orbit of a point and D + is the Γ-orbit of a totally geodesic cocompact submanifold; this result is due to Herrmann [Herr]. For the case in which D ± are Γ-orbits of points and F is a H¨ older-continuous potential, see [PauPS, Thm. 9.1, 9.3], and we refer, for instance, to [BoyM] for an application of this particular case. f, Γ, Fe, D − , D + be as in Corollary 12.3. Then Corollary 12.5. Let M R X e F lim δ kmF k e−δ t e αi, γj ∆π(v− ) ⊗ ∆π(v+ t→+∞

i, γj

i∈I − /∼ , j∈I + /∼ , γ∈Γ 0 0 such that for all nonnegative ψ ± ∈ Ccα (T 1 M ), we have, as t → +∞, δ kmF k eδ t

X

m∂+1 D − (v) m∂−1 D + (w) nt (v, w) ψ − (v) ψ + (w)

v, w∈T 1 M

Z = T 1M

+ ψ − dσD −

Z

0

T 1M

− −κ t ψ + dσD kψ − kα kψ + kα ) . + + O(e

f is a symmetric space, that D± has smooth boundary for every (2) Assume that M k ± k ∈ I , that mF is finite and smooth,7 and that the geodesic flow on T 1 M is mixing with exponential speed for the Sobolev regularity for the potential F . Then there exist ` ∈ N and κ0 > 0 such that for all nonnegative maps ψ ± ∈ Cc` (T 1 M ), we have, as t → +∞, δ kmF k eδ t

X

m∂+1 D − (v) m∂−1 D + (w) nt (v, w) ψ − (v) ψ + (w)

v, w∈T 1 M

Z =

ψ T 1M

−

+ dσD −

Z T 1M

0

− −κ t ψ + dσD kψ − k` kψ + k` ) . + + O(e

Furthermore, if D − and D + respectively have nonzero finite outer and inner f, Γ, Fe) satisfies the conditions of (1) or (2) above, skinning measures, and if (M 7 Recall

that a measure on a smooth manifold N is smooth if in every local chart, it is absolutely continuous with respect to the Lebesgue measure, with smooth Radon–Nikodym derivative.

12.3. Error terms for equidistribution and counting for Riemannian orbifolds

261

then there exists κ00 > 0 such that as t → +∞, ND − , D + , F (t) =

+ − kσD 00 − k kσD + k eδ t 1 + O(e−κ t ) . δ kmF k

f, Γ, F, D ± , and the speeds of mixing. The proof The maps O(·) depend on M is a generalisation to nonzero potential of [PaP17b, Thm. 15]. Proof. We will follow the proofs of Theorem 11.1 and Corollary 12.3 to prove generalisations of Assertions (1) and (2) by adding to these proofs a regularisation process of the test functions φe± η as in the deduction of Theorem 10.3 from Theorem 10.2. We will then deduce the last statement from these generalisations, again using this regularisation process. Let β be either α ∈ ]0, 1] small enough in the Hölder regularity case or ` ∈ N large enough in the Sobolev regularity case. We fix i ∈ I − , j ∈ I + , and 1 we use the notation D± , αγ , λγ and σ e± of Equation (11.2). Let vγ± ∈ ∂∓ D± be the initial and terminal tangent vectors to αγ and γ −1 αγ respectively. Let ψe± ∈ 1 Ccβ (∂∓ D± ). Under the assumptions of Assertion (1) or Assertion (2), we first prove the following avatar of Equation (11.4), indicating only the required changes in its proof: there exists κ0 > 0 (independent of ψe± ) such that as T → +∞, R X e F δ kmF k e−δ T e αγ ψe− (vγ− ) ψe+ (vγ+ ) γ∈Γ, 0 0 and χ± η, R ∈ C (T M ) such that 0

−κ1 • kχ± ), η, R kβ = O(η ± • 1V ∓ (∂ 1 D ± ) 6 χη, R 6 1V ∓ η e− O(η) , R e− O(η)

∓

1 ± η, R (∂∓ D )

,

1 ∂∓ D± ,

• for every w ∈ we have Z ∓ ∓ ± ± − O(η) ± χ± = νw (Vw, ) eO(η) . η, R dνw = νw (Vw, η, R ) e η e− O(η) , R e− O(η) ∓ Vw, η, R

We now define the new test functions (compare with Section 10.1). For every 1 w ∈ ∂∓ D± , let 1 ± Hη, R (w) = R ± ± . χ ∓ η, R dνw V w, η, R

262


1f Let Φ± η : T M → R be the map defined by ± ∓ ± e± Φ± η = (Hη, R ψ ) ◦ fD ± χη, R . 1 The support of this map is contained in Vη,∓R (∂∓ D± ). Since M is compact in Assertion (1) and by homogeneity in Assertion (2), if R is large enough, by the ± ± definitions of the measures νw , the denominator of Hη, R (w) is at least c η, where ± c > 0. The map Hη, R is hence Hölder-continuous under the assumptions of Assertion (1), and it is smooth under the assumptions of Assertion (2). Therefore β 1f 0 Φ± η ∈ C (T M ), and there exists a constant κ2 > 0 such that −κ02 e± kΦ± kψ kβ ) . η kβ = O(η

As in Lemma 10.1, the functions Φ± η are measurable, nonnegative, and satisfy Z f T 1M

Φ± η

Z dm eF =

1 D± ∂∓

ψe± de σ∓ .

As in the proof of Theorem 11.1, we will estimate in two ways the quantity Z

T

Iη (T ) =

X Z

eδ t

0

f T 1M

γ∈Γ

−t/2 t/2 (Φ− ) (Φ+ ◦ γ −1 ) dm e F dt . η ◦g η ◦g

(12.5)

We first apply the mixing property, now with exponential decay of correlations, as in Step 2 of the proof of Theorem 11.1. For all t > 0, let X Z −t/2 t/2 −1 Aη (t) = Φ− v) Φ+ γ v) dm e F (v) . η (g η (g f v∈T 1 M

γ∈Γ

Then with κ > 0 as in the definitions of the exponential mixing for the Hölder or Sobolev regularity, we have Z Z 1 − + Aη (t) = Φη dm eF Φ+ e F + O e−κ t kΦ− η dm η kβ kΦη kβ kmF k T 1 M 1 f f Z ZT M 0 1 − + = ψe de σ ψe+ de σ − + O e−κ t η −2κ2 kψe− kβ kψe+ kβ . 1 D+ kmF k ∂+1 D− ∂− Hence by integrating, eδ T Iη (T ) = δ kmF k

Z

ψe− de σ+

1 D− ∂+

Z

0 ψe+ de σ − + O e−κ T η −2κ2 kψe− kβ kψe+ kβ

.

1 D+ ∂−

(12.6)

12.3. Error terms for equidistribution and counting for Riemannian orbifolds

263

Now, as in Step 3 of the proof of Theorem 11.1, we exchange the integral over t and the summation over γ in the definition of Iη (T ), and we estimate the integral term independently of γ: Iη (T ) =

X Z

T

Z

eδ t

f T 1M

0

γ∈Γ

−t/2 t/2 (Φ− ) (Φ+ ◦ γ −1 ) dm e F dt . η ◦g η ◦g

± ∓ ± ∓ ± e± b± b± Let Φ η = Hη, R ◦ fD ± χη, R , so that Φη = ψ ◦ fD ± Φη . By the last two ± ± properties of the regularised maps χη, R , we have, with φη0 , η00 , Ω± defined as in Equation (10.4), ± O(η) b± φ± e− O(η) 6 Φ . η 6 φη e η e− O(η) , R e− O(η) , ∂ 1 D ±

(12.7)

∓

f belongs to the support of (Φ− ◦ g−t/2 ) (Φ+ ◦ gt/2 ◦ γ −1 ), then If v ∈ T 1 M η η 1 1 we have v ∈ gt/2 Vη,+R (∂+ D− ) ∩ g−t/2 Vη,−R (γ∂− D+ ). Hence the properties (i), (ii), 1 and (iii) of Step 3M of the proof of Theorem 11.1 still hold (with Ω− = ∂+ D− 1 + + + and Ω+ = ∂− (γD ), and where vγ therein now replaced by γvγ ). In particular, + − − + −1 if w− = fD = γ −1 fγD v), we have, by Assertion (iii) − (v) and w + (v) = fD + (γ 8 in Step 3M of the proof of Theorem 11.1, that d(w± , vγ± ) = O(η + e−λγ /2 ) . Hence, with κ03 = α in the Hölder case and κ03 = 1 in the Sobolev case (we may assume that ` > 1), we have 0 | ψe± (w± ) − ψe± (vγ± ) | = O((η + e−λγ /2 )κ3 kψe± kβ ) .

Therefore there exists a constant κ04 > 0 such that X 0 Iη (T ) = ψe− (vγ− )ψe+ (vγ+ ) + O((η + e−λγ /2 )κ4 kψe− kβ kψe+ kβ ) × γ∈Γ

Z 0

T

eδ t

Z f v∈T 1 M

−t/2 −1 t/2 b− b+ Φ v) Φ g v) dm e F (v) dt . η (g η (γ

Now, using the inequalities (12.7), Equation (12.4) follows as in Steps 3M 0 and 4M of the proof of Theorem 11.1, by taking η = e−κ5 T for some κ05 > 0 and using the effective control given by Equation (11.21) in Step 4M. In order to prove Assertions (1) and (2) of Theorem 12.7, we may assume that the supports of ψ ± are small enough, say contained in B(x± , ) for some f) with x± ∈ T 1 M and small enough. Let x e± be lifts of x± and let ψe± ∈ Ccβ (T 1 M 8 See

also the picture at the beginning of the proof of Lemma 11.8.

264


f → M is support in B( x e± , ) be such that ψe± = ψ ± ◦ T p on B( x e± , ), where p : M the universal Riemannian orbifold cover. By a finite summation argument, since Γ\I ± are finite, and by Equation (12.4), we have R

X

δ kmF k e−δ T

e

αγ

e F

ψe− (vγ− ) ψe+ (vγ+ )

i∈I − /∼, j∈I + /∼, γ∈Γ 0 λγ, γ 0 , then αγ, (t) ∈ γ V D is the endpoint t(αγ, γ 0 ) of the edge γ0 9 See

Section 3.5. leave to the reader the extension to more general locally finite families of subtrees, as, for instance, finite unions of those above. 10 We

12.4. Equidistribution and counting for quotient simplicial and metric trees

265

− path αγ, γ 0 , and αγ, γ 0 |[0, λγ, γ 0 ] is the shortest geodesic arc starting from a point of − γD and ending at a point of γ 0 D+ . For all γ, γ 0 in Γ such that γD− and γ 0 D+ are disjoint, we define the multiplicity of the common perpendicular αγ, γ 0 from γD− to γ 0 D+ as

mγ, γ 0 =

1 Card(γΓD

−1 −γ

∩ γ 0 ΓD+ γ 0 −1 )

.

(12.9)

Note that mγ, γ 0 = 1 for all γ, γ 0 ∈ Γ when Γ acts freely on EX (for instance, when Γ is torsion-free). Generalising the definition for simplicial trees in Section 11.4, we set k X e c(α) = e c (ei ) λ(ei ) , i=1

for any edge path α = (e1 , . . . , ek ) in X. For t > 0 in the continuous-time case and t = n ∈ N − {0} in the simplicial case, let X ND− , D+ (t) = me, γ eec (αe, γ ) , (12.10) [γ]∈ ΓD− \Γ/ΓD+ 0 0, giving for every τ ∈ N − {0} an asymptotic on X ND− , D+ , τ (n) = me, γ eec (αe, γ ) . [γ]∈ ΓD− \Γ/ΓD+ n−τ 2 and L+ > 2 respectively, then as N → +∞, the number of common perpendiculars of even length at most 2N from Y− to Y+ is equal to (p + q) L− L+ (pq)N +1 + O(rN ) (12.12) 2 (pq − 1) |EY| √ for some r < pq. Proof. As in the above proof of Example 12.13 (3), let X → Y be a universal cover of Y, with covering group Γ, and let D± be a geodesic line in X mapping to Y± . Let V1 X be the preimage of Vp Y and let V2 X be one of Vq Y. We normalise the Patterson density (µx )x∈V X of Γ so that kµx k = √ degX (x) . By Proposition 8.4 degX (x)−1

(3) with k = 1 and trivial vertex stabilisers, and since a cycle of length λ in a biregular graph of different degrees p + 1 and q + 1 has exactly λ2 vertices of degree either p + 1 or q + 1, we have √ X X √ L∓ p kµx k (degX (x) − k) ± kσD k = = p = . ∓ ,1 degX (x) 2 ∓ ∓ Γx∈Vp Y L∓

y∈Vp Y

√ q

± Similarly, kσD ∓ ,2 k = 2 . The result without the error term then follows from Theorem 12.12, using Proposition 8.1 (2) and Remark 2.11, since the number we are looking for is ND− ,D+ ,1,1 (2N ) + ND− ,D+ ,2,2 (2N ). We refer to Section 12.6 (see Remark (ii) following the proof of Theorem 12.16) for the error term.

12.5

Counting for simplicial graphs of groups

In this section, we give an intrinsic translation “à la Bass–Serre” of the counting result in Theorem 12.9 using graphs of groups (see [Ser3] and Section 2.6 for background information). Let (Y, G∗ ) be a locally finite connected graph of finite groups, and let 13 (Y± , G± Let c : EY → R be a system of ∗ ) be connected subgraphs of subgroups. conductances on Y. 13 See

Section 2.6 for definitions and background.

272


Let X be the Bass–Serre tree of the graph of groups (Y, G∗ ) (with geometric realisation X = |X|1 ) and Γ its fundamental group (for an arbitrary choice of basepoint). Assume that Γ is nonelementary. We denote by G (Y, G∗ ) = Γ\G X and gt : G (Y, G∗ ) → G (Y, G∗ ) t∈Z the quotient of the (discrete-time) geodesic flow on G X, by e c : X → R the (Γ-invariant) lift of c, with δc its critical exponent (assumed to be finite) and Fec : T 1 X → R its associated potential, by mc the Gibbs measure on G (Y, G∗ ) associated with a choice of Patterson densities (µ± x )x∈X for the pairs (Γ, Fc± ), by D± two subtrees in X such that the quotient graphs of groups ± ΓD± \\D± identify with (Y± , G± ∗ ) (see below for precisions), and by σ(Y∓ ,G∓ ) the ∗ associated skinning measures. The fundamental groupoid π(Y, G∗ ) of (Y, G∗ )14 is the quotient of the free product of the groups Gv for v ∈ V Y and of the free group on EY by the normal subgroup generated by the elements e e and e ρe (g) e ρ e (g)−1 for all e ∈ EY and g ∈ Ge . We identify each Gx for x ∈ V Y with its image in π(Y, G∗ ). Let n ∈ N − {0}. A (locally) geodesic path of length n in the graph of groups (Y, G∗ ) is the image α in π(Y, G∗ ) of a word, called reduced in [Bass, 1.7], h0 e1 h1 e2 . . . hn−1 en hn with • ei ∈ EY and t(ei ) = o(ei+1 ) for 1 6 i 6 n − 1 (so that (e1 , . . . , en ) is an edge path in the graph Y); • h0 ∈ Go(e1 ) and hi ∈ Gt(ei ) for 1 6 i 6 n; • if ei+1 = ei , then hi does not belong to ρei (Gei ), for 1 6 i 6 n − 1. Its origin is o(α) = o(e1 ) and its endpoint is t(α) = t(en ). They do not depend on the chosen words with image α in π(Y, G∗ ). + + A common perpendicular of length n from (Y− , G− ∗ ) to (Y , G∗ ) in the graph of groups (Y, G∗ ) is the double coset + [α] = G− o(α) α Gt(α)

of a geodesic path α of length n as above, such that: • α starts transversally from (Y− , G− ∗ ), that is, its origin o(α) = o(e1 ) belongs to V Y− and h0 ∈ / G− ρ (G ) if e1 ∈ EY− , e e 1 1 o(e1 ) • α ends transversally in (Y+ , G+ ∗ ), that is, its endpoint t(α) = t(en ) belongs + to Y+ and hn ∈ / ρen (Gen ) G+ t(en ) if en ∈ EY . Note that these two notions do not depend on the representative of the double + coset G− o(α) α Gt(α) , and we also say that the double coset [α] starts transversally − − from (Y− , G− ∗ ) or ends transversally in (Y , G∗ ). 14 It

is denoted by F (Y, G∗ ) in [Ser3, §5.1], and called the path group in [Bass, 1.5]; see also [Hig].

12.5. Counting for simplicial graphs of groups

273

We denote by Perp((Y± , G± ∗ ), n) the set of common perpendiculars in (Y, G∗ ) + + of length at most n from (Y− , G− ∗ ) to (Y , G∗ ). We denote by c(α) =

n X

c(ei )

i=1

the conductance of a geodesic path α as above, which depends only on the double class [α]. We define the multiplicity mα of a geodesic path α as above by mα =

1 . −1 ) Card(G− ∩ α G+ o(α) t(α) α

It depends only on the double class [α] of α. We define the counting function of the common perpendiculars in (Y, G∗ ) of length at most n from (Y− , G− ∗ ) to (Y+ , G+ ) (counted with multiplicities and with weights given by the system of ∗ conductances c) as + (n) = N(Y− ,G− + ∗ ), (Y ,G∗ )

X

mα ec(α) .

[α]∈Perp((Y± ,G± ∗ ),n)

Theorem 12.14. Let (Y, G∗ ), (Y± , G± ∗ ), and c be as at the beginning of this section. Assume that the critical exponent δc of c is finite and positive, and that the Gibbs measure mc on G (Y, G∗ ) is finite and mixing for the discrete-time geodesic flow. ± Assume that the skinning measures σ(Y are finite and nonzero. Then as n ∈ N ∓ ,G∓ ) ∗ tends to ∞, + (n) ∼ N(Y− ,G− + ∗ ), (Y ,G∗ )

+ − eδc kσ(Y k kσ(Y k − ,G− ) + ,G+ ) ∗

(eδc − 1) kmc k

∗

eδc n .

Proof. Let X be the Bass–Serre tree of (Y, G∗ ) and Γ its fundamental group (for an arbitrary choice of basepoint). As seen in Section 2.6, the Bass–Serre trees ± D± of (Y± , G± ∗ ), with fundamental groups Γ , identify with simplicial subtrees ± ± D of X, such that Γ are the stabilisers ΓD± of D± in Γ, and that the maps (ΓD± \D± ) → (Γ\X) induced by the inclusion maps D± → X by taking the quotient are injective. As seen in Definition 2.10, for all z ∈ V Y ∪ EY and e ∈ EY, we fix a lift g = t(e ze ∈ V X ∪ EX of z and ge ∈ Γ such that ee = e e, ge t(e) e), Gz = Γze, and −1 the monomorphism ρe : Ge → Gt(e) is γ 7→ ge γge . We assume, as we may, that ze ∈ V D± ∪ ED± if z ∈ V Y± ∪ EY± . We assume, as we may using Equation (2.22), that if e ∈ EY± , then ge ∈ ΓD± . We denote by p : X → Y = Γ\X the canonical projection.

274


γD− o(f1 ) γγ0 γ

−1

γ1 ee1 e1 Y−

γ 0 D+ t(fk ) −1 γ 0 γk+1 γ 0 ] γ 0 t(e k)

γk eei

ge1

fk

γi+1

γi

] γ o(e 1) γ ] o(e ) 1

fi+1

fi

f1

ei

gei

hi

eg i+1 gei+1 gi ) t(e

ei+1

h0

eek

gek

γ0 ] t(e k)

ek hk

Y+

For all γ, γ 0 ∈ Γ such that γD− and γ 0 D+ are disjoint, the common perpendicular αγD− , γ 0 D+ from γD− to γ 0 D+ is an edge path (f1 , f2 , . . . , fk ) with ∼ o(f1 ) ∈ γD− and t(fk ) ∈ γ 0 D+ . Note that γ −1 o(f1 ) and p(o(f1 )) are two vertices ∼ −1 of D− in the same Γ-orbit, and that γ 0 t(fk ) and p(t(fk )) are two vertices of D+ in the same Γ-orbit. Hence by Equation (2.22), we may choose γ0 ∈ ΓD− such that ∼ ∼ −1 γ0 γ −1 o(f1 ) = p(o(f1 )) and γk+1 ∈ ΓD+ such that γk+1 γ 0 t(fk ) = p(t(fk )). For ] 1 6 i 6 k, choose γi ∈ Γ such that γi fi = p(f i ). We define (see the above picture) • ei = p(fi ) for 1 6 i 6 k, • hi = ge−1 γi γi+1−1 g ei+1 , which belongs to Γt(e g) = Gt(ei ) for 1 6 i 6 k − 1, i i

• h0 =

γ0 γ −1 γ1−1 g e1

=γ

ge−1 γk γ 0 γk+1−1 k

• hk = Gt(ek ) .

−1

=

(γγ0 γ

−1

)γ1−1 g e1 ,

which belongs to Γo(e ]) = Go(e1 ) ,

−1 ge−1 γk (γ 0 γk+1 γ 0 )−1 γ 0 , k

1

which belongs to Γt(e ]) = k

Lemma 12.15. (1) The word h0 e1 h1 . . . hk−1 ek hk is reduced. Its image α in the fundamental groupoid π(Y, G∗ ) does not depend on the choices of γ1 , . . . , γk , and + + it starts transversally from (Y− , G− ∗ ) and ends transversally in (Y , G∗ ). The double class [α] of α is independent of the choices of γ0 and γk+1 . e from the set of common perpendiculars in X between disjoint images (2) The map Θ − of D and D+ under elements of Γ into the set of common perpendiculars in + + (Y, G∗ ) from (Y− , G− ∗ ) to (Y , G∗ ), sending αγD− , γ 0 D+ to [α], is invariant under the action of Γ at the source, and preserves the lengths and the multiplicities. e from the set of Γ-orbits of common perpendiculars (3) The map Θ induced by Θ in X between disjoint images of D− and D+ under elements of Γ into the set + + of common perpendiculars in (Y, G∗ ) from (Y− , G− ∗ ) to (Y , G∗ ) is a bijection, preserving the lengths and the multiplicities.


275

Proof. (1) If ei+1 = ei , then by the definition of hi , we have hi ∈ ρei (Gei ) = ge−1 Γeei gei ⇐⇒ gei hi ge−1 eei = eei i i ⇐⇒ gei ge−1 γi γi+1−1 g ei+1 ge−1 eei = eei i i ⇐⇒ γi γi+1−1 eg i+1 = eei −1 ⇐⇒ γi+1−1 eg i+1 = γi eei ⇐⇒ fi+1 = fi .

Hence the word h0 e1 h1 . . . hk−1 ek hk is reduced. The element γi for i ∈ {1, . . . , k} is uniquely determined up to multiplication on the left by an element of Γeei = Gei . If we fix15 i ∈ {1, . . . , k} and if we replace γi by γi0 = α γi for some α ∈ Gei , then only the elements hi−1 and hi change, replaced by elements that we denote by h0i−1 and h0i respectively. We have (if 2 6 i 6 k − 1, but otherwise the argument is similar by the definitions of h0 and hk ) −1 −1 h0i−1 ei h0i = g−1 α g ei ei ge−1 α γi γi+1−1 g ei+1 ei−1 γi−1 γi i −1 γi γi+1−1 g ei+1 . = g−1 g ei ρ ei (α)−1 ei ρei (α) ge−1 ei−1 γi−1 γi i

Since ρ ei (α)−1 ei ρei (α) is equal to ei −1 = ei in the fundamental groupoid, the words h0i−1 ei h0i and hi−1 ei hi have the same image in π(Y, G∗ ). Therefore α does not depend on the choices of γ1 , . . . , γk . We have o(α) = o(e1 ) ∈ V Y− and t(α) = t(ek ) ∈ V Y+ ; hence α starts from Y− and ends in Y+ . Assume that e1 ∈ EY− . Let us prove that h0 ∈ G− o(e1 ) ρ e1 (Ge1 ) if and only if f1 ∈ γ ED− . γ −1 f1 ] o(e 1)

ee1

o(γ −1 f1 ) g e1

D−

γ0

By the definition of ρ e1 , we have h0 ∈ G− o(e1 ) ρ e1 (Ge1 ) if and only if there −1 −1 exists α ∈ Γ o(e ∩ ΓD− such that α h0 ∈ g e1 Γ ee1 g e1 . By the definition of h0 ] 1) and since γ1 maps f1 to ee1 , we have α−1 h0 ∈ g e−1 Γ ee1 g e1 ⇐⇒ g e1 α−1 γ0 γ −1 γ1−1 g e1 g e−1 ee1 = ee1 1 1 ee1 . ⇐⇒ f1 = γ γ0−1 α g e−1 1 Since ee1 ∈ ED− and γ0 , α, g e1 all belong to ΓD− , this last condition implies that f1 ∈ γ ED− . Conversely (for future use), if f1 ∈ γ ED− , then (see the above 15 We

leave to the reader the verification that the changes induced by various i’s do not overlap.

276


] picture) γ0 γ −1 f1 is an edge of D− with origin o(e 1 ), in the same Γ-orbit as the −1 − ] edge g e1 ee1 of D , which also has origin o(e1 ). By Equation (2.22), this implies −1 −1 that there exists α ∈ Γ o(e ]) ∩ ΓD− such that f1 = γ γ0 α g e1 ee1 . By the above 1

equivalences, we hence have that h0 ∈ G− o(e1 ) ρ e1 (Ge1 ). Similarly, one proves that if ek ∈ EY+ , then hk ∈ ρek (Gek ) G+ t(ek ) if and 0 + only if fk ∈ γ ED . Since (f1 , . . . , fn ) is the common perpendicular edge path from γ D− to γ 0 D+ , this proves that α starts transversally from Y− and ends transversally in Y− . Note that the element γ0 ∈ ΓD− is uniquely defined up to multiplication on the left by an element of Γo(e ∩ ΓD− = G− ] o(e1 ) , and appears only as the first letter 1) in the expression of h0 . Note that the element γk+1 ∈ ΓD+ is uniquely defined up + −1 to multiplication on the left by an element of Γt(e ]) ∩ ΓD+ = Gt(ek ) ; hence γk+1 k

is uniquely defined up to multiplication on the right by an element of G+ t(ek ) , and appears only as the last letter in the expression of hk . Therefore α is uniquely defined in the fundamental groupoid π(Y, G∗ ) up to multiplication on the left by + an element of G− o(e1 ) and multiplication on the right by an element of Gt(ek ) , that is, the double class [α] ⊂ π(Y, G∗ ) is uniquely defined. (2) Let β be an element in Γ and let x = αγD− , γ 0 D+ be a common perpendicular in X between disjoint images of D− and D+ under elements of Γ. Let us prove that e x) = Θ(x). e Θ(β e x), we may take, instead Since β x = αβ γD− , β γ 0 D+ , in the construction of Θ(β e of the elements γ0 , γ1 , . . . , γk , γk+1 used to construct Θ(x), the elements ] γ0] = γ0 , γ1] = γ1 β −1 , . . . , γk] = γk β −1 , γk+1 = γk+1 . ]

And instead of γ and γ 0 , we now may use γ ] = β γ and γ 0 = β γ 0 . e The only terms involving γ, γ 0 , γ1 , . . . , γk in the construction of Θ(x) come −1 −1 −1 under the form γ γ1 in h0 , γi γi+1 in hi for 1 6 i 6 k − 1, and γk γ 0 in hk . ] Since (γ ] )−1 (γ1] )−1 = γ −1 γ1−1 , (γi] )(γi+1 )−1 = γi γi+1−1 for 1 6 i 6 k − 1, and ] ] e x) = Θ(x), e (γk )(γ 0 ) = γk γ 0 , this proves that Θ(β as desired. It is immediate that if the length of αγD− , γ 0 D+ is k, then the length of [α] is k. Let us prove that the multiplicity, given in Equation (12.9), mγ, γ 0 =

1 Card(γΓD

−1 −γ

∩ γ 0 ΓD+ γ 0 −1 )

of the common perpendicular αγD− , γ 0 D+ in X between γ D− and γ 0 D+ is equal to the multiplicity 1 mα = −1 ) Card(G− ∩ α G+ o(α) t(α) α + + of the common perpendicular α in (Y, G∗ ) from (Y− , G− ∗ ) to (Y , G∗ ).


277

Since the multiplicity mγ, γ 0 is invariant under the diagonal action by left translations of γ0−1 γ −1 ∈ Γ on (γ, γ 0 ), we may assume that γ = γ0 = id. Since the multiplicity mγ, γ 0 is invariant under right translation by γk+1−1 , which stabilises D+ , on the element γ 0 , we may assume that γk+1 = id. In particular, we have ] o(f1 ) = o(e 1)

and

] t(fk ) = γ 0 t(e k) .

