Communications In Mathematical Physics - Volume 281

Commun. Math. Phys. 281, 1–21 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0457-x Communications in Mathe...

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Commun. Math. Phys. 281, 1–21 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0457-x

Communications in

Mathematical Physics

How Smooth is Your Wavelet? Wavelet Regularity via Thermodynamic Formalism Mark Pollicott1 , Howard Weiss2, 1 Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom 2 School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30334, USA.

E-mail: [email protected] Received: 26 April 2006 / Accepted: 3 October 2007 Published online: 6 May 2008 – © Springer-Verlag 2008

Abstract: A popular wavelet reference [W] states that “in theoretical and practical studies, the notion of (wavelet) regularity has been increasing in importance.” Not surprisingly, the study of wavelet regularity is currently a major topic of investigation. Smoother wavelets provide sharper frequency resolution of functions. Also, the iterative algorithms to construct wavelets converge faster for smoother wavelets. The main goals of this paper are to extend, refine, and unify the thermodynamic approach to the regularity of wavelets and to devise a faster algorithm for estimating regularity. The thermodynamic approach works equally well for compactly supported and non-compactly supported wavelets, and also applies to non-analytic wavelet filters. We present an algorithm for computing the Sobolev regularity of wavelets and prove that it converges with super-exponential speed. As an application we construct new examples of wavelets that are smoother than the Daubechies wavelets and have the same support. We establish smooth dependence of the regularity for wavelet families, and we derive a variational formula for the regularity. We also show a general relation between the asymptotic regularity of wavelet families and maximal measures for the doubling map. Finally, we describe how these results generalize to higher dimensional wavelets. 0. Introduction While the Fourier transform is useful for analyzing stationary functions, it is much less useful for analyzing non-stationary cases, where the frequency content evolves over time. In many applications one needs to estimate the frequency content of a nonstationary function locally in time, for example, to determine when a transient event occurred. This might arise from a sudden computer fan failure or from a pop on a music compact disk. The usual Fourier transform does not provide simultaneous time and frequency localization of a function. The work of the second author was partially supported by a National Science Foundation grant DMS0355180.

2

M. Pollicott, H. Weiss

The windowed or short-time Fourier transform does provide simultaneous time and frequency localization. However, since it uses a fixed time window width, the same window width is used over the entire frequency domain. In applications, a fixed window width is frequently unnecessarily large for a signal having strong high frequency components and unnecessarily small for a signal having strong low frequency components. In contrast, the wavelet transform provides a decomposition of a function into components from different scales whose degree of localization is connected to the size of the scale window. This is achieved by integer translations and dyadic dilations of a single function: the wavelet. The special class of orthogonal wavelets are those for which the translations and dilations of a fixed function, say, ψ form an orthonormal basis of L 2 (R). Examples include the Haar wavelet, the Shannon wavelet, Meyer wavelet, Battle-Lemarié wavelets, Daubechies wavelets, and Coiffman wavelets. A standard way to construct an orthogonal wavelet is to solve the dilation equation ∞ √ φ(x) = 2 ck φ(2x − k),

(0.1)

k=−∞

with the normalization k ck = 1. Very few solutions of the dilation equation for wavelets are known to have closed form expressions. This is one reason why it is difficult to determine the regularity of wavelets. 1 Provided the solution φ satisfies some additional conditions, the function ψ defined by ψ(x) =

∞

(−1)k c1−k φ(2x − k),

n=−∞

is an orthogonal wavelet, i.e., the set {2 j/2 ψ(2 j x − n) : j, n ∈ Z} is an orthonormal basis for L 2 (R). Henceforth, the term wavelet will mean orthogonal wavelet. Daubechies [Da1] made the breakthrough of constructing smoother compactly supported wavelets. For these wavelets, the wavelet coefficients {ck } satisfy simple recursion relations, and thus can be quickly computed. However, the Daubechies wavelets necessarily possess only finite smoothness.2 Smoother wavelets provide sharper frequency resolution of functions. Also, the regularity of the wavelet determines the speed of convergence of the cascade algorithm [Da1], which is another standard method to construct the wavelet from Eq. (0.1). Thus knowing the smoothness or regularity of wavelets has important practical and theoretical consequences. The main goals of this paper are to extend, refine, and unify the thermodynamic approach to the regularity of wavelets and to devise a faster algorithm for estimating regularity. In Sect. 1 we recall the construction of wavelets using multiresolution analysis. We then discuss how the L p -Sobolev regularity of wavelets is related to the thermodynamic pressure of the doubling map E 2 : [0, 2π ) → [0, 2π ). The latter is the crucial link between wavelets and dynamical systems. Previous authors have noted that for compactly supported wavelets, the transfer operator preserves a finite dimensional subspace and is thus represented by a d × d matrix. Providing the matrix is small, its maximal eigenvalue, which is related to the pressure, 1 We learned in [Da2] (see references) that this dilation equation arises in other areas, including subdivision schemes for computer aided design, where the goal is the fast generation of smooth curves and surface. 2 Frequently, increasing the support of the function leads to increased smoothness.

Wavelet Regularity via Thermodynamic Formalism

3

can be explicitly computed, and thus the L p -Sobolev regularity can be determined by matrix algebra. Cohen and Daubechies identified the L p -Sobolev regularity in terms of the smallest zero of the Fredholm-Ruelle determinant for the transfer operator acting on a certain Hilbert space of analytic functions [CD1,Da2]. Their analysis leads to an algorithm for the Sobolev regularity of wavelets having analytic filters that converges with exponential speed. In Sect. 2, we study the transfer operator acting on the Bergman space of analytic functions and effect a more refined analysis. This leads to a more efficient algorithm which converges with super-exponential speed to the L p -Sobolev regularity of the wavelet. The algorithm involves computing certain orbital averages over the periodic points of the doubling map E 2 . In Sect. 3 we illustrate, in a number of examples, the advantage of this method over the earlier Cohen-Daubechies method. Running on a fast PC, this algorithm provides a highly accurate approximation for the Sobolev regularity in less than two minutes. A feature of this thermodynamic approach is that it works equally well for compactly supported and non-compactly supported wavelets (e.g., Butterworth filters). In practice, for wavelets with large support, the distinction may become immaterial, and the use of our algorithm which applies to arbitrary wavelets may prove quite useful. In Sect. 4, we use this improved algorithm to study the parameter dependence of the L p -Sobolev regularity in several smooth families of wavelets. In particular, we construct new examples of wavelets having the same support as Daubechies wavelets, but with higher regularity. In Sect. 5, we show that the L p -Sobolev regularity depends smoothly on the wavelet, and we provide, via thermodynamic formalism, a variational formula. This problem has been considered experimentally by Daubechies [Da1] and Ojanen [Oj1] for different parameterized families. Our thermodynamic approach, based on properties of pressure, gives a sound theoretical foundation for this analysis. In Sect. 6, we study the asymptotic behavior of the L p -Sobolev regularity of wavelet families. We establish a general relation between this asymptotic regularity and maximal measures for the doubling map, extending work of Cohen and Conze [CC1]. The study of maximal measures is currently an active research area in dynamical systems. In Sect. 7, we describe the natural generalization of these results to arbitrary dimensions. Finally, in Sect. 8, we briefly describe the natural generalization of these results to wavelets whose filters are not analytic. 1. Wavelets: Construction and Regularity Constructing wavelets. We begin by recalling the construction of wavelets via multiresolution analysis. A comprehensive reference for wavelets is [Da1], which also contains an extensive bibliography. The Fourier transform of the scaling equation (0.1) satisfies 1 (ξ ) = m(ξ/2)φ (ξ/2), where m(ξ ) = √ φ ck e−ikξ . (1.1) 2 k The function m is sometimes called the filter defined by the scaling equation. Iterating this expression, and assuming that m(0) = 1, one obtains that (ξ ) = φ

∞ j=1

m(2− j ξ ).

(1.2)

4


Conditions (1)–(3) below allow one to invert this Fourier transform and obtain the scaling function φ. The wavelet ψ is then explicitly given by ψ(t) =

√

2

∞

dk φ(2t − k), where

dk = (−1)k−1 c−1−k ,

(1.3)

k=−∞

and thus the family

ψ j,k = 2 j/2 ψ(2 j t − k)

j,k∈Z

L 2 (R).3

forms an orthonormal basis of A major result in the theory is the following. Theorem 1.1. [Da1,p.186] Consider an analytic 2π -periodic function m with Fourier expansion m(ξ ) = n cn einξ satisfying the following conditions: (1) m(0) = 1; (2) |m(ξ )|2 + |m(ξ + π )|2 = 1; and (3) |m(ξ )|2 has no zeros in the interval [− π3 , π3 ]. defined in (1.2) is the Fourier transform of a function φ ∈ L 2 (R) Then the function φ and the function ψ defined by (1.3) defines a wavelet. (0) = 0, which is necessary to ensure the completeCondition (1) is equivalent to φ ness in the multiresolution defined by φ. Condition (1) is also necessary for the infinite product (1.2) to converge. Condition (2) is necessary for the wavelet to be orthogonal, and condition (3), called the “Cohen condition”, is a sufficient condition for the wavelet to be orthogonal [Da1, Cor. 6.3.2]. If one wishes to construct compactly supported wavelets, then one needs the additional assumption: (4) There exists N ≥ 1 with cn = 0 for |n| ≥ N . If ck = 0 for k ∈ [N1 , N2 ], then the support of φ is also contained in [N1 , N2 ], and the support of ψ is contained in [(N1 − N2 + 1)/2, (N2 − N1 + 1)/2]. It is frequently useful to work with the function q(ξ ) = |m(ξ )|2 . This function is positive, analytic, and even. Condition (1) is equivalent to q(0) = 1 and (2) is equivalent to q(ξ ) +q(ξ + π ) = 1. A simple exercise shows that (1) is equivalent to k∈Z ck = 1/2 and (2) is equivalent to q(ξ ) = 1/2 + k∈Z c2k+1 cos((2k + 1)ξ ). Thus the set K of positive even analytic functions satisfying both (1) and (2) is a convex subset of the positive analytic functions. Example (Daubechies wavelet family). This family of continuous and compactly supported wavelets φ Daub {N ∈ N}, is obtained by choosing the filter m such that N |m(ξ )|2 = cos2N (ξ/2)PN (sin2 (ξ/2)), where PN (y) =

N −1

N −1+k k

yk .

k= 0

The wavelets in this family have no known closed form expression. Note that |m(ξ )|2 vanishes to order 2N at ξ = π . More generally, a trigonometric ξ 2 2 2N polynomial m with |m(ξ )| = cos 2 u(ξ ) satisfies (2) only if u(ξ ) = Q(sin (ξ/2)), where Q(y) = PN (y) + y N R(1/2 − y), and R is an odd polynomial, chosen such that Q(y) ≥ 0 for 0 ≤ y ≤ 1 [Da1, 171]. 3 Engineers often call the family {c , d } a quadratic mirror filter (QMF). k k


5

Wavelet regularity. Since the Fourier transform is such a useful tool for constructing wavelets, and since wavelets (and other solutions of the dilation equation) rarely have closed form expressions, it seems useful to measure the regularity of wavelets using the Fourier transform. For p ≥ 1, the L p -Sobolov regularity of φ is defined by f (ξ )| p ∈ L 1 (R) . s p ( f ) = sup s : (1 + |ξ | p )s | The most commonly studied case is p = 2, where if s2 ( f ) > 1/2, then f ∈ C s2 −1/2− . One can also easily see that s p ( f ) − sq ( f ) ≤

1 1 − , for 0 < p < q. p q

Thermodynamic approach to wavelet regularity. We now discuss a thermodynamic formalism approach to wavelet regularity. We assume that |m(ξ )|2 has a maximal zero of order M at ±π and write |m(ξ )|2 = cos2M (ξ/2)r (ξ ),

(1.4)

where r ∈ C ω ([0, 2π )) is an analytic function with no zeros at ±π . We further assume that r > 0, and thus log r ∈ C ω ([0, 2π )). Given a function g : [0, 2π ) → R and the doubling map E 2 : [0, 2π ) → [0, 2π ) defined by E 2 (x) = 2x(mod 2π ), the pressure of g is defined by P(g) = sup h(µ) +

2π

g dµ : µ a E 2 -invariant probability measure ,

0

where h(µ) is the measure theoretic entropy of µ. If g is Hölder continuous, then this supremum is realized by a unique E 2 −invariant measure µg , called the equilibrium state for g. The following theorem expresses the relationship between the pressure and the Sobolev regularity s p (φ), and provides the crucial link between wavelets and dynamical systems. Theorem 1.2. The L p -Sobolov regularity satisfies the expression s p (φ) = M −

P( p log r ) . p log 2

Proof. The proof for p = 2 is due to Cohen and Daubechies [CD1] and the general case is due to Eirola [Ei1] and Villemoes [Vi1]. See also [He1,He2,Sun]. In the literature this result is formulated in terms of the spectra radius of the transfer operator on continuous functions; but this is precisely the pressure (see Proposition 2.1). See Appendix I for a proof of the lower estimate in arbitrary dimensions.

