Ergodic Theory and Statistical Mechanics

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

Jean Moulin Ollagnier

anvd Statistical h/lechanics

Spri nger-Verlag Berlin Heidelberg New York Tokyo

Author Jean Moulin Ollagnier Departement de Mathematiques, Universite Paris Nord Avenue J. B, Clement, 93430 Villetaneuse, France

AMS Subject Classification (1980): 20F, 28D, 54H20, 82A05, 82A25 ISBN 3-540-15192-3 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-15192-3 Springer-Verlag New York Heidelberg Berlin Tokyo Th~s work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft WOW', Munich. © by Springer-Verlag Berlin Heidelberg 1985 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210

CONTENTS

INTRODUCTION

....................................................

I. P R E L I M I N A R Y

ANALYSIS

I.I.

Sublinear

1.2.

Compact

1.3.

Radon measures

1.4.

Extremal

1.5.

References

2. D Y N A M I C A L

........................................

functions

convex

and

sets

the H a h n - B a n a c h

theorem

.......

...................................

........................................

points

in c o m p a c t

convex

sets

................

............................................

SYSTEMS

AND AMENABLE

GROUPS

.......................

I I 4 6 10 14

15

2.1.

Dynamical

systems

2.2.

The

point

2.3.

Amenability

2.4.

References

............................................

33

3. E R G O D I C T H E O R E M S

............................................

35

4.

fixed

.....................................

V

property

and the a m e a n i n g

and a l g e b r a i c

constructions

3.1.

Invariant

linear

functionals

3.2.

Invariant

vectors

and mean

3.3.

Individual

ergodic

theorems

3.4.

The

ergodic

theorem

3.5.

References

ENTROPY

saddle

Equivalence

DYNAMICAL

4.2.

Entropy

of p a r t i t i o n s

Entropy

of d y n a m i c a l

4.4.

The

a n d the 4.5.

theorems

...........

41 47

............................

50

ergodic

52

.......................

53

systems ..............

53

................................. systems

subadditive

35

...........................

dynamical

Shannon-McMillan

References

19 30

..........................

ergodic

SYSTEMS

of a b s t r a c t

4.3.

almost

......

...............

............................................

OF ABSTRACT

4.1.

filter

15

..........................

54 59

theorem

theorem

......................

............................................

64 70

IV

5. E N T R O P Y

6.

7.

A

FUNCTION

AND

THE

5.1.

Topological

entropy

5.2.

Pressure

a continuous

5.3.

Entropy

5.4.

References

STATISTICAL

of as

MECHANICS

6.2.

Cocycles

6.3.

Phase

6.4.

Supermodular

6.5.

References

ON

of

and

and

72

.................

82

measure

measures

measures

86

................

91 96

............................

...........................................

SYSTEMS

IN

STATISTICAL

MECHANICS

7.2.

Invariant

Gibbs

measures

7.3.

Mixing

7.4.

Example:

7.5.

References

properties

and

.................

....................... equilibrium

measures

of

Ruelle's

....

.......................

...........................................

COUNTABLE

8.1.

Tiling

amenable

8.2.

Equivalence

8.3.

Rokhlin's

8.4.

References

of

AMENABLE

groups countable

lemmaand

GROUPS

...................

............................... groups

......................

hyperfiniteness of countable amenable groups

...........................................

..................................................

.........................................................

98 104

....................................

a theorem

86

..............

....................................

interactions

specifications

INDEX

85

.........................

Gibbs

local

BIBLIOGRAPHY

71

the

quasi-invariant

transitions

71

the variational principle

A LATTICE

specifications

OF

........

function,and

Invariant

EQUIVALENCE

PRINCIPLE

...........................................

Local

DYNAMICAL

VARIATIONAL

..................................

a function

6.1.

7.1.

8.

AS

105 105 106 108 112 115

117 117 124 130 138

139

145

INTRODUCTION

It can be said that for the study of dynamical the crucial property of the acting group. classical

point of view is not only natural,

applications isometries

in statistical

systems amenability

This generalization

is

of the

but is also related

to the

mechanics where the acting group consists of

of the lattice.

This text, which grew out of a "third cycle" course in Ergodic Theory and Statistical the University

Mechanics

of Paris VI in 1980, deals with both topological

measure-theoretic dynamical

which I gave together with Didier Pinchon at

dynamical

systems,

systems of statistical

The existence of the ameaning

and

the symbolic

mechanics.

filter for an amenable group shall be

proved with the use of strongly dynamical

and in particular

subadditive

system of total orders.

Several

set functions

and the special

ergodic theorems

shall be

given. The entropy theory of measure-theoretic completely

described;

dynamical

and the Shannon-McMillan

corollary of a new ergodic

theorem,

systems shall be

theorem is given as a

the "almost subadditive

ergodic

theorem." A link between topological

and measure-theoretic

be made by way of the variational continuous

principle

function on a compact Hausdorff

dynamical

systems

shall

for the pressure of a space under the action of an

amenable group. A careful

study of subadditivity

use of tiling methods

in proving

of set functions several

allows us to avoid the

important

tiling is essential when proving the equivalence

theorems.

However,

of a countable

amenable

group with Z. This proof is given in the last chapter along with Rokhlin's lemma and the proof of the hyperfiniteness

of countable

(using the tower extension argument of Connes,

amenable groups

Feldman and Weiss).

Vl I would like to express my indebtedness significant portion of the material

to Didier Pinchon,

to whom a

contained in this text is due.

I also would like to thank Jean-Paul Thouvenot for many helpful discussions which stimulated the work on this text, and Tony Frank Paschall for his assistance with the English manuscript.

Jean Moulin Ollagnier Villetaneuse,

September 1984

i. P R E L I M I N A R Y ANALYSIS

I.I. S U B L I N E A R FUNCTIONS AND THE H A H N - B A N A C H THEOREM

I.I.I.

Definition.

Let E be a real vector

space. A real function p on E

is said to be sublinear if it is both subadditive and p o s i t i v e l y homogeneous,

i.e. if the two following conditions hold:

i)

for every pair

p(x+y)

(x,y) of vectors in E

ii) for every x and every positive number

1.1.2.

Remark.

~ p(x)

+ p(y)

p(Ix) = I p(x)

It is quite clear that a linear functional

is a sublinear

function and that the least upper bound of a set of sublinear functions, if one exists,

is still sublinear.

linear functionals

Therefore,

a least upper bound of

is a sublinear function. We shall state that this

p r o p e r t y is characteristic.

1.1.3. Example.

Consider the vector space C(X) of all real continuous

functions on the compact set X. Function s, defined on this space by the formula

s(f)

=

sup f(x) xeX

is sublinear.

1.1.4.

Extension theorem

(Hahn-Banach).

Let E be a real vector

space and

p be a sublinear function on E. Let F be a subspace of E and m be a linear functional b o u n d e d above by p on F, i.e. for every x in F, m(x) Then,

~ p(x).

there exists a linear functional n on the whole space E, still

b o u n d e d above by p, w h i c h is an extension of m. The proof of the theorem e s s e n t i a l l y depends on an extension lemma, and, w h e n the dimension is infinite,

on Zorn's lemma as well.

1.1.5.

Extension

lemma.

Let E be a real vector

function on E, G a vector functional

subspace of E non-equal

p a sublinear

to E, and f a linear

on G bounded by p.

Let a be an element of E\G so that the vector greater

space,

than G. It is then possible

a linear functional

space G 8 Ra is strictly

to find an extension of f which is

on G 8 Ra and is still bounded by p.

Proof of the lemma. We look for a real number ~ such that, in G and every real number ~, the following f(x) + ~ Using homogeneity,

< p(x+~a)

we have only to verify

f(x) + a

~

p(x+a)

and

that, f(x)

The real number ~ must then be chosen greater Sup yeG

for every x

inequality holds:

for every x in G,

- ~

~

p(x-a)

than or equal to

(f(y)-p(y-a))

and less than or equal

to

Inf (p(x+a)-f(x)) xeG Such a choice is possible f(y) - p(y-a) which is equivalent Because

to

if, for every pair


on F and the dual Eucli-

on F'

Call f' the orthogonal

projection

of f on K, i.e.

the unique element

of

K such that Inf geK



Let x be the vector

=

such that

f - f' = <x,.>.

For every g in K and every real number e between 0 and I, belongs

(f'+e.(g-f'))

to K and (f-f')(x)

=< IIf-f'-e.(g-f')II 2

=

(f-f') (x) +

For every strictly positive 0

< 2(f'-g) (x) + EIIf'-gll2

2e. (f'-g) (x)

+

E2 llf'-gll2

When ~ tends to 0, this formula reduces to

(g-f')(x) ~ 0 , from which

we derive f'(x) = Sup g(x) geK Therefore f(x)

is less than or equal to f'(x), and the scalar square of

f-f' is non-positive, which means

1.2.5. Proposition.

that f belongs to K.

Let K be a convex part of the dual space E* of a

real vector space E, and let it be compact in the weak* topology. Let x be a vector in E, and consider the least upper bound p(x) of all numbers

f(x), where f belongs to K. For every x in E, p(x)

So defined, and K(E,p) Proof.

function p is sublinear,

is finite.

and the two compact convex sets K

are equal.

For every x, the function

therefore bounded.

f --> f(x)

is continuous on K and

Function p is then defined on the whole space E and

it is a supremum of linear functionals.

According to remark 1.1.2, p is

sublinear. By definition, K is contained in K(E,p). To prove the converse inclusion,

consider an element f of K(E,p).

For every finite dimensional vector subspace of E, there exists according to lemma 1.2.4 a convex compact n o n - e m p t y subset K F of K whose elements give the same value as f to all elements of F. The family of all compact sets K F has the non-empty finite intersection property.

Therefore,

there exists a convex compact subset K E of K, whose

elements give the same value as f to all elements of E. There is only one element in K, which is indeed f, and the converse inclusion is proven.

1.2.6. Remark.

It is possible to deduce the result of p r o p o s i t i o n 1.2.5

from a geometrical

form of the H a h n - B a n a c h theorem.

1.3. RADON MEASURES

1.3.1.

Definition.

Consider the vector space C(X) of all real continuous

functions on a given compact Hausdorff space X. A Radon measure on X is a linear functional on C(X),

continuous for the s u p r e m u m norm given by

the formula

1.3.2. gical

IIfll = Sup xeX

Definition.

If(x) l.

Let

space and ~ a

(X,~) be a measurable

o-algebra

A positive real measure measure

of all its compact

is said to be outer regular the greatest 1.3.3.

subsets.

space,

theorem.

and let A be a positive

Then there exists

a o-algebra ~ i n

~ represents

real measure on this space

of every measurable

set is

of all open sets containing

linear functional

on Cc(X),

on X with a compact

X which contains

all Borel

support. sets, and

the following properties:

A

feCc(X)

---> ~(f)

=

f f d~ X

second,

~ is inner and outer regular and gives a finite measure

compact

sets; and,

third,

the o-algebra ~ is complete

One proof of this can be found in Rudin 1.3.4.