We use the basepoint x0 = o(e1 ) in the construction of the fundamental group and the Bass–Serre tree of (Y, G∗ ), so that (see in particular [Bass, Eq. (1.3)]) a VX= β Gt(β) β∈π(Y, G∗ ) : o(β)=x0

and Γ = π1 (Y, G∗ ) = {β ∈ π(Y, G∗ ) : o(β) = t(β) = x0 } . Since an element in Γ that preserves D− and γ 0 D+ fixes pointwise its (unique) common perpendicular in X, we have ΓD− ∩ γ 0 ΓD+ γ 0

−1

= ΓD− ∩ Γγ 0 D+ = (Γo(f1 ) ∩ ΓD− ) ∩ (Γt(fk ) ∩ Γγ 0 D+ ) = (Γo(e ]) ∩ ΓD− ) ∩ (Γγ 0 t(e ]) ∩ Γγ 0 D+ ) . 1

k

G− o(e1 ) .

Note that Γo(e ∩ΓD− = By the construction of the edges in the Bass–Serre ] 1) tree of a graph of groups (see [Bass, page 11]), the vertex α Gt(ek ) is exactly the ] vertex t(fk ) = γ 0 t(e k ). By [Bass, Eq. (1.4)], we hence have α Gt(ek ) α−1 = Stabπ1 (Y,G∗ ) (α Gt(ek ) ) = Γγ 0 t(e ]) . k

Therefore mγ, γ 0 = mα . (3) Let [α] = Go(α) α Gt(α) be a common perpendicular in (Y, G∗ ) from (Y− , G− ∗) to (Y+ , G+ ), with representative α ∈ π(Y, G ), and let h e h . . . e h be a ∗ 0 1 1 k k ∗ reduced word whose image in π(Y, G∗ ) is α. We define • γ1 = g e1 h−1 0 , • f1 = γ1−1 ee1 , • assuming that γi and fi for some 1 6 i 6 k − 1 are constructed, let −1 γi+1 = g ei+1 h−1 i gei γi

and

fi+1 = γi+1−1 eg i+1 ,

• with γk and fk constructed by induction, finally let γ 0 = γk−1 gek hk . It is easy to check, using the equivalences in the proof of Lemma 12.15 (1) with γ = γ0 = γk+1 = id, that the sequence (f1 , . . . , fk ) is the edge path of ] a common perpendicular in X from D− to γ 0 D+ with origin o(e 1 ) and endpoint 0 ] γ t(ek ).

278


If h0 is replaced by α h0 with α ∈ G− o(e1 ) , then by induction, f1 , f2 , . . . , fk 0 are replaced by αf1 , αf2 , . . . , αfk and γ is replaced by αγ 0 . Note that the edge path (αf1 , αf2 , . . . , αfk ) is then the common perpendicular edge path from D− = αD− to αγ 0 D+ . If hk is replaced by hk α with α ∈ G+ t(ek ) , then f1 , f2 , . . . , fk are unchanged, and γ 0 is replaced by γ 0 α. Note that γ 0 α D+ = γ 0 D+ . Hence the map that associates to [α] the Γ-orbit of the common perpendicular in X from D− to γ 0 D+ with edge path (f1 , . . . , fk ) is well defined. It is easy to see by construction that this map is the inverse of Θ. Theorem 12.14 now follows from Theorem 12.9.

12.6

Error terms for equidistribution and counting for metric and simplicial graphs of groups

In this section, we give error terms to the equidistribution and counting results of Section 12.4, given by Theorem 12.8 for metric trees (and their continuous-time geodesic flows) and by Theorem 12.9 for simplicial trees (and their discrete-time geodesic flows), under appropriate assumptions on bounded geometry and the rate of mixing. ± Let (X, λ), X, Γ, e c, c, Fec , Fc , δc , D± , D± , D ± , λγ,γ 0 , αγ,γ 0 , αγ,γ 0 , mγ,γ 0 be as in Section 12.4. We first consider the simplicial case (when λ = 1), for the discrete-time geodesic flow.

Theorem 12.16. Let X be a locally finite simplicial tree without terminal vertices, let Γ be a nonelementary discrete subgroup of Aut(X), let e c be a system of conductances on X for Γ, and let D± be nonempty proper simplicial subtrees of X. Assume that the critical exponent δc is finite and positive, that the Gibbs measure mc (for the discrete-time geodesic flow) is finite, and that the skinning measures ± σD ∓ are finite and nonzero. Assume furthermore that (1) at least one of the following holds: • ΓD± \∂D± is compact, • C ΛΓ is uniform and Γ is a tree lattice of C ΛΓ, (2) there exists β ∈ ]0, 1] such that the discrete-time geodesic flow on (Γ\G X, mc ) is exponentially mixing for the β-H¨ older regularity. b

Then there exists κ0 > 0 such that for all ψ ± ∈ Ccβ (Γ\ G X), we have, as n → +∞, eδc − 1 kmc k e−δc n eδc Z =

ψ

−

+ dσD −

X

− me, γ eec (αe, γ ) ψ − (Γαe,γ ) ψ + (Γαγ+−1 ,e )

[γ] ∈ΓD− \Γ/ΓD+ 0 0 (independent of ψe± ) such that as n → +∞, X eδc − 1 −δc n km k e eec (αγ ) ψe− (αγ− ) ψe+ (αγ+ ) c δ ec γ∈Γ, 0 e2 large enough. Let 0 < η < 1. We introduce the following 16 modification of the test functions φ± η: ± ∓ e± Φ± η = (hη, R ψ ) ◦ fD ±

1Vη,∓R (∂∓1 D± ) .

As in Lemma 10.1, the functions Φ± η are measurable and satisfy Z Z ± Φη dm ec = ψe± de σ∓ . GX

(12.14)

1 D± ∂∓

Lemma 12.17. The maps Φ± older-continuous with η are β-H¨ e± kΦ± η kβ = O(kψ kβ ) .

(12.15)

± ± Proof. Since X is a simplicial tree and η < 1, we have Vw, η, R = B (w, R) for 1 every w ∈ ∂± D∓ . By the proof of Lemma 3.3,17 there exists cR > 0 depending only on R such that if `0 ∈ G X satisfies d(`, `0 ) 6 cR , then `0 coincides with ` on 1 ±[0, ln R + 1]. Therefore, if ` ∈ B ± (w, R) for some w ∈ ∂± D∓ and d(`, `0 ) 6 cR , 16 See

± Equation (10.4) for the definition of φ± η and Equation (10.1) for the definition of hη, R ,

± −1 , since X is simplicial, as seen in Equation which simplifies as h∓ η, R (w) = (µW ± (w) (B (w, R))) (11.27). 17 See also the proof of Lemmas 3.4 and 3.8.

280


then `0 coincides with w on ±[ln R, ln R + 1], and thus `0 ∈ B ± (w0 , R), where 1 0 w0 ∈ ∂± D∓ is the geodesic ray with w0 (0) = w(0) and w± = `0± . Hence (see Section 3.1) the characteristic function 1V ± (∂ 1 D∓ ) is cR -locally constant, thus ± η, R β-H¨ older-continuous by Remark 3.11. By Assumption (1) in the statement of Theorem 12.16, the denominator of h∓ η, R (w) =

1 µW ± (w) (B ± (w, R))

is at least a positive constant depending only on R; hence h∓ η, R is bounded by a constant depending only on R. Since the map 1B ± (w,R) is cR -locally constant, so is the map h∓ η, R . The result then follows from Lemma 3.8 and Equation (3.7). In order to prove Equation (12.13), as in the proofs of Theorems 12.7 and 11.9, for all N ∈ N, we estimate in two ways the quantity Iη (N ) =

N X n=0

eδc n

X Z γ∈Γ

−bn/2c dn/2e −1 Φ− `) Φ+ γ `) dm e c (`) . η (g η (g

`∈G X

(12.16)

On the one hand, as in order to obtain Equation (12.6), using now Assumption (2) in the statement of Theorem 12.16 on the exponential mixing for the discrete-time geodesic flow, a geometric sum argument, and Equations (12.14) and (12.15), we have Iη (N ) =

eδc (N +1) δ (e c − 1) kmc k

Z 1 D− ∂+

ψe− de σ+

+ O(e

−κ N

Z 1 D+ ∂−

ψe+ de σ−

kψe− kβ kψe+ kβ ) .

(12.17)

On the other hand, exchanging the summations over γ and n in the definition of Iη (N ), we have Iη (N ) =

N X X γ∈Γ n=0

eδc n

Z GX

−bn/2c dn/2e −1 Φ− `) Φ+ γ `) dm e c (`) . η (g η (g

With the simplifications in Step 3T of the proof of Theorem 11.1 given by the proof of Theorem 11.9, if η < 12 and if ` ∈ G X belongs to the support of the + − −bn/2c dn/2e + function Φ− Φ+ ◦ γ −1 , setting w− = fD = fγD − (`) and w + (`), η ◦g η ◦g − − + + we then have λγ = n, w (0) = αγ (0), w (0) = γαγ (0), and w− (bn/2c) = w+ (−dn/2e) = `(0) = αγ− (bn/2c) = γαγ+ (−dn/2e) .

12.6. Error terms for equidistribution and counting

281

Hence by the triangle inequality, d(w− , αγ− ) =

Z

+∞

d(w− (s), αγ− (s)) e−2s ds 6 e−2bn/2c

bn/2c

= O(e

Z

+∞

0

2s0 e−2s ds0

0

−λγ

).

Similarly, d(w+ , γαγ+ ) = O(e−λγ ). Therefore, since ψe± is β-Hölder-continuous, | ψe− (w− ) − ψe− (vγ− ) | = O(e−βλγ kψe− kβ ) , | ψe+ (γ −1 w+ ) − ψe+ (vγ+ ) | = O(e−βλγ kψe+ kβ ) . ∓ ± e± Note that now Φ± η = ψ ◦ fD ± φη , so that

Iη (N ) =

X

ψe− (αγ− )ψe+ (αγ+ ) + O(e−2βλγ kψe− kβ kψe+ kβ ) ×

γ∈Γ N X n=0

eδc n

Z GX

−bn/2c dn/2e −1 φ− `) φ+ γ `) dm e c (`) . η (g η (g

Now if η < 21 , Equation (12.13) with κ0 = min{2β, κ} follows as in Steps 3T and 4T of the proof of Theorem 11.1 with the simplifications given by the proof of Theorem 11.9. The end of the proof of the equidistribution claim of Theorem 12.16 follows from Equation (12.13) like that of Theorem 12.7 from Equation (12.4). The counting claim follows from the equidistribution one by taking ψ ± = 1ΓVη,R (∂∓1 D± ) , which has compact support, since ΓD± \∂D± is assumed to be compact, and is β-H¨ older-continuous by previous arguments. Remarks. (i) Assume that e c = 0, that the simplicial tree X0 with |X0 |1 = C ΛΓ is uniform without vertices of degree 2, that LΓ = Z, and that Γ is a geometrically finite tree lattice of X0 . Then all assumptions of Theorem 12.16 are satisfied by the results of Section 4.4 and Corollary 9.6. Therefore we have an exponentially small error term in the (joint) equidistribution of the common perpendiculars, and in their counting if ΓD± \∂D± is compact; see Examples 12.11 (2) and 12.13 (4). (ii) Assume in this remark that Assumption (2) of the above theorem is replaced by the assumptions that C ΛΓ is uniform without vertices of degree 2, that LΓ = 2Z, and that there exists β ∈ ]0, 1] such that the square of the discrete-time geodesic flow on (Γ\Geven X, mc ) is exponentially mixing for the β-Hölder regularity, for instance, by Corollary 9.6 (2) if Γ is geometrically finite. Then a similar proof (replacing the references to Theorem 11.9 by references to Theorem 11.11) shows

282


b

0

±

that there exists κ > 0 such that for all ψ ∈ e2δc − 1 kmc k e−δc n 2 e2δc Z =

ψ

−

+ dσD −

Z

X

Ccβ (Γ\ G X),

we have, as n → +∞,

− me, γ eec (αe, γ ) ψ − (Γαe,γ ) ψ + (Γαγ+−1 ,e )

[γ] ∈ΓD− \Γ/ΓD+ 0 0. As in the proof of Theorem 12.7, Z Z ± Φ dm ec = ψe± de σ∓ GX

η

1 D± ∂∓

and there exists κ00 > 0 such that 00

−κ kΦ± kψe± kk, β ) . η kk, β = O(η

We again estimate in two ways as T → +∞ the quantity Z T X Z −t/2 t/2 −1 Iη (T ) = eδc t Φ− `) Φ+ γ `) dm e c (`) dt . η (g η (g 0

γ∈Γ

(12.21)

`∈G X

Note that as T → +∞, e−δc T

Z 1

T

eδc t t−N n dt = e−δc T

Z 1

T /2

eδc t t−N n dt + e−δc T

Z

T

eδc t t−N n dt

T /2

= O(e−δc T /2 ) + O(T −N n+1 ) = O(T −(N −1) n ) . Using Equation (12.18), an integration argument, and the above two properties of the test functions, we hence have Z Z eδc T − + e Iη (T ) = ψ de σ ψe+ de σ− 1 D− 1 D+ δc kmc k ∂+ ∂− (12.22) −(N −1) n −2κ00 e− + e + O(T η kψ kk, β kψ kk, β ) . As in Step 3T of the proof of Theorem 11.1, for all γ ∈ Γ and t > 0 large −t/2 t/2 enough, if ` ∈ G X belongs to the support of Φ− Φ+ ◦ γ −1 (which η ◦g η ◦g

12.6. Error terms for equidistribution and counting

285

−t/2 + is contained in the support of φ− φη ◦ gt/2 ◦ γ −1 ), then we may define η ◦g + − + w − = fD = fγD − (`) and w + (`). By property (iii) in Step 3T of the proof of Theorem 11.1, the generalised geodesic lines w− and αγ− coincide, besides on ] − ∞, 0], at least on [0, 2t − η], and similarly, w+ and γαγ+ coincide, besides on [0, +∞[ , at least on [− 2t + η, 0]. λ Therefore, by an easy change of variable and since | 2t − 2γ | 6 η,

d(w

−

, αγ− )

+∞

Z 6

d(w t 2 −η

−

(s), αγ− (s)) e−2s

ds 6 e

−2( 2t −η)

Z

+∞

2s e−2s ds

0

= O(e−t ) = O(e−λγ ) . Similarly, d(w+ , γαγ+ ) = O(e−λγ ). Hence since ψe± is β-Hölder-continuous, we have | ψe− (w− ) − ψe− (αγ− ) |, | ψe+ (γ −1 w+ ) − ψe+ (αγ+ ) | = O(e−βλγ kψe± kβ ) . Therefore, as in the proof of Theorem 12.16, we have X Iη (T ) = ψe− (αγ− )ψe+ (αγ+ ) + O(e−2βλγ kψe− kβ kψe+ kβ ) × γ∈Γ

Z 0

T

eδ t

Z `∈G X

−t/2 −1 t/2 b− b+ Φ `) Φ g `) dm e c (`) dt . η (g η (γ

Finally, Equation (12.19) follows as at the end of the proof of Equation (12.4), using Equations (12.20) and (11.17) instead of Equations (12.7) and (11.21), by taking η = T −n and N = 2(dκ00 e + 1). The end of the proof of the equidistribution claim of Theorem 12.18 follows from Equation (12.19) like that of Theorem 12.7 from Equation (12.4). The counting claim follows from the equidistribution one by taking ψ ± to be 1 ± β-H¨ older-continuous plateau functions around ΓVη,R (∂∓ D ). We are now in a position to prove one of the counting results in the introduction. Proof of Theorem 1.9. Let X be the universal cover of Y, with fundamental group Γ for an arbitrary choice of basepoint, and let D± be connected components of the preimages of Y± in X. Assertion (1) of Theorem 1.9 follows from Theorem 12.18 and its subsequent remark. Assertion (2) of Theorem 1.9 follows from Theorem 12.16 and its Remarks (ii) and (i) following its proof, respectively, as Y is bipartite or not.

Chapter 13

Geometric Applications In this final chapter of Part II, we apply the equidistribution and counting results obtained in the previous chapters in order to study geometric equidistribution and counting problems for metric and simplicial trees concerning conjugacy classes in discrete isometry groups and closed orbits of geodesic flows.

13.1

Orbit counting in conjugacy classes for groups acting on trees

In this section, we study the orbital counting problem for groups acting on metric or simplicial trees when we consider only the images by elements in a given conjugacy class. We refer to the introduction for motivations and previously known results for manifolds (see [Hub1] and [PaP15]) and graphs (see [Dou] and [KeS]). The main tools we use are Theorem 12.8 for the metric tree case and Theorem 12.9 for the simplicial tree case, as well as their error terms given in Theorems 12.18 and 12.16. In particular, we obtain a much more general version of Theorem 1.12 in the introduction. Let (X, λ) be a locally finite metric tree without terminal vertices, let us denote by X = |X|λ its geometric realisation, let x0 ∈ V X, and let Γ be a nonelementary discrete subgroup of Aut(X, λ).1 Let e c : EX → R be a Γ-invariant system of conductances, let Fec and Fc be its associated potentials on T 1 X and Γ\T 1 X respectively, and let δc = δΓ, Fc± be its critical exponent.2 Let (µ± x )x∈X (respec± tively (µx )x∈V X ) be (normalised) Patterson densities for the pairs (Γ, Fc± ), and let m ec = m e Fc and mc = mFc be the associated Gibbs measures on G X and Γ\G X (respectively G X and Γ\G X) for the continuous-time geodesic flow (respectively the discrete-time geodesic flow, when λ = 1).3 1 See

Section 2.6 for definitions and notation. Section 3.5 for definitions and notation. 3 See Sections 4.3 and 4.4 for definitions and notation. 2 See


287

288

Chapter 13. Geometric Applications

Recall that the virtual center Z virt (Γ) of Γ is the finite (normal) subgroup of Γ consisting of the elements γ ∈ Γ acting by the identity on the limit set ΛΓ of Γ in ∂∞ X; see, for instance, [Cha, §5.1]. If ΛΓ = ∂∞ X (for instance, if Γ is a tree lattice), then Z virt (Γ) = {id}. For any nontrivial element γ in Γ with translation length λ(γ) in X, let Cγ be • the translation axis of γ if γ is loxodromic on X, • the fixed point set of γ if γ is elliptic on X, and let ΓCγ be the stabiliser of Cγ in Γ. In the simplicial case (that is, when λ = 1), Cγ is a simplicial subtree of X. Note that we have λ(γ) = λ(γ 0 γ(γ 0 )−1 ) and γ 0 Cγ = Cγ 0 γ(γ 0 )−1 for all γ 0 ∈ Γ, and that for all x0 ∈ X, d(x0 , Cγ ) =

d(x0 , γx0 ) − λ(γ) . 2

(13.1)

Let D = (γ 0 Cγ )γ 0 ∈Γ/ΓCγ , which is a locally finite Γ-invariant family of nonempty proper (since γ 6= id) closed convex subsets of X.4 By the equivariance properties of the skinning measures, the total mass of the skinning measure5 − − σD depends only on the conjugacy class K of γ in Γ, and will be denoted by kσK k. This quantity, called the skinning measure of K, is positive unless ∂∞ Cγ = ΛΓ, which is equivalent to γ ∈ Z virt (Γ) (and implies in particular that γ is ellip− tic). Furthermore, kσK k is finite if γ is loxodromic, and it is finite if γ is elliptic and ΓCγ \(Cγ ∩ C ΛΓ) is compact. This last condition is in particular satisfied if Cγ ∩ C ΛΓ itself is compact, and this is the case, for instance, if for some k > 0, the action of Γ on X is k-acylindrical (see, for instance, [Sel, GuL]), that is, if every element of Γ fixing a segment of length k in C ΛΓ is the identity. For every γ ∈ Γ − {e}, we define mγ =

1 Card(Γx0 ∩ ΓCγ )

,

which is a natural multiplicity of γ, and equals 1 if the stabiliser of x0 in Γ is trivial (for instance, if Γ is torsion-free). Note that for every β ∈ Γ, the real number mβγβ −1 depends only on the double coset of β in Γx0 \Γ/ΓCγ . The centraliser ZΓ (γ) of γ in Γ is contained in the stabiliser of Cγ in Γ. The index iK = [ΓCγ : ZΓ (γ)] depends only on the conjugacy class K of γ; it will be called the index of K. The index iK is finite if γ is loxodromic (the stabiliser of its translation axis Cγ is then virtually cyclic), and also finite if Cγ is compact (as, for instance, if the action of Γ on X is k-acylindrical for some k > 0). 4 See 5 See

Section 7.2 for definitions and notation. the previous footnote.

13.1. Orbit counting in conjugacy classes for groups acting on trees

289

We define cγ =

k X

e c (ei ) λ(ei ) ,

i=1

where (e1 , . . . , ek ) is the shortest edge path from x0 to Cγ . We finally define the orbital counting function in conjugacy classes, counting with multiplicities and weights coming from the system of conductances, as X NK, x0 (t) = m α ec α , α∈K, d(x0 , αx0 )6t

for t ∈ [0, +∞[ (simply t ∈ N in the simplicial case). When the stabiliser of x0 in Γ is trivial and when the system of conductances c vanishes, we recover the definition of the introduction (above Theorem 1.12). Theorem 13.1. Let K be the conjugacy class of a nontrivial element γ0 of Γ, with − finite index iK , and with positive and finite skinning measure kσK k. Assume that δc is finite and positive. (1) Assume that mc is finite and mixing for the continuous-time geodesic flow on Γ\G X. Then, as t → +∞, − − iK kµ+ x0 k kσK k e NK, x0 (t) ∼ δc kmc k

λ(γ0 ) 2

e

δc 2

t

.

If ΓCγ \(Cγ ∩ C ΛΓ) is compact when γ ∈ K is elliptic and if there exists β ∈ ]0, 1] such that the continuous-time geodesic flow on (Γ\G X, mc ) has superpolynomial δc decay of β-H¨ older correlations, then the error term is O t−n e 2 t for every n ∈ N. (2) Assume that λ = 1 and that mc is finite and mixing for the discrete-time geodesic flow on Γ\G X. Then, as n → +∞, NK, x0 (n) ∼

− n−λ(γ0 ) eδc iK kµ+ x0 k kσK k eδc b 2 c . δ c (e − 1) kmc k

If ΓCγ \(Cγ ∩ C ΛΓ) is compact when γ ∈ K is elliptic and if there exists β ∈ ]0, 1] such that the discrete-time geodesic flow on (Γ\G X, mc ) is exponentially mixing for the β-H¨ older regularity, then the error term is O e(δc −κ)n/2 for some κ > 0. One can also formulate a version of the above result for groups acting on bipartite simplicial trees based on Theorem 12.12 and Remark (ii) following the proof of Theorem 12.16. The error term in Assertion (1) holds, for instance, if e c = 0, X is uniform, and either Γ\X is compact and the length spectrum LΓ is 2-Diophantine or Γ is a geometrically finite tree lattice of X whose length spectrum LΓ is 4-Diophantine, by the remark following Theorem 12.18. When Γ\X is compact and Γ has no torsion (in particular, Γ has then a very restricted group structure, since it is then a free group), we thus recover a result of [KeS].

290


The error term in Assertion (2) holds, for instance, if e c = 0, X is uniform with vertices of degrees at least 3, Γ is a geometrically finite tree lattice of X with length spectrum equal to Z, by Remark (i) following the proof of Theorem 12.16. Theorem 1.12 in the introduction follows from this theorem, using Proposition 4.16 (3) and Theorem 4.17. Proof. We give a full proof only of Assertion (1) of this theorem; Assertion (2) follows similarly using Theorems 12.9 and 12.16 instead of Theorems 12.8 and 12.18. The proof is similar to the proof of [PaP15, Thm. 8]. Let D− = {x0 } and D = Cγ0 . Let D − = (γD− )γ 0 ∈Γ/ΓD− and D + = (γD+ )γ∈Γ/ΓD+ . By Equation (7.15), we have kµ+ x0 k + kσD . −k = |Γx0 | +

By Equation (13.1), by the definition6 of the counting function ND− , D+ , and by the last claim of Theorem 12.8, we have, as t → +∞, X

X

m α ec α =

α∈K, 0 0, if t is large enough, we hence have e− m e c (ψ 0 ) 6 νt000 ⊗ ds(ψ 0 ) 6 e m e c (ψ 0 ) . Note that the support of any continuous function with compact support on G X may be covered by finitely many open sets V (x) × R, where x ∈ X. The induced measure11 of the Γ-invariant measure νt000 ⊗ ds on G X is the measure νt00 on Γ\G X. Since mc is the induced measure of m e c , this proves Equation (13.3), hence gives the first claim of Theorem 13.3. The second claim follows from the first one in the same way as in [PauPS, Thm. 9.11]: Consider the measures m0t = δc e−δc t

X

eLg (F ) Lg and m00t = δc te−δc t

g∈Per0 (t)

eLg (F )

X g∈Per0 (t)

Lg λ(g)

on Γ\G X. Fix a continuous map ψ : Γ\G X → [0, +∞[ with compact support. For every > 0, for every t > 0, we have, since λ(g) > e− t for all g in the second sum below, eLg (F ) Lg (ψ)

X

m00t (ψ) > m0t (ψ) > δc e−δc t

g∈Per0 (t)−Per0 (e− t)

> e− δc te−δc t

eLg (F )

X g∈Per0 (t)−Per0 (e− t)

= e− m00t (ψ) − e− δc t e−δc t

X g∈Per0 (e− t)

Lg (ψ) λ(g)

eLg (F )

Lg (ψ) . λ(g)

By the local finiteness of X, the closed orbits meeting the support of ψ have a positive lower bound on their lengths. Thus, by the first claim of Theorem 13.3, there exists a constant C > 0 such that the second term of the above difference is − at most C t e−δc t eδc e t , which tends to 0 as t tends to +∞. Hence by applying 11 See

the beginning of Chapter 12.

13.2. Equidistribution and counting of closed orbits

295

twice the first claim of Theorem 13.3, we have mc (ψ) = lim m0t (ψ) 6 lim inf m00t (ψ) 6 lim sup m00t (ψ) t→+∞ t→+∞ kmc k t→+∞ mc (ψ) 6 lim e m0t (ψ) = e , t→+∞ kmc k and the result follows by letting go to 0 (and writing any continuous function ψ : T 1 M → R with compact support as the sum of its positive and negative parts). In order to prove the last claim of Theorem 13.3, assume that Γ is geometrically finite. The narrow convergence follows as in [PauPS, Thm. 9.16], using the fact that there exists a compact subset of Γ\G X meeting every periodic orbit of the geodesic flow, and replacing Lemma 3.10, Lemma 3.2, Equation (112), Theorem 8.3 and Corollary 5.15 of [PauPS] by respectively Lemma 4.3 (1), the HC property, Proposition 3.20 (4), Theorem 4.8, and Corollary 4.7 (1). In a similar way, replacing in the above proof Corollary 11.2 of Theorem 11.1 by the similar corollary of Theorem 11.9 with D − = (γx)γ∈Γ and D − = (γy)γ∈Γ for any x, y ∈ V X, we get the following analogous result for simplicial trees. For every n ∈ N, let now Per(n) be the set of periodic orbits of the discrete-time geodesic flow on Γ\G X with length at most n and let Per0 (n) be the subset of primitive ones. Theorem 13.4. Let X be a locally finite simplicial tree without terminal vertices, let Γ be a nonelementary discrete subgroup of Aut(X), and let e c : EX → R be a Γ-invariant system of conductances. Assume that the critical exponent δc of c is finite and positive and that the Gibbs measure mc is finite and mixing for the discrete-time geodesic flow. As n → +∞, the measures eδc − 1 −δc n e eδc

X

ec(g) Lg

and

g∈Per0 (n)

eδc − 1 n e−δc n eδc

X g∈Per0 (n)

ec(g)

Lg λ(g)

mc for the weak-star convergence of measures. If Γ is geometrically converge to km ck finite, the convergence holds for the narrow convergence.

In the special case that Γ\X is a compact graph and F = 0, the following immediate corollary of Theorem 13.3 is proved in [Gui], and it follows from the results of [ParP].12 There are also some works on nonbacktracking random walks with related results. For example, for regular finite graphs, [LuPS1] and [Fri1] (see [Fri2, Lem. 2.3]) give an expression of the irreducible trace that is the number of closed walks of a given length. 12 See

the introduction of [Sha] for comments.


296

Corollary 13.5. Let (X, λ) be a locally finite metric tree without terminal vertices. Let Γ be a geometrically finite discrete subgroup of Aut(X, λ). Let c : EX → R be a Γ-invariant system of conductances, with finite and positive critical exponent δc . (1) If the Gibbs measure mc is finite and mixing for the continuous-time geodesic flow, then X eδc t ec(g) ∼ δc t 0 g∈Per (t)

as t → +∞. (2) If λ = 1 and if the Gibbs measure mc is finite and mixing for the discrete-time geodesic flow, then X eδc eδc n ec(g) ∼ δc e −1 n 0 g∈Per (n)

as n → +∞.