6


2. Determinants and Transfer Operators Given h : [0, 2π ) → R we define the semi-norm |h|β = sup x= y

|h(x) − h(y)| , |x − y|β

and the Hölder norm ||h||β = |h|∞ + |h|β , where 0 < β < 1. Given g ∈ C β [0, 2π ), the transfer operator L g : C β ([0, 2π ) → C β ([0, 2π ) is the positive linear operator defined by x

x

x

x

k + exp g +π k +π . L g k(x) = exp g 2 2 2 2 The following properties of pressure are well known [Ru1,PP1]. Proposition 2.1. (1) The spectral radius of L g is exp(P(g)); (2) The pressure P(g) has an analytic dependence on g ∈ C β ([0, 2π )); and 2π (3) The derivative Dg P( f ) = 0 f dµg . The transfer operator L g C β ([0, 2π )) → C β ([0, 2π )) is not trace class on this class of functions. However, following Jenkinson and Pollicott [JP1,JP2], we consider the transfer operator acting on the Bergman space A2 ( ) of square-integrable analytic functions on an open unit disk ⊂ C containing [0, 2π ). In Appendix I we show that for analytic functions g, the transfer operator L g . A2 ( ) → A2 ( ) is trace class and L g has a welldefined Ruelle-Fredholm determinant. For small z this can be defined by det(I − z L g ) = n k exp(tr(log(I − z L g ))) [Si1]. We can write tr(log(I − z L g )) = ∞ k=1 (z /n)tr(L g ), and k each operator L g is the sum of weighted composition operators having sharp-trace (in the sense of Atiyah and Bott) tr(L kg ) =

exp(Sk g(x)) , 1 − 2−k k

E 2 x=x

Sk (x) = g(x) + g(E 2 x) + · · · + g(E 2k−1 x). The Ruelle-Fredholm determinant for L g can be expressed as ⎛ ⎞ ∞ n z exp(S g(x)) n ⎠, det(I − z L g ) = exp ⎝− n n 1 − 2−n n=1

E 2 x=x

(2.1)

and has the “usual determinant property” det(I − z L g ) = ∞ n=1 (1 − zλn ), where λn denotes the n th eigenvalue of L g . Note that this determinant is formally quite similar to the Ruelle zeta function, and has an additional factor of the form 1/(1 − 2−n ) in the inner sum. More generally, L g need not be trace class for expression (2.1) to be well defined for small z (see Sect. 8). For our applications to wavelet regularity, we consider g = p log r (see (1.1)). To compute the determinant, one needs to compute the orbital averages of the function g over all the periodic points of E 2 . Every root of unity of order 2n − 1 is a periodic point


7

for E 2 , and there are exactly 2n − 1 such roots of unity, hence there are exactly 2n − 1 periodic points of period n for E 2 . If one expands the exponential function in a Taylor series and applies Cauchy’s convolution formula for a power of a Taylor series, one obtains the following formula: d p (z) := det(I − z L plogr ) = 1 +

∞

bn z n ,

n=1

where bn =

n (−1)k k!

k= 0

and p

an =

1 1 − 2−n

p

n 1 +···+n k = n

p

an 1 · · · an k , n1 · · · nk

⎡ ⎤p n −1 n−1 2 jk 2 ⎣ ⎦ . r 2n − 1 k= 0

j=0

The next theorem summarizes some important properties of this determinant d p (z). Theorem 2.2. Assume log r ∈ C ω [0, 2π ). Then (1) The function d p (z) can be extended to an entire function of z and a real-analytic function of p; (2) The reciprocal of the exponential of pressure z p := exp(−P( p log r ))) is the smallest zero for d p (z); (3) The Taylor coefficients bn of d p (z) decay to zero at a super-exponential rate, i.e., 2 there exists 0 < A 3. We see that the parameter value a = 3 corresponding to the maximally regular wavelet lies on the boundary of an allowable parameter interval, and thus could not be detected using variational methods. a -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 2.7 2.8 2.9 3.0

Zero of determinant 0.230 0.239 0.250 0.262 0.276 0.294 0.317 0.350 0.367 0.377 0.389 0.403

Sobolev regularity 0.939 0.967 1.000 1.030 1.071 1.116 1.710 1.242 1.276 1.296 1.318 1.344

(b) Daubechies [Da3] also considered the one-parameter family of functions m(ξ ) =

1 + eiξ 2

2 Q(ξ ),

where |Q(ξ )|2 = P(cos ξ ) and P(x) = 2 − x + a4 (1 − x)2 , with the parameter a chosen such that P(x) > 0 for 0 ≤ x < 1. The value a = 3 corresponds to ψ2Daub , and the associated polynomial has a zero at x = −1. She reparameterizes this wavelet family using the zero at x = −1+δ, and obtains, numerically, a wavelet having the same support as ψ2Daub but with higher regularity. 5. Perturbation Theory and Variational Formulae In this section we observe that the analyticity of pressure implies that the L p Sobolev regularity depends smoothly on the wavelet. It thus seems natural to search for critical points (maximally smooth wavelets) in parametrized families of wavelets by explicitly calculating the derivative of the Sobolev regularity functional. We caution that the example in the previous section shows that maximally regular wavelets can lie on the boundary of allowable parameter intervals. We now restrict |m(ξ )|2 to the subset of analytic functions B satisfying conditions (1) − (3) and let r0 be a tangent vector to this space. It follows from remarks in Sect. 1 that r0 is represented by an analytic function of the form ∞

1 r0 = + d2n+1 cos((2n + 1)ξ ), 2 n= 0

We have the following result.

such that

∞ n= 0

dn =

1 . 2

14


Proposition 5.1. (1) For every p ≥ 1, the regularity mapping K → R defined by |m(ξ )|2 → s p (φ) is analytic; (2) Let |m λ (ξ )|2 = cos2M (ξ/2) (r0 (ξ ) + λ r0 (ξ ) + · · · ) be a smooth parametrized family of filters. Then the change in the Sobolev regularity is given by 1 r0 λ 0 s p (φ ) = s p (φ ) + dµ λ + · · · , p log 2 r0 where µ is the unique equilibrium state for p log r0 . Proof. The analyticity of the Sobolev regularity follows from Proposition 2.1(2), the λ 0 analyticity of the pressure, and the implicit function theorem. Since log r = log(r + r0 0 r0 + · · · ) = log r + r0 λ + · · · and Dg P(·) = ·dµg , this follows from the chain rule. If we restrict to finite dimensional spaces we can consider formulae for the second (and higher) derivatives of the pressure, which translate into formulae for the next term in the expansion of s p (φλ ). Also observe that we can estimate the derivatives of the Sobolev exponent of a family s p (φλ ) using the series for d p (φλ , z). More precisely, by the implicit function theorem ds p (φ λ ) d(det(φ λ , z)) = dλ dλ

d(det(φ λ , z)) , dz

which is possible to estimate for periodic orbit data. We can similarly compute the higher derivatives ! d k s p (φ λ ) !! , for k ≥ 1 ! dλk ! λ = λ0

using Mathematica, say. We can then expand the power series for the Sobolev function: ! ! ds p (φ λ ) !! (λ − λ0 )2 d 2 s p (φ λ ) !! λ λ0 s p (φ ) = s p (φ ) + (λ − λ0 ) + + ··· . dλ !λ = λ0 2 dλ2 !λ = λ0 6. Asymptotic Sobolev Regularity The use of the variational principle to characterize pressure (and thus the Sobolev regularity) leads one naturally to the idea of maximizing measures. More precisely, one can associate to g: 2π

Q(g) = sup g dµ, µ an E 2 -invariant Borel probability measure . 0

Unlike pressure, which has analytic dependence on the function, the quantity Q(g) is less regular, but we have the following trivial bound.


15

Lemma 6.1. For any continuous function g : [0, 2π ) → R, Q(g) ≤ P(g) ≤ Q(g) + log 2. Proof. Since 0 ≤ h(µ) ≤ log 2, this follows from the definitions.

Proposition 6.2. Let φn , n ∈ N be a family of wavelets with filters |m(ξ )|2 = cos2Mn (ξ/2)rn (ξ ), where Mn ≥ 0. Then the asymptotic L p -Sobolev regularity of the family is given by s p (φn ) Mn Q(log rn ) = lim − lim . n→∞ n→∞ n n→∞ n log 2 n lim

Proof. Using Theorem 1.2, Lemma 6.1, and (1.1), one can easily show Q( p log rn ) ≤ p log 2(Mn − s p (φn )) ≤ Q( p log rn ) + log 2. The proposition immediately follows after dividing by n and taking the limit as n → ∞. We now apply Proposition 6.2 to compute the L p -Sobolev regularity of the Daubechies family of wavelets for arbitrary p ≥ 2. Cohen and Conze [CC1] first proved this result for p = 2. Proposition 6.3. Let p ≥ 2. The asymptotic L p -Sobolev regularity of the Daubechies family is given by s p (ψ NDaub ) log 3 =1− . N →∞ N 2 log 2 lim

Proof. If one applies Proposition 6.2 to the Daubechies filter coefficients (see Sect. I), one immediately obtains " # Q(log PN (sin2 ( 2y ))) s p (φ Daub ) N lim = 1 − lim . N →∞ N →∞ N N log 2 Cohen and Conze [CC1, p. 362] show that y log PN (3/4) , Q log2 PN (sin2 ( )) = 2 2 log 2 and the sup is attained for the period two periodic orbit {1/3, 2/3}. Thus s p (φ Daub ) 1 log PN (3/4) N =1− lim . N →∞ N 2 log 2 N →∞ N −1 3 N −1 The exponentially dominant term in PN (3/4) is easily seen to be 2N , and N −1 ( 4 ) a routine application of Stirling’s formula yields lim

log PN (3/4) = log 3. N →∞ N lim

The proposition immediately follows.

16


Remark. As briefly discussed in Sect. 3.2, the Butterworth filter family ψ NButt , N ∈ N is the family of non-compactly supported wavelets defined by the filter |m(ξ )|2 =

cos2N (ξ/2) sin2N (ξ/2) + cos2N (ξ/2)

.

Fan and Sun [FS1] computed the asymptotic regularity for this wavelet family; they showed that s p (ψ NButt ) log 3 = . N →+∞ N log 2 lim

This asymptotic linear growth of regularity is similar to that for the Daubechies wavelet family. 7. Regularity of Wavelets in Higher Dimensions The dilation equation (0.1) has a natural generalization to d-dimensions of the form: φ(x) = det(D) ck φ(Dx − k), (7.1) k∈Zd

where D ∈ G L(d, Z) is a d × d matrix with all eigenvalues having modulus strictly greater than 1. We are interested in solutions φ ∈ L 2 (Rd ). We consider the expanding map T : [0, 2π )d → [0, 2π )d by T (x) = Dx (mod 1). By taking d-dimensional Fourier transformations, (7.1) gives rise to the equation (ξ ) = m(D −1 ξ )φ (D −1 ξ ), where m(ξ ) = φ ck e−i k,ξ . k∈Zd

−n (ξ ) = ∞ Thus by iterating this identity we can write φ n=1 m(D ξ ). β d For g ∈ C ([0, 2π ) ), one can define a transfer operator L g : C β ([0, 2π )d ) → β C ([0, 2π )d ) by

L g k(x) = exp g(y) k(y), T y=x

where the sum is over the det(D) preimages of x. We consider the Banach space of periodic analytic functions A in a uniform neighbourhood of the torus [0, π )d and the associated Bergman space of functions. We have that for g ∈ A the operator L g : A → A and its iterates L kg are trace class operators for each k ≥ 1. We can compute tr(L kg ) =

|det(D)|n exp(Sn g(x)) , |det(D n − I )| n

T x =x

where Sn g(x) = g(x) + g(T x) + · · · + g(T n−1 x).