Remark.

theorem,

the Borel o-algebra 1.3.5.

Corollary.

correspondence,

and also, when necessary,

the restriction

Let ~ be a Radon measure on a compact Hausdorff semi-continuous

Proof.

to

space

function on X. Then the integral

is the supremum of all ~(~), where ~ is a continuous

greater

we shall

derived from a linear functional

of this measure.

X, and f be an upper ~(f)

to

for ~.

(i).

Because of the above one-to-one

call a Radon measure both the measure on C(X) by Riesz's

it.

Let X be a locally compact

real functions

a unique positive measure ~ on ~ w i t h first,

A positive

if the measure

lower bound of the measures

the space of all continuous

if the

subset of X is the least upper bound of the

The Riesz representation

Hausdorff

the Borel o-algebra.

on this space is said to be inner regular

of every measurable

measures

containing

space, where X is a topolo-

function on X

than or equal to f. For every continuous

results

from the positivity

function ~, the inequality

~(f)

~

~(~)

of ~.

Consider now,

for every natural number n, the function fn which is equal

to Sup(f,-n),

and is so upper

Sequence

(fn) then decreases

theorem,

the integral

semi-continuous to f. According

and bounded. to the monotone

of f is the limit of the integral

convergence

of fn when n

tends to infinity. We have only then to prove the results for bounded upper semi-continuous functions.

By adding a constant we can even restrict the proof to the

case of non-negative functions. To achieve such a proof,

let g be a n o n - n e g a t i v e upper semi-continuous

function on X. For every positive real number 8, g~ is the function given by

= g~ In fact,

~

l{g~n6}

6"n= 0

there is a finite number of terms in the sum that are different

from 0 since g is bounded.

The last subscript is

n O = E(Sup(g)/6).

This function is greater than or equal to g, and this inequality holds:

~(g~ - g)


0.

Denote by C + the subspace of C consisting of all points reaches

linear

z at which f

its m a x i m u m value on C.

Because C is compact,

C + is not empty.

It it clearly a convex and closed

subset of C. The open-segment p r o p e r t y still holds for C +. Indeed,

consider a convex combination

M = ~A + BB, where ~ and B are

n o n - n e g a t i v e and whose sum is equal to i; and, if M belongs to C +, A and B also belong to C because the open-segment property holds for C.

11

Moreover,

the inequalities f(A) ~ of(A) + ~f(B)

imply

f(A) = f(B) = f(M)

Therefore

1.4.3.

Example.

extremal

f(B) ~ of(A) + ~f(B)

, and the points

C + is an element

and this concludes

and

A and B then belong to C +.

of I, strictly greater

than C for the order;

the proof.

The following

situation will enable us to characterize

points.

X is a compact Hausdorff (defined in 1.1.3)

space and s is the sublinear

so that K(C(X),s)

all Radon probability measures N is a vector

function on C(X)

is the convex and compact

set of

on X (1.1.8).

subspace of C(X) such that,

for every f in N, s(f)

is non-

negative. The null functional

is then bounded by s on N and it is possible,

ing to theorem 1.1.4,

to find a functional

space E whose restriction The non-empty

to N is 0.

set K(C(X),s,N)

give an integral

equal

the extension p(f)

The following

=

on X that

of N is then a convex and

this compact and convex set is characterized

by its upper bound p according Applying

of all Radon probability measures

to 0 to all elements

compact part of K(C(X),s);

accord-

bounded by s on the whole

to 1.2.5.

lemma 1.1.5 to the space N • Rf, p may be written

Sup meK(C(X),s,N)

m(f)

=

Inf s(f+g) geN

theorem gives a characterization

of the extremal

points

in

K(C(X),p). 1.4.4.

Theorem.

In 1.4.3,

the extremal

the measures m for which the subspace

points N @ R.I

in K(C(X),p)

are precisely

is dense in C(X)

for the

p s e u d o - n o r m m(l.l). Proof. i) Let us first show that a measure with the above property Consider

a strict convex combination

property holds

for m:

(=i.o2 > 0) where

is extremal.

the density

12

m = ~iml + ~2m2 The two linear functionals m I and m 2 are continuous m(I.l)

for the p s e u d o - n o r m

since

Iml(f) l

~

(1/~ 1) m(Ifl)

Im2(f) l ~ (1/~ 2) m(Ifl)

and

They are equal to m on the dense subspace N @ R.I, and therefore are equal on the whole space. Functional m is then an extremal point of K(C(X),p). 2) The converse can be proven by contraposition. If

N ~ R.I

is not a dense subspace of C(X) for the p s e u d o - n o r m m(l.l),

there exist a real continuous

function f and a strictly positive real

number ~ such that, for every g in N and every real number a,

m(l f-g-al )

>

~

>

0

Consider the linear functional 8 on

0(g + a + ~f) =

N • R.I @ R.f

given by

~=

The functional verifies on this subspace the inequality

e ~ m(I.I).

It is then possible to find on the whole space C(X) a linear functional b o u n d e d above by m(I.I) , whose restriction is 0. Denote it by ~. The d e c o m p o s i t i o n

m

=

(1/2)

((m+~) + (m-~))

is non-trivial because #(f) It remains to be shown that

is not 0. m+~

and

m-~

are elements of K(C(X),p);

but the only thing to be proven is that they are positive measures. If h is a continuous non-negative

(m + ~¢)(h)

1o4.5. Remark.

=

m(h)

The dense subspace

function on X, we get (s = jl)

+ E¢(h)

N @ R.I

>

m(lh[)

- [#(h)[

>

0

is then dense in LI(X,Q,m)

the L l - n o r m , where m is the regular measure built from the functional m by theorem 1.3.4. 1.4.6. Remark.

Compact convex sets described in 1.4.3 are simplexes and

for

13

all compact 1.4.7.

simplexes

Example.

of probability measures have such forms.

Let f be a continuous

mapping

from a compact

space Y

onto a compact set X, and let m be a Radon probability measure The set of all probability measures type described Extremal morphism

in this convex set are exactly those for which f is a

of the two measure of the quotient

the o-ideals 1.4.8.

in 1.4.3.

points

conjugacy

spaces

(Y,~,n)

Boole algebras

and

(X,~,m),

i.e. an iso-

of the o-algebras

of events by

of null sets.

Example.

Let X be a compact

A Radon probability measure

set and T a h o m e o m o r p h i s m

of X.

~ on X is said to be invariant under the

action of T if the equality function

on X.

on Y whose image by f is m is of the

~(f)

= ~(foT)

holds

for every continuous

f on X.

The set M(X,T)

of all Radon probability measures

on X, which are invar-

iant under T, is not empty and is also a convex and compact set of the type described Extremal invariant

elements

Let us prove Here,

in 1.4.3.

invariant probabilities of LI(X,~,~)

are ergodic, are constant

which is to say the only functions.

these results.

the space N is generated by the increments

belongs

to C(X).

increment It shall

Every linear

f - foT

combination ~ ~i(fi-fioT)

where

f

is still an

f - foT. suffice to verify

s(f-foT)

~ 0

Let x be a point in X at which f reaches a point the difference therefore,

(f-foT)(x)

s is non-negative

The extremal

points

in M(X,T)

its least upper bound.

is greater

At such

than or equal to 0, and,

on N. are those for which the subspace N @ R.I

is dense in LI(X,~,~ ). Consider

now the sequence of contraction

operators

A n of LI(X,~,~)

where

A n is given by the expression An(f) The sequence

=

(An(f))

n-I (I/n). ~ loT i of averages

converges

to the expectation

~(f)

for

every element of N • R.I; and, by virtue of the density of this subspace, the convergence

result also holds

for every element of LI(X,~,~).

14

If f is an invariant to u(f).

An invariant

if u is extremal 1.4.9.

element,

Remark.

An(f)

is a constant

element of LI(X,Q,u)

In the previous

space.

In Chapter

that converges

is then a constant

function

in M(X,T). example we proved that there always

at least one invariant probability under Hausdorff

sequence

This was demonstrated

2, we shall examine

a homeomorphism

by Bogoliubov

the transformation

are invariant probability measures whenever

and Krylov

groups

exists

of a compact (1).

for which there

they act on a compact

space

by homeomorphisms. To prove the existence we used a convergence ergodic

theorem.

of invariant probabilities theorem of means,

in the present

which is a special

chapter,

case of a mean

Chapt,er 3 deals further with this topic.

1.5 REFERENCES

For further Bourbaki's

study of Hahn-Banach text devoted

Riesz representation Choquet

theorem,

to topological

refer to Meyer

vector

spaces

theorem is proved by Rudin

(i) is recommended

for general

and more in-depth study of simplexes.

(i) or to

(3).

(i, chapter

information

2).

on functional

analysis

2. DYNAMICAL

2.1. DYNAMICAL

2.1.1.

SYSTEMS AND AMENABLE GROUPS

SYSTEMS

Definition.

We call a pair

when X is a compact Hausdorff

(X,G)

a topological

dynamical

system

space and G a group of homeomorphisms

of

this space. When discussing

the action of an abstract group G, we shall denote by T g

the homeomorphism (X,T)

of X associated with the element g of G and denote by

the corresponding

dynamical

system.

When G is the group Z of all relative the generator

integers,

T is simply the image of

i.

2.1.2. Example.

The first example

set, and X the product

is given by shifts.

space I G of all mappings

Endowed with the product topology of the discrete a compact Hausdorff

space which is metrizable

Given an element g of G, consider Tg(~)(a) The mappings

topologies,

X becomes

when G is countable.

the mapping T g from X to X:

= ~ (ag)

T g are homeomorphisms

group of all homeomorphisms corresponding

Let I be a finite

from G to I.

of X; and the mapping from G to the

of K, obtained by :sending every g to the

T g, is a group homomorphism.

Such a group homomorphism We shall sometimes

is called an action of G on X by homeomorphisms.

refer to T g as a translation.

Indeed,

T g is the right

translation by g-I of the graphs r($): (a.g-l,i) The set I is sometimes we have constructed 2.1.3.

Example.

e

r(Tg(¢))




(a,i)

called an alphabet,

is called a symbolic

Here is another example,

e

r(~)

and the dynamical

dynamical

system that

system.

which is as symbolic as the

16

preceding.

Given a group G, consider

the set T(G) of all total orders

on G. If t is a total order on a finite part F of G, O(F,t) the set of all total orders on G whose restriction The set of all O(F,t)

is one of the bases of a topology on T(G).

Endowed with this topology, metrizable to T(G)

T(G)

when G is countable;

of the product

Of course,

topology

is a compact Hausdorff on



x < y

For every element g in G, consider Tg(r)

(a,b)

=

The T g are homeomorphisms

2.1.4.