Part III Arithmetic Applications

Chapter 14

Fields with Discrete Valuations b be a non-Archimedean local field. Basic examples of such fields are the Let K field of formal Laurent series over a finite field and the field of p-adic numbers (see Examples 14.1 and 14.2). In Part III of this book, we apply the geometric equidistribution and counting results for simplicial trees given in Part II in order b The to prove arithmetic equidistribution and counting results in such fields K. link between the geometry and the algebra is provided by the Bruhat–Tits tree of b the construction of which is recalled in Section 15.1. We will use only (PGL2 , K), the system of conductances equal to 0 in this Part III. In the present chapter, before embarking on our arithmetic applications, we recall basic facts on local fields for the convenience of the geometer reader. For more details, we refer, for instance, to [Ser2, Gos]. We refer to [BrPP] for an announcement of the results of Part III, with a presentation different from the one in the introduction. b and We will give results only for the algebraic group G = PGL2 over K b special discrete subgroups Γ of PGL2 (K), even though the same methods give equidistribution and counting results when G is any semisimple connected linear b b of K-rank b algebraic group over K 1 and Γ any lattice in G = G(K).

14.1

Local fields and valuations

Let F be a field and let F × = (F −{0}, ×) be its multiplicative group. A surjective group morphism v : F × → Z to the additive group Z that satisfies v(a + b) > min{v(a), v(b)} for all a, b ∈ F × is a (normalised discrete) valuation v on F . We make the usual convention and extend the definition of v to F by setting v(0) = +∞. Note that v(a + b) = min{v(a), v(b)} if v(a) 6= v(b). When F is an extension of a finite field k, the valuation v vanishes on k × . © Springer Nature Switzerland AG 2019 A. Broise-Alamichel et al., Equidistribution and Counting Under Equilibrium States in Negative Curvature and Trees, Progress in Mathematics 329, https://doi.org/10.1007/978-3-030-18315-8_14

299

300

Chapter 14. Fields with Discrete Valuations

The subring Ov = {x ∈ F : v(x) > 0} is the valuation ring (or local ring) of v (or of F is v is implicit). The maximal ideal mv = {x ∈ F : v(x) > 0} of Ov is principal, and it is generated as an ideal of Ov by any element πv ∈ F with v(πv ) = 1, which is called a uniformiser of F . The residual field of the valuation v is kv = Ov /mv . When kv is finite, the valuation v defines a (normalised, non-Archimedean) absolute value | · |v on F by |x|v = |kv |−v(x) , with the convention that |kv |−∞ = 0. This absolute value induces an ultrametric distance on F by (x, y) 7→ |x − y|v . Let Fv be the completion of F with respect to this distance. The valuation v of F uniquely extends to a (normalised discrete) valuation on Fv , again denoted by v. Example 14.1. Let K = Fq (Y ) be the field of rational functions in one variable Y with coefficients in a finite field Fq of order a positive power q of a positive prime p in Z, let Fq [Y ] be the ring of polynomials in one variable Y with coefficients in Fq , and let v∞ : K × → Z be the valuation at infinity of K, defined on every P/Q ∈ K with P ∈ Fq [Y ] and Q ∈ Fq [Y ] − {0} by v∞ (P/Q) = deg Q − deg P . The absolute value associated with v∞ is |P/Q|∞ = q deg P −deg Q . The completion of K for v∞ is the field Kv∞ = Fq ((Y −1 )) of formal Laurent series in one variable Y −1 with coefficients in Fq . The elements x in Fq ((Y −1 )) are of the form X x= xi Y −i , i∈Z

where xi ∈ Fq for every i ∈ Z, and xi = 0 for i small enough. The valuation at infinity of Fq ((Y −1 )) extending the valuation at infinity of Fq (Y ) is v∞ (x) = sup{i ∈ Z : ∀ j < i,

xj = 0} ,

14.2. Global function fields

301

that is, v∞

X ∞

xi Y −i

= i0

i=i0

if xi0 6= 0. The valuation ring of v∞ is the ring Ov∞ = Fq [[Y −1 ]] of formal power series in one variable Y −1 with coefficients in Fq . The element πv∞ = Y −1 is a uniformiser of v∞ ; the residual field Ov∞ /πv∞ Ov∞ of v∞ is kv∞ = Fq . Example 14.2. For a positive prime p ∈ Z, the field of p-adic numbers Qp is the completion of Q with respect to the absolute value | · |p of the p-adic valuation vp defined by setting a vp pn = n, b when n ∈ Z, a, b ∈ Z − {0} are not divisible by p. Then the valuation ring Ovp , denoted by Zp , of Qp is the closure of Z for the absolute value | · |p , the element πvp = p is a uniformiser, and the residual field is kvp = Zp /pZp = Fp , a finite field of order p. A field endowed with a valuation is a non-Archimedean local field if it is complete with respect to its absolute value and if its residual field is finite.1 Its valuation ring is then a compact open additive subgroup. Any non-Archimedean local field is isomorphic to a finite extension of the p-adic field Qp for some prime p, or to the field Fq ((Y −1 )) of formal Laurent series in one variable Y −1 over Fq for some positive power q of a prime p. These formal Laurent series fields may occur as completions of numerous (global) function fields, which we now define. The basic case is described in Example 14.1 above, and the general case is detailed in Section 14.2 below. The geometer reader may skip Section 14.2 and use only Example 14.1 in the remainder of Part III (using g = 0 when the constant g occurs).

14.2

Global function fields

In this section, we fix a finite field Fq with q elements, where q is a positive power of a positive prime p ∈ Z, and we recall the definitions and basic properties of a function field K over Fq , its genus g, its valuations v, its completion Kv for the associated absolute value | · |v , and the associated affine function ring Rv . See, for instance, [Gos, Ros] for the content of this section. Let K be a (global) function field over Fq , which can be defined in two equivalent ways as 1 There

are also two Archimedean local fields C and R; see, for example, [Cas].

302


(1) the field of rational functions on a geometrically irreducible smooth projective curve C over Fq , or (2) an extension of Fq of transcendence degree 1 in which Fq is algebraically closed. There is a bijection between the set of closed points of C and the set of (normalised discrete) valuations of its function field K, the valuation of a given element f ∈ K being the order of the zero or the opposite of the order of the pole of f at the given closed point. We fix such an element v from now on. We denote by g the genus of the curve C. In the basic Example 14.1, C is the projective line P1 over Fq , which is a curve of genus g = 0, and the closed point associated with the valuation at infinity v∞ is the point at infinity [1 : 0]. We denote by Kv the completion of K for v, and by Ov = {x ∈ Kv : v(x) > 0} the valuation ring of (the unique extension to Kv ) of v. We choose a uniformiser πv ∈ K of v. We denote by kv = Ov /πv Ov the residual field of v, which is a finite field of order qv = |kv | . The field kv is from now on identified with a fixed lift in Ov (see, for instance, [Col, Théo. 1.3]), and is an extension of the field of constants Fq . The degree of this extension is denoted by deg v, so that qv = q deg v . We denote by | · |v the (normalised) absolute value associated with v : for every x ∈ Kv , we have |x|v = (qv )−v(x) = q −v(x) deg v . P Every element x ∈ Kv is2 a (convergent) Laurent series x = i∈Z xi (πv )i in the variable πv over kv , where xi ∈ kv is zero for i ∈ Z small enough. We then have |x|v = (qv )− sup{j∈Z : ∀ i<j, xi =0} , (14.1) P and Ov consists of the (convergent) power series x = i∈N xi (πv )i (where xi ∈ kv ) in the variable πv over kv . We denote by Rv the affine algebra of the affine curve C − {v}, consisting of the elements of K whose only poles are at the closed point v of C. Its field of fractions is equal to K; hence we will often write elements of K as x/y with x, y ∈ Rv and y 6= 0. In the basic Example 14.1, we have Rv∞ = Fq [Y ]. Note that Rv ∩ Ov = Fq , 2 See,

for instance, [Col, Coro. 1.6].

(14.2)


303

since the only rational functions on C whose only poles are at v and whose valuation at v is nonnegative are the constant ones. We have (see, for instance, [Ser3, II.2 Notation], [Gos, page 63]) (Rv )× = (Fq )× .

(14.3)

The following result is immediate when C = P1 , since then Rv + Ov = Kv . Lemma 14.3. The dimension of the quotient vector space Kv /(Rv + Ov ) over Fq is equal to the genus g of C. Proof. (indicated by J.-B. Bost) We refer, for instance, to [Ser1] for background on sheaf cohomology. We denote in the same way the valuation v and the corresponding closed point on C. Let O = K ∩ Ov be the discrete valuation ring of v restricted to K. Since K is dense in Kv and Ov is open and contains 0, we have Kv = K + Ov . Therefore the canonical map K/(Rv + O) → Kv /(Rv + Ov ) is a linear isomorphism over Fq . Let us hence prove that dimFq K/(Rv + O) = g. In what follows, V ranges over the affine Zariski-open neighbourhoods of v in C, ordered by inclusion. Let OC be the structural sheaf of C. Note that by the definition of Rv , since the zeros of elements of K × are isolated and by the relation between valuations of K and closed points of C, Rv = H 0 (C − {v}, OC ), K = lim H 0 (V − {v}, OC ), and O = lim H 0 (V , OC ) . −→ −→ V

V

Since V and C−{v} are affine curves, we have H 1 (C−{v}, OC ) = H 1 (V , OC ) = 0. By the Mayer–Vietoris exact sequence for the covering {C−{v}, V } of C, we hence have an exact sequence H 0 (C, OC ) → H 0 (C−{v}, OC )×H 0 (V , OC ) → H 0 (V −{v}, OC ) → H 1 (C, OC ) . Therefore K/(Rv + O) = lim H 0 (V − {v}, OC )/ H 0 (C − {v}, OC ) + H 0 (V , OC ) −→

V

' H 1 (C, OC ) . Since dimFq H 1 (C, OC ) = g by one definition of the genus of C, the result follows. Recall that Rv is a Dedekind ring.3 In particular, every nonzero Q ideal (respectively fractional ideal) I of Rv may be written uniquely as I = p pvp (I) where p 3 See,

for instance, [Ser3, II.2 Notation]. We refer, for instance, to [Nar, §1.1] for background on Dedekind rings.

304


ranges over the prime ideals in Rv and vp (I) ∈ N (respectively vp (I) ∈ Z), with only finitely many of them nonzero. By convention, I = Rv if vp (I) = 0 for all p. For all x, y ∈ Rv (respectively x, y ∈ K), we denote by hx, yi = x Rv + y Rv the ideal (respectively fractional ideal) of Rv generated by x, y. If I, J are nonzero fractional ideals of Rv , we have Y Y I ∩J = pmax{vp (I), vp (J)} and I + J = pmin{vp (I), vp (J)} . (14.4) p

p

The (absolute) norm of a nonzero ideal I = p pvp (I) of Rv is Y N (I) = [Rv : I] = |Rv /I| = q vp (I) deg p , Q

p

where deg p is the degree of the field Rv /p over Fq , so that N (Rv ) = 1. By convention, N (0) = 0. This norm is multiplicative: N (IJ) = N (I)N (J) , Q and the norm of a nonzero fractional ideal I = p pvp (I) of Rv is defined by the Q vp (I) deg p same formula N (I) = p q . Note that if (a) is the principal fractional ideal in Rv generated by a nonzero element a ∈ K, we define N (a) = N (a) . We have (see, for instance, [Gos, page 63]) N (a) = |a|v .

(14.5)

The Dedekind zeta function of K is (see, for instance, [Gos, §7.8] or [Ros, §5]) X 1 ζK (s) = N (I)s I

if Re s > 1, where the summation is over the nonzero ideals I of Rv . By, for instance, [Ros, §5], it has an analytic continuation on C − {0, 1} with simple poles at s = 0, s = 1. It is actually a rational function of q −s . In particular, if K = Fq (Y ), then (see [Ros, Thm. 5.9]) ζFq (Y ) (−1) =

1 . (q − 1)(q 2 − 1)

(14.6)

We denote by HaarKv the Haar measure of the (Abelian) locally compact topological group (Kv , +), normalised so that HaarKv (Oν ) = 1.4 The Haar measure scales as follows under multiplication: for all λ, x ∈ Kv , we have d HaarKv (λx) = |λ|v d HaarKv (x) . 4 Other

[Tat].

(14.7)

normalisations are useful in considering Fourier transforms; see, for instance, Tate’s thesis


305

Note that any nonzero fractional ideal I of Rv is a discrete subgroup of (Kv , +), and we will again denote by HaarKv the Haar measure on the compact group Kv /I, which is induced by the above normalised Haar measure of Kv . Lemma 14.4. For every nonzero fractional ideal I of Rv , we have HaarKv (Kv /I) = q g−1 N (I) . Proof. By the scaling properties of the Haar measure, we may assume that I is an ideal in Rv . By Lemma 14.3, we have Card Kv /(Rv + Ov ) = q g . By Equation (14.2) and by the normalisation of the Haar measure, we have HaarKv (Rv + Ov )/Rv = HaarKv Ov /(Rv ∩ Ov ) = HaarKv Ov /Fq =

1 . q

Hence HaarKv (Kv /Rv ) = Card Kv /(Rv + Ov ) HaarKv (Rv + Ov )/Rv = q g−1 . Since HaarKv (Kv /I) = N (I) HaarKv (Kv /Rv ), the result follows.

Chapter 15

Bruhat–Tits Trees and Modular Groups In this chapter, we give background information and preliminary results on the main link between the geometry and the algebra used for our arithmetic applications: the (discrete-time) geodesic flow on quotients of Bruhat–Tits trees by arithmetic lattices. We refer to Section 2.6 for the basic notation and terminology for trees and graphs (of groups). a b We denote the image in PGL2 = GL2 /Z(GL2 ) of an element ∈ GL2 c d a b by ∈ PGL2 . c d

15.1

Bruhat–Tits trees

Let Kv be a non-Archimedean local field, with valuation v, valuation ring Ov , choice of uniformiser πv , and residual field kv of order qv (see Section 14.1 for definitions). In this section, we recall the construction and basic properties of the Bruhat– Tits tree Xv of (PGL2 , Kv ), see, for instance, [Tit]. We use its description given in [Ser3], to which we refer for proofs and further information. An Ov -lattice Λ in the Kv -vector space Kv ×Kv is a rank-2 free Ov -submodule of Kv × Kv , generating Kv × Kv as a vector space. The Bruhat–Tits tree Xv of (PGL2 , Kv ) is the graph whose set of vertices V Xv is the set of homothety classes (under (Kv )× ) [Λ] of Ov -lattices Λ in Kv × Kv , and whose nonoriented edges are the pairs {x, x0 } of vertices such that there exist representatives Λ of x and Λ0 of x0 for which Λ ⊂ Λ0 and Λ0 /Λ is isomorphic to Ov /πv Ov . We again denote by Xv the geometric realisation of Xv .1 Two (oriented) edges are naturally 1 We

give length 1 to (the geometric realisation of) each nonoriented edge; see Section 2.6.


307

308

Chapter 15. Bruhat–Tits Trees and Modular Groups

associated with each nonoriented edge. If K is any field endowed with a valuation v whose completion is Kv , then the similarly defined Bruhat–Tits tree of (PGL2 , K) coincides with Xv ; see [Ser3, p. 71]. The graph Xv is a regular tree of degree |P1 (kv )| = qv + 1. In particular, the Bruhat–Tits tree of (PGL2 , Qp ) is regular of degree p + 1, and if Kv = Fq ((Y −1 )) and v = v∞ , then the Bruhat–Tits tree Xv of (PGL2 , Kv ) is regular of degree q +1. More generally, if Kv is the completion of a function field over Fq endowed with a valuation v as in Section 14.2, then the Bruhat–Tits tree of (PGL2 , Kv ) is regular of degree qv + 1 = q deg v + 1. The standard basepoint ∗v of X is the homothety class [Ov × Ov ] ∈ V Xv of the Ov -lattice Ov × Ov , generated by the canonical basis of Kv × Kv . In particular, we have d(∗v , [Ov × xOv ]) = |v(x)| (15.1) for every x ∈ (Kv )× . The link lk(∗v ) = {y ∈ V Xv : d(y, ∗v ) = 1} of ∗v in the tree Xv can be identified with the projective line P1 (kv ). The left linear action of GL2 (Kv ) on Kv × Kv induces a faithful, vertextransitive left action of PGL2 (Kv ) by automorphisms on the Bruhat–Tits tree Xv . The stabiliser in PGL2 (Kv ) of the basepoint ∗v is PGL2 (Ov ). We will hence identify PGL2 (Kv )/ PGL2 (Ov ) with V Xv by the map g PGL2 (Ov ) 7→ g ∗v . The group PGL2 (Ov ) acts projectively on lk(∗v ) = P1 (kv ) by reduction modulo πv Ov of the coefficients. We identify the projective line P1 (Kv ) with the Alexandrov compactification Kv ∪ {∞} using the map Kv (x, y) 7→ xy , so that ∞ = [1 : 0] . The projective action of GL2 (Kv ) or PGL2 (Kv ) on P1 (Kv ) isthe action by homoa b z+b graphies2 on Kv ∪ {∞}, given by (g, z) 7→ g · z = ac z+d if g = ∈ GL2 (Kv ), c d a b or g = ∈ PGL2 (Kv ). As usual we define ∞ 7→ ac and − dc 7→ ∞. c d There exists a unique homeomorphism between the boundary at infinity ∂∞ Xv of Xv and P1 (Kv ) such that the (continuous) extension to ∂∞ Xv of the isometric action of PGL2 (Kv ) on Xv corresponds to the projective action of PGL2 (Kv ) on P1 (Kv ). From now on, we identify ∂∞ Xv and P1 (Kv ) by this homeomorphism. Under this identification, Ov consists of the positive endpoints `+ of the geodesic lines ` of Xv with negative endpoint `− = ∞ that pass through the vertex ∗v (see the picture below). 2 Or

linear fractional transformations.

15.1. Bruhat–Tits trees

309

x ∈ Kv x2

x1 x0 xi0 ∞

0

0

∗v

0

0 Ov

H∞

∂∞ Xv Let H∞ be the horoball centered at ∞ ∈ ∂∞ Xv whose associated horosphere passes through ∗v . There is a unique labeling of the edges of Xv by elements of P1 (kv ) = kv ∪ {∞} such that (see the above picture) • the label of any edge of Xv pointing towards ∞ ∈ ∂∞ Xv is ∞, P • for any x = i∈Z xi (πv )i ∈ Kv , the sequence (xi )i∈Z is the sequence of the labels of the (directed) edges that make up the geodesic line ]∞, x[ oriented from ∞ towards x, • x0 is the label of the edge of ]∞, x[ exiting the horoball H∞ . We refer to [Pau3, Sect. 5] for a detailed treatment of the case Kv = Fq ((Y −1 )) and v = v∞ . For all η, η 0 ∈ Kv = ∂∞ Xv − {∞}, we have |η − η 0 |v = dH∞ (η, η 0 )ln qv

(15.2)

by the definitions of the absolute value | · |v and of Hamenstädt’s distance, see Equation (14.1), the above geometric interpretation, and Equation (2.12). Note that in [Pau3], Hamenstädt’s distance in a regular tree is defined in a different way: in that reference, the distance |η − η 0 |v equals Hamenstädt’s distance between η and η 0 . In particular, the Hölder norms3 kψkβ, | · − · |v and kψkβ 0 , dH∞ of a function ψ : Kv → R, respectively for the distances (x, y) 7→ |x − y|v and dH∞ on Kv , are related by the following formula: 1 ∀ β ∈ 0, , kψkβ, | · − · |v = kψkβ ln qv , dH∞ . (15.3) ln qv The group PGL2 (Kv ) acts simply transitively on the set of ordered triples of distinct points in ∂∞ Xv = P1 (Kv ). In particular, it acts transitively on the 3 See

the definition in Section 3.1.

310


space G Xv of (discrete) geodesic lines in Xv . The stabiliser under this action of the geodesic line (from ∞ = [1 : 0] to 0 = [0 : 1]) `∗ : n 7→ [Ov × (πv )−n Ov ] is the maximal compact-open subgroup ( ) a 0 × A(Ov ) = : a, d ∈ (Ov ) 0 d of the diagonal group ( ) a 0 × A(Kv ) = : a, d ∈ (Kv ) . 0 d We will hence identify the quotient space PGL2 (Kv )/A(Ov ) with G Xv by the e : gA(Ov ) 7→ g `∗ . Define mapping Ξ 1 0 av = , 0 πv−1 e is equivwhich belongs to A(Kv ) and centralises A(Ov ). The homeomorphism Ξ ariant for the actions on the left of PGL2 (Kv ) on PGL2 (Kv )/A(Ov ) and G Xv . It is also equivariant for the actions on PGL2 (Kv )/A(Ov ) under translations on the right by (av )Z and on G Xv under the discrete-time geodesic flow (gn )n∈Z : for all n ∈ Z and x ∈ PGL2 (Kv )/A(Ov ), we have e (x av n ) = gn Ξ e (x) . Ξ

(15.4)

Furthermore, the stabiliser in PGL2 (Kv ) of the ordered pair of endpoints (`∗− = ∞, `∗+ = 0) of `∗ in ∂∞ Xv = P1 (Kv ) is A(Kv ). Therefore any element γ ∈ PGL2 (Kv ) that is loxodromic on Xv is diagonalisable over Kv . By [Ser3, page a 0 108], the translation length on Xv of γ0 = is 0 d λ(γ0 ) = |v(a) − v(d)| .

(15.5)

a 0 ∈ GL2 (Kv ) is a representative of γ0 with det γe0 ∈ (Ov )× , 0 d then we have 0 = v(det γe0 ) = v(ad) = v(a) + v(d), so that v(d) = −v(a) and λ(γ0 ) = 2|v(a)|. Since v(a) 6= v(d) if λ(γ0 ) 6= 0, we have Note that if γe0 =

v(tr γe0 ) = v(a + d) = min{v(a), v(d)} = −|v(a)|. Thus, λ(γ0 ) = 2|v(tr γe0 )| .

(15.6)

By conjugation, this formula is valid if γ0 ∈ PGL2 (Kv ) is loxodromic on Xv and represented by γe0 ∈ GL2 (Kv ) such that det γe0 ∈ (Ov )× . Let H be a horoball in Xv whose boundary is contained in V Xv and whose point at infinity ξ is different from ∞. With β∞ : V Xv × V Xv → Z the Busemann

15.2. Modular graphs of groups

311

function at ∞ (see Equation (2.5)), the height of H is ht∞ (H ) = max{β∞ (x, ∗v ) : x ∈ ∂H } ∈ Z , which is the signed distance between H∞ and H .4 It is attained at the intersection point with ∂H of the geodesic line from ∞ to ξ, which is then called the highest point of H . Note that the height of H is invariant under the action of the stabiliser of H∞ in PGL2 (Kv ) on the set of such horoballs H . The following lemma is a generalisation of [Pau3, Prop. 6.1], which covered only the particular case of K = Fq (Y ) and v = v∞ . Lemma 15.1. Assume that Kv is the completion of a function field K over Fq endowed with a valuation v, with associated affine function ring Rv . For every a b γ= ∈ PGL2 (K) with a, b, c, d ∈ K such that ad − bc ∈ (Ov )× and c 6= 0, c d the image of H∞ by γ is the horoball centered at ac ∈ K ⊂ Kv = ∂∞ Xv − {∞} with height ht∞ (γH∞ ) = −2 v(c) . Proof. It is immediate that γ∞ = ac under the projective action. Up to multiplying a 1 −c γ on the left by the element ∈ PGL2 (K), which does not change c or 0 1 −1 the height of γH∞ , we may assume that a = 0 and thatb has the form c u d 1 −c with u = bc − ad ∈ (Ov )× . Multiplying γ on the right by ∈ PGL2 (K) 0 1 preserves γH∞ and does not change a = 0, b = c−1 u or c. Hence we may assume that d = 0. Since γ then exchanges the points ∞ and 0 in ∂∞ Xv , the highest point of γH∞ is γ∗v . Assuming first that 0, γ∗v , ∗v , ∞ are in this order on the geodesic line from 0 to ∞, we have, by Equation (15.1), ht∞ (γH∞ ) = d(∗v , γ∗v ) = d([Ov × Ov ], [c−1 uOv × cOv ]) = d([Ov × Ov ], [Ov × c2 Ov ]) = −v(c2 ) = −2 v(c) . If 0, γ∗v , ∗v , ∞ are in the opposite order, then the same computation holds, up to replacing the distance d by its opposite −d.

15.2

Modular graphs of groups

Let K be a function field over Fq , let v be a (normalised discrete) valuation of K, let Kv be the completion of K associated with v, and let Rv be the affine function ring associated with v (see Section 14.2 for definitions). The group Γv= PGL2 (Rv ) is a lattice in the locally compact group PGL2 (Kv ), and a tree lattice5 of the Bruhat–Tits tree Xv of (PGL2 , Kv ), called the modular 4 See 5 See

the definition of the signed distance just above Lemma 11.13. Section 2.6 for a definition.

312


group at v of K. The quotient graph Γv \Xv is called the modular graph at v of K, and the quotient graph of groups6 Γv \\Xv is called the modular graph of groups at v of K. We refer to [Ser3] for background information on these objects, and, for instance, to [Pau3] for a geometric treatment when K = Fq (Y ) and v = v∞ . By, for instance, [Ser3], the set of cusps Γv \P1 (K) is finite, and Γv \Xv is the disjoint union of a finite connected subgraph containing Γv ∗v and maximal open geodesic rays hz ( ]0, +∞[), for z = Γv ze ∈ Γv \P1 (K), where hz (called a cuspidal ray; see Section 2.6) is the injective image by the canonical projection Xv → Γv \Xv of a geodesic ray whose point at infinity in P1 (K) ⊂ ∂∞ Xv is equal to ze. Conversely, any geodesic ray whose point at infinity lies in P1 (K) ⊂ ∂∞ Xv contains a subray that maps injectively by the canonical projection Xv → Γv \Xv . The group Γv = PGL2 (Rv ) is a geometrically finite lattice7 by, for instance, [Pau4]. The set of bounded parabolic fixed points of Γv is exactly P1 (K) ⊂ ∂∞ Xv , and the set of conical limit points of Γv is P1 (Kv ) − P1 (K). Let us denote by Γ\ v \Xv = (Γv \Xv ) t Ev Freudenthal’s compactification of Γv \Xv by its finite set of ends Ev ; see [Fre]. This set of ends is indeed finite, in bijection with Γv \P1 (K) by the map that associates to z ∈ Γv \P1 (K) the end towards which the cuspidal ray hz converges. See, for instance, [Ser3] for a geometric interpretation of Ev in terms of the curve C. Let Iv be the set of classes of fractional ideals of Rv . The map that associates to an element [x : y] ∈ P1 (K) the class of the fractional ideal xRv + yRv generated by x, y induces a bijection from the set of cusps Γ\P1 (K) to Iv . The volume8 of the modular graph of groups Γv \\Xv can be computed using Equation (14.3) and Exercice 2 (b) in [Ser3, II.2.3]: Vol(PGL2 (Rv )\\Xv ) = (q − 1) Vol(GL2 (Rv )\\Xv ) = 2 ζK (−1) .

(15.7)

If K = Fq (Y ) is the rational function field over Fq and if we consider the valuation at infinity v = v∞ of K, then the Nagao lattice9 Γv = PGL2 (Fq [Y ]) acts transitively on P1 (K). Its quotient graph of groups Γv \\Xv is the following modular ray (with associated edge-indexed graph) Γ−1

Γ0 Γ00

q+1 q 1 6 See

Γ1 Γ0

Γ2 Γ1

q 1

Γ2 q 1

q 1

Section 2.6 for a definition. Section 2.6 for a definition and, for instance, [BasL] for a profusion of geometrically infinite tree lattices in simplicial trees. 8 See Section 2.6 for a definition. 9 This tree lattice was studied by Nagao in [Nag]; see also [Moz, BasL]. It is called the modular group in [Wei2]. 7 See

15.3. Computations of measures for Bruhat–Tits trees

313

where Γ−1 = PGL2 (Fq ), Γ00 = Γ0 ∩ Γ−1 and, for every n ∈ N, a b × Γn = ∈ PGL2 (Fq [Y ]) : a, d ∈ Fq , b ∈ Fq [Y ], deg b 6 n + 1 . 0 d Note that even though PGL2 (Kv ) has inversions on Xv , its discrete subgroup Γv = PGL2 (Rv ) acts without inversion on Xv (see, for instance, [Ser3, II.1.3]). In particular, the quotient graph Γv \Xv is then well defined.

15.3

Computations of measures for Bruhat–Tits trees

In this section, we compute explicit expressions for the skinning measures10 of horoballs and geodesic lines, and for the Bowen–Margulis measures,11 concerning tree lattices of Bruhat–Tits trees. See [PaP16, Section 7], [PaP17a, Section 4] and [PaP20, Section 7] for analogous computations in the real, complex and quaternionic hyperbolic spaces respectively, and [BrP1] for related computations in the tree case. Let (Kv , v) be as at the beginning of Section 15.1. Let Γ be a tree lattice of the Bruhat–Tits tree Xv of (PGL2 , Kv ). Since Xv is regular of degree qv + 1, the critical exponent of Γ is δΓ = ln qv (15.8) by Proposition 4.16 and Equation (8.1). ∞ xt H∞

∗v

∂H∞

Kv = ∂∞ Xv − {∞} Ov 10 See 11 See


πv Ov

314


We normalise the Patterson density (µx )x∈V Xv of Γ as follows.12 Let H∞ be the horoball in Xv centered at ∞ whose associated horosphere passes through ∗v . Let t 7→ xt be the geodesic ray in Xv such that x0 = ∗v and that converges to ∞. Hamenst¨ adt’s measure13 associated with H∞ , µH∞ = lim eδΓ t µxt = lim qv t µxt , t→+∞

t→+∞

is a Radon measure on ∂∞ Xv − {∞} = Kv , invariant under all isometries of Xv preserving H∞ , since Γ is a lattice. Hence it is invariant under the translations by the elements of Kv . By the uniqueness property of Haar measures, µH∞ is a constant multiple of the chosen Haar measure14 of Kv , and we normalise the Patterson density (µx )x∈V Xv so that µH∞ = HaarKv .