17

M a(Dx) Assume that we can write m(x) = |det(D)|a(x) r (x), where r (x) > 0.8 The Ruelle-Fredholm determinant of L p log r takes the form: ⎛ ⎞ ∞ n n exp( pS log r (x)) z |det(D)| n ⎠. det(I − L p log r ) = exp ⎝− n n |det(D n − I )| n=1

T x =x

The statement of Theorem 1.2 has a partial generalization to d-dimensions. We define the L p -Sobolev regularity of p to be (ξ )| p (1 + ||ξ || p )s ∈ Ł1 (Rd )}. s p (φ) := sup{s > 0 : |φ Proposition 7.1. The L p -Sobolev regularity satisfies the inequality s p (φ) ≥

P( p log r ) Mdet(D) − , log λ p log λ

where λ > 1 is smallest modulus of an eigenvalue of D. n (ξ )| ≤ C|det(D)| Mn i=0 Proof. We can write |φ r (D −i ξ ), for some C > 0 and all n n d ξ ∈ D ([0, 2π ) ). However, we can identify T n D i=0 r (D −i ξ ) p dξ = D L np log r 1(ξ )dξ . Moreover, since L p log r has maximal eigenvalue e P( p log r ) , we can bound this by C en P( p log r ) , for some C > 0. In particular, ∞ p p s psn (ξ )| p dξ |φ (ξ )| (1 + ||ξ || ) dξ ≤ λ |φ Rd

n=1

≤ CC

T n D−T n−1 D

∞

λ psn |det(D)| M pn en P( p log r ) .

n=1

In particular, the right-hand side converges if λ ps |det(D)| M p e P( p log r ) < 1, from which the result follows. In certain cases, for example if D has eigenvalues of the same modulus and some other technical assumptions, then this inequality becomes an equality [CD1]. As in Sect. 2, a routine calculation gives that d p (z) := det(I − z L p log r ) = 1 + bn =

n (−1)k k=0

k!

n=1 p p an 1 · · · an k

n 1 +···+n k =n

n1 · · · nk

and p an

=

|det(D)|n T n x=x

∞

n

T n x=x |det(D n −

bn z n , ,

i=0 r (x)

I )|

where

p .

The simplest case is that of diagonal matrices, but this reduces easily to tensor products (x1 , . . . , xd ) = φ1 (x1 ) · · · φd (xd ) of the one dimensional case. The next easiest example illustrates the “non-separable” case: 8 In particular, if T n x = x we have that n−1 |m(T i x)|2 = 2−n n−1 g(T i x). i =0 i =0

18


Example. For d = 2 we let D =

1

1 1 −1

√ , which has eigenvalues λ = ± 2.

In the d-dimensional setting we have the following version of Theorem 2.2: Theorem 7.2. Assume that log r ∈ A. Then (1) The function d p (z) can be extended to an entire function of z and a real analytic function of p; (2) The exponential of pressure z p := exp(−P( p log r )) is the smallest zero for d p (z); (3) The Taylor coefficients bn of d p (z) decay to zero at a super-exponential rate, i.e., 1+1/d there exists 0 < A 0 such that |cn | = O(|n|−β ). In this case we can only expect that the determinant algorithm for the Sobolevregularity converges with exponential n speed. Recall d p (z) = det(I − z L plogr ) = 1 + ∞ n=1 bn z . Theorem 8.1.

(1) Let 0 < β < 1. If |cn | = O |n|1β , then bn = O(2−nβ(1−) ) for any > 0;

(2) Let k ≥ 1. If |cn | = O |n|1 k , then bn = O(2−nk(1−) ) for any > 0. This follows from a standard analysis of transfer operators, and in particular, studying their essential spectral radii. The radius of convergence of d p (z) is the reciprocal of the essential spectral radius of the operators. In the case that 0 < β < 1, this theorem follows from the Hölder theory of transfer operators [PP1]. But for k > 1 this requires a slightly different analysis [Ta1]. The necessary results are summerized in the following proposition. Proposition 8.2. (1) Assume cn = O(|n|−β ). The transfer operator L f has essential spectral radius (1/2)α e P( f ) (and the function d(z) is analytic in a disk |z| < 2β ). (2) Assume cn = O(|n|−k ). The transfer operator L f has essential spectral radius (1/2)k e P( f ) (and the function d(z) is analytic in a disk |z| < 2k ). The first part is proved in [Po1,PP2]. The second part is proved in [Ta1,Ru2]9 . As in Sect. 2, the speed of decay of the coefficients bn immediately translate into the corresponding speed of convergence for the main algorithm. 9 At least when β ∈ N.


19

Appendix I: Proof of Theorem 2.1 Given a bounded linear operator L : H → H on a Hilbert space H , its i th approximation number (or singular value) 10 si (L) is defined as si (L) = inf{||L − K || : rank(K ) ≤ i − 1}, where K is a bounded linear operator on H . Let r ⊂ C denote the open disk of radius r centered at the origin in the complex plane. The phase space [0, 2π ) for T = E 2 is contained in 2π + for any > 0, and the two inverse branches T1 (x) = 21 x and T2 (x) = 21 x + 21 have analytic extensions to 2π + satisfying T1 ( 2π + ) ∪ T2 ( 2π + ) ⊂ 2π + 2 . Thus T1 and T2 are strict contractions of 2π + onto 2π + 2 with contraction ratio θ = (2π + )/(2π + 2) < 1. We can choose > 0 arbitrarily large (and thus θ arbitrarily close to 1/2). Let A2 ( r ) denote the Bergman Hilbert space of analytic functions on r with inner product f, g := f (z) g(z) d x d y [Ha1]. Lemma A.1. The approximation numbers of the transfer operator L p log r : A2 ( 2π ) → A2 ( 2π ) satisfy s j (L p log r ) ≤

||L p log r || A2 ( 2π ) j θ, 1−θ

for all j ≥ 1. This implies that the operator L p log r is of trace class. k Proof. Let g ∈ A2 ( 2π + ) and write L p log r g = ∞ k (z) = z . k=−∞ lk (g) pk , where p$ $ π π We can easily check that || pk || A2 ( 2π ) = k+1 (2π )k+1 and || pk || A2 ( 2π + ) = k+1 (2π + )k+1 . The functions { pk }k∈Z form a complete orthogonal family for A2 ( 2π + ), and so L p log r g, pk A2 ( 2π + ) = lk (g)|| pk ||2A2 ( 2π + ) . The Cauchy-Schwarz inequality implies that |lk (g)| ≤ ||L p log r g|| A2 ( 2π + ) || pk ||−1 A2 ( 2π + ) . j−1

( j)

We define the rank- j projection operator by L p log r (g) = g ∈ A2 ( θ ) we can estimate

k = 0 lk (g) pk .

∞

( j) || L p log r − L p log r (g)|| A2 ( 2π ) ≤ ||L p log r g|| A2 ( 2π ) θ k+1 . k= j

It follows that ( j)

||L p log r − L p log r || A2 ( 2π ) ≤

||L p log r || A2 ( 2π ) j+1 θ , 1−θ

and so s j+1 (L p log r ) ≤ and the result follows.

||L p log r || A2 ( 2π ) j+1 θ , 1−θ

10 We use these terms interchangeably because one can show that s (L) = k

% λk (L ∗ L).

For any

20


We now show that the coefficients of the power series of the determinant decay to zero with super-exponential speed. Lemma A.2. If one writes det(I − z L p log r ) = 1 + |cm | ≤ B where B =

∞

m=1 (1 − θ

m )−1

||L p log r || A2 ( 2π ) 1−θ

∞

m=1 cm z

m,

then

m θ m(m+1)/2 ,

< ∞.

Proof. By [Si1, Lemma 3.3], the coefficients cn in the power series expansion of the determinant have the form cm = i 1 0 such that A ≤ C B. The second order derivatives are ∂2 H = −α

(y1 ) (y3 )φ(ρ), ∂ y12 ∂2 H = −α (y1 ) (y3 )φ(ρ), ∂ y1 ∂ y3 ∂2 H y2 y4 = −α 2 (y1 ) (y3 ) φ

(ρ) − φ (ρ)ρ −1 , ∂ y2 ∂ y4 ρ 2 ∂ H = −αy j ρ −1 (y1 ) (y3 )φ (ρ), ∂ y1 ∂ y j % $ ∂2 H = −α(y1 ) (y3 ) φ

(ρ)y 2j ρ −2 + φ (ρ)ρ −1 − φ (ρ)y 2j ρ −3 , 2 ∂yj ∂2 H = −α(y1 ) (y3 )φ (ρ)y j ρ −1 , ∂ y3 ∂ y j ∂2 H = −α(y1 )

(y3 )φ(ρ), ∂ y32

(4.9)

Generic Dynamics of 4-Dim Hamiltonians

613

where j = 2, 4. So, D X H = 0 if y1 ≤ 0 or y1 ≥ , and D 2 (H − H0 )C 0 α(1 − ν)−2 .

(4.10)

Hence, there is α0 (1 − ν)2 such that H − H0 C 2 < for all 0 < α ≤ α0 .

Remark 4.3. It is not possible to find α as above if we require C 3 -closeness. This can 3 easily be seen in the proof by computing the third order derivatives. e.g. ∂∂y 3 H contains 2

the term α(y1 ) (y3 )y23 ρ −3 φ

(ρ) that can not be controlled by a bound of smaller order −1 than αr .

4.3. Realizing Hamiltonian systems. In this section we define the central objects for the proof of Proposition 3.1, the achievable or realizable linear flows. These will be constructed by perturbations of tH . We start with a point x ∈ O(H ) with lack of hyperbolic behavior and mix the directions Nx+ and Nx− to cause the decay of the upper Lyapunov exponent. In fact we are interested in “a lot” of points (related to the Lebesgue measure on transversal sections). Therefore, we perturb the Hamiltonian to make sure that “many” points y near x have tH (y) close to tH (x). For this reason we must be very careful in our procedure. Consider a Darboux atlas {h j : U j → R4 } j∈{1,...,} . For each x ∈ R(H ) choose j such that x ∈ U j , and take the 3-dimensional normal section Nx to the flow. In the sequel we abuse notation to write Nx for h j (Nx ∩ U j ), so √ that we work in R4 instead of M. Furthermore, denote by B(x, r ) = {(u, v, w) ∈ R3 : u 2 + v 2 < r, |w| < r } the open ball in Nx about x with small enough radius r . We estimate the distance between linear maps on tangent fibers at different base points by using the atlas and translating the objects to the origin in R4 . That is, At1 − At2 for linear flows Ait : Txi M → Tϕ t (xi ) M, H is given by Dh j1,t (ϕ tH (x1 ))At1 (Dh j1,0 (x1 ))−1 − Dh j2,t (ϕ tH (x2 ))At2 (Dh j2,0 (x2 ))−1 , where ji,t is the indice of the chart corresponding to ϕ tH (xi ). Consider the standard Poincaré map t (x) : U → Nϕ t PH

H (x)

,

where U ⊂ Nx is chosen sufficiently small. Given T > 0, the self-disjoint set & t ' (x) y ∈ M : y ∈ U, t ∈ [0, T ] , F HT (x, U ) = P H is called a T -length flowbox at x associated to the Hamiltonian H . There is a natural way to define a measure µ in the transversal sections by considering the invariant volume form ι X H ωd . We easily obtain an estimate on the time evolution of the measure of transversal sets: for ν, t > 0 there is r > 0 such that for any measurable A ⊂ B(x, r ) we have µ(A) − α(t) µ(P t (x) A) < ν, (4.11) H where α(t) =

X H (ϕ tH (x)) . X H (x)

614

M. Bessa, J. Lopes Dias

Definition 4.4. Fix a Hamiltonian H ∈ C s+1 (M, R), s ≥ 2 or s = ∞, T, > 0, 0 < κ < 1 and a non-periodic point x ∈ M (or with period larger than T ). The flow L of symplectic linear maps: L t (x) : Nx → Nϕ t

H (x)