As in example

Definition.

of T(G)

the mapping

and mapping

is a one-to-one

An automorphism

bimeasurable mapping

Definition.

to T(G):

are called translations.

if T and its inverse

of a probability

space

from X to X that preserves

i.e. for every A in the u-algebra (T -I(A))

from T(G)

g to T g gives an action of G

A mapping T is bimeasurable

are both measurable.

the measure,

for

2.1.2 these homeomorphisms

(X,~,~)

4, the following holds:

= ~(A)

Given a probability

action of G on (X,~,~) system.

{0,I} GxG.

~(ag,bg)

mapping

2.1.5.

space, which is

this topology is simply the restriction

we identify an order T with its graph: T (x,y)=l

on T(G).

stands for

to F is t.

space

by automorphisms

(X,~,~)

and a group G, an

is called an abstract

Such an action is then a family of automorphisms

lity space, which is indexed by the elements

dynamical

of the probabi-

of G, and for which the

equality T gh

holds 2.1.6.

=

for every pair Remark.

TgoT h

(g,h) of elements

The question of modulo

spaces will be investigated In these first chapters, pological

of G. 0 automorphisms

and Lebesgue

later.

abstract

dynamical

ones and the automorphisms

systems are derived from to-

are therefore well-defined

mappings.

17

2.1.7.

Proposition.

Let

(X,G) be a topological

be a Radon probability measure of G. This means

every element g of G, the following

Then,

=

function f on X, and for

equality holds:

of the probability

space

is the o - a l g e b r a built by the Riesz representation

lar,

Indeed,

and

~ (f)

the T g are automorphisms

Proof.

system,

on X, which is invariant under the action

that for every continuous

u (foT g)

dynamical

the T g are measurable

for ~; and,

(X,~,~) where

theorem

(1.3.3).

since ~ is inner regu-

for every A in Q, (A)

Of course, compact

=

Sup KcA

~ (K)

here we take the least upper bound of the measures

of all

subsets of A.

A similar result holds for (Tg)-I(A). On the other hand,

according

to corollary

the compact set K, is the greatest continuous Then,

functions

1.3.5, ~(K),

the measure

lower bound of the measures

above the indicator

of

of all

of K.

for every element A in the o-algebra

~, the following

equality

holds: ~(A)

2.1.8.

Example.

=

~((Tg)-I(A))

Taking the topological

dynamical

duced in 2.1.2 we can define a probability Let

system that we intro-

on it in the following way.

(Pl .... ,pn ) be a n-uple of strictly positive

is I, where n is the number of elements If f is a function only depending example

the coordinates

(f) If we take another

=

real numbers whose

sum

in I.

on a finite number of coordinates

in the finite part F), ~(f)

(for

is defined by

i=n ~ F f(a).( ~ Pa. ) aeI i=l l finite part F' that contains

F, the value of u(f)

is

the same. Then the above formula defines value

1 to the constant

a positive

function

linear functional,

I, on the vector

giving the

space of all functions

that only depend on a finite number of coordinates.

18

According

to the Stone-Weierstrass

the uniform norm on C(X), can be extended

theorem,

this subspace

and the uniformly continuous

to the whole

space C(X).

is dense for

linear functional

Here we get a Radon probability

measure ~ on X. It is easy to see that ~ is invariant under tions.

Such an abstract

2.1.9.

Example.

dynamical

Consider

the action of G by transla-

system is called a Bernoulli

the dynamical

system T(G)

introduced

scheme. in 2.1.3

of all total orders on G. All O(F,t)

are open and closed

Let us define a positive which

consists

subsets

of T(G).

linear functional ~ on the subspace of C(T(G)),

of all functions

depending

solely on the restriction

of

the order to a finite part of G, by giving in a coherent way the mass of all O(F,t). It is possible

to do this by setting

The above subspace

~(O(F,t))

is dense in C(T(G))

the Stone-Weierstrass

theorem;

for the uniform norm according

and the functional

of a unique Radon probability measure still

= I/IFI! to

~ is the restriction

on T(G) which we shall nonetheless

call simply ~.

One can easily ensure that ~ is invariant under the action of G on T(G) by translations;

in fact, ~ is invariant under the action of the wider

group of all homeomorphisms According

of T(G)

2.1.10.

Remark.

variant

Radon probability measures

systems.

In the two previous

However,

probability

topological

to find invariant

topological

dynamical

2.1.12.

Radon

system.

for every action on a compact Haus-

there is at least one invariant

A group G is said to have the fixed-point

acting by affine continuous

one-to-one mappings

pact and convex subset of a locally convex Hausdorff space,

dynamical

Radon

As we saw in 1.4.9 the group Z has this property.

Definition.

if, whenever

of G.

system.

examples we were able to build in-

in an arbitrary

dorff space by homeomorphisms, probability.

dynamical

in particular

it is not always possible

measures

But there exist groups such that,

2.1.11.

induced by the permutations

to 2.1.7, we then get an abstract

this compact convex space contains The Markov-Kakutani

theorem.

property on a com-

topological

vector

at least one invariant point.

An Abelian group has the fixed-

point property. The study done in 2.3 leads to one proof of this result,

cf. 2.3.6.

19

2.1.13.

Remark.

Concerning vocabulary:

the study of dynamical

systems

b e g a n with some actions of R and Z, which represented r e s p e c t i v e l y the e v o l u t i o n of a mechanical

system through time, and the d i s c r e t i z a t i o n

b e t w e e n regular time intervals of this evolution; hence the adjective "dynamical". In this text the most significant examples of dynamical be found in statistical mechanics.

systems shall

In these examples the group G does

not connote evolution but consists of isometries.

However,

not prevent us from calling such an action a dynamical

that will

system.

2.2. THE FIXED POINT PROPERTY AND THE AMEANING FILTER

2.2.1 Definition. We call a group G amenable if it has the fixed-point p r o p e r t y described in 2.1.11, however used, 2.2.2.

cf. Greenleaf Definition.

several other definitions

can be

(i). A group is said to give an invariant version of the

H a h n - B a n a c h theorem if the following extension result holds: Let E be a real vector space,

s a sublinear function on E, and T a

right action of a group G on E by linear one-to-one mappings preserving s, i.e. for every f in E and every g in G,

s(f) = s(f.Tg).

Let

F be a subspace of E invariant under the action of G, and m a linear functional on F bounded by s and invariant under the action of G. There exists a linear functional on E bounded by s and invariant under the action of G whose r e s t r i c t i o n to F is m.

2.2.3.

Proposition.

A group G gives an invariant version of the Hahn-

Banach theorem if and only if it has the fixed-point property.

Proof. Let us first show that the fixed-point p r o p e r t y leads to an invariant version of the H a h n - B a n a c h theorem. A c c o r d i n g to the H a h n - B a n a c h theorem, cribed in 2.2.2,

there exist,

in the s i t u a t i o n des-

linear functionals on E bounded by s, whose r e s t r i c t i o n

to F is m. The set of all these linear functionals of E* for the weak* topology.

is a convex and compact subset

20

The group G acts on this set by the affine one-to-one mappings

Tg(n)(f)

=

given by

n(f.T g)

There is an invariant point in this compact and convex set, i.e. a linear functional bounded by s and invariant under the action of G on E whose r e s t r i c t i o n to F is m.

In order to demonstrate the converse,

let K be a convex and compact

subspace of a locally convex Hausdorff topological vector space E, and let T be an action of a group G on K by affine continuous one-to-one mappings. Let E be the real vector space of all continuous real functions on K, and s the usual

sublinear function on E (defined in 1.1.3).

From the left action of G on K we can deduce a right action of G on E and still denote it by T

(f.Tg)(x)

=

f(Tg(x))

Next we take the subspace reduced to the function 0 as the subspace F. Thanks

to the fixed point property,

there exists a Radon p r o b a b i l i t y

m e a s u r e on K invariant under the given action of G. Because K is a compact and convex part of E*, the center of gravity of this p r o b a b i l i t y is an invariant point of K.

2.2.4.

Example.

We can prove that finite groups are amenable.

Indeed,

if

the finite group F acts on the compact and convex set K, the m e a n value

(IlIFI). [ Tg(x) geF is a fixed point of this action for every x in K.

2.2.5.

Example.

To prove that Z is amenable,

let K be a compact convex

set and T be an affine continuous one-to-one mapping of K. Consider a point x in K and the sequence

(Mn) of affine continuous oper-

ators on K given by

Mn

=

n-i (l/n). ~ Ti

Let x' be a limit point of the sequence

x n = Mn(X).

And let p be any of the continuous pseudo-norms

that define the topology

21 of the locally convex topological

vector

space in which K lies.

Because p is continuous

p(x'-T(x'))

~

Inf Sup P(Mn(X)-T.Mn(X))

N And because

Mn(X)

- T.Mn(X)

p(x'-T(x'))

= (i/n).(x - Tn(x))

~ Inf Sup (2/n).p(x)

N The value p(x'-T(x'))

n~N

n~N

is then equal to 0 for every continuous

n o r m on E; and because E is a Hausdorff

space this means

pseudo-

that x' is an

invariant point. 2.2.6.

Remark.

property

In the above example,

we demonstrated

the fixed-point

for Z by finding a sequence of finite parts of Z, the segments

A n = {0 ..... n-l},

such that the ratio

(I/IAnl).IAnAAn T I tends to 0

when n tends to infinity. All of which 2.2.7. F(G),

leads us to the following

Definition.

definition.

Given a finite part D of G, m D is the function on

the set of all finite parts of G, defined by mD(A )

=

l{xeA,~deD,dx~A}l

Intuitively we can see that A is as invariant under the right translations by the elements 2.2.8.

Definitions.

of D as the ratio

mD(A)/IA I

is small.

A group G is said to have an ameaning

filter

if, for

every finite part D of G and every positive real number 6, there exist finite non-empty parts of G such that the ratio

mD(A)/IA 1

is less

than o. The non-empty

set of all these finite parts of G is then denoted by

M(D,6 ). It is clear that if D' contains contained

The parts M(D,~) positive

of F(G)\{~}

real number)

we call the ameaning 2.2.9.

Proposition.

amenable.

D, and if 6' is less than 6, M(D',6')

is

in M(D,~). (where D is a finite part of G, and 6 is a

are then a basis of a filter M on F(G)\{~},

which

filter of G. If the group G has an ameaning filter,

it is then

22 This proof shall be similar

to the one in example

2.2.5.

Let K be a compact and convex subset of a locally convex Hausdorff

topolo-

gical vector space, and T an action of the group G on K by affine continuous one-to-one mappings. For every finite and non-empty part of G, consider

the average operator

M A given by MA

=

For every pair

(I/IAI).