(15.9)

We summarise the various measure computations in the following result. Proposition 15.2. Let Γ be a tree lattice of the Bruhat–Tits tree Xv of (PGL2 , Kv ), with Patterson density normalised as above. (1) The outer/inner skinning measures of the singleton {∗v } are given by −2 ± dσ e{∗ (ρ) = dµ∗v (ρ± ) = max{1, |ρ± |v } d HaarKv (ρ± ) v} 1 on the set of ρ ∈ ∂± {∗v } such that ρ± 6= ∞.

(2) The total mass of the Patterson density is kµx k =

qv + 1 qv

for all x ∈ V Xv . (3) The skinning measure of the horoball H∞ is the projection of the Haar measure 1 of Kv : for all ρ ∈ ∂± H∞ , we have ± de σH (ρ) = dµH∞ (ρ± ) = d HaarKv (ρ± ) . ∞

(4) If ∞ is a bounded parabolic fixed point of Γ, with Γ∞ its stabiliser in Γ, if D = (γH∞ )γ∈Γ/Γ∞ , we have ± kσD k = HaarKv (Γ∞ \Kv ) = Vol(Γ∞ \\∂H∞ ) .

(5) Let L be a geodesic line in Xv with endpoints L± ∈ Kv = ∂∞ Xv − {∞}. Then 1 on the set of ρ ∈ ∂+ L such that ρ+ ∈ Kv = ∂∞ Xv − {∞} and ρ+ 6= L± , the outer skinning measure of L is + dσ eL (ρ) =

|L+ − L− |v d HaarKv (ρ+ ) . |ρ+ − L− |v |ρ+ − L+ |v

− the system of conductances is 0, note that µ+ x = µx = µx is the standard Patterson measure of Γ. 13 See Equation (7.5). 14 Recall that we normalise the Haar measure of (K , +) such that Haar v Kv (Ov ) = 1. 12 Since


315

(6) Let L be a geodesic line in Xv , let ΓL be the stabiliser in Γ of L, and assume that ΓL \L has finite length. Then with D = (γL)γ∈Γ/ΓL , we have ± kσD k=

qv − 1 Vol(ΓL \\L) . qv

Proof. (1) For every ξ ∈ Kv , by the description of the geodesic lines in the Bruhat– Tits tree Xv starting from ∞ given in Section 15.1, we have ξ ∈ Ov if and only if PH∞ (ξ) = ∗v .15 ∞

H∞ −v(ξ) ∗v

PH∞ (ξ)

0

∂H∞

Kv = ∂∞ Xv − {∞}

ξ

For every ξ ∈ Kv − Ov , by Equations (15.2) and (2.12), we have 1

|ξ|v = dH∞ (0, ξ)ln qv = qv 2

d(∗v , PH∞ (ξ))

.

(15.10)

1 On the set of geodesic rays ρ ∈ ∂± {∗v } such that ρ± 6= ∞, by Equation (7.2), by the last claim of Proposition 7.2, by Equation (15.8),16 since PH∞ (ρ± ) belongs to the geodesic ray [∗v , ρ± [ (even when ρ± ∈ Ov ), and by Equation (15.9), we have ± de σ{∗ (ρ) = dµ∗v (ρ± ) = eCρ± (PH∞ (ρ± ), ∗v ) dµH∞ (ρ± ) v}

= eδΓ βρ± (PH∞ (ρ± ), ∗v ) dµH∞ (ρ± ) −d(PH∞ (ρ± ), ∗v )

= qv

d HaarKv (ρ± ) .

that PH∞ : ∂v Xv − {∞} → ∂H∞ is the closest-point map to the horoball H∞ ; see Section 2.4. 16 Since the potential is zero, the Gibbs cocycle is the critical exponent times the Busemann cocycle. 15 Recall

316


1 Therefore, if ρ ∈ ∂± {∗v } is such that

ρ± ∈ Ov = {ξ ∈ Kv : |ξ|v 6 1} = {ξ ∈ Kv : PH∞ (ξ) = ∗v } , ± then d σ e{∗ (ρ) = d HaarKv (ρ± ). If ρ± ∈ Kv − Ov , Equation (15.10) gives the v} claim.

(2) This assertion follows from Assertion (1) by a geometric series argument, but we give a direct proof. Since Γ is a tree lattice, the family (µx )x∈V Xv is actually equivariant under Aut(Xv ),17 which acts transitively on the vertices of Xv , and the stabiliser in Aut(Xv ) of the standard basepoint ∗v acts transitively on the edges starting from ∗v . Since Xv is (qv + 1)-regular, since the set of points at infinity of the geodesic rays starting from ∗v , whose initial edge has terminal vertex 0 ∈ lk(∗v ) = P1 (kv ), is equal to πv Ov , since all geodesic lines from ∞ ∈ ∂∞ Xv to points of πv Ov ⊂ ∂∞ Xv pass through ∗v , and by the normalisation of the Patterson density and of the Haar measure, we have kµ∗v k = (qv + 1) µ∗v (πv Ov ) = (qv + 1) µH∞ (πv Ov ) = (qv + 1) HaarKv (πv Ov ) qv + 1 qv + 1 = HaarKv (Ov ) = . qv qv (3) This follows from Equation (7.4), and from the normalisation µH∞ = HaarKv of the Patterson density. (4) This follows from Assertion (3) and from Equation (8.11) (where the normalisation of the Patterson density was different from the one given by Assertion (2)).

∗v ρ(0) L−

∂∞ X 17 See

Proposition 4.16 (2).

ρ(+∞)

L+


317

(5) Let L, L+ , L− be as in the statement of Assertion (5); see the above picture. The result follows from Lemma 8.6 applied with H = H∞ , from Equations (15.9), (15.8) and (15.2). (6) This follows from Equation (8.12) (where the normalisation of the Patterson density was different), since Xv is (qv + 1)-regular, and from Assertion (2): ± kσD k = kµ∗v k

qv − 1 qv − 1 Vol(ΓL \\L) = Vol(ΓL \\L) . qv + 1 qv

We now turn to measure computations for arithmetic lattices Γ in Xv in the function field case. We still assume that the Patterson density of Γ is normalised so that µH∞ (Ov ) = 1, and we denote by mBM the Bowen–Margulis measure of Γ associated with this choice of Patterson density. Proposition 15.3. Let K be a function field over Fq of genus g and let v be a valuation of K. Let Γ be a finite-index subgroup of Γv = PGL2 (Rv ), with Patterson density normalised such that µH∞ = HaarKv . (1) We have kmBM k =

(qv + 1) [Γv : Γ] 2 (qv + 1) ζK (−1) [Γv : Γ] Vol(Γv \\Xv ) = , qv qv

and if K = Fq (Y ) and v = v∞ is the valuation at infinity of C = P1 , then kmBM k =

2 [PGL2 (Fq [Y ]) : Γ] . q (q − 1)2

(2) Let Γ∞ be the stabiliser in Γ of ∞ ∈ ∂∞ Xv , and let D = (γH∞ )γ∈Γ/Γ∞ . We have q g−1 [(Γv )∞ : Γ∞ ] ± . kσD k= q−1 Proof. (1) Recall that Γv = PGL2 (Rv ) acts without inversion on Xv . By Equation (8.4) (which uses a different normalisation of the Patterson density of Γ), and by Proposition 15.2 (2), we have kmBM k = kµ∗v k2

qv (qv + 1) [Γv : Γ] Vol(Γ\\Xv ) = Vol(Γv \\Xv ) . qv + 1 qv

The first claim of Assertion (1) hence follows from Equation (15.7). If K = Fq (Y ) and v = v∞ , then the second claim of Assertion (1) follows either from the first claim, in which the value of ζK (−1) is given by Equation (14.6), or from the fact that qv = q and that the covolume Vol(PGL2 (Fq [Y ])\\Xv∞ ) of the Nagao lattice PGL2 (Fq [Y ]) is Vol(PGL2 (Fq [Y ])\\Xv∞ ) =

2 , (q − 1)(q 2 − 1)

(15.11)

as an easy geometric series computation shows using the description of the modular ray given in Section 15.2 (see also [BasL, Sect. 10.2]).

318


(2) Let us prove that HaarKv ((Γv )∞ \Kv ) =

q g−1 . q−1

(15.12)

The result then follows by Proposition 15.2 (4), since ± kσD k = HaarKv (Γ∞ \Kv ) = [(Γv )∞ : Γ∞ ] HaarKv ((Γv )∞ \Kv ) .

The stabiliser of ∞ = [1 : 0] in Γv acts on Kv exactly by the set of transformations z 7→ az + b with a ∈ (Rv )× and b ∈ Rv . Since18 (Rv )× = (Fq )× acts freely by multiplication on the left on (Kv − Rv )/Rv , and by Lemma 14.4, we have HaarKv ((Γv )∞ \Kv ) =

1 q g−1 HaarKv (Kv /Rv ) = . q−1 q−1

This proves Equation (15.12).

15.4

Exponential decay of correlation and error terms for arithmetic quotients of Bruhat–Tits trees

As at the beginning of Section 15.1, let Kv be a non-Archimedean local field, with valuation v, valuation ring Ov , choice of uniformiser πv , and residual field kv of order qv . Let Γ be a tree lattice of the Bruhat–Tits tree Xv of (PGL2 , Kv ). In this section, we discuss the error-term estimates that we will use in Part III. Partly in order to simplify the references, we start by summarising in the next statement the only results from the geometric Part II of this book, on geometric equidistribution and counting problems, that we will use in this algebraic Part III. We state it with the normalisation introduced in Section 15.3, which will be useful in what follows. Theorem 15.4. Let Γ be a lattice of Xv whose length spectrum LΓ is equal to 2Z. for every Assume that the Patterson density of Γ is normalised so that kµx k = qvq+1 v x ∈ V Xv . Let D± be nonempty proper simplicial subtrees of Xv with stabilisers ΓD± in Γ, such that the families D ± = (γD± )γ∈Γ/ΓD± are locally finite in Xv . For every − γ ∈ Γ such that D− and γD+ are disjoint, let αe, γ be the generalised geodesic line, − + isometric exactly on [0, d(D , γD )], whose image is the common perpendicular − between D− and γD+ . If the measure σD + is nonzero and finite, then lim

n→+∞

(qv 2 − 1)(qv + 1) Vol(Γ\\Xv ) −n qv − 2 qv 3 kσD +k

X

∆α− = σ eD+− , e, γ

γ∈Γ/ΓD+ 0 0. As recalled at the end of Section 2.6, lattices in PGL2 (Kv ) are geometrically finite; see [Lub1]. We will hence be able to use the error term in Theorem 15.4 in particular when • Kv is the completion of a function field K over Fq with respect to a (normalised discrete) valuation v of K and Γ is a finite-index subgroup of PGL2 (Rv ) with Rv the affine function ring associated with v,19 as in Chapters 16 and 19 and in Sections 17.2 and 18.2; • when Kv = Qp and Γ is an arithmetic lattice in PGL2 (Kv ) derived from a quaternion algebra; see Sections 17.3 and 18.2. Proof. In order to prove the first claim, we apply Corollary 11.12 with X = Xv and p = q = qv . Since LΓ = 2Z, the lattice Γ leaves invariant the partition of V Xv into vertices at even distance from a basepoint x0 ∈ V Xv and vertices at odd distance from x0 . Since the Patterson density is now normalised so that kµx0 k = qvq+1 v ∓ (instead of kµx0 k = q√v +1 eD ± are now qv in Corollary 11.12), the skinning measures σ √1 qv

times the ones in the statement of Corollary 11.12. Hence the second assertion of Corollary 11.12 gives lim

n→+∞

qv 2 − 1 TVol(Γ\\Xv ) −n qv √ − 2 qv 2 qv kσD +k

X

∆α− = e, γ

√ qv σ eD+− .

γ∈Γ/ΓD+ 0 0 such that for every ∈ ]0, 1], for all -locally constant maps φ, ψ : Γ\G Xv → R and n ∈ Z, we have Z Z Z 1 −n φ ◦ g ψ d m − φ d m ψ d m BM BM BM kmBM k Γ\G Xv Γ\G Xv Γ\G Xv 6 C e−κ|n| kφk lc, β kψk lc, β . (2) If LΓ = 2Z, then there exist C, κ > 0 such that for every ∈ ]0, 1], for all -locally constant maps φ, ψ : Γ\Geven Xv → R and n ∈ Z, we have Z φ ◦ g−2n ψ d mBM Z Z Γ\Geven X 1 − φ d mBM ψ d mBM mBM (Γ\Geven X) Γ\Geven X Γ\Geven X 6 C e−κ|n| kφk lc, β kψk lc, β .

We will not use the following result in this book, but its Assertion (2) is used in the announcement [BrPP], which considers only locally constant regularity. Theorem 15.8. Assume that Γ is a lattice of PGL2 (Kv ), and let β ∈ ]0, 1]. b exists κ > 0 such that for every ∈ ]0, 1] and every -locally (1) If LΓ = Z, there e constant map ψ : G Xv → R, we have, as n → +∞, (qv − 1) Vol(Γ\\Xv ) −n qv − (qv + 1) qv kσD +k

X γ∈Γ/ΓD+ 0 0 such that for every ∈ ]0, 1] and every -locally (2) If LΓ = 2Z, there e constant map ψ : G Xv → R, we have, as n → +∞, (qv 2 − 1)(qv + 1) Vol(Γ\\Xv ) −n qv − 2 qv 3 kσD +k

X γ∈Γ/ΓD+ 0 ln qv instead of β ∈ ]0, 1] for the locally constant norm. For simplicity, we consider only Assertion (1) of Theorem 15.8, though Assertion (2) may be treated similarly. The group Gv acts (on the left) on the real vector space of maps ψ from Γ\Gv to R, by right translation on the source: for every g ∈ Gv , we have gψ : x 7→ ψ(xg). A function ψ : Γ\Gv → R is algebraically locally constant if there exists a compactopen subgroup U of PGL2 (Ov ) that leaves ψ invariant: ∀ g ∈ U,

gψ = ψ ,

or equivalently, if ψ is constant on each orbit of U under the right action of Gv on Γ\Gv . Note that ψ is then continuous, since the orbits of U are compact-open subsets. We define dψ = dim VectR (PGL2 (Ov )ψ) as the dimension of the real vector space generated by the images of ψ under the elements of PGL2 (Ov ), which is finite, and even satisfies dψ 6 [PGL2 (Ov ) : U ] . We define the alc-norm of every bounded algebraically locally constant map ψ : Γ\Gv → R by p kψkalc = dψ kψk∞ . Though the alc-norm thus defined does not satisfy the triangle inequality, we have kλ ψkalc = |λ| kψkalc for every λ ∈ R. We denote by alc(Γ\Gv ) the vector space of bounded algebraically locally constant maps ψ from Γ\Gv to R. For every n ∈ N, let Un be the compact-open subgroup of PGL2 (Ov ) that is the kernel of the morphism PGL2 (Ov ) → PGL2 (Ov /πv n Ov ) of reduction modulo πv n . Note that any compact-open subgroup U of PGL2 (Ov ) contains Un for some n ∈ N. Hence ψ : Γ\Gv → R is algebraically locally constant if and only if there exists n ∈ N such that ψ is constant on each right orbit of Un . For every n ∈ N, since the order of PGL2 (Ov /πv n Ov ) is at most the order of (Ov /πv n Ov )4 , which is qv 4n , if ψ : Γ\Gv → R is constant on each right orbit of Un , then kψkalc 6 qv 2n kψk∞ .

(15.15)

Recall24 that we have a natural homeomorphism Ξ : ΓgA(Ov ) 7→ Γg `∗ between the double coset space Γ\Gv /A(Ov ) and the space Γ\G Xv . We denote by 23 This

assumption is introduced in order to apply the following Theorem 15.10. Note that the existence of a nonuniform lattice in Gv = PGL2 (Kv ) forces the characteristic to be positive; see, for instance, [Lub1]. 24 See Section 15.1.

15.4. Exponential decay of correlation and error terms

323

pG : Γ\Gv → Γ\G X the composition map of the canonical projection Γ\G → v v Γ\Gv /A(Ov ) and of Ξ. By Equation (15.4), for all x ∈ Γ\Gv and n ∈ N, we have pG (x av n ) = gn pG (x) . (15.16) Lemma 15.9. For every -locally constant function ψ : Γ\G Xv → R, and for every ∈ ]0, 1], if n = − 12 ln , then the map ψ ◦ pG : Γ\Gv → R is Un -invariant and kψ ◦ pG kalc 6 qv 2 kψk lc, ln qv .

(15.17)

Proof. Let , ψ, n be as in the statement. Let us first prove that if `, `0 ∈ G Xv satisfy `[−n,+n] = `0[−n,+n] , then d(`, `0 ) 6 . If `[−n,+n] = `0[−n,+n] , then d(`(t), `0 (t)) = 0 for t ∈ [−n, n], and by the triangle inequality, we have d(`(t), `0 (t)) 6 2(|t| − n) if |t| > n. Hence Z

0

d(`, ` ) 6 2 6e

+∞

2 (t − n) e

n −2(− 12 ln )

−2 t

dt = 2 e

−2 n

Z 0

+∞

u e−u

du = e−2 n 2

=,

as desired. In order to prove that ψ ◦ pG : Γ\Gv → R is Un -invariant, let x, x0 ∈ Γ\Gv be such that x0 ∈ x Un . Since Un acts by the identity map on the ball of radius n in the Bruhat–Tits tree Xv , the geodesic lines pG (x) and pG (x0 ) in Γ\G Xv coincide (at least) on [−n, n]. Hence, as we saw at the beginning of the proof, we have d(pG (x), pG (x0 )) 6 . Therefore ψ(pG (x)) = ψ(pG (x0 )), since ψ is -locally constant. Now using Equation (15.15), we have kψ ◦ pG kalc 6 qv 2n kψ ◦ pG k∞ 1

6 qv 2(1− 2 ln ) kψk∞ = qv 2 − ln qv kψk∞ = qv 2 kψk lc, ln qv .

Now we will use an algebraic result of exponential decay of correlations, Theorem 15.10 (see, for instance, [AtGP]). We first recall some definitions and notation, useful for its statement. Recall that the left action of the locally compact unimodular group Gv on the locally compact space G Xv is continuous and transitive, and that its stabilisers are compact hence unimodular. Since Γ is a tree lattice, the (Borel, positive, regular) Bowen–Margulis measure m e BM on G Xv is Gv -invariant (see Proposition 4.16 (2)). Hence by [Wei1] (see also [GoP, Lem. 5]), there exists a unique Haar measure on Gv , which disintegrates by the evaluation map pf G : Gv → G Xv defined by g 7→ g`∗ , with conditional measure on the fiber over ` = g`∗ ∈ G Xv the probability Haar measure on the stabiliser gA(Ov )g −1 of ` under Gv . Hence, taking the quotient

324


under Γ and normalising in order to have probability measures, if µv is the right Gv -invariant probability measure on Γ\Gv , we have (pG )∗ µv =

mBM . kmBM k

(15.18)

For every g ∈ Gv , we denote by Rg : Γ\Gv → Γ\Gv the right action of g, e ψe0 on Γ\Gv , we define and for all bounded continuous functions ψ, Z Z Z e ψe0 ) = covµv , g (ψ, ψe ◦ Rg ψe0 dµv − ψe dµv ψe0 dµv . Γ\Gv

Γ\Gv

Γ\Gv

Note that by Equations (15.18) and (15.16), for all bounded continuous functions ψ, ψ 0 : Γ\G Xv → R and n ∈ Z, we have25 cov

mBM ,n kmBM k

(ψ, ψ 0 ) = covµv , av n (ψ ◦ pG , ψ 0 ◦ pG ) .

(15.19)

Recall that the adjoint representation of Gv = PGL2 (Kv ) is the continuous morphism Gv → GL(M2 (Kv )) defined by [h] 7→ {x 7→ hxh−1 }, which is independent of the choice of the representative h ∈ GL2 (Kv ) of [h] ∈ PGL2 (Kv ). For every g ∈ Gv , we denote by |g|v the operator norm of the adjoint representation 1 0 of g. For instance, recalling that av = , we have, for all n ∈ Z, 0 πv−1 |av n |v = qv |n| .

(15.20)

We refer, for instance, to [AtGP] for the following result of exponential decay of correlations. Theorem 15.10. Let Γ be a nonuniform lattice of Gv . There exist C 0 , κ0 > 0 such e ψe0 : Γ\Gv → R and g ∈ Gv , that for all bounded locally constant functions ψ, 0 e ψe0 ) 6 C 0 kψk e alc kψe0 kalc |g|v −κ . covµv , g (ψ,

(15.21)

Proposition 15.7 (1), and therefore Theorem 15.8 (1) with β > ln qv , follows from this result applied to ψe = ψ ◦pG , ψe0 = φ◦pG , and g = av n by using Equations (15.19), (15.17), and (15.20) and taking C = C 0 qv4 and κ = κ0 ln qv . Remark. There is a similar relationship between locally constant functions on Kv in an algebraic sense and those in the metric sense. The additive group (Kv , +) acts on the real vector space of functions from Kv to R by translations on the source: for all y ∈ Kv and ψ : Kv → R, the function y · ψ is equal to x 7→ ψ(x + y). A function ψ : Kv → R is algebraically locally constant if there exists k ∈ N such that ψ is invariant under the action of the 25 See

Section 9.2 for a definition of covµ, n .

15.5. Geometrically finite lattices with infinite Bowen–Margulis measure

325

compact-open subgroup (πv )k Ov of Kv , that is, if for all x ∈ Kv and y ∈ (πv )k Ov , we have ψ(x + y) = ψ(x). Note that a locally constant function from Kv to R is continuous. For any locally constant function ψ : Kv → R, the real vector space VectR (Ov · ψ) generated by the images of ψ under the elements of Ov is finite-dimensional. Its dimension dψ satisfies, with k as above, dψ 6 [Ov : (πv )k Ov ] = qv k . We define the alc-norm of every bounded algebraically locally constant function ψ : Kv → R by p kψkalc = dψ kψk∞ . Though the alc-norm thus defined does not satisfy the triangle inequality, we have kλψkalc = |λ| kψkalc for every λ ∈ R, and the set of bounded algebraically locally constant maps from Kv to R is a real vector space. Actually, a function ψ : Kv → R is algebraically locally constant if and only if it is locally constant. More precisely, for every ∈ ]0, 1], since the closed balls of radius qv−k in Kv are the orbits by translations under (πv )k Ov , every -locally constant function ψ : Kv → R is constant under the additive action of (πv )k Ov for k = d −lnlnqv e, whence kψkalc 6 kψk lc, 12 .

15.5

Geometrically finite lattices with infinite Bowen–Margulis measure

This section is a digression from the theme of arithmetic applications, in which we use the Nagao lattice defined in Section 15.2 in order to construct a geometrically finite discrete group of automorphisms of a simplicial tree that has infinite Bowen– Margulis measure. This example was promised after Proposition 4.16, where we saw that such examples do not exist for uniform trees. We will equivariantly change the lengths of the edges of a simplicial tree X endowed with a geometrically finite (nonuniform) tree lattice Γ in order to turn X into a metric tree for which the group Γ remains a geometrically finite lattice, but now has a geometrically finite subgroup with infinite Bowen–Margulis measure. This example is an adaptation of the negatively curved manifold example of [DaOP, §4]. The simplicial example is obtained as a modification of the metric tree example. Theorem 15.11. There exists a geometrically finite discrete group of automorphisms of a metric tree with constant finite degrees whose Bowen–Margulis measure is infinite.

326


There exists a geometrically finite discrete group of automorphisms of a simplicial tree with uniformly bounded degrees whose Bowen–Margulis measure is infinite. Proof. Let v = v∞ be the valuation at infinity of K = Fq (Y ), Kv = Fq ((Y −1 )), Ov = Fq [[Y −1 ]], and Rv = Fq [Y ]. Let Xv be the Bruhat–Tits tree of (PGL2 , Kv ) with basepoint ∗v = [Ov × Ov ]. Let Γv = PGL2 (Rv ), which is a tree lattice of Xv , with quotient graph of groups the modular ray Γv \\Xv described in Section 15.2. We denote by (yi )i=−1,0,... the ordered vertices along Γv \\Xv with vertex stabilisers (Γi )i=−1,0,... , and by (ei )i∈N the ordered edges along Γv \\Xv (pointing away from the origin of the modular ray). The subgroup [ a Q × P = Γi = : Q ∈ Fq [Y ], a, d ∈ Fq 0 d i>0

is the stabiliser in Γ of ∞∈ ∂∞Xv . Let P0 be the finite-index subgroup of P 1 Q consisting of the elements with Q(0) = 0. Observing that d(γ∗v , ∗v ) is 0 1 equal to 2(i + 1) for any element γ ∈ Γi − Γi−1 and that the cardinality of the set (Γi − Γi−1 ) ∩ P0 is (q − 1)q i+1 , it is easy to see that the Poincaré series QP0 , 0, ∗v , ∗v (s) =

X

e−s d(∗v ,γ∗v )

γ∈P0

of the discrete (though elementary) subgroup P0 of Isom(Xv ) is (up to a multiP∞ plicative constant) equal to i=0 q i e−2si , which gives δP0 = ln2q for the critical exponent of P0 on Xv . Let h be an element of Γv that is loxodromic on Xv and whose fixed points belong to the open subset Y −1 Ov of Kv = ∂∞ Xv − {∞}. Hence the horoball H∞ centered at ∞ ∈ ∂∞ Xv , whose horosphere contains ∗v , is disjoint from the translation axis Axh of h. Note that the stabiliser of H∞ in Γv is P and that P0 acts freely on the edges exiting H∞ . Let x0 ∈ V Xv be the closest point on Axh to H∞ , let e∗ be the edge with origin ∗v pointing towards x0 , and let e− , e+ be the two edges with origin x0 on Axh (see the figure on top of the next page). Let Uh be the set of points x in V Xv − {x0 } such that the geodesic segment from x0 to x starts either by the edge e− or by e+ . Let UP0 be the set of points y in V Xv − {t(e∗ )} such that the geodesic segment from t(e∗ ) to y starts by the edge e∗ . We have (1) Uh ∩ UP0 = ∅ and x0 ∈ / Uh ∪ UP0 , (2) hk (V Xv − Uh ) ⊂ Uh for every k ∈ Z − {0} and w(V Xv − UP0 ) ⊂ UP0 for every w ∈ P0 − {id}, (3) d(x, y) = d(x, x0 ) + d(x0 , y) for all x ∈ Uh and y ∈ UP0 .