, 0 ≤ t ≤ T,

is (, κ)-realizable of length T at x if the following holds: For γ > 0 there is r > 0 such that for any open set U ⊂ B(x, r ) ⊂ Nx we can find (1) K ⊂ U with µ(U \ K ) ≤ κ µ(U ), and ∈ C s (M, R) -C 2 -close to H , verifying (2) H outside F T (x, U ), (a) H = H H T (x) U , and (b) D X H (y) = D X H(y) for y ∈ U ∪ P H T T (c) H(y) − L (x) < γ for all y ∈ K . Let us add a few words about this definition: (2a) and (2b) guarantee that the support of the perturbation is restricted to the flowbox and it C 1 “glues” to its complement; (2c) says that a large percentage of points (given numerically by (1)) have the transversal (as in (2)) very close to the abstract linear action of the central linear Poincaré flow of H point x along the orbit. Notice that the realizability is with respect to the C 2 topology. Remark 4.5. Using Vitali covering arguments we may replace any open set U of Definition 4.4 by open balls. That turns out to be very useful because the basic perturbation Lemma 4.2 works for balls. It is an immediate consequence of the definition that the transversal linear Poincaré flow of H is itself a realizable linear flow. In addition, the concatenation of two realizable linear flows is still a realizable linear flow as it is shown in the following lemma. Lemma 4.6. Let H ∈ C s+1 (M, R), s ≥ 2 or s = ∞, and x ∈ M non-periodic. If L 1 is T1 (x) so (, κ1 )-realizable of length T1 at x and L 2 is (, κ2 )-realizable of length T2 at ϕ H that κ = κ1 + κ2 < 1, then the concatenated linear flow L t1 (x), 0 ≤ t ≤ T1 L t (x) = T1 T1 1 L t−T (ϕ (x)) L (x), T1 < t ≤ T1 + T2 2 1 H is (, κ)-realizable of length T1 + T2 at x. Remark 4.7. Notice that concatenation of realizable flows worsens κ. 1 , H 2 the obvious variables in the definition Proof. For γ > 0, take r1 , r2 , K 1 , K 2 , H for L satisfying the for L 1 and L 2 . We want to find the corresponding ones r, K , H T1 properties of realizable flows. Let x2 = ϕ H (x). • First, choose r ≤ r1 such that T1 U2 := P H (x) U ⊂ B(x2 , r2 )

with U = B(x, r ).


as • Now, we construct H

615

⎧ 1 on F T1 (x, U ) ⎪ ⎨H H = H 2 on F T2 (x2 , U2 ) H H ⎪ ⎩ H otherwise.

Notice that F HT1 +T2 (x, U ) = F HT1 (x, U ) ∪ F HT2 (x2 , U2 ). −T1 • Consider K = K 1 ∩ P H (x) (K 2 ∩ U2 ). Hence, −T1 µ(U \ K ) ≤ µ(U \ K 1 ) + µ(U \ P H (x) (K 2 ∩ U2 )) −T1 (x) (K 2 ∩ U2 )). ≤ (κ1 + 1) µ(U ) − µ(P H −T1 (x) (K 2 ∩ U2 ) we know that Now, by (4.11) applied to A = P H −T1 µ(P H (x) (K 2 ∩ U2 )) ≥ α(T1 ) µ(K 2 ∩ U2 ) = α(T1 ) [µ(U2 ) − µ(U2 \ K 2 )] ≥ α(T1 )(1 − κ2 ) µ(U2 ).

On the other hand, using (4.11) for A = U , µ(U2 ) ≥ α(T1 )−1 µ(U ). Combining all the above estimates we get µ(U \ K ) ≤ (κ1 + κ2 ) µ(U ). yields that D X H = D X on U because that is true for H 1 . The • The choice of H H T1 +T2 2 . same holds on P H (x) U related to H is C s it is enough to look at U2 . That follows from the same • In order to check that H reason as the previous item. T1 • Finally, there is C > 0 verifying for y ∈ K and writing y2 = P H (x) y, TH1 +T2 (y) − L T1 +T2 (x) ≤ TH2 (y2 )[ TH1 (y) − L T1 (x)] + [ TH2 (y2 ) − L T2 (x2 )]L T1 (x) 0 and 0 < κ < 1. Then, there exists α0 = α0 (H, , κ) > 0 such that for any non-periodic point x ∈ M (or with period larger than 1) and 0 < α ≤ α0 , the linear flow tH (x) ◦ Rα : Nx → Nϕ t (x) is H (, κ)-realizable of length 1 at x. Proof. Let γ > 0. We start by choosing r > 0 sufficiently small such that:

616


• B(x, r ) is inside the neighbourhood given by Lemma 4.1. Notice that by taking the transversal section = B(x, r ) in the proof of the lemma, such neighbourhood can be extended along its orbit to an open set A containing FH1 (x, r ), where FHτ (x, r ) = 0≤t≤τ ϕ tH (B(x, r )). So, a C s -symplectomorphism g : A → R4 exists satisfying: g(B(x, r )) is an orthogonal section to X H0 at g(x) = 0, H = H0 ◦ g, and all norms of the derivatives are bounded. Moreover, the derivatives of g and g −1 are of order r -close to the identity tangent map I on local coordinates; 1 (x, B(x, r )) is not self-intersecting; • FH 1 (x, B(x, r )) (recall that 0 < < 1 is a fixed constant introduced • FH (x, r ) ⊂ F H before Lemma 4.2). > 0 and 0 < κ < 1. Define gx = g − g(x ) and Let U = B(x , r ) ⊂ B(x, r ), ⊂ B(0, r2 ) and U = gx (U ). For r small we find r1 , r2 > 0 such that B(0, r1 ) ⊂ U r2 /r1 = [(1 − κ )−1/3 + 1]/2 > 1. Setting ν = [1 + (1 − κ )1/3 ]/2 < 1, Lemma 4.2 gives us that there is α0 = α0 (H, , κ ) > 0 such that for any 0 < α ≤ α0 and using the radius r1 we have that tH0 (0) ◦ Rα is ( , κ )-realizable of length-1 at the origin. Take and H such that K = B(0, r1 ν) with the obvious variables in the definition K ⊂ U ) = π(r1 ν)3 and H − H0 C 2 < ) ≥ (1 − ). µ( K . Then, µ( K κ )µ(U ) ⊂ U and H = H ◦ gx . If Define K = gx ( K and κ are small enough (depending on , κ and the norms of the derivatives of g), we get that Definition 4.4 (1) is satisfied and − H C 2 = ( H − H0 ) ◦ gx C 2 H < . We use the same notation as in the proof of Lemma 4.2. By construction it is simple to check that Definition 4.4 (2a) and (2b) also hold. As discussed before, Lemma 4.1 determines the existence of a neighborhood at each regular point of M and a C s -symplectomorphism straightening the flow. By compacity of M the derivatives of the symplectomorphism up to the order s are uniformly bounded on small length 1 flowboxes. For this reason α0 given above (depending on and κ ) was chosen to be independent of x ∈ M. It remains to check (c) in Definition 4.4. This will require further restrictions on r , depending on γ . By definition, the time-1 transversal linear Poincaré flow on N y ⊂ Ty H −1 (H (y)) is y) Dϕ 1H(g(y)) Dg(y)

1H(y) = ϕ 1 (y) Dg −1 ( H

for y ∈ K , and in x yields

1H (x) ◦ Rα = ϕ 1 (x) Dg −1 ( x ) Dg(x) Rα , H

where x = ϕ 1H0 ◦ g(x) = (1, 0, 0, 0) and y = ϕ 1H ◦ g(y) are of order r -close. Notice that ϕ 1 (y) − ϕ 1 (x) r and H

H

Dg −1 ( y) − Dg −1 ( x ) r. Therefore, 1H(y) − 1H (x) ◦ Rα r + ϒ, where

% $ x ) Dϕ 1H(g(y)) Dg(y) − Dg(x) Rα . ϒ = ϕ 1 (x) Dg −1 ( H


617

Moreover, Dg −1 ( x ) − I r . So, ϒ r + ϕ 1 (x) (Rα Dg(y) − Dg(x) Rα ) , H

where we have also used ϕ 1 (x) − π0 r and π0 Dϕ 1H(0, y2 , 0, y4 ) = Rα . Finally, H since Rα Dg(y) − Dg(x) Rα = Rα (Dg(y) − I ) + (I − Dg(x))Rα , we obtain the bound 1H(y) − 1H (x) ◦ Rα r < γ for r γ small enough.

Remark 4.9. A similar result holds true also for Rα ◦ tH (x) using essentially the same proof. 5. Proof of Proposition 3.1 We present here a sketch of how to complete the proof of Proposition 3.1; see [1] for full details. We would like to highlight the fact that our result does not hold for a C 2 has to be one degree of differentiability less. Hamiltonian H , since the perturbed one H The differentiability loss comes from the symplectomorphism obtained in Theorem 4.1 that rectifies the flow. 5.1. Local. The lemma below states that the absence of dominated splitting is sufficient to interchange the two directions of non-zero Lyapunov exponents along an orbit segment by the means of a realizable flow. Lemma 5.1. Let H ∈ C s+1 (M, R), s ≥ 2 or s = ∞, > 0 and 0 < κ < 1. There exists m ∈ N, such that for every x ∈ R(H ) ∩ O(H ) with a positive Lyapunov exponent and satisfying mH (x)|Nx− 1 ≥ , mH (x)|Nx+ 2 there exists a (, κ)-realizable linear flow L of length m at x such that L m (x) Nx+ = Nϕ−m (x) . H

Proof. The proof is the same as for Lemma 3.15 of [1] in which the constructions of Lemma 4.8 are used, namely the concatenation of rotated Poincaré linear maps. Now we aim at locally decaying the upper Lyapunov exponent. Lemma 5.2. Let H ∈ C s+1 (M, R), s ≥ 2 or s = ∞, and , δ > 0, 0 < κ < 1. There is T : m (H ) → R measurable, such that for µ-a.e. x ∈ m (H ) and t ≥ T (x), we can find a (, κ)-realizable linear flow L at x with length t satisfying 1 log L t (x) < δ. t

(5.1)

618


Proof. We follow Lemma 3.18 of [1]. Notice that for µ-a.e. x ∈ m (H ) with λ = λ+ (H, x) > 0 and due to the nice recurrence properties of the function T (see Lemma 3.12 of [2]) we obtain for every (very large) t ≥ T (x) that mH (y)|N y− mH (y)|N y+

≥

1 2

for y = ϕ sH (x) with s ≈ t/2. + Now, by Lemma 5.1 we obtain a (, κ)-realizable linear flow L t2 such that L m 2 Ny = − t t Nϕ m (y) . We consider also the realizable linear flows L 1 : Nx → N y and L 3 : Nϕ mH (y) → H

Nϕ t (x) given by tH for 0 ≤ t ≤ s and t ≥ m, respectively. Then we use Lemma 4.6 H and concatenate L 1 → L 2 → L 3 as L t , which is a (, κ)-realizable linear flow at x with length t. The choice of t m and the exchange of the directions will cause a decay on the norm of L t . Roughly that is: • in Nx+ the action of L 1 is approximately eλt/2 , • in Nϕ−m (y) the action of L 3 is approximately e−λt/2 , and H • L 2 exchange these two rates. Therefore, L t (x) < etδ .

5.2. Global. Notice that, in Lemma 5.2, we obtained L t (x) < etδ . However, we still need to get an upper estimate of the upper Lyapunov exponent. Due to (3.2) this can be done without taking limits, say in finite time computations. In other words, we will be using the inequality

1 + ˜ log tH˜ (x)dµ(x), λ ( H , x)dµ(x) ≤ (5.2) m (H ) m (H ) t which is true for all t ∈ R. Therefore, δ is larger than the upper Lyapunov exponent of at least most of the points near x. To prove Proposition 3.1 we turn Lemma 5.2 global. This is done by a recurrence argument based in the Kakutani towers techniques entirely described in [1], Sect. 3.6. In broad terms the construction goes as follows: • Take a very large m ∈ N from Lemma 5.1. Then Lemma 5.2 gives us a measurable function T : m (H ) → R depending on κ and δ. Let δ 2 = κ. • For x1 ∈ m (H ), the realizability of the flow L t (x1 ) guarantees that we have a 1 . t-length flowbox at x1 (a tower T1 ) associated to the perturbed Hamiltonian H If we take a point in the measurable set K 1 (cf. (1) of Definition 4.4) contained in the base of the tower, then by (2c) of Definition 4.4 and Lemma 5.2, we have tH (y) < e2δt for all y ∈ K 1 . 1 • Now, for x2 , . . . , x j ∈ m (H ), where j ∈ N is large enough, we define self-disjoint towers Ti , i = 1, . . . , j, which (almost) cover the set m (H ) in the measure theoretical sense. We take these towers such that their heights are approximately the same, say h. is defined by glueing together all perturbations H i , • The C s Hamiltonian H i = 1, . . . , j.