(D,~),

~ Tg geA

F(D,~)

stands for the closed subset of K, which is

the closure of the union of all images of K by the operators

M A where A

is an element of M(D,~). Because

the set of all M(D,~)

has the non-empty

is a filter basis,

finite intersection

Next let F be the common intersection

the family of the F(D,~)

property. of all F(D,~),

and let y be a point

in F. For every h in G and every continuous

pseudo-norm

p on E, the following

inequality holds: p(y - Th(y))

Then,

Inf (D,~)

Sup AeM(D,~)

Inf (D,d)

Sup AeM(D,~)

P(MA(X)

Th.MA(X))

(2/IAl).m{h}(A).p(x)

for every h in G and every p,

p(y - Th(Y))

= 0

And y is invariant under the action of G. The rest of part 2.2 is devoted to the proof of the converse of the previous result:

every amenable group has an ameaning

This theorem was first proved by E. F~Iner tained in our work 2.2.10.

filter.

(I). We give here the proof ob-

(2) which comes from the study of invariant

Definition.

An invariant

capacity

capacities.

is a real function on F(G) with

the four following properties: i)

m(~)

=

0

ii)

for every pair

(A,B),

(strong subadditivity)

m(AUB)

+ m(A~B)

~ m(A) + m(B)

23 iii)

for every finite part A and every g in G,

iv)

there exists

m(A)

= m(Ag)

(right invariance) a positive

and a, the increment

2.2.11.

Example.

the cardinal

A -->

functions m D for instance, 2.2.12.

m(A~{a})

The first example

function

Proposition.

constant K such that,

of an invariant

Properties

used to verify

than -K.

capacity is given by

as the following

proposition

examples,

shows.

For every finite part D of G, the real function m D

(i) and

(iii)

capacity.

evidently hold.

The constant

I can be

(iv).

Next we prove the decisive Therefore,

is greater

IAI. There are some less trivial

on F(G) defined in 2.2.7 is an invariant Proof.

- m(A)

for every A

calculate

strong

subadditivity

the difference

which is the integral

mD(A)

property

+ mD(B)

for the counting measure

(iii).

mD(AnB)

- mD(AUB),

on AUB of the function

IA.I(D-IAC ) + IB.I(D-IB c) - !(ANB).I(D-I(A~B)C ) - I(D-I(AuB)c ) This function

is always non-negative,

dering every possibility, 2.2.13.

Definition.

greatest

The mean value of an invariant

capacity.

capacity m is the

lower bound q(m) of the ratio m(A)/IA I where A is a finite non-

empty part of G. Due to the property ties,

as one can see by carefully consi-

and thus m D is an invariant

q(m)

2.2.14.

Remark.

whereas

the function

homogeneous

(2.2.10

(iv)) of invariant

capaci-

is finite. The set ~ of all invariant m-->

q(m)

capacities

is a convex cone,

is clearly increasing

and positively

on E.

The existence

of the ameaning

filter

is proven when,

for every finite

part D, the mean value q(m D) is equal to 0. It is clear that the following

mD ~ Therefore, point,

d~D

subadditivity

result holds:

m{d}

we must now state the result for the parts reduced to a

and show the subadditivity

property of q on the cone E.

24

Let then d be an element of G. If d generates Otherwise,

a finite group F, one has

when n tends to infinity,

(i/IFl).m{d}(F)

= 0 .

the limit of the ratio

(i/I Anl ) .m{ d} (An) where

A n = {di,i=O,..,n-l},

The subadditivity

is equal to O.

property of q is the most important point of the proof.

In order to demonstrate

it we will rely on several

But first we must state a definition 2.2.15.

Definition.

support,

Let f be a non-negative

tive coefficients,

of f as a combination

we select a special

0 = s 0 < ~i < "'" < o k

values of f in ascending The following

special

2.2.16.

of indicators

one as follows.

be the finite sequence of the different

i=k ~ (~i-~i_l) I i= 1 " (f>=i) of f is called its pyramidal

Let m be a strongly

subadditive

decomposition.

function on F(G),

and f

function with a finite support on G.

Among all possible decompositions with positive

coefficients,

est lower bound to the sum Consider

f =

of f as a combination

the pyramidal

decomposition

of indicators gives the great-

[ ~Am(A).

an arbitrary

decomposition

of f:

~ ~AIA Ael

Let J be the set of all finite parts of G which is obtained by making repeated unions

and intersections

Then J is finite and the finite parts Choose a maximal of elements

(f~i)

element among all finite

in J containing

Every indicator were not,

with posi-

order.

decomposition

Lemma.

a positive

Proof.

with a finite

equality holds:

f = This

function,

lemma.

on a set G.

Among all decompositions Let

theorems.

and prove a combinatorial

the

I A is constant

(f~i),

of elements

from I

of I.

belong to it.

strictly increasing

and call this sequence

on every non-empty

the finite part K' = (AnKi+I)UK i

set Ki+ ~ K i ;

sequences (KI,.,Kn). if it

would be strictly between K i

25

and Ki+ 1 for the order, and this would contradict the maximality of the sequence. We use an Abel transform to obtain

(~i-ai_l) m(f>~ i)

=

~km(f=~k) +

k-I ~ ~i(m(f_->~i) - m(f>~i+l)) 1

Inserting the other K i of the sequence, we see that the first sum is equal to n f(i) .(m(Ki)-m(Ki_l))

+

f(1) .m(K I)

where f(i) is the constant value of f on Ki\Ki_ 1 (on K 1 for f(1)). Replace f by the given decomposition to get the following expression of the sum relative to the pyramidal decomposition: n

A~ei ~A

~ (A(i)'(m(Ki)-m(Ki-l))

+ A(1)'m~KI)

)

Because of the strong subadditivity property of m, the difference m(Ki)-m(Ki_l) equal to i.

is less than or equal to

m(A~Ki)-m(A~Ki_l) , when A(i) is

We then get the inequality

(~i_~i_l).m(f>~i)


~.Tnk.~.T -nk

and its image is contained

for every k, the kernel Ker(h)

in the kernel

contains

.

is a one-to-one of h.

a subgroup

isomorphic

to

S K and is therefore not solvable.

2.4. REFERENCES

For the existence and Krylov

of invariant probability measures,

to Bogoliubov

(I).

Invariant means on groups are examined by Greenleaf Markov-Kakutani also by Bourbaki groups).

refer

theorem is discussed (3)

(I).

of course by its two authors but

(with a generalization

to the case of solvable

34

Regarding paper

the existence

of the ameaning filter,

(I) or refer to Moulin Ollagnier

see F~Iner's

and Pinchon

original

(2,7) for an alter-

nate proof and a study of locally compact amenable groups.

3. ERGODIC THEOREMS

3.1. INVARIANT LINEAR FUNCTIONALS

3.1.1. Ergodic theorem. Let E be a real vector space, s a sublinear function, and T a right action of an amenable group G by linear one-toone mappings preserving s on E. Then, for every f of E, the following holds: Sup ~(f) ~eK(E,s,G)

=

Inf AeF(G)

(I/IAI).S(gYAf.Tg)

=

lim sup (i/IAI).s( [ f.T g) M geA

Proof. Since G is amenable, there exist linear invariant functionals bounded by s on E. Denote by K(E,s,G) the convex and compact set consisting of all these functionals and let ~ be one of them. Then, for every f in E, ~(f)

=

(I/IAI).~( ~ f.T g) geA

and the two inequalities easily follow Sup ~(f) ~eK(E,s,G)

~

Inf AeF(G)

(I/nAl).s( X f.T g) geA

lim sup (I/IAI).s( ~ f.T g) M geA Consider the function p on E given by p(f)

=

lim sup (i/IAI).s( X f'Tg) M geA

So defined, p is a sublinear function and, according to 1.1.7, is the least upper bound of all linear functionals below it.

36

In order to complete

the proof,

convex compact sets K(E,s,G) The first inclusion,

we have to show the equality of the two

and K(E,p).

K(E,s,G) ~ K(E,p),

tivity property of s gives

is already proven.

the inclusion of K(E,p)

The subaddi-

in K(E,s).

Now the only thing to be proven is that a linear functional, p on E, is invariant,

i.e. equal

to 0 on the elements

bounded by

f-f.T h with f in E

and h in G. What is true for f is also true for -f and we have only to show that p(f-f.T h) is equal

to 0.

Let us calculate p(f-f.T h)

=

lim sup (I/IAl).s( M

~ (f-f.Th).T g) geA

lim sup (i/IAl).(s(f)+s(-f)).m{h}(A) M 3.1.2.

Corollary.

As the upper

limit along the ameaning

the theorem above is equal to the greatest tity on the set of all non-empty

filter used in

lower bound of the same quan-

finite parts of G, this upper

limit is

in fact a limit. 3.1.3.

Example.

continuous

Consider

the usual vector

near function

s defined

for every continuous

a right action that preserves

function

to the limit p(f) of the ergodic

3.1.4.

Corollary.

number p(f), For an upper

Proof.

on X, is equal

along M

I foT ). geA g

In the above situation,

functions

it is possible

functions

on X with values

semi-continuous

measure ~ on X, ~(f) eventually

averages

not only for continuous

semi-continuous

s on C(X).

f on X, the least upper bound of all

~(f), where ~ is an invariant Radon probability measure

(I/IAl)'s(

of all

space X, and the subli-

in 1.1.3. When acting on X by homeomorphisms,

the amenable group G induces Then,

space C(X) consisting

real functions on a compact Hausdorff

to define the

on X, but also for upper

in the interval

E-~, +~]

function f and an invariant Radon probability

is well defined by regularity;

and p(f), which is

equal to -~, is still the least upper bound of all ~(f).

Consider

the upper

the value ~ (f) to ~.

semi-continuous

function on K(C(X),s,G)

giving

37 Because

function p is non-decreasing,

corollary

1.3.7 of Dini's

lemma

gives us Sup ~eK(C(X),s,G) The converse 3.1.5.

inequality

Definition.

~(f) holds

A real

=

Inf ~eC(X),~f

and the proof

function

p(~)

~

p(f)

is achieved.

c on the set F(G) of all finite parts

of G is said to be subadditive

if c(~)

decomposition

of a finite part A as a combination

indicators following

of the indicator

of subsets inequality c(A)

Function

F(G)

Remark.

coefficients,

of

1 A = ~ ~BIB,

the

~ ~BC(B)

The subadditive there

Remark.

to 0 and if, for every

holds:

g of G, the numbers

for which

3.1.7.

of it with positive

c is said to be invariant

every element 3.1.6.

g

is equal

if, for every finite part A of G, and c(A)

functions

is a sublinear

According

and c(Ag)

are equal.

are exactly

extension

to lemma 2.2.15,

the functions

on

to the cone C~(G).

strongly

subadditive

functions

are subadditive. 3.1.8.

Lemma.