15.5. Geometrically finite lattices with infinite Bowen–Margulis measure

327

∞ UP0

H∞

∗v e∗ e−

e+

Axh

x0 Uh

Uh

Let Γ0 be the subgroup of Γv generated by P0 and h. By a ping-pong argument, Γ0 is a free product of P0 and the infinite cyclic group generated by h, and Γ0 is geometrically finite (see, for instance, [Mask, Theorem C.2 (xi)] for Kleinian groups). Hence every element γ in Γ0 − {e} may be written uniquely as a word w0 hn0 w1 hn1 . . . wk hnk with k ∈ N, wi ∈ P0 , ni ∈ Z with wi 6= e if i 6= 0 and ni 6= 0 if i 6= k. Using the above properties, we have by induction X X d(x0 , w0 hn0 w1 hn1 . . . wk hnk x0 ) = d(x0 , hni x0 ) + d(x0 , wi x0 ) . (15.22) 06i6k

06i6k

Let λ : EXv → R+ be the Γ0 -invariant length map on the set of edges of Xv Ssuch that for every i ∈ N, the length of e ∈ EXv is 1 if e is not contained in γ∈Γ0 γH∞ , and otherwise, if e maps to ei or to ei under the canonical map Xv → Γv \Xv , then λ(e) = 1 + ln i+1 i if i > 1 and λ(e) = 1 if i = 0. Note that the distance in the metric graph |Xv |λ from ∗v to the vertex on the geodesic ray from ∗v to ∞ originally at distance i from ∗v is now i + ln i. The distances along the translation axis of h have not changed. Equation (15.22) remains valid with the new distance. Let us denote by d0 this (new) distance on |Xv |λ , and by QΓ0 0 (s) = QΓ0 0 , 0, x0 , x0 (s) and QP0 0 (s) the Poincaré series for the actions of Γ0 and P0 on (|Xv |λ , d0 ) (taking (x0 , x0 ) as a pair of basepoints and the zero system of conductances). Let us now prove that the discrete subgroup Γ0 of automorphisms of the metric tree (Xv , λ) (with degrees all equal to q + 1) satisfies the first claim of Theorem 15.11. By Γ0 -invariance of λ, the group Γ0 remains a discrete subgroup of Aut(Xv , λ). The elements of Γ0 ∞, or equivalently, the points at infinity of the horoballs in the Γ0 -equivariant family of horoballs (γH∞ )γ∈Γ0 /Γ0∞ with pairwise disjoint interiors

328


in Xv , remain bounded parabolic fixed points of Γ0 , and the other limit points remain conical limit points of Γ0 . Hence Γ0 remains a geometrically finite discrete subgroup of Aut(Xv , λ). The Poincaré series QP0 0 is (up to a multiplicative and additive constant) P∞ i −2si −2s i , which has the same critical exponent δP0 = ln2q as previously, i=1 q e but it is easy to see that the discrete group P0 is now of convergence type if q > 3. Let λ(h) be the (old and new) translation length of h. Using Equation (15.22) and partitioning the above words by the number of nonzero powers of h they contain, we have, for every s > 0, X 0 QΓ0 0 (s) = e−s d (x0 ,γx0 ) γ∈Γ0

6

1+

+∞ X k=1

X i∈Z−{0}

e

−s |i| λ(h)

X

e

−s d0 (x0 ,wx0 )

w∈P0

k ! X

0

e−s d (x0 ,wx0 ) .

w∈P0

At s = δP0 , the above sums over i ∈ Z − {0} and over w ∈ P0 converge, and hP by a high enough power if necessary, we may assume that P by replacing −s |i|λ(h) −s d0 (x0 ,wx0 ) e < 1 if s = δP0 , which makes the Poincaré i∈Z−{0} w∈P0 e 0 series of Γ converge at s = δP0 . Since P0 is a subgroup of Γ0 with critical exponent δP , we have that QΓ0 0 (s) > QP0 0 (s) = +∞ if s < δP0 . We conclude that the critical exponent of Γ0 is equal to δP0 and that Γ0 is of convergence type. By Corollary 4.7 (1), the Bowen–Margulis measure of Γ0 is infinite. In order to prove the second claim of Theorem 15.11, we first define a new length map λ : EXv → R+ that coincides with the previous one on every edge e of Xv , unless e maps to ei or to ei for every i ∈ N under the canonical map Xv → Γ\Xv , in which case we set λ(e) = 1 + bln(i + 1)c − bln ic if i > 1 and λ(e) = 1 if i = 0 (where b·c is the largest previous integer map). This map λ now has values in {1, 2}, and we subdivide each edge of length 2 into two edges of length 1. The tree Y thus obtained has uniformly bounded degrees (although it is no longer a uniform tree), and the group Γ0 defines a geometrically finite discrete subgroup of Aut(Y) with infinite Bowen–Margulis measure. To see that P0 is again of convergence type, observe that each distance in the simplicial case appearing in the Poincaré series differs by at most 1 from the corresponding distance in the metric tree case. The remainder of the argument is the same as in the metric tree case.

Chapter 16

Equidistribution and Counting of Rational Points in Completed Function Fields Let K be a (global) function field over Fq of genus g, let v be a (normalised discrete) valuation of K, let Kv be the associated completion of K, and let Rv be the affine function ring associated with v.1 In this chapter, we prove analogues of the classical results on the counting and equidistribution towards the Lebesgue measure on R of the Farey fractions pq with (p, q) ∈ Z×(Z−{0}) relatively prime.2 In particular, we prove various equidistribution results of locally finite families of elements of K towards the Haar measure on Kv , using the geometrical work on equidistribution of common perpendiculars done in Section 11.4 and recalled in Section 15.4.

16.1

Counting and equidistribution of non-Archimedean Farey fractions

The first result of this section is an analogue in function fields of the equidistribution of Farey fractions to the Lebesgue measure in R; see the introduction, and, for example, [PaP14b, p. 978] for the precise statement and a geometric proof. For every (x0 , y0 ) ∈ Rv × Rv − {(0, 0)}, let mv, x0 , y0 = Card{a ∈ (Rv )× : ∃ b ∈ x0 Rv ∩ y0 Rv , (a − 1)x0 y0 − bx0 ∈ y02 Rv } . For future use, note that by Equation (14.3), mv, 1, 0 = q − 1 .

(16.1)

1 See

Section 14.2 for definitions and background. for instance, [Nev], as well as [PaP14b] for an approach using methods similar to those in this text. 2 See,


329

330

Chapter 16. Equidistribution and Counting of Rational Points

For every (a, b) ∈ Rv × Rv and every subgroup H of GL2 (Rv ), let H(a,b) be the stabiliser of (a, b) for the linear action of H on Rv × Rv . Theorem 16.1. Let G be a finite-index subgroup of GL2 (Rv ), and let (x0 , y0 ) be an element in Rv × Rv − {(0, 0)}. Let c=

(qv 2 − 1) (qv + 1) ζK (−1) mv, x0 , y0 (N hx0 , y0 i)2 [GL2 (Rv ) : G] . (q − 1) q g−1 qv3 [GL2 (Rv )(x0 ,y0 ) : G(x0 ,y0 ) ]

Then, as s → +∞, c s−2

X

∆ xy

∗

* HaarKv .

(x, y)∈G(x0 , y0 ), |y|v 6s

For every β ∈ ]0, ln1qv ], there exists κ > 0 such that for every β-Höldercontinuous function ψ : Kv → R with compact support,3 as, for instance, if the function ψ : Kv → R is locally constant with compact support (see Remark 3.11), there is an error term in the equidistribution claim of Theorem 16.1 evaluated on ψ of the form O(s−κ kψkβ ). It is remarkable that due to the general nature of our geometrical tools, we are able to work with any finite-index subgroup G of GL2 (Rv ), and not only with its congruence subgroups. In this generality, the usual techniques (for instance, involving analysis of Eisenstein series) are not likely to apply. Also note that the H¨ older regularity for the error term is a much weaker assumption than the locally constant one that is usually obtained by analytic number theory methods. Theorem 1.13 in the introduction follows easily from this result, by taking K = Fq (Y ) (so that g = 0), v = v∞ , qv = q, (x0 , y0 ) = (1, 0), and s = q t , and using Equations (14.6) and (16.1) in order to simplify the constant. Before proving Theorem 16.1, let us give a counting result that follows from this equidistribution result. The additive group Rv acts on Rv × Rv by horizontal shears (transvections), ∀ z ∈ Rv , ∀ (x, y) ∈ Rv × Rv ,

z · (x, y) = (x + zy, y) ,

and this action preserves the absolute value |y|v of the vertical coordinate y. Let G be a finite-index subgroup of GL2 (Rv ), and let x0 , y0 ∈ Rv . Let Rv, G be the finite-index additive subgroup of Rv consisting of the elements x ∈ Rv such that 1 x ∈ G, acting by horizontal shears on G(x0 , y0 ). Note that Rv, G = Rv if 0 1 G = GL2 (Rv ). We may then define a counting function ΨG, x0 ,y0 of the elements of K in an orbit by homographies under G, as ΨG, x0 , y0 (s) = Card Rv, G \ (x, y) ∈ G(x0 , y0 ), |y|v 6 s} . 3 Where

Kv is endowed with the distance (x, y) 7→ |x − y|v .

16.1. Counting and equidistribution of non-Archimedean Farey fractions

331

Corollary 16.2. Let G be a finite-index subgroup of GL2 (Rv ), and let (x0 , y0 ) be an element in Rv × Rv − {(0, 0)}. Then there exists κ > 0 such that as s → +∞, ΨG, x0 ,y0 (s) =

(q − 1) q 2g−2 qv3 [GL2 (Rv )(x0 ,y0 ) : G(x0 ,y0 ) ] [Rv : Rv, G ] s2 + O(s2−κ ) . (qv − 1) (qv + 1) ζK (−1) mv, x0 , y0 (N hx0 , y0 i)2 [GL2 (Rv ) : G] 2

Proof. This follows from Theorem 16.1, by considering the locally constant characteristic function of a closed and open fundamental domain of Kv modulo the action by translations of Rv, G , and using Lemma 14.4 with I = Rv . Let us fix some notation for this section. For every subgroup H of GL2 (Rv ), we denote by H its image in Γv = PGL2 (Rv ). Let Xv be the Bruhat–Tits tree4 of (PGL2 , Kv ), which is regular of degree qv + 1. Let r=

x0 ∈ K ∪ {∞} . y0

If y0 = 0, let gr = id ∈ GL2 (K), and if y0 6= 0, let r 1 gr = ∈ GL2 (K) . 1 0 Proof of Theorem 16.1. We apply Theorem 15.4 with Γ = G, D− = H∞ and D+ = gr H∞ . Recall that H∞ is the horoball in Xv centered at ∞ whose boundary contains ∗v (see Section 15.1). Note that Γ has finite index in Γv , and in particular, it is a tree lattice of Xv . By [Ser3, II.1.2, Cor.], for all x ∈ V Xv and γ ∈ GL2 (Rv ), the distance d(x, γx) is even, since v(det γ) = 0. Hence by the equivalence in Equation (4.17), the length spectrum LΓv of Γv is 2Z. The length spectrum of Γ is also 2Z, since it is contained in LΓv . Note that D+ is a horoball in Xv centered at r = xy00 ∈ ∂∞ Xv , by Lemma 15.1. The stabiliser ΓD− of D− (respectively ΓD+ of D+ ) coincides with the stabiliser Γ∞ of ∞ ∈ ∂∞ Xv (respectively the stabiliser Γr of r) in Γ. Note that the Γ-equivariant families D ± = (γD± )γ∈Γ/ΓD± are locally finite, since Γv , and hence its finite-index subgroup Γ, is geometrically finite,5 and since ∞ and r ∈ K are bounded parabolic limit points of Γv , hence of its finite-index subgroup Γ. − For every γ ∈ Γ/Γr such that D− and γD+ are disjoint, let αe, γ be the − + generalised geodesic line, isometric exactly on [0, d(D , γD )], whose image is the common perpendicular between D− and γD+ , and let ργ be the geodesic ray − starting (at time t = 0) from αe, γ (0) and ending at the point at infinity γ · r of + − γD . Note that ργ and αe, γ coincide on [0, d(D− , γD+ )]. 4 See 5 See


332


Since the Patterson densities of tree lattices of Xv have total mass qvq+1 by v Proposition 15.2 (2), they are normalised as in Theorem 15.4. Then by Equation (15.14), we have lim

n→+∞

(qv 2 − 1)(qv + 1) Vol(Γ\\Xv ) −n qv − 2 qv3 kσD +k

X

∆(ργ )+ = (∂ + )∗ σ eD+− .

γ∈Γ/Γr 0 0 in the above formula when evaluated on ψ ∈ Ccβ ln qv (∂∞ Xv − {∞}), where ∂∞ Xv − {∞} is endowed with Hamenstädt’s distance6 dH∞ . Hence, by using Equation (15.3), we have an error term O(e−κ n kψkβ ) for some κ > 0 in the above formula when evaluated on ψ ∈ Ccβ (Kv ), where Kv = ∂∞ Xv − {∞} is endowed with the distance (x, y) 7→ |x − y|v . By Proposition 15.2 (3), we have (∂ + )∗ σ eD+− = HaarKv . Hence Equation (16.2) gives, with the appropriate error term, lim

n→+∞


X

∆γ·r = HaarKv .

γ∈Γ/Γr 0−t

∗

* HaarKv , where the product ranges over the prime factors p of the ideal I. Furthermore, if Ψ(t) = Card Rv \ (x, y) ∈ Rv × I : hx, yi = Rv , v(y) > −t , then there exists κ > 0 such that as t → +∞, Ψ(t) =

q 2g−2 qv3 Q (qv 2 − 1) (qv + 1) ζK (−1) N (I) p|I (1 +

1 N (p) )

qv 2t + O(qv (2−κ)t ) .

336


For every β ∈ ]0, ln1qv ], there exists κ > 0 such that for every ψ ∈ Ccβ (Kv ) there is an error term in the above equidistribution claim evaluated on ψ of the form O(qv−κ t kψkβ ). Proof. The counting claim is deduced from the equidistribution claim in the same way that Corollary 16.2 is deduced from Theorem 16.1, noting that the action of Rv by horizontal shears preserves Rv × I. In order to prove the equidistribution claim, we apply Theorem 16.1 with (x0 , y0 ) = (1, 0) and with G the Hecke congruence subgroup a GI = c

b ∈ GL2 (Rv ) : c ∈ I , d

(16.11)

which is the preimage of the upper triangular subgroup of GL2 (Rv /I) by reduction modulo I. In this case, the constant mv,x0 ,y0 appearing in the statement of Theorem 16.1 is equal to q − 1 by Equation (16.1). The group GI has finite index in GL2 (Rv ). The following result is well known to arithmetic readers (see, for instance, [Shi, page 24] when Rv is replaced by Z). We give only a sketch of proof (indicated to us by J.-B. Bost) for the sake of the geometer readers. Lemma 16.5. We have Y [GL2 (Rv ) : GI ] = N (I) 1+ p|I

1 N (p)

,

where the product ranges over the prime factors p of the ideal I. Proof. Recall that we denote by |E| the cardinality of a finite set E. For every commutative ring A with finite group of invertible elements A× , we have a disjoint union [ a 0 GL2 (A) = SL2 (A) . 0 1 × a∈A

a 0 Hence [GL2 (A) : SL2 (A)] = |A |. Since belongs to GI for all a ∈ (Rv )× , 0 1 we have [ a 0 GI = GI ∩ SL2 (Rv ) , 0 1 × ×

a∈(Rv )

so that [GL2 (Rv ) : GI ] = [SL2 (Rv ) : GI ∩ SL2 (Rv )]. The group morphism of reduction modulo I from SL2 (Rv ) to SL2 (Rv /I) is onto, by an argument of further reduction to the various prime power factors of I and lifting elementary matrices. The order of the upper triangular subgroup of SL2 (Rv /I) is |(Rv /I)× | |Rv /I|, where (Rv /I)× is the group of invertible elements

16.1. Counting and equidistribution of non-Archimedean Farey fractions

337

of the ring Rv /I (which we will see again below). Hence [GL2 (Rv ) : GI ] = [SL2 (Rv ) : GI ∩ SL2 (Rv )] = =

| SL2 (Rv /I)| |(Rv /I)× | |Rv /I|

| GL2 (Rv /I)| . |(Rv /I)× |2 |Rv /I|

(16.12)

By the multiplicativity of the norm and by the Chinese remainder theorem,7 one reduces the result to the case that I = pn is the nth power of a fixed prime ideal p with norm N (p) = N , where n ∈ N. Note that since Rv /p is a field, we have | GL1 (Rv /p)| = |(Rv /p)× | = |Rv /p| − 1 = N − 1 and 1 | GL2 (Rv /p)| = (|Rv /p|2 − 1)(|Rv /p|2 − |Rv /p|) = N 2 (N − 1)2 1 + . N For k = 1 or k = 2, the kernel of the morphism of reduction modulo p from 2 GLk (Rv /I) = GLk (Rv /pn ) to GLk (Rv /p) has order N k (n−1) . Hence 1 , | GL2 (Rv /I)| = N 4(n−1) N 2 (N − 1)2 1 + N and |(Rv /I)× | = N n−1 (N − 1) . Therefore, by Equation (16.12), we have [GL2 (Rv ) : GI ] =

N 4(n−1) N 2 (N − 1)2 1 + N 2(n−1) (N − 1)2 N n

1 N

1 . = Nn 1 + N

This proves the result.

We can now conclude the proof of Theorem 16.4. Note that GL2 (Rv )(1,0) = (GI )(1,0) . The result then follows from Theorem 16.1 and its Corollary 16.2, using the change of variables s = (qv )t , since GI (1, 0) = {(x, y) ∈ Rv × I : hx, yi = Rv } .

The following result is a particular case of Theorem 16.4. Corollary 16.6. Let P0 be a nonzero element of the polynomial ring R = Fq [Y ] Qk over Fq , and let P0 = a0 i=1 (Pi )ni be the prime decomposition of P0 . Then as t → +∞, Qk X (q + 1) i=1 q ni deg Pi (1 + q − deg Pi ) −2t ∗ q ∆ P * HaarFq ((Y −1 )) . 2 Q (q − 1) q (P,Q)∈R×(P0 R) P R+QR=R, deg Q6t

7 Saying

that the rings Rv /I and

Q

p

Rv /pvp (I) are isomorphic; see, for instance, [Nar, page 11].

338


For every β ∈ ]0, ln1qv ], there exists a constant κ > 0 such that for every ψ ∈ Ccβ Fq ((Y −1 )) there is an error term in the above equidistribution claim evaluated on ψ of the form O(q −κ t kψkβ ). Proof. In this statement, we use the standard convention that k = 0 if P0 is constant, a0 ∈ (Fq )× , and Pi ∈ R is monic. We apply the first claim of Theorem 16.4 with K = Fq (T ) and v = v∞ so Qk that g = 0, qv = q, and Rv = R, and with I = P0 R, so that N (I) = i=1 q ni deg Pi . The result follows from Equation (14.6).

16.2

Mertens’s formula in function fields

In this section, we recover the function field analogue of Mertens’s classical formula on the average order of the Euler function. We begin with a more general counting and equidistribution result. Let m be a (nonzero) fractional ideal of Rv , with norm N (m). Note that the action of the additive group Rv on Kv × Kv by the horizontal shears z · (x, y) = (x + zy, y) preserves m × m. We consider the counting function ψm : [0, +∞[ → N defined by ψm (s) = Card Rv \ (x, y) ∈ m × m : 0 < N (m)−1 N (y) 6 s, hx, yi = m . Note that ψm depends only on the ideal class of m, and thus we can assume in the computations that m is integral, that is, contained in Rv . Corollary 16.7. There exists κ > 0 such that as s → +∞, ψm (s) =

(qv

2

(q − 1) q 2g−2 qv 3 − 1) (qv + 1) ζK (−1) mv, x0 , y0

s2 + O(s2−κ ) ,

where m = hx0 , y0 i. Furthermore, as s → +∞, (qv 2 − 1) (qv + 1) ζK (−1) mv, x0 , y0 −2 s (q − 1) q g−1 qv 3

X

∗

∆ xy * HaarKv .

(x,y)∈m×m N (m)−1 N (y)6s, hx,yi=m

For every β ∈ ]0, ln1qv ], there exists κ > 0 such that for every ψ ∈ Ccβ Kv ) there is an error term in the above equidistribution claim evaluated on ψ of the form O(s−κ kψkβ ). Theorem 1.14 in the introduction follows easily from this result, by taking K = Fq (Y ) (so that g = 0) and v = v∞ (so that qv = q). In order to simplify the constant, we use Equation (14.6) and the fact that the ideal class number of K, which equals the number of orbits of PGL2 (Fq [Y ]) on P1 (Fq (Y )), is 1. Thus,

16.2. Mertens’s formula in function fields

339

if m = hx0 , y0 i, then the constant mv, x0 , y0 is equal to mv, 1, 0 , which is q − 1 by Equation (16.1). Proof. Every nonzero ideal I in Rv is of the form I = xRv + yRv for some (x, y) in Rv × Rv − {(0, 0)}; see, for instance, [Nar, Cor. 5, page 11]. For all (x, y) and (z, w) in Rv ×Rv , we have x Rv +y Rv = z Rv +w Rv if and only if (z, w) ∈ GL2 (Rv )(x, y). The ideal class group of K corresponds bijectively to the set PGL2 (Rv )\P1 (K) of cusps of the quotient graph of groups PGL2 (Rv )\\Xv ,8 by the map induced by I = x Rv + y Rv 7→ [x : y] ∈ P1 (K). Given a fixed ideal m in Rv , we apply Theorem 16.1 with G = GL2 (Rv ) and (x0 , y0 ) ∈ Rv × Rv − {(0, 0)} a fixed pair such that x0 Rv + y0 Rv = m, so that G(x0 , y0 ) = {(x, y) ∈ m × m : hx, yi = m}. Using therein the change of variable s 7→ N (m)s and Equation (14.5), the result follows from Theorem 16.1 and its Corollary 16.2. As already encountered in the proof of Lemma 16.5, the Euler function ϕRv of Rv is defined on the set of (nonzero, integral) ideals I of Rv by setting9 ϕRv (I) = Card((Rv /I)× ) , and we set ϕRv (y) = ϕRv (y Rv ) for every y ∈ Rv . Thus, by the definition of the action of Rv on Rv × Rv by horizontal shears, we have X ψRv (s) = Card{x ∈ Rv /yRv : hx, yi = Rv } y∈Rv , 0 0 , x∈V Xv

and the subset Axγ = {x ∈ V Xv : d(x, γx) = λ(γ)} is the image of a (discrete) geodesic line in Xv , which we call the (discrete) translation axis of γ. The points at infinity of Axγ are denoted by γ − and γ + , chosen so that γ translates away from γ − and towards γ + on Axγ . Note that for every γ 0 ∈ PGL2 (Kv ), we have γ 0 Axγ = Axγ 0 γ (γ 0 )−1

and

γ 0 γ ± = (γ 0 γ (γ 0 )−1 )± .

If Γ is a discrete subgroup of PGL2 (Kv ) and if α is one of the two fixed points of a loxodromic element of Γ, we denote the other fixed point of this element by ασ . Since Γ is discrete, the translation axes of two loxodromic elements of Γ coincide if they have a common point at infinity. Hence ασ is uniquely defined. For every γ ∈ Γ, we hence have3 (γ · α)σ = γ · (ασ ) . (17.1) We define the complexity h(α) of the loxodromic fixed point α by h(α) =

1 |α − ασ |v

(17.2)

if α, ασ 6= ∞, and by h(α) = 0 if α or ασ is equal to ∞. We define ια ∈ {1, 2} by ια = 2 if there exists an element γ ∈ Γ such that γ · α = ασ , and ια = 1 otherwise. Following [Ser3, II.1.2], we denote by PGL2 (Kv )+ the kernel of the group morphism PGL2 (Kv ) → Z/2Z defined by γ = [g] 7→ v(det g) mod 2. The definition does not depend on the choice of a representative g ∈ GL2 (Kv ) of an element γ ∈ PGL2 (Kv ), since λ 0 v det = 2 v(λ) 0 λ is even for every λ ∈ (Kv )× . Note that when Kv is the completion of a function field over Fq endowed with a valuation v, with associated affine function ring Rv , 2 See

Section 2.1. that the groups GL2 (Kv ) and PGL2 (Kv ) act on P1 (Kv ) = Kv ∪ {∞} by homographies, and that these actions are denoted by · (see Section 15.1). 3 Recall

17.1. Counting and equidistribution of loxodromic fixed points

343

the group Γv = PGL2 (Rv ) is contained in PGL2 (Kv )+ : for every g ∈ GL2 (Rv ), since det g ∈ (Rv )× = (Fq )× ,4 we have v(det g) = 0. The following result proves the equidistribution in Kv of the loxodromic fixed points with complexity at most s in a given orbit by homographies under a lattice in PGL2 (Kv ) as s → +∞, and its associated counting result. If ξ is a point of ∂∞ Xv = P1 (Kv ) and if Γ is a subgroup of PGL2 (Kv ), we denote by Γξ the stabiliser in Γ of ξ. Theorem 17.1. Let Γ be a lattice in PGL2 (Kv )+ , and let γ0 ∈ Γ be a loxodromic element of Γ. Then as s → +∞, (qv + 1)2 Vol(Γ\\Xv ) −1 s 2 qv2 Vol(Γγ − \\ Axγ0 ) 0

X

∗

∆α * HaarKv ,

α ∈ Γ·γ0− , h(α)6s

and there exists κ = κΓ > 0 such that Card{α ∈ (Γ · γ0− ) ∩ Ov : h(α) 6 s} =

2 qv2 Vol(Γγ − \\ Axγ0 ) 0

(qv + 1)2 Vol(Γ\\Xv )

s + O(s1−κ ) .

For every β ∈ ]0, ln1qv ], there is an error term of the form O(s−κ kψkβ ) for some κ > 0 in the equidistribution claim evaluated on any β-Hölder-continuous function ψ : Ov → C. Proof. The second result follows from the first one by integrating on the characteristic function of the compact-open subset Ov , whose Haar measure is 1. In order to prove the equidistribution result, we apply Theorem 15.4 with D− = {∗v } and D+ = Axγ0 . The families D ± = (γD± )γ∈Γ/ΓD± are locally finite, since Γ is discrete and the stabiliser ΓD+ of D+ acts cocompactly on D+ . Fur− thermore, kσD + k is finite and nonzero by Equation (8.12). Since Γ is contained in PGL2 (Kv )+ , the length spectrum LΓ of Γ is contained in 2Z by [Ser3, II.1.2, Cor.]. Hence, it is equal to 2Z by the equivalence given by Equation (4.17). − For every γ ∈ Γ such that d(D− , γD+ ) > 0,5 let αe, γ be the generalised − + geodesic line, isometric exactly on [0, d(D , γD )], whose image is the common perpendicular between D− and γD+ , and let ργ be the geodesic ray starting at − − time 0 from the origin of αe, γ (which is ∗v ) with point at infinity γ · γ0 . Since Xv is a tree and γ · γ0− is one of the two endpoints of γD+ , the geodesic segment − 6 αe, γ |[0,d(D− ,γD+ )] is an initial subsegment of ργ . Therefore, by Equation (15.14), 4 See

Equation (14.3). means that ∗v ∈ / γD+ . 6 It connects ∗ to its closest point P + v γD+ (∗v ) on γD , with P · (·) defined in Section 2.4. 5 This

344

Chapter 17. Equidistribution and Counting of Quadratic Irrational Points

1 − for the weak-star convergence of measures on ∂+ D , we have

lim

n→+∞


X

∆γ·γ − = (∂ + )∗ σ eD+− . 0

γ∈Γ/ΓD+ 0 0 in the above formula when evaluated on ψ ∈ Ccβ ln qv (∂∞ Xv ), where ∂∞ Xv is endowed with the visual distance d∗v .8 Note that on Ov , the visual distance d∗v and the distance (x, y) 7→ |x − y|v are related by |x − y|v = dH∞ (x, y)ln qv = d∗v (x, y)ln qv , using Equation (15.2) for the first equality. Hence we have an error term of the form O(e−κ n kψkβ ) for some κ > 0 in the above formula when evaluated on ψ ∈ Ccβ (Ov ), where Ov is endowed with the distance (x, y) 7→ |x − y|v . ∞

H∞ k

k

pH∞ (πv−k )

∗v

∂H∞

γD+ Kv = ∂∞ Xv − {∞} Ov

πv−k + Ov

γ · γ0−

γ · γ0+

Let us fix for the moment k ∈ N. For every ξ ∈ πv−k + Ov , we have |ξ|v = = qvk if k > 1 and |ξ|v 6 1 if k = 0. By restricting the measures to the

−v(ξ) qv 7 See

the end of Section 2.6. that when D = {x} is a singleton, the distance-like map dD used in Remark 15.6 coincides with the visual distance dx , as stated after Equation (3.8). 8 Note

17.1. Counting and equidistribution of loxodromic fixed points

345

compact-open subset πv−k + Ov and by Proposition 15.2 (1), we have, with the appropriate error term when k = 0, lim

n→+∞


X

∆γ·γ − 0

γ∈Γ/ΓD+ γ·γ0− ∈πv−k +Ov 0 0 if i > 0 such that 1

f = a0 +

.

1

a1 + a2 +

1 a3 +

1 .. .

See, for instance, the surveys [Las, Sch2], as well as [Pau3] for a geometric interpretation. We say that the continued fraction expansion of f is eventually periodic

17.2. Counting and equidistribution of quadratic irrationals

353

if there exist n ∈ N and N ∈ N − {0} such that an+i = an+N +i for every i ∈ N, and we write f = [a0 , . . . , an−1 , an , . . . , an+N −1 ] . Such a sequence an , . . . , an+N −1 is called a period of f , and if of minimal length, it is well defined up to cyclic permutation. Two elements β, β 0 ∈ Fq ((Y −1 )) are in the same PGL2 (Fq [Y ])-orbit if and only if their continued fraction expansions have equal tails up to an invertible element of Fq [Y ] by [Sch2, Thm. 1] or [BerN, Thm. 1] (and even before that by [deM, Sect. IV.3]). More precisely, β, β 0 ∈ Kv are in the same PGL2 (Fq [Y ])-orbit if and only if there exist m, n ∈ N and x ∈ F× q such that for every k ∈ N, we have 0 (−1)k an+k (β ) = x am+k (β). Proposition 17.8. Assume that K = Fq (Y ) and v = v∞ . (1) An element α ∈ Kv − K is quadratic irrational over K if and only if its continued fraction expansion of β is eventually periodic, and if and only if it is a fixed point of a loxodromic element of PGL2 (Fq [Y ]). (2) A quadratic irrational α ∈ Kv is reciprocal if and only if the period a0 , . . . , aN −1 of the continued fraction expansion of α is palindromic up to cyclic permutation and invertible elements, in the sense that there exist x ∈ F× q and p ∈ N such (−1)k that for k = 0, . . . , N − 1, we have ak+p = x aN −k−1 (with indices modulo N ). Proof. (1) The equivalence of being quadratic irrational and having an eventually periodic continued fraction expansion is well known; see, for instance, the survey [Las, Thm. 3.1]. The second part of the claim follows from Proposition 17.3. (2) The proof is similar to the Archimedean case in [Per, §23].12 For every quadratic irrational f ∈ Fq ((Y −1 )), up to the action of GL2 (Fq [Y ]), we may assume that f, (f σ )−1 ∈ Y −1 Fq [[Y −1 ]] and f = [0, a1 , a2 , . . . , an ]. Then we may define by induction quadratic irrationals f2 , . . . , fn ∈ Fq ((Y −1 )) over Fq (Y ) such that 1 1 1 1 = a1 + f2 , = a2 + f3 , . . . , = an+1 + fn , = an + f . f f2 fn+1 fn Passing to the Galois conjugates, we have 1 1 1 = a1 + f2σ , σ = a2 + f3σ , . . . , σ = an + f σ . fσ f2 fn Taking these equations in the reverse order, we have 1 1 1 1 1 1 = an − σ , = an−1 − σ , . . . , = a1 − σ , fn − f1σ fn−1 f − f1σ − f1σ n

12 See

2

also [BerN, Cor. 1] by relating, using twice the period, what the authors call the − continued fraction expansion to the standard expansion.