619

• Consider T = ∪i Ti , U = ∪i Ui and K = ∪i K i . Clearly K ⊂ U . Note that for points in U \ K we may not have tH (·) < e2δt . 1

• Denote by T K the subtowers of T with base K instead of U . By (1) of Definition 4.4 we obtain that µ(U \ K ) ≤ κµ(U ), hence µ(T \ T K ) < µ(T ) ≤ δ 2 .

We claim that it is sufficient to take t = hδ −1 in (5.2). It follows from (5.1) that we only control the iterates that enter the base of T K . Since the height of each tower is approximately h the orbits leave T K at most δ −1 times. For each of those times the chance of not re-entering again is less than δ 2 . So, the probability of leaving T K along t iterates is less than δ. In conclusion, most of the points in m (H ) satisfy the inequality (5.1) and Proposition 3.1 is proved. Acknowledgements. We would like to thank Gonzalo Contreras and the anonymous referee for useful comments. MB was supported by Fundação para a Ciência e a Tecnologia, SFRH/BPD/20890/2004. JLD was partially supported by Fundação para a Ciência e a Tecnologia through the Program FEDER/POCI 2010.

References 1. Bessa, M.: The Lyapunov exponents of generic zero divergence 3-dimensional vector fields. Erg. Theor. Dyn. Syst. 27, 1445–1472 (2007) 2. Bochi, J.: Genericity of zero Lyapunov exponents. Erg. Theor. Dyn. Syst. 22, 1667–1696 (2002) 3. Bochi, J., Fayad, B.: Dichotomies between uniform hyperbolicity and zero Lyapunov exponents for S L(2, R) cocycles. Bull. Braz. Math. Soc. 37, 307–349 (2006) 4. Bochi, J., Viana, M.: The Lyapunov exponents of generic volume preserving and symplectic maps. Ann. Math. 161, 1423–1485 (2005) 5. Bochi, J., Viana, M.: Lyapunov exponents: How frequently are dynamical systems hyperbolic? In: Advances in Dynamical Systems. Cambridge: Cambridge Univ. Press, 2004 6. Bonatti, C., Díaz, L., Viana, M.: Dynamics beyond uniform hyperbolicity. A global geometric and probabilistic perspective. Encycl. of Math. Sc. 102. Math. Phys. 3. Berlin: Springer-Verlag, 2005 7. Bowen, R.: Equilibrium states and ergodic theory of Anosov diffeomorphisms. Lect. Notes in Math. 470, Berlin-Heidelberg New York: Springer-Verlag, 1975 8. Doering, C.: Persistently transitive vector fields on three-dimensional manifolds. In: Proceedings on Dynamical Systems and Bifurcation Theory, Pitman Res. Notes in Math 160, 1987, pp. 59–89 9. Hunt, T., MacKay, R.S.: Anosov parameter values for the triple linkage and a physical system with a uniformly chaotic attractor. Nonlinearity 16, 1499–1510 (2003) 10. Mañé, R.: Oseledec’s theorem from generic viewpoint. In: Proceedings of the international Congress of Mathematicians, Warszawa, Vol. 2, Warszawa: PWN,1983, pp. 1259–1276 11. Mañé, R.: The Lyapunov exponents of generic area preserving diffeomorphisms. International Conference on Dynamical Systems (Montevideo, 1995), Res. Notes Math. Ser. 362, London: Pitman, pp. 1996, 110–119 12. Markus, L., Meyer, K.R.: Generic Hamiltonian Dynamical Systems are neither Integrable nor Ergodic. Memoirs AMS Providence, RI: Amer. Math. Soc. 144, 1974 13. Moser, J.: On the volume elements on a manifold. Trans. Amer. Math. Soc. 120, 286–294 (1965) 14. Newhouse, S.: Quasi-elliptic periodic points in conservative dynamical systems. Am. J. Math. 99, 1061–1087 (1977) 15. Oseledets, V.I.: A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19, 197–231 (1968) 16. Robinson, C.: Lectures on Hamiltonian Systems. Monograf. Mat. Rio de Janeiro IMPA, 1971 17. Vivier, T.: Robustly transitive 3-dimensional regular energy surfaces are Anosov. Preprint Dijon, 2005, available at http://math.u-bourgogne.fr/topolog/prepub/HamHyp11.pdf, 2005 Communicated by G. Gallavotti

Commun. Math. Phys. 281, 621–653 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0490-9

Communications in


Enumerative Geometry of Calabi-Yau 4-Folds A. Klemm1 , R. Pandharipande2 1 Department of Physics, Univ. of Wisconsin, Madison, WI 53706, USA.

E-mail: [email protected]

2 Department of Mathematics, Princeton University, Princeton, NJ 08544, USA.

E-mail: [email protected] Received: 11 May 2007 / Accepted: 23 November 2007 Published online: 30 May 2008 – © Springer-Verlag 2008

Abstract: Gromov-Witten theory is used to define an enumerative geometry of curves in Calabi-Yau 4-folds. The main technique is to find exact solutions to moving multiple cover integrals. The resulting invariants are analogous to the BPS counts of Gopakumar and Vafa for Calabi-Yau 3-folds. We conjecture the 4-fold invariants to be integers and expect a sheaf theoretic explanation. Several local Calabi-Yau 4-folds are solved exactly. Compact cases, including the sextic Calabi-Yau in P5 , are also studied. A complete solution of the Gromov-Witten theory of the sextic is conjecturally obtained by the holomorphic anomaly equation. 0. Introduction 0.1. Gromov-Witten theory. Let X be a nonsingular projective variety over C. Let M g,n (X, β) be the moduli space of genus g, n-pointed stable maps to X representing the class β ∈ H2 (X, Z). The Gromov-Witten theory of primary fields1 concerns the integrals n X (γ1 , . . . , γn ) = evi∗ (γn ), (1) N g,β [M g,n (X,β)]vir i=1

where evi : M g,n (X, β) → X is the ith evaluation map and γi ∈ H ∗ (X, Z). The notation X N g, = 1 β [M g (X,β)]vir

is used in case there are no insertions. 1 We consider only primary Gromov-Witten theory in the paper.

622

A. Klemm, R. Pandharipande

Since the moduli space M g,n (X, β) is a Deligne-Mumford stack, the Gromov-Witten invariants (1) are Q-valued. 0.2. Enumerative geometry. The relationship between Gromov-Witten theory and the enumerative geometry of curves in X is straightforward in three cases: (i) X is convex (in genus 0), (ii) X is a curve, (iii) X is P2 or P1 × P1 . For (i–iii), the Gromov-Witten theory with primary insertions equals the classical enumerative geometry of curves. A discussion of convex varieties (i) in genus 0 can be found in [11]. Examples (ii) and (iii) hold for all genera and recover the étale Hurwitz numbers and the classical Severi degrees respectively. Case (iii) certainly extends in some form to all rational surfaces viewed as generic blow-ups. The genus 0 case is treated in [16]. The first nontrivial cases occur for irrational surfaces. When X is a minimal surface of general type, Taubes’ results exactly determine the primary Gromov-Witten invariant for the adjunction genus in the canonical class, N gXX ,K X = (−1)χ (X,O X ) , see [36–39]. While much is known about surfaces of general type [23,28], surfaces in between are more mysterious. For example, many questions about the relationship of Gromov-Witten theory to the enumerative geometry of the K 3 and Enriques surfaces remain open [5,20,22,28]. The enumerative significance of Gromov-Witten theory in dimension 3 has been studied since the beginning of the subject. For Calabi-Yau 3-folds, essentially all GromovWitten invariants, even in genus 0, have large denominators. The Aspinwall-Morrison formula [1] was conjectured to produce integer invariants in genus 0. A full integrality conjecture for the Gromov-Witten theory of Calabi-Yau 3-folds in terms of BPS states was formulated by Gopakumar and Vafa [14,15]. Later, integral invariants for all 3-folds were conjectured in [32,33]. Various mathematical attempts to capture the BPS counts in terms of the cohomologies of associated moduli of sheaves on X were put forward without a definitive treatment. However, the integral invariants of [14,15,33] can be conjecturally interpreted in terms of the sheaf enumeration of Donaldson-Thomas theory [25,26,40]. Our main point here is to show the integrality of Gromov-Witten theory persists in higher dimensions as well. We speculate there exist universal transformations in every dimension which express Gromov-Witten theory in terms of Z-valued invariants. We conjecture the exact form of the transformation for Calabi-Yau 4-folds. A sheaf theoretic interpretation of the resulting invariants remains to be found.

0.3. Calabi-Yau 4-folds. Let X be a nonsingular, projective, Calabi-Yau 4-fold, and let β ∈ H2 (X, Z) be a curve class. Since vir dim M g (X, β) = c1 (X ) + (dim X − 3)(1 − g) = 1 − g, β

Gromov-Witten theory vanishes for g ≥ 2. We need only consider genus 0 and 1.

Enumerative Geometry of Calabi-Yau 4-Folds

623

We measure the degree of β with respect to a fixed ample polarization L on X , deg(β) = c1 (L). β

All effective curve classes2 satisfy deg(β) > 0. We abbreviate the latter condition by β > 0. We are only interested here in Gromov-Witten invariants for classes satisfying β > 0. Integrality in genus 0 is expressed by the following generalization of the AspinwallMorrison formula. Invariants n 0,β (γ1 , . . . , γn ) virtually enumerating rational curves of class β incident to cycles dual to the classes γi are uniquely defined by

N0,β (γ1 , . . . , γn )q β =

β>0

n 0,β (γ1 , . . . , γn )

β>0

∞

d −3+n q dβ .

(2)

d=1

A justification for the definition via multiple coverings is given in Sect. 1.1. Conjecture 0. The invariants n 0,β (γ1 , . . . , γn ) are integers. Let S1 , . . . , Ss be a basis of H 4 (X, Z) mod torsion. Let Si ∪ S j gi j = X

be the intersection form, and let g i j [Si ⊗ S j ] ∈ H 8 (X × X, Z) i, j

be the H 4 (X, Z) ⊗ H 4 (X, Z) part of the Künneth decomposition of the diagonal (mod torsion). For β1 , β2 ∈ H2 (X, Z), we define invariants m β1 ,β2 virtually enumerating rational curves of class β1 meeting rational curves of class β2 . The meeting invariants are uniquely determined by the following rules: (i) The invariants are symmetric, m β1 ,β2 = m β2 ,β1 . (ii) If either deg(β1 ) ≤ 0 or deg(β2 ) ≤ 0, then m β1 ,β2 = 0. (iii) If β1 = β2 , then, m β1 ,β2 = n 0,β1 (Si ) g i j n 0,β2 (S j ) + m β1 ,β2 −β1 + m β1 −β2 ,β2 . i, j 2 Integrality constraints for Gromov-Witten theory always exclude constant maps. The constant contributions are easily determined in terms of the classical cohomology of X . For D ∈ H 2 (X, Z), the genus 1 invariant 1 c3 (X ) ∪ D N1,0 (D) = − 24 X

has denominator bounded by 24.

624


(iv) In case of equality, m β,β = n 0,β (c2 (TX )) +

n 0,β (Si ) g i j n 0,β (S j ) −

m β1 ,β2 .