Let B be a finite non-empty

part A of G, the following I{ geG,BgcA} I The positive


exp(-[t[ ~) is of a positive

transform of a probability,

Because

i.e.

It is possible,

of the "El6ments

1 at 0, it is the Fourier

to Bochner~s

result

theorem.

in Probability for instance,

d'Analyse"

transform is one-to-one,

type.

Theory that can be to solve exercise

by Dieudonn4.

the stability of p~ comes

of the function exp(-]t]~).

The law of the average

of n independent

random variables with a law pe

is then that of the homothetic (n ((I-~)/~) ) .X

of a random variable whose

law is ~ .

If we denote by ~ the Fourier the Fubini's

transform of p~, a simple application

theorem shows that the integral (I

- Re~(t))/t

of

of

B+I

is the product of the integral

over R for the Lebesgue measure

of

(I - cos u)/u B+I

by the expectation

for p~ on R of Ix[ B

If the real number ~ strictly for the Lebesgue measure

lies between 0 and 2, the integral

of the function

over R

2

50

(i - cos u)/u B+I

is finite and the two other integrals It is then easy to see that is strictly The series

are then simultaneously

Ixl B is integrable

for ~

finite

if and only if B

less than ~. that we are interested

as the integral

for ~

in has,

for every a, the same nature

of the function

Ixl (=-II~) and therefore

diverges.

The probability ~ 3.3.3.

Remark.

amenable tion.

group,

then provides

The existence

us with the counterexample

of ameaning

for which Birkhoff's

Only partial

results,

sequences

sought.

of finite parts of an

result holds,

is still an open ques-

such as for groups with a polynomial

growth,

are known.

3.4. THE SADDLE ERGODIC THEOREM

This section is devoted pological 3.4.1.

dynamical

to the proof of a minimax

sytems; we give this result in a very general

Saddle ergodic

theorem.

subset of a real locally convex Hausdorff

following

to toform.

Let K be a convex and compact non-empty

T be a left action of an amenable one mappings

theorem related

topological

group G by affine

on K, and let f be a real function on

vector

space;

continuous F(G) x K

let

one-towith the

properties:

i)

for every x in K, the partial

function

in the sense given in definition ii)

for every finite part A of G, the partial is concave and upper semi-continuous

iii)

f(.,x)

this function

is invariant,

=

function

f(A,.)

on K

i.e. for every g in G, every

finite part A of G and every point x of K, f(Ag-l,Tg(x))

is subadditive

3.1.5

f(A,x)

51

Then,

the following

Inf AeF(G)\{~}

minimax

(iii AI ) .Sup

f(A,x)

for the subset

action

of G.

Proof.

For every element inequality f(A,x)

of all invariant

of K under

the

Sup f(A,t) teK

of these subadditive

the limits

points

x in K G and every finite part A of G, the

along M of these averages taking

(ill AI ) .f(A,x)

Sup Inf xeK G AeF(G)\{~}

holds: ~

Also the averages Because

=

xeK

where K G stands

following

equality holds:

invariant

functions

and the limits

are in the same order.

along M are equal

to the greatest

lower bounds,

the least upper bound on all x in KG, we get the inequality

in

one direction Sup xeK G

Inf (I/IAl).f(A,x) AeF(G)\{ ~}

In order

to prove

function

on F(G)

the converse defined

T(A)

=

decomposition

lower bound,

=

call ~ the subadditive

and also the limit along M, of the

is nothing

to prove.

fix a finite part B of G and consider

of the indicator

iA

inequality,

by

ratio (I/IA I).~(A). If ~ is equal to -~, there case,

Inf (I/IAI).Sup f(A,t) AeF(G)\{ ~} teK

Sup f(A,t) teK

Let = be the greatest

In any other

=
> P.

4.2.5. Remark. The previous relation ">>" is a preorder relation on the set of all finite partitions of a given probability space; and the equivalence classes of the corresponding equivalence relation (P>>Q and Q>>P) can be identified with the partitions of the unity between non-null idempotents in the quotient algebra ~/~. We shall identify a partition with its equivalence class for the previous equivalence relation. 4.2.6. Proposition.

For the previous order relation, every two parti-

tions have a least upper bound. Proof. The partition R, which is defined by its atoms

Rij = Pi~Qi, is

clearly a least upper bound for the set (P,Q). 4.2.7. Definition. The entropy of the partition P = (Pi) is the entropy of the probability vector (~(Pi)); this entropy is denoted by H(P). 4.2.8. Proposition. Entropy is a strictly increasing function on the ordered set of all partitions of a given probability space. Proof. Suppose Q finer than P and use the subadditivity of n to derive H(Q)

=

~ ( ~ n(~(Qj))) l Qj Pi

=> ~ n(~(Pi)) l

Equality can only hold if, for every atom Pi of P, the number n(~(Pi)) is the sum of all n(~(Qj)) where Qj is contained in Pi" Strict concavity of the logarithm then implies that this last equality is only possible when one of the ~(Qj) is equal to ~(Pi); and entropy is then strictly increasing.

57 4.2.9. Proposition. Entropy is strongly subadditive on the set of all finite partitions, i.e. verifies, for every triple (P,Q,R) H(PfQVR)

+

H(P)

__< H (PVQ)

+

H (PVR)

Proof. This proof is apparent from the strong subadditivity property of H on the set of all probability vectors (proposition 4.2.2). 4.2.10. Definition.

Given a o-algebra

6 contained in ~ a n d

a finite par-

tition P of (X,~,~), the conditional expectations E(IPiI~) constitute a partition of the unity almost everywhere. The integral of the entropy of this random probability vector is called the conditional entropy of P with respect to D and is denoted by H(PI6): H(PI~)

=

4.2.11. Proposition.

[ I n(E(ip.I~)) l V(~) and V - - > ~(V), defined in the two previous propositions between positive local specifications and cocycles, are inverse of each other. There is no difficulty in calculating the proof of this outcome.

94 6.2.7.

Proposition.

the Gibbs measures

Given a pair

(V,~), where V = V(~)

and

for ~ are exactly the quasi-invariant

~ = ~(V),

measures

for

the cocycle V. Proof.

Let us first prove that a Gibbs measure

for V; therefore,

we have to show that,

for ~ is quasi-invariant

for every element

g in C(X) and

every point x of S,

~ (g.exp (Vx))

=

~ (go~x)

Because p is a Gibbs measure

~(f)

=

for ~, we get

~(~x(f))

=

~((f + fOTx.exp(Vx))/(l

+ exp(Vx)))

Then

p((f.exp(Vx))/(l Choosing

+ exp(Vx)))

=

f = g.(l + exp(Vx)) , we get the quasi-invariance

In order to prove that a quasi-invariant measure

p((fo~x.exp(Vx))/(l

measure ~ for V is a Gibbs

=

~(( ~ fOTB.exp(VB))/( BaA

local

Remark.

specification

B~A ((f°~B)/(CcA [ exp(Vc°~B))

=

~ BcA

=

p (f)

((f.exp(VB))/(

of the Gibbs measures

that a Gibbs measure

a mass which is proportional

are interesting:

probabilities;

and the

of the energy

(we shall

of some measures

under the

to prove a result of Ruelle's).

Given a cocycle V = (V A) on X, there exists

J = (J(A)) of real numbers

the

gives to the configurations

to the exponential

the quasi-invariance

group of modifications

Proposition.

~ exp(Vc))) CCA

involves given conditional

means

use in the next chapter action a wider

~ exp(Vc))) CcA

=

Two properties

quasi-invariance

6.2.9.

relation.

for ~, let us simply calculate ~(~A(f))

~(HAf))

6.2.8.

+ exp(Vx)))

a family

indexed by the finite parts of the set S, such

95 that the Fourier

transform

of the continuous

function

V

is given by X

Proof.

ix(A)

=

-2J(A)

if xeA

ix(A)

=

0

if

Using

Fourier

the cocycle

coefficients

x~A

relations

Vx(A)

V x + VxO~ x = 0, we discover

are equal

that the

to 0 when the finite part A does not

contain x. The other

cocycle

that contains

relations

ix(A)

the existence

Imagine

Fourier

=

Vy(A)

ix(A)

-

= Vy(A).

of the family J, the "-2" being related

that there exists coefficients

the increments

Notice

for a part A

to the

interpretation.

the Fourier with

Vy(A)

-

that is to say to following

, lead,

both x and y, to the equality

ix(A)

Hence

V x + Vyo~ x = Vy + Vxo~y

coefficients that J(@)

E representing

-2J(A)

a cocycle

You would

and that on X

then find the

that we have just built.

function,

of energy

the energy,

and then build

V A = Eo~ A - E of the energy.

is not determined,

tation of an energy the differences

a function

of E are J(A);

which

is coherent

with our interpre-

only defined up to an additive

are actually

An interaction

constant;

significant.

6.2.10.

Definition.

indexed

by the set of all finite non-empty

is a family J = (J(A)) parts

of real numbers

of S, such that, for

every point x of S, the mapping

A --> J(A).ixe A is the Fourier

transform

These functions 6.2.11.

Definition.

if J(A)

is equal

6.2.12.

Definition.

J(A)

is equal

of a continuous

then verify

the cocycle

An interaction

function

on X.

relations.

is said to satisfy

Ising's

condition

to 0 for every finite part A with an odd cardinal An interaction

to 0 whenever

number.

J is said to be a pair interaction

the cardinal

number

of A is greater

if

than or

96 equal to 3. Notice that a function J on the set of all finite parts of S which verifies this condition,

is an interaction if and only if, for every point x

of S, the series J({x,y}) where y runs in S converges. This results, 6.2.13.

for instance,

Definition.

from proposition 6.1.2.

An interaction J is said to be attractive

if the

function J is non-negative. A non-negative function J on the set of all finite parts of S is an interaction if and only if, for every point x of S, the series

(J(A),xeA)

of real numbers converges. To prove this equivalence,

apply proposition 6.1.2 to the unit configu-

ration.

6.3. PHASE TRANSITIONS

6.3.1. Remark.

Certainly,

one of the most important problems

tical mechanics on a lattice is the phase transition problem,

in statisi.e. deci-

ding whether there are many different Gibbs measures for a given specification. When J is an interaction,

~J is also an interaction for every positive

real number 8. For some types of interactions, the comparison theorem:

it is possible to state

if there is a phase transition for BJ, there is

also a phase transition for ~'J~ whenever B' is greater than B. On the other hand,

the Kirkwood-Salsburg

little further on, postulates

theorem,

that we shall prove a

that in some cases there is no phase

transition for B adequately minute. Therefore, verifies orems,

the specific phase transition problem for an interaction which

the hypotheses of the comparison and the Kirkwood-Salsburg

the-

is knowing if there are several Gibbs measures for ~ adequately

large. Classical

interactions,

like attractive pair interactions,

verify

all these hypotheses. The problem of phase transition is a sizable one and we only give here the general 6.3.2.

theorems used to simplify this particular work.