354


so that since − f1σ ∈ Y −1 Fq [[Y −1 ]], we have −

1 = [0, an , . . . , a2 , a1 ] . fσ

Therefore f σ = [−an , . . . , −a2 , −a1 ]. Thus, if f and f σ are in the same orbit, the periods are palindromic up to cyclic permutation and invertible elements by [Sch2, Thm. 1], [BerN, Thm. 1].

17.3

Counting and equidistribution of quadratic irrationals in Qp

There are interesting arithmetic (uniform) lattices of PGL2 (Qp ) constructed using quaternion algebras. In this section, we study equidistribution properties of loxodromic fixed-point elements of these lattices. See, for instance, [LedP] for an equidistribution result of the eigenvalues of the loxodromic elements. We use [Vig] as our standard reference on quaternion algebras. Let F be a field and let a, b ∈ F × . Let D = a,b be the quaternion algebra F over F with basis 1, i, j, k as an F -vector space such that i2 = a, j 2 = b, and ij = ji = −k. If x = x0 + x1 i + x2 j + x3 k ∈ D, then its conjugate is x = x0 − x1 i − x2 j − x3 k , its (reduced) norm is N(x) = x x = x x = x20 − a x21 − b x22 + ab x23 , and its (reduced) trace is Tr(x) = x + x = 2 x0 . Let us fix two negative rational integers a, b and let D = a,b Q . For every field extension E of Q, we denote by DE the quaternion algebra D ⊗Q E over E, and we say that D splits over E if the E-algebra D ⊗Q E is isomorphic to M2 (E). The assumption that a, b are negative implies that D does not split over R. Furthermore, when p ∈ N is an odd prime, D splits over Qp if and only if the equation a x2 + b y 2 = 1 has a solution in Qp ; see [Vig, page 32]. The reduced discriminant of D is Y DiscD = q, q∈Ram(D)

where Ram(D) is the finite set of primes p such that D does not split over Qp . For instance, the quaternion algebra D = −1,−1 splits over Qp if and only Q if p 6= 2; hence it has reduced discriminant 2.

17.3. Counting and equidistribution of quadratic irrationals in Qp

355

Assume from now on that p ∈ N is a positive rational prime that D √ such √ splits over Qp and, for simplicity, that Qp contains square roots a and b of a and b. For example, if a = b = −1, this is satisfied if p ≡ 1 mod 4. We then have an isomorphism of Qp -algebras θ = θa, b : DQp → M2 (Qp ) defined by θ(x0 + x1 i + x2 j + x3 k) =

√ ! √ √ x0 + x1 a b (x2 + a x3 ) , √ √ √ b (x2 − a x3 ) x0 − x1 a

(17.9)

so that det(θ(x)) = N(x) and

tr(θ(x)) = Tr(x) .

If the assumption on the existence of the square roots in Qp is not satisfied, we can replace Qp by an appropriate finite extension, and prove equidistribution results in this extension. Let O be a Z p1 -order in DQp , that is, a finitely generated Z p1 -submodule of DQp generating DQp as a Qp -vector space, which is a subring of DQp . Let O 1 be the group of elements of norm 1 in O. Then the image Γ1O of θ(O 1 ) in PGL2 (Qp ) is a cocompact lattice; see, for instance, [Vig, Sect. IV.1]. In fact, this lattice is contained in PSL2 (Qp ), hence in PGL2 (Qp )+ . In this section, we denote by Xp the Bruhat–Tits tree of (PSL2 , Qp ), which is (p + 1)-regular. The next result computes the covolume of this lattice.13 Proposition 17.9. Let D be a quaternion algebra over Q that splits over Qp and does not split over R, and let O be a Z p1 -order in DQp . If Omax is a maximal Z p1 -order in DQp containing O, then 1 Vol(Γ1O \\Xp ) = [Omax : O 1]

p 12

Y

(q − 1) .

q| DiscD

Proof. We refer to [Vig, page 53] for the (common) definition of the discriminant Disc(Qp ) of the local field Qp and Disc(DQp ) of the quaternion algebra DQp over the local field Qp . We will use only the facts that Disc(Qp ) = 1, as easily follows from the definition, and that 2 Disc(DQp ) = Disc(Qp )4 N (pZp ) = p2 ,

(17.10)

which follows by [Vig, Lem. 4.7, page 53] and [Vig, Cor. 1.7, page 35] for the first equality and N (pZp ) = Card(Zp /pZp ) = Card(Z/pZ) = p for the second one. We refer to [Vig, Sect. II.4] for the definition of the Tamagawa measure µT on X × when X = DQp or X = Qp . It is a Haar measure of the multiplicative locally compact group X × , and understanding its explicit normalisation is the main point of this proposition. By [Vig, Lem. 4.6, page 52],14 with dx the Haar measure on 13 The 14 See

index q ranges over the primes dividing DiscD , that is, over the elements of Ram(D) . more precisely the top of page 55 in op. cit.

356


the additive group X,15 with kxk the module of the left multiplication by x ∈ X × on the additive group X,16 we have 1 dµT (x) = p dx . Disc(X) kxk By [Vig, proof of Lem. 4.3, page 50], identifying DQp with M2 (Qp ) by θ, the mass 1 −2 of GL2 (Zp ) for the measure (1−p−1 . Hence, by scaling and by ) kxk dx is 1 − p Equation (17.10), we have µT (GL2 (Zp )) =

(1 − p−2 )(1 − p−1 ) (p2 − 1)(p − 1) p = . p4 Disc(DQp )

By [Vig, Lem. 4.3, page 49], the mass of Z× p for the measure is 1; hence by scaling,

1 (1−p−1 ) kxk

dx on Q× p

1 − p−1 p−1 . µT (Z× = p)= p p Disc(Qp ) By [Vig, pages 53–54], since we have an exact sequence det

1 −→ SL2 (Qp ) −→ GL2 (Qp ) −→ Q× p −→ 1 , the Tamagawa measure of GL2 (Qp ) disintegrates by the determinant over the Tamagawa measure of Q× p with conditional measures the translates of a measure on SL2 (Qp ), called the Tamagawa measure of SL2 (Qp ) and again denoted by µT . Thus, µT (GL2 (Zp )) p2 − 1 µT (SL2 (Zp )) = = . p3 µT (Z× p) By Example 3 on page 108 of [Vig], since the Z p1 -order Omax is maximal, 1 we have, with G = θ(Omax ), Y 1 µT (G\ SL2 (Qp )) = (1 − p−2 ) (q − 1) . 24 q| DiscD

Since GL2 (Qp ) acts transitively on V Xp with stabiliser of the basepoint ∗ = [Zp × Zp ] the maximal compact subgroup GL2 (Zp ),17 and by the centered equation midpage 116 of [Ser3], we have X 1 µT (G\ GL2 (Qp )) µT (G\ SL2 (Qp )) Vol(G\\Xp ) = = = |Gx | µT (GL2 (Zp )) µT (SL2 (Zp )) [x]∈G\V Xp Y p (q − 1) . = 24 q| DiscD

15 With

a normalisation that does not need to be made precise. have (Mx )∗ dx = kxk dx, where Mx : y 7→ xy is the left multiplication by x on X. 17 See Section 15.1. 16 We


357

1 The natural homomorphism G = θ(Omax ) → Γ1Omax is 2-to-1, so that

Vol(Γ1Omax \\Xp ) = 2 Vol(G\\Xp ) . 1 Since [Γ1Omax : Γ1O ] = [Omax : O 1 ], Proposition 17.9 follows.

Note that the fixed points z for the action on P1 (Qp ) = Qp ∪ {∞}√by homo√ graphies of the elements in the image of θ(D) are quadratic over Q( a, b). More √ precisely, √zb is quadratic over Q( a). An immediate application of Theorem 17.1, using Proposition 17.9, √ gives √ the following result of equidistribution of quadratic elements in Qp over Q( a, b). Theorem 17.10. Let Γ be a finite-index subgroup of Γ1O , and let γ0 ∈ Γ be a loxodromic element of Γ. Then as s → +∞, (p + 1)2

Q

q| DiscD (q

1 − 1) [Omax : O 1 ] [Γ1O : Γ]

24 p Vol(Γγ − \\ Axγ0 )

X

s−1

α ∈ Γ·γ0− ,

0

∆α

h(α)6s

∗

* HaarQp , where Omax is a maximal Z p1 -order in DQp containing O, and there exists κ > 0 such that as s → +∞, Card{α ∈ (Γ · γ0− ) ∩ Zp : h(α) 6 s} =

24 p Vol(Γγ − \\ Axγ0 ) 0

(p +

1)2

Q

q| DiscD (q

1 − 1) [Omax : O 1 ] [Γ1O : Γ]

s + O(s1−κ ) .

Assume furthermore that the positive rational prime p ∈ N is such that p ≡ 1 2 mod 4 and that the integer p 4−1 is not of the form 4a (8b + 7) for a, b ∈ N (for instance, p = 5). By Legendre’s three squares theorem (see, for instance, [Gros]), 2 2 2 2 there exist x01 , x02 , x03 ∈ Z such that p 4−1 = x01 + x02 + x03 . Hence there are 2 2 2 2 x1 , x2 , x3 ∈ 2Z such that p − 1 = x1 + x2 + x3 . A standard consequence of Hensel’s theorem says that when p is odd, a number n ∈ Z has a square root in Zp if n is relatively prime to p and has a square root modulo p; see, for instance, [Kna, page 351]. Thus, 1 − p2 has a square root in p 2 Zp , which we denote by 1 − p . As noted above, since p ≡ 1 mod 4, the element −1 has a square root in Qp , which we denote by ε. The element p ε x1 + 1 − p2 α0 = x3 + ε x2 is a quadratic irrational in Qp over Q(ε).

358


The following result is a counting and equidistribution result of quadratic irrationals over Q(ε) in Qp . We denote by ασ the Galois conjugate of a quadratic irrational α in Qp over Q(ε), and by h(α) =

1 |α − ασ |p

the complexity of α. Theorem 17.11. Let D = −1,−1 be the Hamiltonian quaternion algebra over Q. Q Let p ∈ N be a positive rational prime with p ≡ 1 mod 4 such that there exist x1 , x2 , x3 ∈ 2Z with p2 − 1 = x1 2 + x2 2 + x3 2 , and let O be the Z p1 -order18 O = y ∈ Z p1 + Z p1 i + Z p1 j + Z p1 k : y ≡ 1 mod 2 in DQp . Let Γ be a finite-index subgroup of Γ1O . Then as s → +∞, (p + 1)2 [Γ1O : Γ] −1 s 2 p2 kΓ

∗

X

∆α * HaarQp ,

α ∈ Γ·α0 , h(α)6s

1 + ε x1 x3 + ε x2 Furthermore, there exists κ > 0 such that as s → +∞, where kΓ is the smallest positive integer such that

Card{α ∈ (Γ · α0 ) ∩ Zp : h(α) 6 s} =

−x3 + ε x2 1 − ε x1

kΓ ∈ Γ.

2 p2 kΓ s + O(s1−κ ) . (p + 1)2 [Γ1O : Γ]

Proof. The group O × of invertible elements of O is O × = x ∈ O : N(x) ∈ pZ . The center of O × is Z(O × ) = {±pn : n ∈ Z}, and the center of O 1 is reduced to Z(O 1 ) = {±1}. We identify O 1 /Z(O 1 ) with its image in O × /Z(O × ). The quotient group O × /Z(O × ) is a free group on s = p+1 2 generators γ1 , γ2 , . . . , γs , which are the images modulo Z(O × ) of some elements of O of norm p; see, for instance, [Lub2, Cor. 2.1.11].19 Since N(p) = p2 , any reduced word of even length in S = {γ1± , γ2± , . . . , γs± } belongs to O 1 /Z(O 1 ). Two distinct elements in S differ by a reduced word of length 2, and γ1 does not belong to O 1 /Z(O 1 ). Hence {1, γ1 } is a system of left coset representatives of O 1 /Z(O 1 ) in O × /Z(O × ), and the index of O 1 /Z(O 1 ) in O × /Z(O × ) is [O × /Z(O × ) : O 1 /Z(O 1 )] = 2 . (17.11) 18 This

order plays an important role in the construction of Ramanujan graphs by Lubotzky, Phillips, and Sarnak [LuPS1, LuPS2] (see also [Lub2, §7.4]), and in the explicit construction of free subgroups of SO(3) in order to construct Hausdorff–Banach–Tarsky paradoxical decompositions of the 2-sphere; see, for instance, [Lub2, page 11]. 19 The group O × /Z(O × ) is denoted by Λ(2) in [Lub2, page 11].


Let g0 =

1+ε x1 p x3 +ε x2 p

−x3 +ε x2 p 1−ε x1 p

359

! .

√ √ By the definition of the isomorphism θ in Equation (17.9) (with a = b = ε) and of the integers x1 , x2 , x3 , the element g0 belongs to θ(O), since x1 , x2 , x3 are even (and p is odd), and det g0 = 1. Hence g0 ∈ θ(O 1 ). Its fixed points for its action by homography on P1 (Qp ) are, by an easy computation, p ε x1 ± 1 − p2 . x3 + ε x2 In particular, α0 is one of these two fixed points. Note that tr g0 = p2 , hence |vp (tr g0 )| = 1, and the image [g0 ] of g0 in PGL2 (Qp ) is a primitive loxodromic element of Γ1O . Let us define γ0 = [g0 ]u kΓ where u ∈ {±1} is chosen so that γ0− = α0 and where kΓ is defined in the statement of Theorem 17.11. Since Γ has finite index in Γ1O , some power of [g0 ] belongs to Γ; hence kΓ exists (and note that kΓ = 1 if Γ = Γ1O ). By the minimality of kΓ , the element γ0 is a primitive loxodromic element of Γ. We will apply Theorem 17.1 to this γ0 . The algebra isomorphism θ induces a group isomorphism from O × /Z(O × ) 20 onto its image in PGL2 (Qp ), which we denote by Γ× By [Lub2, Lem. 7.4.1], the O. × group ΓO acts simply transitively on the vertices of the Bruhat–Tits tree Xp . In particular, Γ1O acts freely on Xp , and by Equation (17.11), we have × 1 Vol(Γ1O \\Xp ) = [Γ× O : ΓO ] Vol(ΓO \\Xp )

= [O × /Z(O × ) : O 1 /Z(O 1 )] Card Γ× O \V Xp = 2 .

(17.12)

Again since Γ1O (hence Γ) acts freely on Xp , since γ0 is primitive loxodromic in Γ, and by Equation (15.6), we have Vol(Γγ − \\ Axγ0 ) = Card Γγ − \V Axγ0 = λ(γ0 ) 0

0

= kΓ λ([g0 ]) = 2 kΓ |vp (tr g0 )| = 2 kΓ .

(17.13)

Using Equations (17.12) and (17.13), the result now follows from Theorem 17.1.

20 This

group is denoted by Γ(2) in [Lub2, page 95].

Chapter 18

Equidistribution and Counting of Cross-ratios We use the same notation as in Chapter 17: Kv is a non-Archimedean local field, with valuation v, valuation ring Ov , choice of uniformiser πv , residual field kv of order qv , and Xv is the Bruhat–Tits tree of (PGL2 , Kv ). Let Γ be a lattice in PGL2 (Kv ). In this chapter, we give several counting and equidistribution results inside Kv = ∂∞ Xv − {∞} of orbit points under Γ, using a complexity defined using cross-ratios, which is different from the one in Chapter 17. We refer to [PaP14b] for the development when Kv is R or C with its standard absolute value. Recall that the cross-ratio of four pairwise distinct points a, b, c, d in P1 (Kv ) = Kv ∪ {∞} is (c − a) (d − b) ∈ (Kv )× , [a, b, c, d] = (c − b) (d − a) with the standard conventions when one of the points is ∞. Adopting Ahlfors’s terminology in the complex case, the absolute cross-ratio of four pairwise distinct points a, b, c, d ∈ P1 (Kv ) = Kv ∪ {∞} is |a, b, c, d|v = |[a, b, c, d]|v =

|c − a|v |d − b|v , |c − b|v |d − a|v

with conventions analogous to the previous ones when one of the points is ∞. As in the classical case, the cross-ratio and the absolute cross-ratio are invariant under the diagonal projective action of GL2 (Kv ) on the set of quadruples of pairwise distinct points in P1 (Kv ).1 1 The

logarithm in base qv of this absolute cross-ratio is up to the order equal to the (logarithmic) cross-ratio [[·, ·, ·, ·]] introduced in Section 2.6: more precisely, if ξ1 , ξ2 , ξ3 , ξ4 are pairwise distinct points in the boundary of Xv , then [[ξ1 , ξ2 , ξ3 , ξ4 ]] = logqv |ξ1 , ξ4 , ξ3 , ξ2 |v . We will not use this relationship in this book.


361

362

Chapter 18. Equidistribution and Counting of Cross-ratios

18.1

Counting and equidistribution of cross-ratios of loxodromic fixed points

Let α, β ∈ Kv be loxodromic fixed points of Γ. Recall that ασ , β σ are the other fixed points of a loxodromic element of Γ fixing α, β, respectively. The relative height of β with respect to α is2 hα (β) =

|α −

1 max |β − α|v |β σ − ασ |v , |β − ασ |v |β σ − α|v . σ |β − β |v

ασ |v

When β ∈ / {α, ασ }, we have hα (β) = max{|α, β, β σ , ασ |v , |α, β σ , β, ασ |v } 1 . = σ σ min{|α, β, α , β |v , |α, β σ , ασ , β|v }

(18.1)

We will use the relative height as a complexity when β varies in a given orbit of Γ (and α is fixed). The following properties of relative heights are easy to check using the definitions, the invariance properties of the cross-ratio, and Equation (17.1). Lemma 18.1. Let α, β ∈ Kv be loxodromic fixed points of Γ. Then (1) hαρ (β τ ) = hα (β) for all ρ, τ ∈ {id, σ}. (2) If β ∈ {α, ασ }, then hα (β) = 1. (3) hγ·α (γ · β) = hα (β) for every γ ∈ Γ. (4) hα (γ · β) = hα (β) for every γ ∈ StabΓ ({α, ασ }).

The following result relates the relative height of two loxodromic fixed points to the distance between the two translation axes in Xv . Proposition 18.2. Let α, β ∈ Kv be loxodromic fixed points of Γ such that β does not belong to {α, ασ }. Then σ

hα (β) = qvd(]α,α

[ , ]β,β σ [)

.

In particular, we have hα (β) > 1 if and only if the geodesic lines ]α, ασ [ and ]β, β [ in Xv are disjoint, and hα (β) = 1 otherwise (using Lemma 18.1 (2) when β ∈ {α, ασ }). σ

Proof. Up to replacing α, β, ασ , β σ by their images under a large enough power γ of a loxodromic element in Γ with attracting fixed point in Ov , we may assume that these four points belong to Ov . Note that γ exists, since ΛΓ = ∂∞ Xv , and Γ preserves the relative height by Lemma 18.1 (3) as well as the distances between translation axes. 2 The

factor |α − ασ |v in the denominator, which did not appear in [PaP14b] in the analogous definition for the case in which Kv is R or C, is there in order to simplify the statements below.

18.1. Counting and equidistribution of cross-ratios of loxodromic fixed points

363

Let A = ]α, ασ [ and B = ]β, β σ [ . Let u be the closest point to ∗v on A, so that by the geometric interpretation of elements in Ov given in Section 15.1, we have v(α − ασ ) = d(u, ∗v ) . We will consider five configurations. ∗v

∗v

u

u0

a a=u

b α

ασ

β βσ

b

ασ β

α

βσ

Case 1. First assume that A and B are disjoint. Let [a, b] be the common perpendicular from A to B, with a ∈ A, so that d(A, B) = d(a, b) . First assume that u 6= a. Up to exchanging α, ασ (which does not change d(A, B) or hα (β) by Lemma 18.1 (1)), we may assume that a ∈ [u, α[ . Then (see the picture on the left above), v(β − β σ ) = d(b, ∗v ),

v(α − β) = v(α − β σ ) = d(a, ∗v ),

and v(ασ − β) = v(ασ − β σ ) = d(u, ∗v ) . Therefore σ

|α, β, ασ , β σ |v = |α, β σ , ασ , β|v = qvv(α−β)+v(α

−β σ )−v(α−ασ )−v(β−β σ )

= qvd(a, ∗v )−d(b, ∗v ) = qv−d(a,b) = qv−d(A,B) , which proves the result by Equation (18.1). Assume on the contrary that u = a. Let u0 ∈ V Xv be the vertex of Xv such that [a, ∗v ] ∩ [a, b] = [a, u0 ]. Note that u0 ∈ [∗v , b], since β, β σ ∈ Ov . Then (see the picture on the right above), v(β −β σ ) = d(b, ∗v ),

v(α−β) = v(α−β σ ) = v(ασ −β) = v(ασ −β σ ) = d(u0 , ∗v ) .

Therefore σ

|α, β, ασ , β σ |v = |α, β σ , ασ , β|v = qvv(α−β)+v(α 0


= qv2d(u , ∗v )−d(a, ∗v )−d(b, ∗v ) = qv−d(a, b) = qv−d(A, B) , which proves the result by Equation (18.1).

364


∗v

∗v

∗v

u

u

u0

b

a

a α

β

βσ

ασ

α

b

u=b

ασ

β βσ

a α

β

ασ β σ

Case 2. Now assume that A and B are not disjoint, so that d(A, B) = 0 . Since β ∈ / {α, ασ }, the intersection A ∩ B is a compact segment [a, b] (possibly with a = b) in Xv . Up to exchanging a and b, α and ασ , as well as β and β σ (which does not change d(A, B) or hα (β) by Lemma 18.1 (1)), we may assume that α, a, b, ασ and β, a, b, β σ are in this order on A and B respectively, and that a ∈ [u, α[ . Assume first that b ∈ ]u, α[ . Then (see the picture on the left above), v(α − β) = d(a, ∗v ),

v(α − β σ ) = v(β − β σ ) = d(b, ∗v ),

and v(β − ασ ) = v(β σ − ασ ) = d(u, ∗v ) . Therefore

σ

|α, β σ , ασ , β|v = qvv(α−β)+v(α


= qvd(a, ∗v )−d(b, ∗v ) = qvd(a,b) > 1 , and

|α, β, ασ , β σ |v = qvv(α−β

σ

)+v(ασ −β)−v(α−ασ )−v(β−β σ )

= qv0 = 1 = qv−d(A,B) , which proves the result by Equation (18.1). Assume that b ∈ ]u, ασ [ . Then (see the picture in the middle above), v(α − β) = d(a, ∗v ),

v(ασ − β σ ) = d(b, ∗v ),

and v(α − β σ ) = v(β − ασ ) = v(β − β σ ) = d(u, ∗v ) . Therefore σ



= qvd(a, ∗v )+d(b, ∗v )−2d(u,∗v ) = qvd(a,b) > 1 ,


365

and |α, β, ασ , β σ |v = qvv(α−β

σ


= qv0 = 1 = qv−d(A,B) ,

which proves the result by Equation (18.1). Assume finally that b = u. Let u0 ∈ V Xv be such that [b, ∗v ] ∩ [b, β σ [ = [b, u0 ]. Then (see the picture on the right above), v(α − β) = d(a, ∗v ),

v(ασ − β) = d(u, ∗v ),

and v(α − β σ ) = v(β − β σ ) = v(ασ − β σ ) = d(u0 , ∗v ) . Therefore σ



= qvd(a, ∗v )−d(u,∗v ) = qvd(a,b) > 1 , and |α, β, ασ , β σ |v = qvv(α−β

σ


which proves the result by Equation (18.1).

= qv0 = 1 = qv−d(A,B) ,

The next result says that the relative height is an appropriate complexity on a given orbit under Γ of a loxodromic fixed point, and that the counting function we will study is well defined. We denote by Γξ the stabiliser in Γ of a point ξ ∈ ∂∞ Xv = P1 (Kv ). Lemma 18.3. Let α, β ∈ Kv be loxodromic fixed points of Γ. Then for every s > 1, the set Es = {β 0 ∈ Γα \Γ · β : 1 < hα (β 0 ) 6 s} is finite. Proof. The set Es is well defined by Lemma 18.1 (4). Recall that a loxodromic fixed point is one of the two points at infinity of a unique translation axis. By local finiteness, there are, up to the action of the stabiliser of a fixed translation axis A, only finitely many images under Γ of another translation axis B at distance at most lnlnqsv from A. Since the stabiliser of A contains the stabiliser of either of its points at infinity with index at most 2, the result then follows from Proposition 18.2. We now state our main counting and equidistribution result of orbits of loxodromic fixed points when the complexity is the relative height with respect to a fixed loxodromic fixed point.

366


Theorem 18.4. Let Γ be a lattice in PGL2 (Kv )+ . Let α0 , β0 ∈ Kv be loxodromic fixed points of Γ. Then for the weak-star convergence of measures on Kv −{α0 , α0σ }, as s → +∞, X (qv + 1)2 Vol(Γ\\Xv ) s−1 ∆β σ σ 2 2 qv |α0 − α0 |v Vol(Γβ0 \\ ]β0 , β0 [) β∈Γ·β0 : hα0 (β)6s

∗

*

d HaarKv (z) . |z − α0 |v |z − α0σ |v

Furthermore, there exists κ > 0 such that as s → +∞, Card Γα0 \{β ∈ Γ · β0 : hα0 (β) 6 s} =

2 qv (qv − 1) Vol(Γα0 \\ ]α0 , α0σ [) Vol(Γβ0 \\ ]β0 , β0σ [) s + O(s1−κ ) . (qv + 1)2 Vol(Γ\\Xv )

For every β 0 ∈ ]0, 1], there exists a constant κ > 0 such that for every function 0 ψ ∈ Ccβ (Kv − {α0 , α0σ }), where Kv − {α0 , α0σ } is endowed with the distance-like map d]α0 , ασ0 [ ,3 there is an error term in the equidistribution claim of Theorem 18.4 when evaluated on ψ of the form O(s−κ kψkβ 0 ). This result applies, for instance, if ψ : Kv − {α0 , α0σ } → R is locally constant with compact support; see Remark 3.11. Proof. The proof of the equidistribution claim is similar to that of Theorem 17.1. We now apply Theorem 15.4 with D− := ]α0 , α0σ [ and D+ := ]β0 , β0σ [ . Since Γ is contained in PGL2 (Kv )+ , the length spectrum LΓ of Γ is equal to 2Z. The families − D ± = (γD± )γ∈Γ/ΓD± are locally finite, and kσD + k is finite and nonzero. Arguing as in the proof of Theorem 17.1,4 we have X (qv 2 − 1)(qv + 1) Vol(Γ\\Xv ) −n lim qv ∆γ·β0 = (∂ + )∗ σ eD+− − 3 n→+∞ 2 qv kσD + k γ∈Γ/Γ D+

0 0 such that for every β-Hölder-continuous 0 function ψ ∈ Ccβ (∂∞ Xv − ∂∞ D− ), where ∂∞ Xv − ∂∞ D− is endowed with the distance-like map dD− , there is an error term in the equidistribution statement of Equation (18.2) when evaluated on ψ of the form O(s−κ kψkβ 0 ). By Proposition 18.2, we have hα0 (γ · β0 ) = qvd(D

−

, γD+ )

.

By Proposition 15.2 (5), we have (∂ + )∗ σ eD+− (z) = 3 See 4 See


|α0 − α0σ |v d HaarKv (z) |z − α0 |v |z − α0σ |v


367

for z in the full-measure subset Kv − {α0 , α0σ } of ∂∞ Xv . Hence, using the change of variable s = qvn , we have, with the appropriate error term, lim

s→+∞

(qv 2 − 1)(qv + 1) Vol(Γ\\Xv ) s−1 − 2 qv3 |α0 − α0σ |v kσD +k

=

X

∆γ·β0

γ∈Γ/ΓD+ 1 0 such that as s → +∞, Card Γα0 \{α ∈ Γ · α0 : hα0 (α) 6 s} =

−x3 + ε x2 1 − ε x1

kΓ ∈ Γ.