β1 +β2 =β

i, j

A geometric derivation of the rules (i-iv) is presented in Sect. 1.2. The conjectural integrality of the invariants n 0,β (γ ) implies the integrality of the meeting invariants m β1 ,β2 . In genus 1, we need only consider Gromov-Witten invariants N1,β of X with no insertions since the virtual dimension is 0. The invariants n 1,β virtually enumerating elliptic curves are uniquely defined by

N1,β q β =

β>0

n 1,β

β>0

+

∞ σ (d) d=1

d

q dβ

(3)

1 n 0,β (c2 (TX )) log(1 − q β ) 24 β>0

1 m β1 ,β2 log(1 − q β1 +β2 ). − 24 β1 ,β2

The function σ is defined by σ (d) =

i.

i|d

The number of automorphism-weighted, connected, degree d, étale covers of an elliptic curve is σ (d)/d. Conjecture 1. The invariants n 1,β are integers. The explicit form of (3) is derived from studying a particular solvable local Calabi-Yau 4-fold in Sect. 2. 0.4. Examples. The last four sections of the paper are devoted to the calculation of basic examples of Calabi-Yau 4-folds. The two local cases, OP2 (−1) ⊕ OP2 (−2) → P2 , OP1 ×P1 (−1, −1) ⊕ OP1 ×P1 (−1, −1) → P1 × P1 , are solved in closed form by virtual localization in Sect. 3. The local case OP3 (−4) → P3 and the compact Calabi-Yau 4-fold hypersurfaces X 6 ⊂ P5 , X 10 ⊂ P5 (1, 1, 2, 2, 2, 2),

X 2,5 ⊂ P1 × P4 ,

X 24 ⊂ P5 (1, 1, 1, 1, 8, 12) are solved by the conjectural holomorphic anomaly equation3 in Sects. 4–6. The compact cases are much more interesting than the local toric examples. In all calculations, the integralities of Conjectures 0 and 1 are verified. 3 While mathematical approaches to the genus 1 invariants in the compact case are available [12,24,27], the methods are much less effective than the anomaly equation.


625

0.5. Physical interpretation. Type IIA string compactifications on Calabi-Yau 4-folds give rise to massive theories with (2, 2) supergravity in 2 dimensions. Such theories and their BPS states were extensively studied in general [6,7] and in particular for type IIA 4-folds in [13,18]. The effective action, worked out in [13], contains an on Calabi-Yau d2 z R (2) term, and the topological string at genus 1 calculates a 1-loop correction to this term. The latter comes from the famous 1-loop term in 10 dimensional type IIA theory that was discovered in the context of heterotic type II duality in [41] and gives the following contribution to the 10 dimensional effective action: δS = − d 10 x B Y8 (R). (4) Here, B is the NS − NS 2-form of type IIA coupling to the string and Y8 (R) is an 8-form constructed as a quartic polynomial in the curvature. In 10 dimensions, the term can be directly calculated from the 1-loop amplitude with 4 gravitons and the antisymmetric B-field as external legs. If the latter is in the 2 non-compact dimensions, in the absence (X ) of further flux terms, the tadpole condition that − χ24 vanishes is obtained. The topo 2 (2) logical string computes the correction to the d z R term calculated from a loop with 1 external graviton, 3 internal gravitons, and the B-field.4 As in [14,15], the loop integral receives only contributions from BPS states. The behavior of the topological string amplitude in the large volume limit appears as the zero mass contribution and supports the claim that the amplitude computes the reduction of (4). BPS states with a D2-brane charge β contribute ∼ log(1−q β ) to the integral. The integer expansion (3) can be alternatively written as N1,β q β = − n˜ 1,β log(1 − q β ) β>0

β>0

1 n 0,β (c2 (TX )) log(1 − q β ) + 24 β>0

−

1 m β1 ,β2 log(1 − q β1 +β2 ). 24 β1 ,β2

The integrality condition for the invariants n˜ 1,β is equivalent to the conjectured integrality for n 1,β . We intepret the n˜ 1,β as counting BPS states. Futhermore, the structure of the m β1 ,β2 log(1 − q β1 +β2 ) term suggests that the invariants m β1 ,β2 count bound states at the threshold of BPS states with D2-brane charge β1 and β2 respectively. 0.6. Outlook. The meeting invariants make integrality in genus 1 for Calabi-Yau 4-folds considerably more subtle than the corresponding integrality for Calabi-Yau 3-folds. The integrality transformations in the higher dimensional Calabi-Yau cases should include all genus 0 meeting configurations. In the non-Calabi-Yau cases, higher genus meeting configurations should occur as well. Finding the correct coefficients for such a universal transformation is an interesting problem. 4 Work in progress.

626


1. Genus 0 1.1. Aspinwall-Morrison. Let π and ι denote the universal curve and map over the moduli space, π : C → M 0,0 (P1 , d), ι : C → P1 . The Aspinwall-Morrison formula is 1 ctop R 1 π∗ ι∗ (OP1 (−1) ⊕ OP1 (−1)) = 3 . 1 d M 0,0 (P ,d) By the divisor equation, we obtain n ctop R 1 π∗ f ∗ OP1 (−1) ⊕ OP1 (−1) ∪ evi∗ ([P]) = d −3+n , M 0,n (P1 ,d)

(5)

i=1

where [P] ∈ H 2 (P1 , Z) is the class of a point. Let X be a Calabi-Yau 4-fold, and let V1 , . . . , Vn ⊂ X be cycles imposing a 1-dimensional incidence constraint for curves. Let C⊂X be a nonsingular rational curve transversely incident to the cycles Vi . If the rational curve has generic normal bundle splitting, ∼

N X/C = OP1 (−1) ⊕ OP1 (−1) ⊕ OP1 , the contribution of C to the genus 0 Gromov-Witten theory of X is ∞

d −3+n q d[C]

d=1

by (5). The constraints kill the trivial normal direction. The justification for definition (2) for the virtually enumerative invariants n 0,n (γ1 , . . . , γn ) is complete. Of course, since transversality and genericity were assumed in the justification, we do not have a proof of Conjecture 0. 1.2. Meeting invariants. 1.2.1. Rules (i) and (ii). Let X be a Calabi-Yau 4-fold. The meeting invariant m β1 ,β2 virtually enumerates rational curves of class β1 meeting rational curves of class β2 . Rules (i) and (ii) have clear geometric motivation. In fact, rule (i) is consequence of rules (ii–iv). Rule (ii) may be viewed as a boundary condition. Ultimately, m β1 ,β2 is defined by rules (i–iv). Rules (iii) and (iv) are derived by assuming the best possible behavior for rational curves. However, the ideal assumptions are typically false. As in Sect. 1.1, our derivation can be viewed, rather, as a justification for the definitions.


627

1.2.2. Boundary divisor. For nonzero classes β1 , β2 ∈ H2 (X, Z), let β1 ,β2 denote the virtual boundary divisor

β1 ,β2 → M 0,0 (X, β1 + β2 ) corresponding to reducible nodal curves with degree splitting of type (β1 , β2 ). In the balanced case β1 = β2 , an ordering is taken in β1 ,β2 , and is of degree 2. The virtual dimension of β1 ,β2 is 0. Let Mβ1 ,β2 = 1 ∈Q [ β1 ,β2 ]vir

be the associated Gromov-Witten invariant. By the splitting axiom of Gromov-Witten theory, Mβ1 ,β2 = N0,β1 (Si ) g i j N0,β2 (S j ), i, j

following the notation of Sect. 0.3. The meeting invariants m β1 ,β2 , defined by rules (i-iv), may be viewed as an integral version of Mβ1 ,β2 . 1.2.3. Rule (iii). Ideally, the embedded rational curves in X of class βi occur in complete, nonsingular, 1-dimensional families Fi ⊂ M 0,0 (X, βi ). Let πi and ιi denote the universal curve and map over Fi , πi : Si → Fi , ιi : Si → X. Since β1 = β2 , the families F1 and F2 are distinct. Ideally, the surfaces Si are nonsingular and the morphisms πi are smooth except for finitely many 1-nodal fibers. The meeting number m β1 ,β2 is related to the intersection ι1∗ (S1 ) ∩ ι2∗ (S2 ) ⊂ X.

(6)

However, the intersection (6) is not transverse (even ideally). A fiber of π1 may be a component of a reducible fiber of π2 or vice versa. The meeting number m β1 ,β2 is defined to count the ideal number of isolated points of the intersection (6). Hence, ι1∗ [S1 ] ∩ ι2∗ [S2 ], m β1 ,β2 + δ = X

where the correction δ is determined by the non-transversal intersection loci. The number of times a fiber of π1 occurs as a component of a reducible fiber of π2 is simply m β1 ,β2 −β1 . Similarly, the opposite event occurs m β1 −β2 ,β2 times. The contribution to δ of each non-transversal is easily determined. Let C⊂X

628


be a fiber of π1 and a component of a reducible fiber of π2 . Then δ(C) = c1 (E), C

where 0 → N S1 /C ⊕ N S2 /C → N X/C → E → 0 is the normal bundle sequence. Certainly N S1 /C is trivial and N S2 /C has degree −1. By the Calabi-Yau condition, N X/C is of degree −2. Hence, δ(C) = −1. Rule (iii) is obtained by expanding the intersection (6) via the Künneth decomposition of the diagonal. We have ι∗ [S1 ] ∩ ι∗ [S2 ] − δ m β1 ,β2 = X n 0,β1 (Si ) g i j n 0,β2 (S j ) + m β1 ,β2 −β1 + m β1 −β2 ,β2 . = i, j

1.2.4. Rule (iv). In case of equality, the meeting number is more subtle. While the surface Sβ is ideally nonsingular and ι : Sβ → X is ideally an immersion, ι is not (even ideally) an embedding. The correct interpretation of m β,β is twice the number of ideal double points of ι. The factor of 2 arises from ordering. The double point formula [10] yields a calculation of m β,β as a correction to the self-intersection, c(TX ) , m β,β = ι(Sβ ) ∩ ι(Sβ ) − X Sβ c(TSβ ) where c(TX ) and c(TSβ ) denote the total Chern classes of the respective bundles. Expanding the correction term (and using the Calabi-Yau condition) we find c(TX ) = c2 (TX ) + c1 (TSβ )2 − c2 (TSβ ). Sβ c(TSβ ) S Certainly, n 0,β (c2 (TX )) =

c2 (TX ). Sβ

There is a decomposition c1 (TSβ ) = −ψ + c1 (Fβ ),


629

where ψ is the cotangent line on Sβ viewed as the 1-pointed space. Hence c1 (TSβ )2 = ψ 2 + 4χ (Fβ ). Sβ

Sβ

An elementary geometric argument shows 1 ψ2 = − m β1 ,β2 , 2 Sβ β+β2 =β

where the right side is the number of reducible fibers of πβ . Since 1 c2 (TSβ ) = χ (Sβ ) = 2χ (Fβ ) + m β1 ,β2 , 2 Sβ β+β2 =β

only a calculation of the topological Euler characteristic χ (Fβ ) remains. The formula χ (Fβ ) = −n 0,β (c2 (TX )) + m β1 ,β2 β+β2 =β

is obtained via a Grothendieck-Riemann-Roch calculation applied to the deformation theoretic characterization of TFβ , 0 → R 0 π∗ (ωπ∨ ) → R 0 π∗ ι∗ (TX ) → TFβ → O Fβ (D) → 0, where D ⊂ Fβ is the divisor corresponding to nodal fibers of π : S → Fβ , see [31] for a similar discussion. Rule (iv) is obtained by expanding the self-intersection of ι∗ [Sβ ] via the Künneth decomposition of the diagonal and putting together the above surface calculations. We have c(TX ) ι∗ [Sβ ] ∩ ι∗ [Sβ ] − m β,β = X S c(TSβ ) ij n 0,β (Si ) g n 0,β (S j ) + n 0,β (c2 (TX )) − m β1 ,β2 . = β1 +β2 =β

i, j

2. Genus 1 2.1. Step I. The equation defining the virtually enumerative invariants n 1,β , β>0

N1,β q β =

n 1,β

β>0

∞ σ (d) d=1

d

q dβ

1 + n 0,β (c2 (TX )) log(1 − q β ) 24 β>0

1 m β1 ,β2 log(1 − q β1 +β2 ), − 24 β1 ,β2

is justified in three steps—one for each term on the right.

(7)

630


The first term is the easiest since the contribution of an embedded, super-rigid, elliptic curve E ⊂ X of class β to the genus 1 Gromov-Witten theory of X is ∞ σ (d) d=1

d

q d[E] .

See [32] for a discussion of super-rigidity and multiple covers of elliptic curves. 2.2. Step II. The second term of (7) is obtained from the contributions of families of rational curves to the genus 1 Gromov-Witten invariants of X . Let F ⊂ M 0,0 (X, β) be the ideal nonsingular family of embedded rational curves (as considered in Sect. 1.2.3). We make two further hypotheses. Condition 1. The family F contains no nodal elements. Hence, the morphism π:S→F is a P1 -bundle. In fact, Condition 1 is rarely valid, even ideally, and will be corrected in Step III. The contribution of F to N1,dβ is expressed as an excess integral over the moduli π and space of maps M 1 (S, d) to the P1 -bundle representing d times the fiber class. Let ι denote the universal curve and map over the moduli space, π : C → M 1 (S, d), ι : C → S. Condition 2. R 0 π∗ ι∗ (N X/S ) vanishes. With the vanishing of Condition 2, Cont F (N1,β ) =

[M 1 (S,d)]vir

ctop R 1 π∗ ι∗ N X/S .