Definition.

of all continuous convergent.

Consider the vector subspace A(X) of C(X) consisting functions on X whose Fourier series is absolutely

97

Define

then a real

function

]IIflll This

function

=

is less

bound of the norms

Proof.

measure

function

lity ~(f.(l + th(Vx/2))

V consists

of functions

and if the least upper quasi-

measure

on X that verifies

f + f°~x = 0, the equa-

=

~(f.(2exp(Vx/2)/(exp(Vx/2)+exp(-Vx/2)))

=

~(f.2exp(Vx)/(l

=

~(2fo~x/(l

-

then an operator

(I/]AI).

K(1)

=

0

Because

the °A generate and continous

the iden-

on the elements

OA

~ oA.th(Vx/2) xeA

a dense

subspace

on A(X) whenever

l[Ith(Vx/2) lll is finite.

verifies

(A,x) where x is in A.

K on A(X) by its values

=

+ exp(Vx)))

~(V) necessarily

= 0 for every pair

K(o A)

defined

+ exp(Vxo~x)))

0

for the specification

+ th(Vx/2)))

+ exp(Vx)))

~(2f.exp(Vx)/(l

=

norm of the operator

for V. For every point x of S,

this measure:

~(f.(l + th(Vx/2))

Define

of the norms.

= 0 holds.

we can calculate

tity ~(OA.(l

of A(X),

a subalthat the

for V.

Let ~ be a quasi-invariant

A Gibbs measure

to the product

If the cocycle

are elements

is, moreover,

]II.lll, meaning

lllth(Vx/2)II I is less than I, there is a single

and every continuous Indeed,

for the norm

than or equal

theorem.

V x such that all th(Vx/2)

than the supremum norm on C(X),

for it. Space A(X)

and a Sanach algebra

Kirkwood-Salsburg

invariant

it is finer

is complete

norm of the product 6.3.3.

I If(A) l AeF(S)

lll.III is a norm;

and the space A(X) gebra of C(X)

on A(X) by

If the condition

K is strictly

of A(X),

the operator

K is well

the least upper bound of the norms of the theorem

less than i.

is fulfilled,

the

g8

The restriction verifies

of ~ to the subspace A(X)

the equation

The operator

(I + K*)(~)

is a linear functional

(I + K*) has then an inverse

6.3.4.

of X.

in the dual space of A(X); and

there is only one solution for the previous thus,

that

= h, where h is the Haar measure equation

in this space and,

in the set of measures. Corollary.

are elements

If the continuous

of A(X),

functions V x that define the cocycle

and if the least upper bound of their norms in this

space is less than 7/2,

there is no phase transition

for this cocycle.

For every element ~ of A(X), whose norm is less than i, we can define th(~)

as the sum of a power

series;

ded by tg(lll~Ill). The conclusion 6.3.5.

Example.

Let us describe

S = Z n and the attractive

The interaction

and we can calculate

depend on x because When n is equal

constant)

J is equal

It is clear that the functions A(X),

the Ising model.

pair interaction

(where J is a strictly positive the lattice.

and the norm lllth(~)IIl is then boun-

is clear.

the norm

the interaction

6.4.1.

cocycle belong to invariant. immediately

in the Ising model

confirms

in dimension i for

B.

llIth(Vx/2)lll = 2 th(2J),

that there is no phase transition when th(2J)

which

shows

is less than I, i.e. if J

(Log3)/4.

6.4. SUPERMODULAR

INTERACTIONS

Definitions.

Denote by ~ the order on the product

nl ~ n2

t. .~

in

lIIth(Vx/2)III ; this norm does not is translation

any v a l u e of the inverse temperature is 2, we get

= J

to 0 otherwise.

to i, lllth(Vx/2)lll = th(2J), which

When the dimension

the lattice

for the pairs of neighbors

of the corresponding

that there is no phase transition

is less than

Consider

J defined by J({x,y})

VxeS,nl(X)

set X = {-I,+I} S defined by

=< n2(x )

99

It is the product

order of all natural

orders

of the factors;

it is not

a total order. A continuous decreasing

function

for this order.

monotonic

continuous

functions

belong

6.4.2.

V x + VyOT x = Vy + VxOTy

function ~

the cone of all these

on X; to be specific,

Given two different

VxoTy - V x This

Let us call then M(X)

functions

if it is non-

all coordinate

to M(X).

Definition.

relation

f on X is said to be non-decreasing

xy

=

points

x and y of S, the cocycle

leads to the equality

VyOT x - Vy

has the following

=

~xy

Fourier

transform:

A

~xy(A)

=

4J(A) . IxeA. lyeA

When all functions

+xy.Ox.Oy

to be supermodular.

This

are non-negative,

is the same as saying

of S, the function V x is non-decreasing 6.4.3.

Example.

A typical

vided by an attractive

example

the interaction that,

for every point x

on the subset {o x = -i} of X.

of a supermodular

pair interaction.

is said

interaction

is pro-

The function Sxy then reduces

to 4J({x,y})OxOy. 6.4.4. modular

Proposition.

Let A be a finite part of S and # be a local

specification.

that n 1 ~ n 2. Denote by ~+n

Let n 1 and n 2 be two elements

the configuration,

which

tion of the element

$ of {-i,+i} A and of the element measures

on {-i,+i} A, verify

the Holley relation:

for every pair

#l(~iV$2).~2(~lh$2 )

Proof. product

~ --> ~A($+nl)

(EI,$2)

We have to prove,

~

of {-I,+I} S A such

is the result

The two probability

of elements

super-

of the concatena-

n of {-I,+I} S A

and ~ --> #A(~+n2),

defined

of {-i,+I} A

#i(~i)-~2(~2 )

for every pair

(~i,~2)

of elements

of the

space {-i,+I} A, the inequality

~A((~IV~2)+nl ).~A((¢IA~2)+n2 )

>

#A($2+n2 ).~A($1+nl )

1O0

This

inequality

is equivalent

to

~A( (~ I W 2)+n i)/~A(~ l+n 1 ) According equal

to proposition

6.2.6,

to exp(VB(~l+nl)) , where

is greater points

than $1(x).

at which ~2(x)

The right-hand

= +i and $1(x)

to be demonstrated

the value

This results

B is the subset is equal

is supermodular;

Theorem.

product

[~,~

to the product

or-

of X of all configurations

of B.

of B in an arbitrary

Then, 7' is greater is non-decreasing

~,~

and V B is also non-decreasing

Let 7' and 7" be two probability

and that verify

order.

on the subset

space {-I,+I} A which give a positive

this set,

to exp(VB((~iA$2)+n2 ) .

that, with respect

term in the sum is non-decreasing

6.4.5.

of A of all

Vbl + Vb2°~ b I + • • + VbnO~bl o • • oT bn_ 1

=

where bl, .... b n are the points interaction

is

= -I.

on the subset

-i at all points

side of the inequality

for the part of A on which ~2(x)

from the decomposition

VB

Every

the left-hand

In other words,

der, V B is non-decreasing

~ A(~ 2+n 2)/~A( (~ IAE 2)+n 2 )

B stands

side of the inequality

It then remains taking

>

the Holley

than or equal

for the product

mass

relation

to 7",

measures

because

the

on [ ~ , ~ . on the finite

to every element

of

(6.4.4).

i.e.

for every function

order, ~'(f)

is greater

f which

than or equal

to ~"(f). The proof of this result

can be found in Holley

6.4.6.

to the previous

Corollary.

Thanks

the least upper bound of ~A(f) is equal 6.4.7.

to 1 at every point

of ~

= HA(f)(T) , weakly

is a Gibbs measure +

(f)

=

results,

when f belongs

at the configuration

to M(X),

T, which

of S.

Proposition • The family

where ~ ( f )

is reached

(I).

(~+A ) of Radon probability converges

when A tends

for ~ and verifies,

Sup ~ (f) eG(~ )

measures

on X,

to S. The limit ~+

for every f in M(X),

101

Proof. When f belongs functions

to the cone M(X) of all continuous

on X, the following

~A(f) (I)

=

equality holds

non-decreasing

for every finite part A of S:

S(~A(f))

And this directed family of real numbers has a limit when A tends to S; this

limit is the least upper bound of all ~(f), where ~ is a Gibbs

measure

for ~.

This family of probability measures ments

(fl

then simply converges

for all ele-

f2 ) of C(X), where fl and f2 are in M(X).

The subspace,

consisting

of these differences,

is dense in C(X)

uniform norm and the family is equicontinuous.

for the

Hence the existence

of

the weak limit ~+ is proven. For every f in M(X), ~+(f) this

infimum is equal

= Inf(s(~A(f));

according

to theorem 6.1.7,

to the least upper bound of all ~(f), where ~ is

in G(~). + . In order to verify that p is actually a Gibbs measure for =, notice + + simply that ~A(~B(f)) = ~A(f) whenever A contains B, and take the limit when A tends to S. 6.4.8. weak

Remark.

The probability measure ~- is similarly defined as the

limit of the ~A(.)(---T). It gives the least upper bound of all Gibbs

measures 6.4.9.

for the non-increasing

Proposition.

specification

continuous

functions.

There is a phase transition

if and only if the two particular

for a supermodular + measures ~ and ~

local are

different. Proof.

The condition

is obviously

sufficient because ~

and ~

are Gibbs

measures. If they agree,

proposition

6.4.7 and remark 6.4.8 show that a Gibbs

measure

for ~ takes necessary

because

this subspace

Gibbs measure 6.4.10.

values on the subspace

is dense in the uniform norm,

for ~.

Proposition.

is a phase transition

Let ~ be a local

The condition

is also necessary,

supermodular

specification.

There

for ~ if and only if there exists a point x in S

such that u+(~x ) is strictly greater Proof.

generated by M(X); there is only one

is obviously

than ~-(~x ).

sufficient.

we have only to demonstrate

In order to prove that it the following

inequality:

102

(u+ - ~-)(pA ) where PA is the positive

PA In this case,

=


+® Because

that Vxorx_ n

tends to V--x when n tends to infinity.

the interaction

is invariant,

and we use now the strong mixing

=

p+(for x)

Vx_ n is the translate

property

VxoT n of Vx;

of ~+ to get

+ (for x) In order

to conclude

is equal

to i.

=

~+(f.exp(Vx)).~+(exp(Vx ))

the proof,

we still have to show that ~+(exp(Vx))

Hence we must note that the previous function, proof. 7.4.7.

and in particular

Corollary

i.e.

equal

sition

to ~-

for the constant

(proof of theorem

for the U-cocycle

equality

7.4.2).

V, it is in particular , and there

is no phase

holds

for every continuous

function

Because ~

invariant

+

i, achieving

is quasi-invariant

under

transition

the

the reversal

according

3,

to propo-

6.4.9.