4 p (p − 1) (kΓ )2 s + O(s1−κ ) . (p + 1)2 [Γ1O : Γ]

Proof. This follows, as in the proof of Theorem 17.11, from Theorem 18.4 using Equations (17.12) and (17.13), as well as again Equation (17.13) for the counting claim.

Chapter 19

Equidistribution and Counting of Integral Representations by Quadratic Norm Forms In the final chapter of this book, we give another arithmetic equidistribution and counting result of rational elements in non-Archimedean local fields of positive characteristic, again using our geometric equidistribution and counting results of common perpendiculars in trees summarised in Section 15.4. We use here a complexity defined using the norm forms associated with fixed quadratic irrationals. In particular, the complexity in this chapter is different from that used in the Mertens type of results in Section 16.1. We refer, for instance, to [PaP14b, §5.3] for motivations and results in the Archimedean case, and also to [GoP] for higherdimensional norm forms. Let K be a (global) function field over Fq of genus g, let v be a (normalised discrete) valuation of K, let Kv be the associated completion of K, and let Rv be the affine function ring associated with v.1 Let α ∈ Kv be a quadratic irrational over K. The norm form nα associated with α is the quadratic form K × K → K defined by (x, y) 7→ n(x − yα) = (x − yα)(x − yασ ) = x2 − xy tr(α) + y 2 n(α) . See Proposition 17.4 (3) for elementary transformation properties under elements of GL2 (Rv ) of this norm form. A pair (x, y) ∈ Rv × Rv is an integral representation of an element z ∈ K by the quadratic norm form nα if nα (x, y) = z. The following result describes the projective equidistribution as s → +∞ of the integral representations by nα of elements with absolute value at most s. For every (x0 , y0 ) ∈ Rv × Rv , let H(x0 ,y0 ) be the stabiliser of (x0 , y0 ) for the linear action of any subgroup H of GL2 (Rv ) 1 See

Section 14.2.


371

372

Chapter 19. Equidistribution and Counting of Integral Representations

on Rv × Rv . We use the notation N hx0 , y0 i for the norm of the ideal hx0 , y0 i generated by x0 , y0 (see Section 14.2) and the notation mv, x0 , y0 introduced above Theorem 16.1. Theorem 19.1. Let G be a finite-index subgroup of GL2 (Rv ), let α ∈ Kv be a quadratic irrational over K, and let (x0 , y0 ) ∈ Rv × Rv − {(0, 0)}. Let c0 =

(qv − 1) (qv + 1)2 ζK (−1) mv, x0 , y0 (N hx0 , y0 i)2 [GL2 (Rv ) : G] . qv3 (q − 1) q g−1 [GL2 (Rv )(x0 , y0 ) : G(x0 , y0 ) ]

Then for the weak-star convergence of measures on Kv − {α, ασ }, we have lim

s→+∞

c0 s−1

X

∆ xy =

(x, y)∈G(x0 , y0 ), | n(x−yα)|v 6s

d HaarKv (z) . |z − α|v |z − ασ |v

For every β ∈ ]0, 1], there exists a constant κ > 0 such that for every function ψ ∈ Ccβ (Kv − {α, ασ }), where Kv − {α, ασ } is endowed with the distance-like map d]α, ασ [ ,2 there is an error term in the equidistribution claim of Theorem 19.1 when evaluated on ψ, of the form O(s−κ kψkβ ). This holds, for instance, if ψ : Kv −{α, ασ } → R is locally constant with compact support (see Remark 3.11). Examples 19.2. (1) Let (x0 , y0 ) = (1, 0), K = Fq (Y ), and v = v∞ (so that g = 0 and qv = q). Theorem 1.17 in the introduction follows from Theorem 19.1, using Equations (14.6) and (16.1) to simplify the constant c0 . (2) Let (x0 , y0 ) = (1, 0) and let G = GI be the Hecke congruence subgroup of GL2 (Rv ) defined in Equation (16.11). The index in [GL2 (Rv ) : GI ] is given by Lemma 16.5 and GI satisfies (GI )(1,0) = GL2 (Rv )(1,0) . For every nonzero ideal I of Rv , for the weak-star convergence of measures on Kv − {α, ασ }, we have lim

s→+∞

X

cI s−1

∆ xy =

(x, y)∈Rv ×I, hx, yi=Rv , | n(x−yα)|v 6s

where cI =

(qv − 1) (qv + 1)2 ζK (−1) N (I)

Q

qv3 q g−1

p|I (1

d HaarKv (z) , |z − α|v |z − ασ |v +

1 N (p) )

.

(3) This third example is interesting only when the ideal class number is larger than 1. Given any fractional ideal m of Rv , taking (x0 , y0 ) ∈ Rv × Rv such that the fractional ideals hx0 , y0 i and m have the same ideal class and G = GL2 (Rv ), using the change of variables s 7→ sN (m)2 in the statement of Theorem 19.1, for the weak-star convergence of measures on Kv − {α, ασ }, with the same error term as for Theorem 19.1, we have X d HaarKv (z) lim cm s−1 ∆ xy = , s→+∞ |z − α|v |z − ασ |v −2 (x, y)∈m×m, hx, yi=m, N (m)

2 See

Equation (3.8).

| n(x−yα)|v 6s


where cm =

373

(qv − 1) (qv + 1)2 ζK (−1) mv, x0 , y0 . qv3 (q − 1) q g−1

Before proving Theorem 19.1, let us give a counting result that follows from it. Any subgroup of G acts on the left on any orbit of G. Furthermore, the stabiliser Gα of α in G preserves the map (x, y) 7→ | n(x − yα)|v , by Proposition 17.4 (3). We may then define a counting function Ψ0 (s) = Ψ0G, α, x0 ,y0 (s) of elements of Rv × Rv in a linear orbit under a finite-index subgroup G of GL2 (Rv ) on which the absolute value of the norm form associated with α is at most s, by Ψ0 (s) = Card Gα \ (x, y) ∈ G(x0 , y0 ), | n(x − yα)|v 6 s} . Corollary 19.3. Let G be a finite-index subgroup of GL2 (Rv ), let α ∈ Kv be a quadratic irrational over K, and let (x0 , y0 ) ∈ Rv × Rv − {(0, 0)}. Let g0 ∈ Gα with v(tr g0 ) 6= 0 and let m0 be the index of g0Z in Gα . Let c00 =

2 qv2 (q − 1) q g−1 |Z(G)| |v(tr g0 )| [GL2 (Rv )(x0 , y0 ) : G(x0 , y0 ) ] . (qv + 1)2 ζK (−1) |α − ασ |v m0 mv, x0 , y0 (N hx0 , y0 i)2 [GL2 (Rv ) : G]

Then there exists κ > 0 such that as s → +∞, Ψ0 (s) = c00 s + O(s1−κ ) . Proof. Using Equation (18.3) (with Γ the image of G in PGL2 (Rv )) and Equation (17.8), we have Z d HaarKv (z) 2 (qv − 1) |Z(G)| |v(tr g0 )| = . σ| |z − α| |z − α qv |α − ασ |v m0 σ v v Gα \(Kv −{α, α }) The corollary then follows by applying the equidistribution claim stated in Theorem 19.1 to the characteristic function of a compact-open fundamental domain of Kv − {α, ασ } modulo the action by homographies of Gα . Example 19.4. Let (x0 , y0 ) = (1, 0), K = Fq (Y ), v = v∞ (so that g = 0 and qv = q), and G = GL2 (Fq [Y ]). Using Equations (14.6) and (16.1), Proposition 17.4 (1), the change of variable s = q t , and the fact that |Z(G)| = q − 1 in order to simplify the constant c00 of Corollary 19.3, and recalling the expression of the absolute value at ∞ in terms of the degree from Section 14.2, we get the following counting result: for every integral quadratic irrational α ∈ Fq ((Y −1 )) over Fq (Y ), there exists κ > 0 such that as t → +∞, / hx, yi = Fq [Y ], CardGL (F [Y ]) (x, y) ∈ Fq [Y ] × Fq [Y ] : 2 q α deg(x2 − xy tr(α) + y 2 n(α)) 6 t =

2 1 2 q (q − 1)3 deg(tr g0 ) q − 2 deg(tr(α) −4 n(α)) q t + O(q t−κ ) , m0 (q + 1)

where g0 ∈ GL2 (Fq [Y ]) fixes α with deg(tr g0 ) 6= 0 and m0 is the index of g0Z in the stabiliser GL2 (Fq [Y ])α of α in GL2 (Fq [Y ]).

374


Proof of Theorem 19.1. The proof is similar to that of Theorem 16.1. We consider again r = xy00 ∈ K ∪ {∞}. If y0 = 0, let gr = id ∈ GL2 (K), and if y0 6= 0, let r 1 gr = ∈ GL2 (K) . 1 0 We apply Theorem 15.4 with Γ = G the image of G in PGL2 (Rv ), D− = ]α, ασ [ the (image of any) geodesic line in Xv with points at infinity α and ασ , and D+ = γr H∞ , where γr is the image of gr in PGL2 (Rv ). We have LΓ = 2Z, and the family D + = (γD+ )γ∈Γ/ΓD+ is locally finite, as seen at the beginning of the proof of Theorem 16.1. The Γ-equivariant family D − = (γD− )γ∈Γ/ΓD− is locally finite, as seen at the beginning of the proof of Theorem 17.1. By Proposition 15.2 (5), we have (on the full-measure subset Kv − {α, ασ } of ∂∞ Xv ) |α − ασ |v (∂ + )∗ σ eD+− = d HaarKv (z) . |z − α|v |z − ασ |v As in the derivation of Equation (16.3), since the point at infinity of γD+ is γ · r, we have, with an error term for every β ∈ ]0, 1] of the form O(s−κ kψkβ ) for some κ > 0 when evaluated on ψ ∈ Ccβ (∂∞ Xv − ∂∞ D− ), where ∂∞ Xv − ∂∞ D− is endowed with the distance-like map dD− X (qv 2 − 1)(qv + 1) Vol(Γ\\Xv ) −n lim qv ∆γ·r − 3 n→+∞ 2 qv kσD + k γ∈Γ/Γr (19.1) 0 0, then 1 h(α) d(D− , γD+ ) = ln | n(x − yα)|v . ln qv |z0 |v2 Proof. We begin by showing that g gr (1, 0) =

x z0

,

y . z0

Indeed, if y0 6= 0, we have g gr (1, 0) = g(r, 1) =

1 g(x0 , y0 ) y0


375

and otherwise g gr (1, 0) = g(1, 0) =

1 1 g(x0 , 0) = g(x0 , y0 ) . x0 x0

In particular, (g gr )

−1

=

∗

∗

− zy0

x z0

.

Note that g gr ∈ GL2 (K) and | det(g gr )|v = | det g|v | det gr |v = 1, since g belongs to GL2 (Rv ). By Proposition 17.4 (2), we hence have x y h(α) h((g gr )−1 · α) = n − α h(α) = | n(x − y α)|v . z0 z0 |z0 |v2 v

(19.2)

With β · (·, ·) the Busemann function defined in Equation (2.5), we use the signed distance d(L, H) = minx∈L βξ (x, xH ) between a geodesic line L and a horoball H centered at ξ 6= L± , where xH is any point of the boundary of H. Now, by Equations (15.2) and (2.12), we have d(D− , γD+ ) = d(]α, ασ [ , γγr H∞ ) = d ](g gr )−1 · α, (g gr )−1 · ασ [ , H∞ = v (g gr )−1 · α − (g gr )−1 · ασ − ln (g gr )−1 · α − (g gr )−1 · ασ v ln h((g gr )−1 · α) = = . (19.3) ln qv ln qv Combining Equations (19.2) and (19.3) gives the result.

By discreteness, the set of double classes [g] ∈ Gα \G/G(x0 ,y0 ) such that D− = ]α, α [ and gD+ = g gr H∞ are not disjoint is finite . Let Z(G) be the center of G, which is finite. Since Z(G) acts trivially on P1 (Kv ), the map G/G(x0 ,y0 ) → Γ/Γr induced by the canonical map GL2 (Rv ) → PGL2 (Rv ) is |Z(G)|-to-1. Using the change of variable 2 |z0 |v s= qv n h(α) σ

and Lemma 19.5, since γ · r = gives lim

s→+∞

x y

(qv 2 − 1) (qv + 1) |z0 |v 2 qv 3 |Z(G)|

2

with the notation of this lemma, Equation (19.1) Vol(Γ\\Xv ) −1 s − kσD +k (x, y)∈G(x

X 0 , y0 ),

∆ xy

| n(x−yα)|v 6s

d HaarKv (z) , = |z − α|v |z − ασ |v − with the appropriate error term. Replacing Vol(Γ\\Xv ) and kσD + k by their values respectively given by Equation (16.6) and Lemma 16.3, the claim of Theorem 19.1 follows.

Appendix by J. Buzzi A Weak Gibbs Measure is the Unique Equilibrium In this appendix, we consider a transitive topological Markov shift endowed with a H¨ older-continuous potential and we prove that its unique equilibrium measure is also its unique weak Gibbs measure.

A.1

Introduction

Let σ : Σ → Σ be a topological Markov shift (possibly one- or two-sided); see, for instance, Section 5.1. More precisely, we consider the one-sided and two-sided vertex-shifts defined by a countable oriented graph G with set of vertices VG and set of arrows AG ⊂ VG × VG . We assume that Σ is transitive, that is, that G is connected (as an oriented graph). We denote by P(Σ) the set of σ-invariant probability measures on Σ and by Perg (Σ) the subset of ergodic ones. Recall that for all n ∈ N, the n-cylinders are the following subsets of Σ, where x varies in Σ: Cn (x) = [x0 , . . . , xn−1 ] = {y ∈ Σ : ∀ k ∈ {0, . . . , n − 1}, yk = xk } , so that the 1-cylinders are [v] = {y ∈ Σ : y0 = v} for all v ∈ VG . The points of Σ admitting n ∈ N as a period under the shift σ form the set Fixn (Σ) = {x ∈ Σ : σ n x = x} . We fix a potential on Σ, that is, a continuous function φ : Σ → R. We do not assume that φ is bounded. We define φ− = max{−φ, 0} and, for all n ∈ N − {0}, varn (φ) =

sup

|φ(y) − φ(x)|

x,y∈Σ : ∀ k∈{0,...,n−1}, xk =yk

if (Σ, σ) is one-sided and otherwise varn (φ) =

sup

|φ(y) − φ(x)| .

x,y∈Σ : ∀ k∈{−n+1,...,n−1}, xk =yk

© Springer Nature Switzerland AG 2019 A. Broise-Alamichel et al., Equidistribution and Counting Under Equilibrium States in Negative Curvature and Trees, Progress in Mathematics 329, https://doi.org/10.1007/978-3-030-18315-8

377

378

Appendix: A Weak Gibbs Measure is the Unique Equilibrium

P We say that φ has summable variations if n>1 varn (φ) < ∞. This is in particular Pn−1 the case if φ is H¨ older-continuous. Let Sn φ = i=0 φ ◦ σ i for all n ∈ N − {0}. Definition A.1. A weak Gibbs measure for the potential φ is a σ-invariant Borel probability measure m on Σ such that there exists a number c(m) ∈ R such that for every v ∈ VG , there exists C > 1 with ∀ n > 1, ∀x ∈ Fixn (Σ) ∩ [v],

C −1 6

m(Cn (x)) 6C. exp (Sn φ(x) − c(m)n)

(A.1)

Note that c(m) is then unique; it is called the Gibbs constant of m. Let us stress that we do not assume the so-called Big Image Property [Sar1] and hence using the above weakened Gibbs property (that is, allowing C to depend on v) is necessary. Note that if Σ is locally compact, that is, if every vertex of G has finite degree (the number of arrows arriving or leaving from the given vertex), then the above condition is equivalent to the fact that for every nonempty compact subset K in Σ, there exists C > 1 with m(Cn (x)) 6C. exp (Sn φ(x) − c(m)n) R The pressure P (φ, ν) of an element ν ∈ P(Σ) such that φ− dν < +∞ is Z P (φ, ν) = hν (σ) + φ dν . ∀ n > 1, ∀ x ∈ Fixn (Σ) ∩ K,

C −1 6

An equilibrium measure µeq for (Σ, φ) is an element µeq ∈ P(Σ) such that Z φ− dµeq < +∞ and

Z P (φ, µeq ) = sup{P (φ, ν) : ν ∈ P(Σ) and

φ− dν < +∞} .

The Gureviˇc pressure is PG (φ) = lim sup n→∞

1 log n

X

eSn φ(x)

x∈Fixn (Σ)∩[v]

for any vertex v ∈ VG . Note that the Gureviˇc pressure does not depend on v. Let us recall a few results on the above notions. Theorem A.2 (Iommi–Jordan [IJ, Theorem 2.2]). If φ has summable variations, the following variational principle holds: Z PG (φ) = sup{P (φ, ν) : ν ∈ P(Σ) and φ− dν < +∞} .

A.1. Introduction

379

Theorem A.3 (Buzzi–Sarig [BuS, Theorem 1.1]). Assume that φ has summable variations. If PG (φ) < ∞, then there exists at most one equilibrium measure. Assume that (Σ, σ) is one-sided. If there exists an equilibrium measure µ, then dµ = h dν, where h : Σ → R is a continuous, positive function and ν is a positive measure with full support on Σ such that • Lφ h = ePG (φ) h and L∗φ ν = ePG (φ) ν, where Lφ is the transfer operator defined P by Lφ u (x) = y∈σ−1 x eφ(y) u(y). • ν is finite on each cylinder. We note that [BuS] assumed sup φ < ∞, but this was used only to justify the variational principle and so this condition can be removed using Theorem A.2. We now state the main result of this appendix. Theorem A.4. Let (Σ, σ) be a one-sided transitive topological Markov shift and let φ : Σ → R be a potential with summable variations. Let m be a σ-invariant R probability measure on Σ such that φ− dm < +∞. Then m is a weak Gibbs measure if and only if it is an equilibrium measure. In this case, the Gibbs constant c(m) is equal to the Gureviˇc pressure, and the equilibrium measure is unique. By a classical argument that follows, this result extends to two-sided topological Markov shifts (up to a slight strengthening of the regularity assumption on φ, still satisfied if φ is Hölder-continuous). Corollary A.5. Let (Σ, σ) be a two-sided transitive topological Markov shift and P let φ : Σ → R be a potential with n>1 n varn (φ) < ∞. Let m be a σ-invariant R probability measure on Σ such that φ− dm < +∞. Then m is a weak Gibbs measure if and only if it is an equilibrium measure. In this case, the Gibbs constant c(m) is equal to the Gureviˇc pressure, and the equilibrium measure is unique. Remark. The case of the full shift NZ has been treated in [PeSZ, Sec. 3]. More generally, assuming the Big Image Property, the above result follows from [Sar1] and [BuS] along the lines of [PeSZ]. Proof of Corollary A.5. Let (Σ, σ), φ, and m be as in the statement of this corollary. Let π : Σ → Σ+ with (xn )n∈Z 7→ (xn )n∈N be the obvious factor map onto the onesided topological Markov shift (Σ+ , σ+ ) defined by the same graph G as for (Σ, σ), called the natural extension. First, we replace φ by a potential φ depending only on future coordinates. The proof of [Bowe3, Lemma 1.6] applies to our noncompact setting without changes. To be more precise, for each vertex a ∈ VG , choose z a ∈ Σ with z0a = a. Define

380


r : Σ → Σ by r(x) = y with yn = xn for n > 0 and yn = znx0 for n 6 0. For every x ∈ Σ, let X u(x) = (φ ◦ σ k − φ ◦ σ k ◦ r)(x) . k>0

This defines a bounded real function on Σ, since |φ ◦ σ k − φ ◦ σ k ◦ r| 6 vark+1 (φ) and φ has summable variations. Moreover, u itself has summable variations, since given x, y ∈ Σ with xk = yk for |k| < n, we have X |u(x) − u(y)| 6 |φ(σ k x) − φ(σ k y)| + |φ(σ k (rx)) − φ(σ k (ry))| 06kbn/2c

X

64

vark+1 (φ) ,

k>bn/2c

so that X

varn (u) 6 8

n>1

X

k vark (φ) < ∞ .

k>1

Now define φ : Σ → R by φ=φ+u◦σ−u. The function φ is continuous with summable variations. Following [Bowe3], let us prove that φ = φ ◦ r. We have X X φ=φ+ (φ ◦ σ k+1 − φ ◦ σ k ◦ r ◦ σ) − (φ ◦ σ k − φ ◦ σ k ◦ r) k>0

=φ−φ−

k>0

X

k

k

φ◦σ ◦r◦σ−φ◦σ ◦r

k>0

=

X

φ ◦ σk ◦ r − φ ◦ σk ◦ r ◦ σ .

k>0

Now, r2 = r and r ◦ σ ◦ r = r ◦ σ. Hence φ ◦ r = φ, as claimed. Thus, φ induces on the one-sided shift a function φe : Σ+ → R defined by x0 x0 φ˜ : (xn )n∈N 7→ φ(. . . z−2 z−1 x0 x1 . . . ) ,

satisfying φ = φe ◦ π. To conclude, observe that Sn φ(x) − Sn φ(x) = Sn (u ◦ σ − u)(x) = 0 for every x ∈ Fixn (σ), and that cylinders defined by the same finite words have the same measure for an invariant probability measure m on the two-sided shift (Σ, σ) and for its image π∗ m on the one-sided shift (Σ+ , σ+ ). Therefore m is a weak Gibbs e and their Gibbs measure for φ if and only if π∗ m is a weak Gibbs measure for φ, constants are then equal.

A.2. Proof of the main result, Theorem A.4

381

e = m(φ) = m(φ), since m is invariant. As it is well By construction, π∗ m(φ) known, the natural extension π defines a bijection between P(Σ) and P(Σ+ ) that preserves the entropy. Thus, the measure m is an equilibrium measure with respect e to φ if and only if π∗ m is an equilibrium measure with respect to φ. The reduction to one-sided topological Markov shifts is thus complete.

A.2

Proof of the main result, Theorem A.4

The uniqueness of the equilibrium state is given by Theorem A.3. We need to prove that weak Gibbs measures and equilibrium measures coincide under the integrability assumption on φ− and that the number c(m) is equal to the Gureviˇc pressure. Step 1. If m is an equilibrium measure, then it is a weak Gibbs measure. This is a routine consequence of Theorem A.3. Our definition of an equilibR − rium measure m enforces φ dm < +∞ (hence excludes the concomitance of R hm (σ) = +∞ and φ dm = −∞). Recall from Theorem A.3 that dm = h dν with h and ν as mentioned. For v ∈ VG and x ∈ Fixn (Σ) ∩ [v], we have Z Z m(Cn (x)) = h 1Cn (x) dν = e−n PG (φ) h 1Cn (x) d((L∗φ )n ν) Z −n PG (φ) =e Lnφ (h 1Cn (x) ) dν . By definition, Lnφ (h 1Cn (x) )(z) = exp(Sn φ(x0 . . . xn−1 z)) h(x0 . . . xn−1 z) for all z ∈ σ n (Cn (x)) = σ([v]) (and Lnφ (h 1Cn (x) )(z) = 0 otherwise). Hence3 n Z X m(Cn (x)) = e−n PG (φ) exp Sn φ(x) ± vark (φ) k=1

h(x0 · · · xn−1 z)dν(z) .

σ([v])

P∞ By assumption, C = exp k=1 vark (φ) < +∞. The proof of Lemma 6 in [Sar2] shows that h(y)/h(y 0 ) 6 C for all y, y 0 ∈ [v]. Hence, picking any yv ∈ [v] depending only on v, we have m(Cn (x)) = C ±2 h(yv )ν(σ([v])) e−nPG (φ) exp(Sn φ(x)) . Therefore the measure m is a weak Gibbs measure for φ with Gibbs constant c(m) = PG (φ). 3 We

use, for all u, v, d > 0 and c > 0, the notation u = c±d v if

1 v cd

6 u 6 cd v.

382


We now turn R to the converse implication. Let m be a weak Gibbs measure for φ such that φ− dm < +∞. The weak Gibbs condition controls only the cylinders that start and end with the same symbol. Passing to an induced system (that is, considering a first return map on a 1-cylinder) will remove this restriction. More precisely, let a ∈ VG be a vertex of G and let µ be an invariant probability measure on (Σ, σ) with µ([a]) > 0. The induced system on the 1-cylinder [a] = {x ∈ Σ : x0 = a} is the map σ : [a] → [a] (almost everywhere) defined as follows: • let τ (x) = inf{n > 1 : σ n x ∈ [a]} be the first-return time in [a], which we also denote by τ[a] (x) when we want to emphasise [a]; • let σ(x) = σ τ (x) (x) if τ (x) < ∞; • let µ(B) = µ(B ∩ [a])/µ([a]) for every Borel subset B of Σ be the restriction of µ to [a] normalised to be a probability measure. We also define τ 0 (x) = 0 and by induction τ n+1 (x) = τ (x) + τ n (σx) for every n ∈ N. Note that σ can be iterated only on the subset {x ∈ [a] : ∀ n > 1, τ n (x) < ∞} . By Poincaré’s recurrence theorem, this is a full-measure subset of [a]; hence the distinction will be irrelevant for our purposes. The induced partition is β = {[a, ξ1 , . . . , ξn−1 , a] 6= ∅ : n > 1, ξi 6= a} . We note that σ : [a] → [a] is topologically Bernoulli with respect to the partition β (that is, σ : b → [a] is a homeomorphism for each b ∈ β). For every integer N > 1, we define the N th iterated partition β N of β by β N = {b0 ∩ σ −1 b1 ∩ · · · ∩ σ −N +1 bN −1 6= ∅ : b0 , . . . , bN −1 ∈ β}, and we write β N (x) for the element of the partition β N that contains x. Step 2. The topological Markov shift may be assumed to be topologically mixing. This follows from the spectral decomposition for topological Markov shifts; see, for instance, [BuS, Lem. 2.2]. Step 3. The Gibbs property implies full support and ergodicity. Let A be a σ-invariant (σ −1 (A) = A) measurable subset of Σ with m(A) > 0 and let us prove that m(A) = 1. Observe that the Gibbs property, together with the transitivity of Σ, implies that every cylinder has positive measure for m, hence that m has full support. Let a ∈ VG be such that m(A ∩ [a]) > 0.


383

Since m([a]) > 0, we may consider the induced system on the 1-cylinder [a]. Let N > 1. When f is a homeomorphism between topological spaces, let f ∗ denote the pushforwards of Borel measures by f −1 . First note that for almost every x ∈ [a] and every N ∈ N − {0}, since σ N is a homeomorphism from β N (x) onto [a], we have R d(σ N )∗ m dm m(A ∩ [a]) m(σ N (A ∩ β N (x))) dm β N (x)∩A = = . R N ∗ N d(σ ) m m([a]) m(σ (β N (x))) dm N β (x)

dm

Now observe that for m-almost every y ∈ β N (x), N

N m(σ τ (y) [y0 , . . . , yn ]) d(σ N )∗ m (y) = lim = C ±2 e−Sτ N (y) φ(y)+τ (y) c(m) . n→∞ dm m([y0 , . . . , yn ])

Hence, since τ N is constant on β N (x), P m(A ∩ [a]) m(A ∩ β N (x)) = C ±4 e± k>1 vark (φ) . m([a]) m(β N (x))

By Doob’s increasing martingale convergence theorem (see, for instance, [Pet]), for m-almost every x ∈ [a] − A, the ratio on the right-hand side S converges to 0 as N → ∞. Thus [a] is contained in A modulo m. Therefore A = a∈W [a] modulo m for some subset W of VG . Since (Σ, σ) is topologically mixing, for any vertex b, the triple intersection [a] ∩ σ −i [b] ∩ σ −j [a] is not empty for some integers 0 < i < j. Pick some point x in that set. By invariance, m([b]) > m(σ i (Cj (x))) > m(Cj (x)). But this last number is positive by the weak Gibbs property. Thus [b] is contained in A modulo m. Hence m(A) = 1, proving the ergodicity of m. Step 4. The Gureviˇc pressure PG (φ) is equal to c(m), hence is finite. Furthermore, hm (σ) < ∞ and φ ∈ L1 (m). Fix v ∈ VG and let K = [v]. Note that m(K) > 0. The ergodicity of m gives a Ces` aro convergence: as n → ∞, we have n−1 1X m(K ∩ σ −k K) −→ m(K)2 > 0 . n k=0

The Gibbs property implies that for all n > 1, X m(K ∩ σ −n K) = C ±1 eSn φ(x)−c(m)n x∈Fixn (Σ)∩K

= C ±1

X x∈Fixn (Σ)∩K

eSn φ(x) e−c(m)n .