Lemma 1. Under the above hypotheses, Cont F (N1,β ) = −

1 24d

c2 (TX ). S

Proof. Consider the relative moduli space of maps to the fibers of π, M 1 ( π , d) → F.

(8)

We will use the isomorphism ∼

M 1 (S, d) = M 1 ( π , d). The two virtual classes are easily compared, [M 1 (S, d)]vir = −λ ∩ [M 1 ( π , d)]vir + χ (F) · [M 1 (P1 , d)]vir ∈ H∗ (M 1 (S, d), Q). (9)


631

On the right, λ is the Chern class of the Hodge bundle, χ (F) is the topological Euler characteristic, and M 1 (P1 , d) a fiber of (8). Consider first the integral − (10) λ · ctop R 1 π∗ ι∗ N X/S . [M 1 ( π ,d)]vir

Using the basic boundary relation5 λ=

1

0 ∈ H 2 (M 1,1 , Q), 12

and the normalization sequence, we can rewrite (10) as 1 − (ev1 × ev2 )∗ ([ Diag ]) · ev∗1 (c2 (N X/S )) · ctop R 1 π∗ ι∗ N X/S . 24 [M 0,2 ( π ,d)]vir Finally, using the Aspinwall-Morrison formula, −

[M 1 ( π ,d)]vir

d −3+2 λ · ctop R 1 π∗ ι∗ N X/S = − c2 (N X/S ). 24 S

For the second integral, we use the formula from [17] for genus 1 contributions, χ (F) ctop R 1 π∗ ι∗ N X/S = [M 1 (P1 ,d)]vir

χ (F)

χ (F) ctop R 1 π∗ ι∗ (OP1 (−1) ⊕ OP1 (−1)) = . 12d M 1 (P1 ,d)

Summing the first and second integrals, we obtain, by (9),

1 − c2 (N X/S ) + 2χ (F) . Cont F (N1,β ) = 24d S Finally, using the Calabi-Yau condition and the geometry of P1 -bundles, c2 (N X/S ) = c2 (TX ) + c1 (TS )2 − c2 (TS ) = c2 (TX ) + c2 (TS ) = c2 (TX ) + 2χ (F), concluding the lemma.

Modulo the corrections from nodal elements of F to be discussed in Step III, the derivation of the second term of (7) is complete since n 0,β (c2 (TX )) = c2 (TX ). S 5 The required marked point can be added and removed by the divisor equation.

632


2.3. Step III. We now relax Condition 1 of Sect. 2.2, but keep Condition 2 in the following stronger form. Let π:S→F

(11)

) be the moduli be the ideal family of embedded rational curves of class β. Let M 1 (S, β space of maps to S representing a π -vertical curve class ∈ H2 (S, Z). β The morphism (11) is the blow-up of a P1 -bundle over finitely many points corresponding to the 1 m β1 ,β2 2 β1 +β2 =β

need not be a multiple of the fiber class. As nodal fibers. Since π is not a P1 -bundle, β ). before let π and ι denote the universal curve and map over the moduli space M 1 (S, β satisfying Condition 2 . R 0 π∗ ι∗ (N X/S ) vanishes for every class β − [Fiber(π )] > 0. β The inequality is required in Condition 2 . Ideally, the inequality is violated for con equals a multiple of a single component of a reducible nodal nected curves only if β fiber of π . Then, R 0 π∗ ι∗ (N X/S ) = 0. We view F as not contributing at all to the Gromov-Witten invariants in classes violating the inequality (as these curves deform away from F). With the vanishing of Condition 2 , Cont F (N1,β ) = ctop R 1 π∗ ι∗ N X/S )]vir [M 1 (S,β

satisfying the inequality. for classes β Since the family F may contain nodal elements, Lemma 1 must be modified. We have ⎛ ⎞ 1 n 0,β (c2 (TX )) log(1 − q β ) Cont F ⎝ N1,β q β ⎠ = (12) 24 β−[Fiber ( π )]>0

∞ ∞ 1 + 2

cd1 ,d2 m β1 ,β2 q d1 β1 +d2 β2

d1 =1 d2 =1 β1 +β2 =β

for universal constants cd1 ,d2 . The first term on the right is the uncorrected answer of Lemma 1. The second term is the correction. The factor of 2 is included for the double counting induced by the ordering. The universal form of the correction terms follows from the canonical local analytic geometry near the nodal fibers . Let π −1 ( p) = E 1 ∪ E 2


633

be a nodal fiber. The local geometry of S near π −1 ( p) is the total space of the node smoothing deformation. The restriction of N X/S splits in the form O S (E 1 ) ⊕ O S (E 2 ). The universality of the correction terms then follows. Lemma 2. We have ∞ ∞ 1 1 log(1 − q1 q2 ). cd1 ,d2 q1d1 q2d2 = − 2 24 d1 =1 d2 =1

Lemma 2 concludes Step III and completes the justification of definition (7) of the invariants n 1,β . 2.3.1. Proof of Lemma 2 By universality, we can prove Lemma 2 by considering any exactly solved geometry that is sufficiently rich to yield all the constants cd1 ,d2 . The simplest is the following local geometry. Let S be the blow-up of P1 × P1 at the point (∞, ∞), ν

S = BL(∞,∞) (P1 × P1 ) → P1 × P1 . Let L 1 and L 2 be line bundles on S, L 1 = ν ∗ OP1 ×P1 (−1, −1) , L 2 = ν ∗ OP1 ×P1 (−1, −1) (E), where E is the exceptional divisor. Let X be the Calabi-Yau total space X = L 1 ⊕ L 2 → S. Of course, X is non-compact. The homology H2 (X, Z) is freely spanned by H2 (P1 × P1 , Z) = Z[C1 ] ⊕ Z[C2 ] and [E]. Let β = [C1 ] ∈ H2 (X, Z). ∼

Certainly, Fβ = P1 and the associated universal family is π : S → P1 , obtained by composing ν with the projection onto the second factor. The morphism π has a unique nodal fiber over ∞ ∈ F which splits as β1 = [C1 ] − [E], β2 = [E]. Hence, the only nonzero meeting numbers for X are m β1 ,β2 = m β2 ,β1 = 1. Condition 2 is easily verified for the family F.

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Proposition 1. We have Cont F (N1,d1 β1 +d2 β2 ) =

δd1 ,d2 12d1

for d1 , d2 > 0. Proof. Let T2 = C∗ × C∗ act on P1 × P1 by (ξ1 , ξ2 ) · ([x0 , x1 ], [y0 , y1 ]) = ([ξ1 x0 , x1 ], [ξ2 y0 , y1 ]) with fixed points (0, 0), (0, ∞), (∞, 0), (∞, ∞). The action of T2 lifts canonically to S. We calculate ctop R 1 π∗ ι∗ (L 1 ⊕ L 2 ) Cont F (N1,d1 β1 +d2 β2 ) = [M 1 (S,d1 β1 +d2 β2 )]vir

(13)

(14)

by T2 -localization. With the correct T2 -equivariant linearizations of L 1 and L 2 , the integral is possible to evaluate explicitly. Let s1 and s2 denote the weights of the two torus factors of T2 . The tangent weights of the T-action on P1 × P1 are (−s1 , −s2 ), (−s1 , s2 ), (s1 , −s2 ), (s1 , s2 ) at the respective fixed points (13). (i) Let T2 act on OP1 ×P1 (−1, −1) with weights s1 + s2 , s1 , s2 , 0 at the respective fixed points (13). The choice induces a canonical T2 -linearization on L 1 . (ii) Let T2 act on OP1 ×P1 (−1, −1) with weights s1 , s1 − s2 , 0, −s2 at the fixed points (13). Together with the canonical linearization on O S (E), the choice induces a canonical T2 -linearization on L 2 . The T2 -localization contributions of the integral (14) over 0 ∈ F must first be calculated. The contribution over 0 ∈ F certainly vanishes unless d1 = d2 . An unravelling of the formulas shows Cont0∈F (N1,dβ1 +dβ2 ) =

−s1 − λ ctop −s1 ⊗ R 1 π∗ ι ∗ (OP1 (−1) ⊕ OP1 (−1)) . s1 M 1 (P1 ,d) Then, by a straightforward expansion similar to the proof of Lemma 1, we obtain the vanishing Cont0∈F (N1,dβ1 +dβ2 ) = 0.


635

Fig. 1. A double comb

By the vanishing over 0 ∈ F, the contribution over ∞ ∈ F must be a constant,6 Cont∞∈F (N1,d1 β1 +d2 β2 ) ∈ Q. The T2 -action on S has 3 fixed points p0 , p∞ , p∞

over ∞ ∈ F. Here, p0 is the fixed point lying over (0, ∞), p∞ is the node of π −1 (∞), is the remaining fixed point. With the linearizations (i) and (ii), L has weight and p∞ 1 , and L has weight 0 over p , p . 0 over p∞ , p∞ 2 0 ∞ is 0, each node of the fixed map over Since the T2 -weight of L 1 at p∞ and p∞ 2 1 ∗ these produces a T -trivial factor of R π∗ ι (L 1 ) by the normalization sequence. Each T2 -fixed component mapping to β2 produces a cancelling T2 -trivial factor of R 1 π∗ ι∗ (L 1 ). Similarly for L 2 . The only localization graphs7 which survive the T2 -trivial factors from the 0 weights of L 1 and L 2 are double combs. A double comb is a connected graph with a single vertex over p , and a single path v0 over p0 , a single vertex v∞ ∞ v0 −v∞ −v∞ , through p . connecting p0 to p∞ ∞ Since a double comb has no loops, one of the vertices must have genus 1. The localization contribution of the double comb is understood to include all possible genus 1 vertex assignments. The final part of the analysis requires taking the nonequivariant limit

Lims1 →0 Cont∞∈F (N1,d1 β1 +d2 β2 ).

(15)

Since the contribution on the right is a constant, no information is lost. Nonequivariant limits are often difficult to study, but for double combs the analysis is simple. A factor of s1 in the denominator of the localization contribution of double have equal comb can only occur if the two edges of the unique path connecting p0 to p∞ degrees e

e

v0 − v∞ − v∞ . 6 By definition, the contribution over ∞ is a rational function in s and s . 1 2 7 We follow [17] for the graphical terminology for the virtual localization formula.

636


The s1 factor occurs here from the node smoothing deformation at v∞ . Even then, the s1 factor in the denominator is cancelled by s1 factors in the numerator if either v0 or has valence greater than 1. v∞ In case d1 = d2 , the latter valence condition must be satisfied, and the nonequivariant limit (15) can be taken for each double comb. In fact, each nonequivariant limit is easily seen to vanish, proving the proposition in the unequal case. If d1 = d2 , there is a unique double comb which does not satisfy the valence condition, d1

d2

v0 − v∞ − v∞ .

(16)

However, since the nonequivariant limit Lims1 →0 exists for all other double combs, the limit must exist as well for (16). As in the unequal case, the nonequivariant limit vanishes for all double combs except (16). The limit for (16) is explicit calculated to equal 1 12d1 in the equal case.

To complete the proof of Lemma 2, we expand (12) for the local Calabi-Yau X . Since c2 (TX ) = 0, ∞ ∞ 1 Cont F N1,d1 β1 +d2 β2 = 2

cd1 ,d2 m β1 ,β2 q d1 β1 +d2 β2

d1 =1 d2 =1 β1 +β2 =β

= cd1 ,d2 . Hence,

cd1 ,d2 =

δd1 .d2 12d

by Lemma 1. The justification for definition (7) of the invariants n 1,β is based on ideal geometry. Since the ideal hypotheses are typically false in algebraic geometry, Conjectures 0 and 1 are not proven. In fact, one may be suspicious of their validity. In the remaining sections, we will compute many examples and find the conjectures to always be valid. 3. Local Examples I 3.1. Solutions. Proposition 1 already describes an exactly solved local Calabi-Yau 4-fold geometry. However, a complete solution is not given by Proposition 1 since only special curve classes of BL(∞,∞) (P1 × P1 ) are considered. The two simplest nontrivial local Calabi-Yau 4-folds are studied here. The examples may be viewed as the analogues of the OP1 (−1) ⊕ OP1 (−1) → P1

(17)

local Calabi-Yau 3-fold. As in (17), we find closed form solutions for all curve classes.