7.5. REFERENCES

The characterization measures

of the invariant

for a suitable

by this author The original

continuous

proof

of the uniqueness by Ruelle

The study of attractive condition

described

invariant

way

in the last sec-

pair interactions

greater

in dimension

for which J is proportional

is then the summability

is necessarily

in a general

(5).

(i).

been done for the interactions exponent

result

as the equilibrium

is proven

in a joint work with D. Pinchon

tion was achieved

interaction

Gibbs measures

function

to n

of the series

than I. If it is greater

i has ; the

and the

than 2, Ruelle's

116

theorem shows that there is no phase transition verse temperature

stated that there is a phase transition therefore

for

a comparison with the hierarchical

= 2, known as Anderson's model, and Spencer

for any value of the in-

B. For an ~ strictly between 1 and 2, Dyson B adequately model.

large.

He used

The borderline

case,

was only solved recently by FrShlich

(I); they showed that there is a phase transition

values of the parameter.

(1,2)

for large

8. EQUIVALENCE OF COUNTABLE AMENABLE GROUPS

8.1. TILING AMENABLE GROUPS

8.1.1. Definitions. The sequence of segments in Z, (An = {0 .... n}), encountered at the beginning of this text is the prototype for all F~Iner sequences in countable amenable groups. But the elements of this sequence,

the segments An, have an interesting

property which is a decisive tool for solving some problems of ergodic theory:

they are tiling sets.

A finite part of a group,

for which it is possible to tile the whole group

with some pair-wise disjoint right translates of it, is called a tiling set. This notion is therefore in fact related to equivalence classes of finite parts under the right translations which can be called forms. Some authors,

Ornstein and Weiss in particular,

have used F~iner sequences

consisting of tiling sets in groups other than Z. Notice that, in all known amenable groups,

it is always possible to con-

struct explicit F~Iner sequences consisting of tiling sets. This means that the class of groups with arbitrarily invariant tiling sets is stable, like the class of amenable groups, under the constructions we described in section 2.3

(see Moulin Ollagnier and Pinchon

that

(3)).

But, as we said in chapter 2, no structure theorem for amenable groups is stated herein. Connes,

Ornstein and Weiss have introduced the notion of almost tiling:

with a finite number of different forms, each of them being as invariant as desired,

it is possible to cover the greatest part of the group with

almost disjoint right translates of these forms. This property is true for all amenable groups and is the subject of the next lemma.

118

8.1.2. Tiling

lemma. Let A, A and D be three finite parts of a group G

and let the unit element e of G belong Then,

there exists a partition

right

translates

of subsets of A, such that the proportion

in A for which all left translates atom,

to D.

of A, whose atoms are contained

is greater

dg by elements

Let A' be the subset of A consisting

is contained

of D are in the same

than

(I - m D ( A ) / I A I ) . ( I A I / I D - I A I ) . ( I

Proof.

in some

of all points

- m D ( A ) / I A I)

of all points

g such that Dg

in A.

By definition,

we get

IA - A'I = mD(A)

Denote by T the set of all total orders on A-~A = {geG,Ag~A~@}. Consider P(T)

the elements

gl .... gk''"

as the partition of A whose

following

A1

=

AnAg 1

A2

=

(A~Ag2)\A 1 k-i (A~Agk)\( U A i) 1

The notation P(T,x) measure

then stands

for the class of x in the partition P(~).

the uniform probability measure

~ on T, i.e. the probability

giving to every point ~ of T the mass

For a given element x in A', the probability P(~,x)

is exactly the probability

the first element This

of the

finite sequence:

"'" Ak =

Consider

of A-~A in the order T, and define atoms are the non-empty elements

I/(IA-~AI)!. that Dx is contained

that Dx is contained

in the order T such that Ag meets Dx.

last event depends only on the restriction

subset A-~D.x; probability Therefore,

in

in Ag, where g is

and its probability

of the order • to the

can be easily obtained because

for a given element g being the first in A - ~ the probability

is equal

of the event that we are searching

the ratio l{g,AgcDx}I/I{g,Ag~Dx#@}l

the to

is equal

to

119

i.e.

to the ratio

(i

Summing

mD(A)/IAI).(IAI/ID-I.A

-

I)

on all x in A', we get the expectation

E( [ iDxcP(~,x )) xeA

~

for

E( ~ IDxcP(~ x)) xeA' ' (IA'I).(IAI/ID-~AI).(I

The proportion positive

(I

There

of all x of A such that Dx is contained

function

of T whose expectation

is bounded

mD(A)/;AI).(IA I/ID-I.AI).(I

-

then exists

the above number.

an order T for which The corresponding

expected inequality. triple (A,A,D).

in P(T,x)

is a

below by

- mD(A)/IA ])

this proportion

partition

Such a partition

- mD(A)/IAI)

P(T)

in greater

thus verifies

is said to be adapted

than the

to the

8.1.3. Iterated tiling lemma. Let A, A 1 . . . . , An, and D 1 .... , D n be finite parts of a group G such that

eeD I c D 2 ~ There

then exists I) 2)

a family

for every right

... c D n (PI ..... Pn ) of partitions

subscript

translates

for every i from 1 to n-l, tion of Pi' i.e.

3)

the proportion is contained ponding

(i

Proof.

i between

of subsets

Consider

1 and n, the atoms

of Ao l the partition

Pi+l is a subparti-

the atoms of Pi+l are unions

of the points

of Pi are

of atoms of Pi

x of A for which every set D.x l containing x of the corres-

in the atom Pi(x)

partition

-

of A such that

m D (A)/i n

Pi is greater

than

i=n AI. ~ (IAil/ID?llAil).(l i=l

the finite probability

space

- mDi(Ai)/IAil)

(T,~), which

of the (Ti,~i) where T i is the set of all total orders the uniform probability on it.

is the product

on A?~AI and ~i is

120

!

For every n-uple

• = (~i ..... ~n ), consider

partitions

where every P! is built as it was in the proof of the

of h

preceding lemma. by setting

A new sequence

p

Pn-I P1

Conditions

of partitions

=

p, n

=

p,

=

Pi V P2

n

Call A' the subset of A consisting

E

i=n ~ ({TeT, i=l

where

Pi(T,x)

built

from the n-uple

stands

By construction

is then defined

of all x such that DnX is contained of the event

Dix(Pi(~,x)}) x of the partition

Pi(~),

~.

of the sequence

of all Pn from the one of the P'n' the

defined

i=n j=n ( ~ ({reT,

as the following

J

of D i :increases, definition

intersection:

D~x~P'(T,x)})) J

i=l j=i

Since the sequence

of

by such a sequence.

for the atom that contains

event E can be equivalently

get the following

(PI,...Pn)

for an x in A', the probability

=

(PI ..... Pn)

n-I V Pn

1 and 2 are then fulfilled

in A; and calculate,

the sequence

this expression

can be simplified

to

of E:

i=n E

({TeT,Dix c P~(r ,x) })

=

i=l

Because

the factors

get the probability

of the product

space T are mutually

independent,

we

of this event as the product

i=n 1 H (I - mD(Ai)/IAil).(IAil/ID.7~.Ail).,,,,, ~ , i=l l Conclude

as in the previous

A family of partitions tiling

lemma by summing

that verifies

and is said to be adapted

over all x in A'

the inequality

is called

to the given 2n+l-uple.

an iterated

121

8.1.4. Definition.

Denote by T O the subset of all total orders on G which

are isomorphic to the natural order of Z, i.e. the orders without a first element, without a last one, and for which there are only finitely m a n y elements b e t w e e n two given ones. Set T O is a subspace of the Hausdorff compact space T of all total orders on G, and it is invariant under the action of G d e s c r i b e d in 2.1.3. When the amenable group G is countable,

this set is not empty.

From now on, we shall work under the a s s u m p t i o n that G is countable. N o w we want to know if there exists, lity measures

among all invariant Radon probabi-

on T, a p r o b a b i l i t y measure ~ such that ~(T 0) = I, when

the countable group G is amenable.

Because T O is not compact, we cannot

apply the fixed-point p r o p e r t y to get the answer.

Since every Radon p r o b a b i l i t y measure on T is inner regular, we get

(T 0)

=

Sup ~ (K) KcT 0

w h e r e K runs in the set of all compact subspaces of T contained in T O .

R e v e r s i n g the order in which we take the least upper bounds, we get

Sup ~(T 0) ~eK(C(T),s,G)

=

Sup Sup u(K) KCT 0 ~eK(C(T),s,G)

But the indicator of a compact set is an upper semi-continuous

function,

and we can use the ergodic theorem 3.1.3 to get the least upper bound of all invariant p r o b a b i l i t y measures on such a compact set.

Next, we have to study the compact sets contained in T O .

8.1.5.

Definition.

Given an element • of TO, d r stands for the function

on GxG w i t h integer values defined by

d (g,h)

=

i +

=

0

I{keG, g~k~h or h~k~g}[

if

g

=

if

g # h

h

It is a distance on G.

8.1.6. Proposition.

Let d be a distance on G with integer values.

K d of all total orders such that d

The set

is less than or equal to d is compact.

122

Proof. equal

Consider the function dg,h on T with values to the n u m b e r

is

of elements between g and h plus one if this number

is finite and to += otherwise.

This function is lower semi-continuous.

The set of all total orders such that dg,h(T) d(g,h)

in N where dg,h(T)

is less than or equal

to

is then a closed subset of T.

The set K d is the intersection of all these closed sets when g and h run in G, and it is therefore compact.

8.1.7.

Theorem.

Let G be a countable amenable group.

Then,

there exists

a Borel probability, w h i c h is invariant under the right translations,

on

the set T O of all total orders on G that are isomorphic to the natural order of Z.

Proof.

It shall be sufficient to find,

a distance d on G with integer values

for every positive real number e, such that the following holds for

the c o r r e s p o n d i n g compact set Kd:

Sup ~ (K d) ueK(C(T),s,G)

>

I -

We then find an invariant Radon p r o b a b i l ~ y measure on T that gives the whole mass to the union of all K d. This union is p r e c i s e l y the set of all total orders for which there are only finitely m a n y elements b e t w e e n two given ones.

On the other hand,

thanks to an easy invariance argument,

the set of all

total orders with a first or a last element has a measure 0 for every invariant Radon p r o b a b i l i t y measure on T. The difference is the set T O of all orders that are isomorphic to the order of Z. Thus we must look for distances on G with integer values,

in particular

ones that have the following form:

d(g,h)

=

d(g) + d(h)

if

w h e r e d is a integer function on G equal

Take a converging sequence

(l-e i)

>

to 0 at e.