(A.2)

384


If we write Zn for the term between the parentheses, we have, by the definition of the Gureviˇc pressure, 1 PG (φ) = lim sup log Zn . n→∞ n Since the value of the left-hand side of Equation (A.2) is less than one, we see that c(m) > PG (φ). If this were a strict inequality, then the left-hand side of Equation (A.2) would converge to zero, contradicting its Cesàro convergence to m(K)2 > 0. Therefore PG (φ) = c(m). Since c(m) Ris finite, so is PG (φ). Hence Theorem A.2 implies that for any ν ∈ P(Σ) with φ− dν < +∞, we have hν (σ) < ∞ and φ is ν-integrable. In particular, this holds for ν = m, which finishes the proof of Step 4. P Step 5. If the mean entropy Hµ (β) = − b∈β µ(b) log µ(b) is finite, then Z dµ hµ (σ) = − log ∗ dµ , dσ µ P where σ ∗ µ is the measure on Σ defined by B 7→ b∈β µ(σ(B ∩ b)), with respect to which µ is absolutely continuous: µ Î σ ∗ µ. This is a classical formula, sometimes called the Rokhlin formula (see, for instance, [BuS]), which follows from the computation of the entropy in terms of the information function when the mean entropy is finite, Z X 1b (x) lim log Eµ (1b | σ−1 β ∨ · · · ∨ σ−n β)(x) dµ(x), hµ (σ) = − b∈β

n→∞

and from the identity, for x ∈ b, Eµ (1b | σ −1 β ∨ · · · ∨ σ −n β)(x) =

µ(β n+1 (x)) µ(β n+1 (x)) . = σ ∗ µ(β n+1 (x)) µ(β n (σx))

The absolute continuity follows from a direct computation and ensures that the integral above is well defined. R Step 6. For all a ∈ VG , N > 1, and µ ∈ Perg (Σ) with φ− dµ < +∞, we have Z 1 dµ hµ (σ) = −µ([a]) log d µ. (A.3) d((σ N )∗ µ) [a] N We use arguments from the proof of [BuS, Theorem 1.1]: the key is to see that the induced partition β of [a] has finite mean entropy for the induced measure µ using a Bernoulli approximation. Let us consider the Bernoulli measure µB for ([a], σ) defined by µB

n−1 \ i=0

−i

σ ¯ Bi

=

n−1 Y i=0

µ(Bi )


385

for all Bi ∈ β. We construct from it an invariant measure µB on (Σ, σ): For every Borel subset A, let Z µB (A) = µ([a])

τ[a] −1

X

1A ◦ σi d µB .

[a] i=0

S Note that µB is ergodic, since µB is ergodic and i>1 σ −i ([a]) has full measure. Pτ[a] −1 Define φ = i=0 φ ◦ σ i . Let C 0 = supk>1 vark (φ), which is finite, since φ has summable variations. Since every b ∈ β is a cylinder of length τ[a] (x) + 1 for every x ∈ b, the conditional expectation Z X 1 φ d¯ µ Eµ ( φ | β) = 1b µ(b) b b∈β

satisfies k φ − Eµ ( φ | β)k∞ 6 C 0 . Hence Z Z Z φ d µB > µ([a]) Eµ ( φ | β) − C 0 d µB φ dµB = µ([a]) [a] [a] Z Z > µ([a]) φ d µ − C 0 = φ dµ − C 0 > −∞ . [a]

Therefore, the last paragraph of the proof of Step 4 applies to ν = µB and hµB (σ) < +∞. Since µB is ergodic, Abramov’s formula yields hµB (σ) = µ([a]) hµB (σ) . Since µB is Bernoulli, the right-hand side of this equality is equal to µ([a]) HµB (β) = µ([a])Hµ (β) , so that Hµ (β) is proven to be finite. Thus, Step 5 applies: Z dµ hµ (σ) = − log dµ . ∗ d(σ µ) [a] This formula extends to hµ (σ N ) for all integers N > 1. Using Abramov’s formula this time for µ and µ (since µ is ergodic), we have Z µ([a]) 1 dµ hµ (σ) = µ([a]) hµ (σ) = hµ (σ N ) = −µ([a]) log dµ , N d((σ N )∗ µ) [a] N as claimed. Step 7. The entropy of m is equal to c(m) −

R

φ dm.

In order to prove this, we apply Step 6 with µ = m (which is possible, since m has been proven to be ergodic in Step 3). As in the proof of Step 3, the

386


Radon–Nikodym derivative is almost everywhere dm m(β n (x)) (x) = lim = C ±2 exp Sτ N (x) φ(x) − c(m)τ N (x) . N N ∗ n n→∞ m(σ (β (x))) d((σ ) m) Therefore, using Step 6 and the fact that m [a] = m([a]) m, we have 1 N →∞ N

Z

c(m) τ N (x) − Sτ N (x) φ(x) dm ± 2 log C

hm (σ) = lim

.

(A.4)

[a]

Note that τ N (x) can be seen as a Birkhoff sum for the induced system on [a] and the function τ and that by Kac’s theorem (see, for instance, [Pet, Sect. 2.4]), Z τ d m = m([a])−1 . [a]

Therefore, Birkhoff’s ergodic theorem yields, with convergence in L1 (m), c(m) τ N (x) c(m) = . N →∞ N m([a]) lim

Pτ (x)−1 ˆ In order to analyze the second term in Equation (A.4), let φ(x) = k=0 φ(σ k x) and observe that by a variation of the proof of Kac’s theorem, φˆ ∈ L1 (m) with ˆ = m([a])−1 m(φ). Indeed, passing to the natural extension, one can assume m(φ) the system to be invertible and use the partition modulo m given by [ σ k ({x ∈ [a] : τ (x) = n}) . n>1, 06k 0 with center any geodesic line extension of w ∈ G± X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 B(`; T, T 0 , r) dynamical ball in the space of geodesic lines G X . . . . . . . . . . . . 88 C geometrically connected smooth projective curve over Fq . . . . . . . . . . . . . 302 c A complementary set of a subset A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 e cF system of conductances on X associated with a potential Fe . . . . . . . . . . . . 80 cF system of conductances on Γ\X associated with a potential F . . . . . . . . . 80 c(g) period for a system of conductances c of a closed orbit g for the geodesic flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 © Springer Nature Switzerland AG 2019 A. Broise-Alamichel et al., Equidistribution and Counting Under Equilibrium States in Negative Curvature and Trees, Progress in Mathematics 329, https://doi.org/10.1007/978-3-030-18315-8

387

388

List of Symbols

Cc (Z)

space of real-valued continuous maps with compact support on Z . . 19

Cbα (Z) Cbk, α (Z)

space of bounded α-Hölder-continuous real-valued functions on Z . 62

space of real-valued functions on Z with bounded α-Höldercontinuous derivatives of order at most k along the flow . . . . . . . . . . . . . . 194

Ccα (Z) space of α-Hölder-continuous real-valued functions with compact support on Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Cck, α (Z) space of real-valued functions on Z with bounded α-Höldercontinuous derivatives of order at most k along the flow and compact support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 Cc` (T 1 M ) space of real-valued C ` -smooth functions with compact support on T 1 M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 CAT(−1) metric space satisfying the Alexandrov–Topogonov comparison property with the real hyperbolic space of constant curvature −1 . . . . . . 24 ± CΓ,F ±

(normalised) Gibbs cocycle associated with the group Γ and the potential Fe± . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

codegD (x)

codegree of a vertex x with respect to a subtree D . . . . . . . . . . . . 175

codegD (x)

codegree of a vertex x with respect to a family of subtrees D . 176

covm, n (φ, ψ) nth correlation coefficient of two observables φ, ψ for the measure m under a transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 covµ, t (ψ, ψ 0 ) correlation coefficient at time t of ψ, ψ 0 for the measure µ under a flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 correlation coefficient for g ∈ Gv and a measure µv on Γ\Gv . . . . . 324

covµv , g C ΛΓ

convex hull in X of the limit set ΛΓ of Γ . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

∂∞ X

space at infinity of X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2 ∂∞ X

space of distinct ordered pairs of points at infinity of X . . . . . . . . . . . . 31

∂e X ∂V D ∂D

set of points at infinity of the geodesic rays whose initial (oriented) edge is e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 boundary of set of vertices of a simplicial subtree D . . . . . . . . . . . . . . . . 175 maximal subgraph with set of vertices ∂V D . . . . . . . . . . . . . . . . . . . . . . . . . 175

1 ∂− D 1 ∂+ D

outer unit normal bundle of a closed convex subset D . . . . . . . . . . . . . . . 36

deg v

degree of valuation v, equal to dimFq kv . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

inner unit normal bundle of a closed convex subset D . . . . . . . . . . . . . . . 36

δ = δΓ, F

critical exponent of (Γ, F ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

∆ec ±

Laplacian operator associated with a system of conductances e c± on a simplicial tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

∆c

Laplacian operator associated with a system of conductances c on a graph of groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

List of Symbols

389

∆x unit Dirac mass at a point x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 DiscD reduced discriminant of a quaternion algebra D over Q . . . . . . . . . . . 354 dCyg Cygan distance on Heis2n−1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 dD distance-like map on ∂∞ X − ∂∞ D associated with a closed convex subset D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 b d = dG X distance on the space of generalised geodesic lines . . . . . . . . . . . . . . . 29 d = dT 1 X distance on the space of germs of geodesic lines . . . . . . . . . . . . . . . . . 31 dH Hamenst¨ adt’s distance at infinity associated with a horoball H . . . . . . 34 dW ± (w) Hamenst¨ adt’s distance on the strong stable/unstable ball of w ∈ G± X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 dx visual distance on ∂∞ X seen from x ∈ X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 EX ϕRv

set of edges of a graph X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Euler function of function ring Rv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

f − negative part of a real-valued map f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ± 1 fD fibration over ∂± D with fibers the stable/unstable leaves . . . . . . . . . . . . . . Fe potential on T 1 X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F potential on Γ\T 1 X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fec potential on T 1 X associated with a system of conductances e c ......... Fc Fq

19 37 65 65

80 potential on Γ\T X associated with a system of conductances c . . . . . . . 80 finite field of order a prime power q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 1

Γv modular group at a valuation v of a function field . . . . . . . . . . . . . . . . . . . . 311 g genus of the smooth projective curve C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 Gb X space of geodesic lines in X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Gb X Gb X

space of generalised geodesic lines in X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 space of generalised discrete geodesic lines in a simplicial tree X . . . . . . 43

Geven X space of generalised discrete geodesic lines ` in X with d(`(0), x0 ) even . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Geven X space of discrete geodesic lines ` in X with `(0) at even distance from x0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 G± X space of generalised positive/negative geodesic rays in X . . . . . . . . . . . . 30 G±, 0 X space of generalised geodesic lines in X isometric exactly on ±[0, +∞[ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 t b (g )t∈R (continuous-time) geodesic flow on space of generalised geodesics G X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 b (gt )t∈R (continuous-time) geodesic flow on the quotient space of generalised geodesics Γ\ G X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

390

List of Symbols

b b (g )t∈Z (discrete-time) geodesic flow on space of generalised geodesics G X, as well as on Γ\ G X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 t

hm (T ) metric entropy of a transformation T with respect to a probability measure m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 hm (φ1 ) metric entropy of a flow (φt )t∈R with respect to a probability measure m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 h(α)

complexity of a loxodromic fixed point α . . . . . . . . . . . . . . . . . . . . . . . . . . . 342

h(α)

complexity of a quadratic irrational α in Kv . . . . . . . . . . . . . . . . . . . . . . . 347

h(α)

complexity of a quadratic irrational α in Qp . . . . . . . . . . . . . . . . . . . . . . . 358

hα (β)

relative height of a quadratic irrational β with respect to α . . . . . . . 368

hα (β) relative height of a loxodromic fixed point β with respect to α in Kv over K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 hα (β) relative height of a quadratic irrational β with respect to α in Qp over Q() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 HaarKv ht∞

normalised Haar measure of (Kv , +) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

height of a horoball in the Bruhat–Tits tree Xv . . . . . . . . . . . . . . . . . . . . . 311

Heis2n−1 H [t]

Heisenberg group of dimension 2n − 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

horoball contained in H whose boundary is at distance t from the boundary of H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

HB+ (w) HB− (w)

stable horoball of w ∈ G+ X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 unstable horoball of w ∈ G− X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

H+ (w)

stable horosphere of w ∈ G+ X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

H− (w)

unstable horosphere of w ∈ G− X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

HnC ι

complex hyperbolic space of dimension n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 b

antipodal map w 7→ {t 7→ w(−t)} on G X, asb well as on T 1 X . . . . . . . . . . . . 29

ι

antipodal map Γw 7→ {t 7→ Γw(−t)} on Γ\ G X, as well as on Γ\T 1 X . . . . 30

ια

reciprocity index of a loxodromic fixed point α . . . . . . . . . . . . . . . . . . . . . . . 342

ιG (β)

G-reciprocity index of a quadratic irrational β . . . . . . . . . . . . . . . . . . . . 350

Isom(X)

isometry group of X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Iv

set of classes of fractional ideals of Rv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

K

global function field over Fq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

Kv

completion of function field K for the valuation v . . . . . . . . . . . . . . . . . . . 302

kv

residual field of the valuation v on K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

λ(γ)

translation length of an isometry γ of X . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

λ(g)

length of a periodic orbit g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

List of Symbols

391

ΛΓ limit set of a discrete group of isometries Γ of X . . . . . . . . . . . . . . . . . . . . . . 24 Λc Γ conical limit set of a discrete group of isometries Γ of X . . . . . . . . . . . . . 25 L2 (Y, G∗ ) Hilbert space of square integrable maps on V Y for the measure vol(Y,G∗ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Lg Lebesgue measure along a periodic orbit g . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 lk links of vertices in simplicial trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 LΓ length spectrum of action of Γ on X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 ln natural logarithm (with ln(e) = 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 `± positive/negative endpoint of geodesic line ` . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 `∗ standard basepoint in space of geodesic lines G Xv . . . . . . . . . . . . . . . . . . . . 310 mD (x) multiplicity of a vertex x with respect to an equivariant family D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 m eF Gibbs measure on the space of geodesic lines G X . . . . . . . . . . . . . . . . . . . . 87 mF Gibbs measure on the quotient space of geodesic lines Γ\G X . . . . . . . . . 87 m e F Gibbs measure on the space of discrete geodesic lines G X . . . . . . . . . . . 100 mF Gibbs measure on the quotient space of discrete geodesic lines Γ\G X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 mF renormalised Gibbs measure mF /||mF || on Γ\G X . . . . . . . . . . . . . . . . . . 181 mc renormalised Gibbs measure mc /||mc || on Γ\G X . . . . . . . . . . . . . . . . . . . . . 183 (µ± )x∈X (normalised) Patterson density for the pair (Γ, Fe± ) . . . . . . . . . . . . . 83 x

µW ± (w) skinning measures on the strong stable or strong unstable leaf W ± (w) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 (µHaus )x∈X Hausdorff measures of the visual distances dx on ΛΓ . . . . . . . . . 104 x N (I) (absolute) norm of a nonzero ideal I in a Dedekind ring . . . . . . . . . . . 304 N A closed -neighbourhood of a subset A of a metric space . . . . . . . . . . . . . . 19 N− A set of points of A at distance at least from the complement of A . 19 Nw± homeomorphism between stable/unstable leaves and inner/outer normal bundles of horoballs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 n(β) relative norm of quadratic irrational β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 N(x) reduced norm of a quaternion x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 ∓ νw conditional measure on the (weak) stable/unstable leaf W 0± (w) of w ∈ G± X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 o initial vertex map EX → V X in a graph X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Ox A shadow of a subset A of X seen from x ∈ X ∪ ∂∞ X . . . . . . . . . . . . . . . . . 25 Ov valuation ring of v in Kv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 Ωc two-sided recurrent set for the geodesic flow in Γ\G X . . . . . . . . . . . . . . . . . 90

392

List of Symbols

footpoint projection w 7→ w(0) of (generalised) geodesic lines, and of their germs at 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

π π+

natural extension from one-sided to two-sided shifts . . . . . . . . . . . . . . . . . . 184

πv

uniformiser of a field endowed with a valuation v . . . . . . . . . . . . . . . . . . . . . 300

πv

uniformiser of a valuation v of a function field K over Fq . . . . . . . . . . . . . 302

π(Y, G∗ )

fundamental groupoid of a graph of groups (Y, G∗ ) . . . . . . . . . . . . 272

PD

closest point map to a convex subset D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

± PD

closest point homeomorphism from ∂∞ X − ∂∞ D to outer/inner normal bundle of D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

φeµ±

total mass function of Patterson density (µ± x )x∈X . . . . . . . . . . . . . . . . . . . 100

PGL2 (Kv )+ kernel of morphism [g] 7→ v(det g) mod 2 from PGL2 (Kv ) to Z/2Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 Pφ

pressure of a potential φ under a transformation . . . . . . . . . . . . . . . . . . . . . 133

Pψ

pressure of a potential ψ under a flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4, 135

Pφ (m) metric pressure for a potential φ of a probability measure m invariant under a transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Pψ (m) metric pressure for a potential ψ of a flow-invariant probability measure m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4, 135 Q = QΓ, F, x, y

Poincaré series of (Γ, F ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

qv

order of residual field kv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

Rv

affine algebra of the affine curve C − {v} . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

σ+

one-sided shift in symbolic dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

σ

other fixed point than α of a loxodromic element . . . . . . . . . . . . . . . . . . . . 342

βσ

Galois conjugate of a quadratic irrational β in Kv . . . . . . . . . . . . . . . . . . . 346

ασ

Galois conjugate of a quadratic irrational α in Qp . . . . . . . . . . . . . . . . . . . . 358

± σ eD

skinning measure on outer/inner normal bundle of convex subset D . . 155 b

α

± σ eD − σD

outer/inner skinning measure on G X of a family of closed convex subsets D . . . . . . . . . . . . . . . . . . . .b. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 inner skinning measure on Γ\ G X of a family of closed convex subsets D . . . . . . . . . . . . . . . . . . . .b. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

+ σD

outer skinning measure on Γ\ G X of a family of closed convex subsets D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

± σ eΩ

outer/inner skinning measure of a family Ω = (Ωi )i∈I of subsets 1 of (∂± Di )i∈I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

t Tπ

terminal vertex map EX → V X in a graph X . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 first edge map of a discrete geodesic line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

List of Symbols

393

T 1 X space of germs at t = 0 of geodesic lines in X . . . . . . . . . . . . . . . . . . . . . . . tr(β) relative trace of quadratic irrational β . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tr(x) reduced trace of a quaternion x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tvol(Y, G∗ ) volume form on the set of edges of a graph of finite groups (Y, G∗ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tvol(Y, G∗ ,λ) volume form of the set of edges of a metric graph of finite groups (Y, G∗ , λ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TVol(Y, G∗ ) total volume of the set of edges of a graph of finite groups (Y, G∗ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TVol(Y, G∗ , λ) total volume of the set of edges of a metric graph of finite groups (Y, G∗ , λ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31 346 354

UD±

domain of the fibration

± Vw, η, η 0

± fD

46 46 46 46

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

dynamical neighbourhoods of a point w ∈ G± X . . . . . . . . . . . . . . . . . . . 38

Vη,±η0 (Ω∓ )

dynamical neighbourhood of a subset Ω∓ of G± X . . . . . . . . . . . . . . 39 v∞ valuation at infinity of Fq (Y ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 v` germ at t = 0 of a geodesic line ` . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 vol(Y, G∗ ) volume form on the set of vertices of a graph of finite groups (Y, G∗ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Vol(Y, G∗ ) volume of a graph of finite groups (Y, G∗ ) . . . . . . . . . . . . . . . . . . . . 46 V X set of vertices of a graph X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Veven X set of vertices of a pointed graph (X, x0 ) at even distances from x0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Vodd X set of vertices of a pointed graph (X, x0 ) at odd distances from x0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 w± positive/negative endpoint of generalised geodesic line w . . . . . . . . . . . . . W + (w) strong stable leaf of w ∈ G+ X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W 0+ (w) stable leaf of w ∈ G+ X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W − (w) strong unstable leaf of w ∈ G− X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W 0− (w) unstable leaf of w ∈ G− X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30 31 35 31 35

|X|λ geometric realisation of a metric tree (X, λ) . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Xv Bruhat–Tits tree of (PGL2 , Kv ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 x

conjugate of a quaternion x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354

ζK (s)

Dedekind’s zeta function of a function field K . . . . . . . . . . . . . . . . . . . . . 304

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Index absolute cross-ratio, 361 value, 300 acylindrical, 122, 216, 288 adjacency matrix, 149 admissible, 112 alc-norm, 322, 325 algebraic lattice, 48 algebraically locally constant, 322, 324 almost precisely invariant, 176 antipodal map, 29, 31 antireversible, 79 attractive, 25 bipartite, 42 biregular, 42 boundary, 175 bounded parabolic limit point, 25 Bowen ball, 88 Bowen–Margulis measure, 87 Bowen–Walters distance, 126 Busemann cocycle, 28 closest point map, 36 cocycle Busemann, 28 cohomologous, 95 Gibbs, 74 codegree, 175, 176 cohomologous, 69, 79, 95, 112, 136 common perpendicular, 40, 272 ending transversally, 272 endpoint, 272 multiplicity, 265 origin, 272 starting transversally, 272 comparison triangle, 23 complex geodesic line, 158

complexity, 342, 347, 358 conductance, 79, 273 reversible, 79 conical limit point, 24 set, 25 conjugate, 354 continued fraction, 352 eventually periodic, 352 convergence narrow, 253 weak, 253 convergence type, 70 convex, 24 convex hull, 24 correlation coefficient, 183, 194 counting function, 255, 273, 330, 373 critical exponent, 70, 82 cross-ratio, 44, 361 absolute, 361 cross-section, 126 cusp, 312 cuspidal ray, 47 Cygan distance, 93 cylinder, 112, 185 tree, 101 decay of correlations exponential, 181 polynomial, 181 superpolynomial, 194 Dedekind zeta function, 304 degree, 42 Diophantine, 11, 195 2-Diophantine, 195 4-Diophantine, 195

© Springer Nature Switzerland AG 2019 A. Broise-Alamichel et al., Equidistribution and Counting Under Equilibrium States in Negative Curvature and Trees, Progress in Mathematics 329, https://doi.org/10.1007/978-3-030-18315-8

409

410 discriminant, 355 reduced, 354 distance Bowen–Walters, 126 Cygan, 93 Hamenst¨ adt’s, 32, 34 signed, 251 visual, 27 distance-like map, 63 divergence type, 70 doubling measure, 86 uniformly, 86 dynamical ball, 88 neighbourhood, 39 of a point, 38 edge length map, 42 nonoriented, 42 opposite, 42 edge-indexed graph, 45 lc-norm, 65 elliptic, 25 ending transversally, 272 endpoint, 272 negative, 30 positive, 30 equilibrium measure, 378 equilibrium state, 4, 133, 135 equivariant family, 165 locally finite, 166 Euler function, 339 exponential decay H¨ older, 181, 183 Sobolev, 182 exponentially mixing H¨ older, 181, 183 Sobolev, 182 extendible geodesics, 23 extension, 30 first return map, 127 time, 126 footpoint projection, 29, 31 full, 113

Index function field, 301 fundamental groupoid, 272 Galois conjugate, 346 generalised geodesic line, 29 discrete, 43 ray, 30 segment, 30 geodesic, 23 current, 87 flow, 29 discrete-time, 43 line, 23 generalised, 29 generalised discrete, 43 path, 272 ray, 23 asymptotic, 24 segment, 23 geodesically complete, 23 geometric realisation, 42 geometrically finite, 25 Gibbs cocycle, 74 constant, 112, 378 measure, 87, 100 weak, 133, 378 property, 88, 101, 112 graph, 41 bipartite, 42 edge-indexed, 45 of groups, 44 metric, 45 quotient, 45 Green function, 148 kernel, 148 growth linear, 50 subexponential, 50 Gureviˇc pressure, 378 Hamenst¨ adt’s distance, 32, 34 measure, 158 harmonic measure, 149

Index height, 311 Heisenberg group, 93 highest point, 311 H¨ older continuity, 50 local, 50 H¨ older-equivalent, 50 H¨ older norm, 62 homogeneous locally, 216 homography, 308 Hopf parametrisation, 31 discrete, 43 Hopf–Tsuji–Sullivan–Roblin theorem, 89 horoball, 29 stable, 35 unstable, 35 horosphere, 29 stable, 35 unstable, 35 horospherical coordinates, 93 index, 288 induced partition, 382 system, 382 inner unit normal bundle, 36 inseparable, 347 inversion, 42 isomorphism, 127 iterated partition, 382 Kac formula, 139 Laplacian, 142, 143 lattice, 47 algebraic, 48 tree, 47 uniform, 47 Ov -lattice, 307 leaf stable, 35 strong stable, 31 strong unstable, 31 unstable, 35 length of common perpendicular, 40 spectrum, 91, 104

411 limit point bounded parabolic, 25 conical, 24 limit set, 24 linear growth, 50 link, 308 Liouville’s measure, 92 Lipschitz, 50 local field non-Archimedean, 301 locally constant, 65 finite, 166 H¨ older-continuous, 50 Lipschitz, 50 loxodromic, 25, 342 reciprocal, 350 Markov chain, 122, 148 shift one-sided, 187 transitive, 112 two-sided, 111 Markov-good, 122 measure Bowen–Margulis, 87 doubling, 86 Gibbs, 87 Hamenst¨ adt’s, 158 induced, 253 Liouville, 92 Patterson, 83 satisfying the Gibbs property, 88, 101, 112 skinning, 155, 161 smooth, 260 Tamagawa, 355, 356 measured metric space, 86 metric graph of groups, 45 pressure, 4, 133, 135 tree, 42 mixing, 105 modular graph, 312

412 graph of groups, 312 group, 312 ray, 312 multiplicity, 175, 254, 265, 273 Nagao lattice, 312 narrow topology, 253 natural extension, 184, 379 suspended, 198 negative endpoint, 30 part, 19 non-Archimedean local field, 301 non-backtracking, 216 nonelementary, 24 norm, 304 lc, 65 alc, 322, 325 form, 371 H¨ older, 62 of a quaternion, 354 of quadratic irrational, 347 opposite edge, 42 order, 355 origin, 216, 272 ortholength spectrum, 7 marked, 7 outer unit normal bundle, 36 parabolic, 25 Patterson density, 83 period, 66, 292, 353 Poincaré map, 127 series, 70 pointing away, 171 towards, 171 polynomially mixing, 181 positive endpoint, 30 potential, 65, 135 associated with, 80 cohomologous, 69 reversible, 69 precisely invariant, 176 almost, 176

Index pressure, 4, 101, 133, 135, 378 Gureviˇc, 378 metric, 4, 133, 135 primitive, 25 proper, 23, 36 proper nonempty properly immersed closed locally convex subset, 255 properly immersed, 5 property HC, 3, 67 quadratic irrational, 346 complexity, 347 reciprocal, 350 radius-continuous ball masses, 236 radius-H¨ older-continuous ball masses, 236 random walk, 148 non-backtracking, 216 rapid mixing, 195 ray cuspidal, 47 generalised geodesic, 30 reciprocal, 350 reciprocity index, 350 recurrent, 148 reduced, 272 regular, 42 relative height, 362, 368, 369 repulsive, 25 residual field, 300 reversible, 69 Rokhlin formula, 384 roof function, 126 Sasaki’s metric, 54 separable, 347 shadow, 25 lemma, 86 for trees, 98 shift, 100, 112, 187 Siegel domain, 93 signed distance, 251 simple, 176 simplicial tree, 42 skinning measure, 155, 161, 288 inner, 167 outer, 167 space at infinity, 24

Index special flow, 126 spherically symmetric, 169 splitting over, 354 stable horoball, 35 horosphere, 35 leaf, 35 standard basepoint, 308 starting transversally, 272 state space, 147 strong stable leaf, 31 unstable leaf, 31 subexponential growth, 50 subgraph of subgroups, 45 subshift of finite type, 112 subtree, 24 suspension, 126 system of conductances, 3, 79 antireversible, 79 cohomologous, 79 reversible, 79 Tamagawa measure, 355, 356 terminal vertex, 42 Theorem of Hopf–Tsuji–Sullivan–Roblin, 89 thermodynamic formalism, 4 topological Markov shift one-sided, 187 transitive, 112 two-sided, 111 topologically mixing, 91, 104 transitive, 112 topology narrow, 253 weak, 253 trace of a quaternion, 354 of quadratic irrational, 347 transient, 148 transition kernel, 147 transitive, 112 translation axis, 25, 342

413 length, 25 tree biregular, 42 metric, 42 uniform, 43 regular, 42 simplicial, 42 R-tree, 24 tree cylinder, 101 uniform tree, 43 uniformiser, 300 uniformly doubling, 86 unit normal bundle inner, 36 outer, 36 tangent bundle, 31 unstable horoball, 35 horosphere, 35 leaf, 35 valuation, 299 p-adic, 301 at infinity, 300 ring, 300 n-variation, 133 vertex initial, 42 terminal, 42 nth vertex of a random walk, 217 virtual center, 288 visual distance, 27 volume form of a graph of groups, 46 of a graph of groups (edge-), 46 of a metric graph of groups, 46 of a graph of groups, 46 weak Gibbs measure, 133 weak topology, 253 zeta function, 304

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