(~n) of p o s i t i v e real numbers

such that

i=l

g#h

1 -

E

less than i

123

and an increasing

sequence

(D n) of finite parts,

containing

the unit

element e of G, whose union is G. Because G is amenable,

there exists,

for every subscript

i, a finite

part A i of G such that

(IAil/IDi~Ail).(I

- mD!Ai)/IAil)

>

I - ~i

i

Define

Next,

therefore

the function d in the following way:

d(e) =

0

d(g) =

IAII - i

if

geDl\{e}

d(g) =

IAnl

if

geDn\Dn_ I

I

denote by K n the following

Kn

=

compact

subset of T:

{meT, VgeD n d (e,g)

OF COUNTABLE

GROUPS

Definitions.

Endowed with the product topology of discrete

topologies,

the set H G of

all mappings from the countable Polish space.

group G in the countable group H is a

Its subspace A(G,H),

of all elements

consisting

for which the image of

the unit element of G is the unit element of H, is a closed subspace, and thus a Polish space for the induced topology. Let g be an element of G. The right translation by g does not preserve the subspace A(G,H). We must now define a mapping T g from A(G,H) following

definition Tg(a)

to itself by giving it the

on the graphs: =

{(glg-l,hlh-l),

(gl,hl)ea}

where h is chosen in such a way that (e,e) belongs h is the image of g by a. Hence, using the functional Tg(a) (g ')

notation, =

to Tg(a),

i.e. that

T g is defined by

a(g'g) .a(g) -i

It is easy to verify that the mappings

T g are continuous

and that they

constitute

an action of G on the Polish space A(G,H) by homeomorphisms.

We denote,

too, by A(G,H)

the corresponding

dynamical

system.

125

The subset I(G,H)

consisting

of all one-to-one

elements

closed and invariant under the action of G. Once again, stands

for the corresponding

The set B(G,H) mappings

But,

B(G,H)

also denotes

(a G6 in a Polish

in this system,

possible

I(G,H)

is

also

system.

is the subspace of I(G,H)

from G onto H. This subspace

of G. Once more, system

dynamical

of A(G,H)

consisting

of the one-to-one

is also invariant under the corresponding

the action

standard

dynamical

space is a Polish space).

the two groups play symmetrical

to define an action of H on B(G,H)

roles and it is

in a way very similar

to

that of G. It is not obvious, bility measure

however,

that there exists

for this dynamical

an invariant

system when G is amenable.

dard space is not a compact one and it is not possible point property because

Borel probaThe stan-

to use the fixed-

the convex set of all Borel probability measures

is also not compact. The next proposition makes

clear the symmetry between

the roles of the

two groups when acting on the set B(G,H). 8.2.2.

Proposition.

Let B(G,H)

be the set of all one-to-one mappings

from G onto H, giving to the unit element of G the unit element

of H as

its image. Consider

the action T of G on this space given by Tg(x) (g ' )

=

x(g'g) .x(g) -I

and the action U of H on it defined by uh(x)(g ') Then,

=

triction of the product Moreover,

Proof.

where

g = x-l(h)

for the topology of a Polish space on B(G,H),

mappings under

x(g'g).x(g) -I

topology of discrete

ones,

obtained

as the res-

the T g are continuous

and the U h are Borel ones. every Borel probability measure

the action of G, is also invariant The continuity

on the space which is invariant under

that of H.

of the T g is straightforward.

In order to state the Borel m e a s u r a b i l i t y

of the transformation

U h, we

126

need only show that, elements

for every pair

This makes

obvious

O ({x,x(g)=h geG

that this set is a Borel

the invariance

for the cylinder A because

set because

it is a F

O

in the

=

.o. and X(gn)=h n}

sets generate

the Borel o-algebra.

U ({x(glg)x(g)-l=hl}~...~{X(gng)X(g)-l=hn}N{x(g)=h}) geG equality where

(uh)-I (A)

=

Using now the invariance

the union is disjoint:

0 (Tg) -i (A~{ x(g-l)=h-l} ) geG of ~ under T, we obtain

~((uh)-I(A))

8.2.3.

=

~ geG

=

,(A)

(Afl{x(g-l)=h-l})

Remark.

The mapping

from B(G,H)

ment of B(G,H)

its inverse

mapping

Nevertheless,

to the Borel

tions becomes

the standard structure,

is invariant

is invariant under

Remark.

the actions

space

fact,

giving

to every ele-

is not a continuous it is a Borel

B(G,H)

the postulated

and forget symmetry

one°

isomorphism. the topology

of the two ac-

clear.

is then an easy corollary

probability

to B(H,G),

as image,

and this is an important

if we consider

leading

this property

set, we get

the following

8.2.4.

and x(g'g)=h'.x(g)})

of ~ under U, we must only verify

{x(gl)=h I and

these cylinder

(uh)-I(A)

This

as follows:

sets

=

For such a Borel

Thus,

the set of all

space B(G,H).

To state

Hence

in GxH,

such that uh(x)(g ') = h' can be decomposed {xeB(G,H),uh(x)(g')=h '} =

Polish

(g',h')

under

of the previous

proposition

one of the two actions

that a Borel

if and only if it

the other one.

A probability

measure

on B(G,H)

T and U of G and H, is ergodic

which

is invariant

for one action

under

if and only if

127

it is ergodic

for the other.

Indeed,

the measurable

because

the orbits under the two actions

8.2.5.

Definition.

if there exists

invariant

sets are the same for the two actions

Two countable

groups G and H are said to be equivalent

a probability measure

on the Polish space B(G,H) which

is invariant under the two previously This relation

is clearly a reflexive

term "equivalence" 8.2.6.

Theorem.

is a Borel

by proving

the transitivity

tensor product

Proof.

one. We justify the

of this relation.

Let G, H and K be three countable

groups.

on B(G,H)

Suppose that

and v a Borel

invar-

on B(H,K).

the probability measure ~ v

actions

defined actions of the groups. and symmetrical

invariant probability measure

iant probability measure Then,

are the same.

of ~ and v under

on B(G,K) which

is the image of the

the composition

is invariant under

from the product

space B(G,H)xB(H,K)

the

of G and K on B(G,K). The composition mapping

B(G,K),

defined by (x,y) --> z = xoy,

Borel o-algebras,

In order to state the invariance of G on B(G,K),

is measurable

allowing us to define

the image ~ v

of the measure ~ v

it is sufficient,

with respect

to

to the

on B(G,K). under

as we have previously

the action V

noted,

to prove

the equality ~ v ( ( V g)-l(A))

when the Borel

A

We can write

=

set A is a cylinder

=

the following

equalities:

n

(vg)-l({z,z(gi)=ki})) 1

set

{z(gl)=kl}~... N{Z(gn)=kn}

n

~v(~

~v(A)

= ~v(~({z,z(gig)z(g)-l=ki})) 1 n = ~ ~v(({z,z(g)=k})~A({z,z(gig)=kik})) keK 1

128

Using the definition of the measure ~ v with the tensor product ,®v, the above number is equal to the following sum: n

~ ~ {~(({x,x(g)=h}) keK heH (hi)eH n

O N({x,x(gig)=hih})). 1 n (({y,y(h)=k}) ~ ~({y,y(hih)=kik}))} 1

Employing the actions T and U, we see that this sum is equal to n

~ ~ {~((Tg)-l(({x,x(g-l)=h-l}) keK heH (hi)eH n

0 ~({x,x(gi)=hi}))). I n ~ ~({y,y(hi)=ki})))} 1

((U h)-l(({y,y(h-l)=k-1}) Next u s i n g t h e i n v a r i a n c e we find that the number

of u u n d e r T and the i n v a r i a n c e

o f v u n d e r U,

n ~ v ( ~ ( v g ) -I ( { z , z ( g i ) = k i } ) ) 1 is equal to n

k~eK h~eH (hileH n {~(({x,x(g-l)=h-l}) A n({x,x(gi)=hi})).l n v (({y,y(h-l)=k -I}) ~ ~({y,y(hi)=k i}))} I The following equality then results from the o-additivity product: n

~ v ((vg) -I (N({ z ,z (gi) =ki} ) ) I

of the tensor

n

=

~ v (N({ z,z(gi)=ki} ) ) I

and the "convolution product" is then invariant under the action V of G on B(G,K). 8.2.7. Proposition. All countable amenable groups are equivalent sense of definition 8.2.5.

in the

Proof. Since the involved relation is an equivalence one, we need only state the equivalence of any amenable group with a fixed one, Z. The measurable space B(G,Z) can be imbedded in the compact space of all total orders on G by identifying an element x of B(G,Z) with the total order on G, which is the image by x of the natural order of Z.

129

This embedding

is clearly a one-to-one mapping

space T O of T. In addition, two measurable B(G,Z)

this mapping

from B(G,Z)

is an isomorphism between

spaces and gives a conjugacy between

and the action of G by right translations

The existence results

of an invariant

from the existence

onto the subthe

the action of G on

on T O .

Borel probability measure

on B(G,Z)

then

of a similar measure on TO, which we proved

in section 8.1. 8.2.8.

Proposition.

an invariant

Given two equivalent

countable groups G and H, and

Borel probability measure ~ on B(H,G),

for every element m of the cone E(G),

we can construct,

a real function m

on the set F(H)

of all finite parts of H by setting m (A)

f

=

m(x(A))

d~(x)

B(H,G) This function m Proof.

belongs

to the cone Z(H)

and we get q(m ) ~ q(m).

These results are easily obtained by integration,

invariance 8.2.9.

follows

from the invariance

Proposition.

Given two equivalent

countable

groups G and H, and

an element m of the cone E (G), denote by i e the upper function on the compact

set T(G)

and the right

of ~.

semi-continuous

of all total orders on G defined

example

3.1.13;

denote by Je the upper

semi-continuous

defined

in the same way from the invariant

in

function on T(H)

capacity m , built

from m

with the invariant measure ~ on B(H,G). Let ~H be a Radon probability measure

on T(H) which is invariant under

the action of H by right translations. Consider

the measurable mapping

where x(T)

(x,r) --> x(r)

from B(H,G)xT(H)

to T(G)

is defined by (a,b)e~

This mapping

-~ 3

(x(a),x(b))ex(T)

sends the tensor product ~®~H to a probability measure ~G

on T(G). Then, Proof.

the integral

of Je for ~H is equal

The following

ie(~) fT(G)

equality results

d~G(T)

=

f

to the integral

of i e for ~G.

from the very definition of ~G:

i (x(T)) B (H,G)xT (H) e

d~ (x) d~H(~)

130

The integral

for p of ie(X(~))

is equal

to je(~)

and the result is

obtained. 8.2.10.

Corollary.

A countable

one is also amenable. amenable

group,

The mean value of an invariant

and the mean value of the invariant

it on another equivalent Indeed,

group which is equivalent

group,

if H is an amenable

to an amenable

capacity

on an

capacity built from

are equal.

group and,

if G is equivalent

=

=

to H, we get,

for every element m of t(G),

q(m)