637

3.2. Local P2 . Let Y be the local Calabi-Yau determined by the total space of the rank 2 bundle OP2 (−1) ⊕ OP2 (−2) → P2 . Let P denote the point class on P2 . Proposition 2. We have

(−1)d 2d , d 2d 2 ⎛ ⎞

2d 1 d d d q ⎠ log ⎝ (−1) N1,d q = d 12 N0,d (P) =

d>0

d≥0

1 = − log(1 + 4q). 24 The proposition is proven by localization. Let T2 act on P2 with fixed points [1, 0, 0], [0, 1, 0], [0, 0, 1]

(18)

and respective tangent weights (s1 , s2 ), (−s2 , s1 − s2 ), (s2 − s1 , −s1). Let P be the equivariant class of the fixed point [1, 0, 0]. Let T2 act on OP1 (−1) with weights 0, s2 , s1 at the respective fixed points (18). Similarly, let T2 act on OP1 (−2) with weights −s1 − s2 , s2 − s1 , s1 − s2 . The above choices kill the localization contributions to N0,d (P) and N1,d of all graphs with either a node over [1, 0, 0] or an edge connecting [0, 1, 0] and [0, 0, 1]. The sum over remaining comb graphs is not difficult and left to the reader. The integral invariants n 0,d (P) and n 1,d can be easily calculated from the GromovWitten invariants by the defining formulas (2) and (3). n 0,d (P) n 1,d

1 −1 0

2 1 0

3 −1 −1

4 2 2

5 −5 −8

6 13 27

7 −35 −90

8 100 314

9 −300 −1140

10 925 4158

The underlying moduli space of maps (with the point condition imposed) for the invariants n 0,1 (P) and n 0,2 (P) are projective spaces of dimension 1 and 4 respectively. It appears when the underlying moduli space is Pk , the invariant is (−1)k reminiscent of Seiberg-Witten theory for surfaces. The genus 1 invariants vanish in case there are no embedded genus 1 curves. The underlying moduli space for n 1,3 is P9 .

638


3.3. Local P1 × P1 . Let Z be the local Calabi-Yau determined by the total space of the rank 2 bundle OP1 ×P1 (−1, −1) ⊕ OP1 ×P1 (−1, −1) → P1 × P1 . Appropriate localization formulas8 for Y in genus 0 and 1 yield

N0,(d1 ,d2 ) (P) =

(−1)d1 +d2 · z(m)z(n)

m∈P(d1 ) n∈P(d2 )

M 0,(m)+(n)+1

(m) i=1

(1 + m i ψi )

1 (n)

j=1 (1 − n j ψ(m)+ j )

and

N1,(d1 ,d2 ) =

(−1)d1 +d2 · z(m)z(n)

m∈P(d1 ) n∈P(d2 )

M 1,(m)+(n)

(m) i=1

(1 + m i ψi )

1 (n)

j=1 (1 − n j ψ(m)+ j )

.

Here, P is the point class on P1 × P1 , and P(d) denotes the set of partitions of d. For p ∈ P(d), the length is denoted by ( p). The function z( p) = |Aut( p)| ·

( p)

pi

i=1

is the usual factor. By evaluating the above localization sums, we obtain the following exact solutions. Proposition 3. We have,

(d1 ,d2 ) =(0,0)

d1 + d2 2 1 N0,(d1 ,d2 ) (P) = , (d1 + d2 )2 d1 ⎛ ⎞

2 d1 + d2 1 log ⎝ N1,(d1 ,d2 ) q1d1 q2d2 = q1d1 q2d2 ⎠. 12 d1 d1 ≥0 d2 ≥0

The integral invariants n 0,(d1 ,d2 ) (P) and n 1,(d1 ,d2 ) can be easily calculated from the Gromov-Witten invariants by the defining formulas (2) and (3). n 0,(d1 ,d2 ) (P) 0 1 2 3 4 5 6

0 * 1 0 0 0 0 0

1 1 1 1 1 1 1 1

2 0 1 2 4 6 9 12

3 0 1 4 11 25 49 87

4 0 1 6 25 76 196 440

8 We now leave the optimal weight choice for the reader to discover.

5 0 1 9 49 196 635 1764

6 0 1 12 87 440 1764 5926


n 1,(d1 ,d2 ) 0 1 2 3 4 5 6

0 * 0 0 0 0 0 0

1 0 0 0 0 0 0 0

639

2 0 0 1 2 5 8 14

3 0 0 2 10 28 68 144

4 0 0 5 28 112 350 922

5 0 0 8 68 350 1370 4426

6 0 0 14 144 922 4426 17220

For n 0,(1,d) (P) the underlying moduli space is P2d . The elliptic invariants vanish in classes in which there are no embedded elliptic curves. For n 1,(2,2) the moduli space is P8 . 4. 1-Loop Amplitude and Ray-Singer Torsion Let X be a nonsingular Calabi-Yau n-fold. The string amplitude which contains information about the genus 1 Gromov-Witten theory of X is the twisted 1-loop amplitude d2 τ 1 Tr H (−1) F FL FR Q HL Q¯ H R . (19) F1 = 2 F Imτ Here, the integral is over the fundamental domain F of the mapping class group of the world-sheet torus with respect to the SL(2, Z)-invariant measure. The trace is over the Ramond sector H of the twisted non-linear σ -model on X . The operators FL and FR are the left and right fermion number operators, F = FL + FR , and HL and H R are the left and right moving Hamilton operators. The parameter τ is the complex modulus of the world-sheet torus, and Q = exp(2πiτ ). The object F1 is an index which depends either only on the complexified Kähler structure moduli of X in the A-model or only on the complex structure moduli of Xˆ in the B-model. The dependence on the moduli is via the spectra of HL and H R . We will use the B-model analysis to evaluate F1 on the mirror Xˆ of X . Predictions for the genus 1 invariants of X are then made by the mirror map. By the world-sheet analysis of [7,2], F1 satisfies the holomorphic anomaly equation 1 1 ∂i ∂¯j¯ F1 = Tr H (−) F Ci C¯ j¯ − Tr H (−) F G i j¯ . (20) 2 24 Here G i j¯ is the Zamolodchikov metric and the derivatives are with respect to the N = 2 moduli. For N = 2σ -models on Calabi-Yau n-folds, we can specialize to the complex moduli on Xˆ . Then, Tr H (−) F becomes the Euler number χ of the Xˆ and G i j¯ becomes the Weil-Peterson metric on the complex structure moduli space of Xˆ . The Ci are genus 0, 3-point functions in the A-model. In the B-model on Xˆ , the Ci can be calculated from the Picard-Fuchs equation for periods of the holomorphic (n, 0) form on Xˆ . The indices i, j¯ run from 1 to h n−1,1 ( Xˆ ).

640


There are two methods to integrate Eq. (20). One can use the integrability conditions of special geometry for Calabi-Yau n-folds or, somewhat more generally, the tt ∗ -equations. The latter apply to any N = 2 conformal world-sheet theory. If the central charge satisfies 3c = n ∈ Z, then the tt ∗ -equations imply the special geometry relations for Calabi-Yau n-folds. The tt ∗ equations are used in [7,2] to obtain p+q 1 χ Tr p,q [log(g)] − K + log | f |2 . F1 = (−1) p+q (21) 2 p,q 2 24 The sum here is over the Ramond-Ramond degenerate lowest energy states labeled by p, q which range for the σ -model case in the left and the right moving sector as follows: n n n − , − + 1, . . . , . 2 2 2 By the usual argument [42], the states are identified in the A-model with harmonic (k, l) = ( p +

n n ,q + ) 2 2

forms. For 4-folds the (2, 2) forms correspond to ( p, q) = (0, 0) and decouple from the sum in (21). Finally, g is the tt ∗ metric, K is the Kähler potential for Weil-Peterson metric, and f is the holomorphic ambiguity. The metric g is related to the Weil-Peterson metric by G i j¯ =

gi j¯ = ∂i ∂¯j¯ K , g00¯

The Kähler potential K is given by e−K =

Xˆ

g00¯ = e K .

¯ , ∧

(22)

(23)

where is the holomorphic (n, 0)-form on Xˆ — e−K can be calculated from the periods on Xˆ . In summary, specializing to 4-folds9 with h 21 = 0, we evaluate (21) to

χ (X ) K − log det G + log | f |2 . (24) F1 = 2 + h 11 (X ) − 24 For 3-folds, in our normalization,10

1 1 χ (X ) (3) 3 + h 11 (X ) − K − log det G + log | f (3) |2 . F1 = 2 12 2

(25)

The Gromov-Witten invariants are extracted in the holomorphic limit of (24) in the large volume of X corresponding to the point P of maximal unipotent monodromy of 9 All of our compact examples will satisfy h = 0. 21 10 This is, up to the normalization factor 1 , the result in [2]. 2


641

Xˆ . Taking the holomorphic limit is very similar for all dimensions. We introduce the flat coordinates near P ti =

X i (z) , X 0 (z)

(26)

which are identified with the complexified Kähler parameters of X . As the coordinates z are the complex structure moduli of Xˆ , Eq. (26) defines the mirror map between the complex structure on Xˆ and the complexified Kähler structure on X . The function X 0 is the unique holomorphic period at P, which we chose to lie at z i = 0. The functions 1 0 X log(z) + holomorphic Xi = 2πi are the h 11 (X ) = h n−1,1 ( Xˆ ) single logarithmic periods. The existence of X 0 and X i satisfying the above conditions is part of the defining property of P. Using further the structure of the periods in an integer symplectic basis at P, we conclude limt¯→i∞ K = − log(X 0 ), limt¯→i∞ G i j¯ =

(27)

∂ti j δ . ∂z j j¯

After substitution in (24), we obtain the holomorphic limit of F1 at P,

χ ∂z 0 + log | f |2 . F1 = − h 11 − 2 log(X ) + log det 24 ∂t

(28)

Here, ∂z/∂t is the Jacobian of the inverse mirror map, and f (z) is the holomorphic ambiguity at genus 1. The latter is restricted by the space time modular invariance of F1 (t, t¯). The first two terms in (24) can be shown to be modular invariant. Therefore f (z) must be modular invariant as well. The modular constraints together with the large volume behavior, which in physical terms comes from a zero mode analysis, F1 →

h n−1,1 (−1)n+1 ti cn−1 (T ) ∧ Hi , 24 X

t → ∞,

(29)

i=1

and the expected universal local behavior at other singular limits in the complex structure moduli space fix the holomorphic ambiguity f (z). As explained in [3], the the genus 1 free energy F1 is related to the Ray-Singer torsion [35]. The latter describes aspects of the spectrum of the Laplacians of V,q = ∂¯ V ∂¯ V† + ∂¯ V† ∂¯ V of a del-bar operator ∂¯ V : ∧q T¯ ∗ ⊗ V → ∧q+1 T¯ ∗ ⊗ V coupled to a holomorphic vector bundle V over M. More precisely, with a regularized determinant over the non-zero mode spectrum of V,q , one defines11 [35] I R S (V ) =

n q (−1)q+1 det V,q 2 .

(30)

q=0 11 [34] reviews these facts and relates the Ray-Singer torsion to Hitchin’s generalized 3-form action at one loop.

642


The case V = ∧ p T ∗ with p,q = ∧ p T ∗ ,q leads to the definition of a family index F1 =

n n (−1) p+q pq 1 det pq log 2

(31)

p=0 q=0

depending only on the complex structure of Xˆ . 5. Local Examples II We now consider the local Calabi-Yau geometry O(−n) → Pn−1 . Since the space is toric, Batyrev’s reflexive cone construction produces the mirror geometry: a compact Calabi-Yau (n − 1)-fold together with a meromorphic (n − 1, 0)-form λ. The latter can be obtained as a reduction of the holomorphic (n, 0)-form to the Calabi-Yau (n − 1)-fold and has a non-vanishing residuum.12 The n periods of λ fulfill the Picard-Fuchs equation n−1 n−1 n LX = θ − (−1) nz (nθ + k) θ X = 0, (32) k=1 d where θ = z dz . The discriminant of the Picard-Fuchs equation is

˜ = (1 − (−n)n z). Equation (32) has a constant solution corresponding to the residuum of λ, a general property of non-compact Calabi-Yau manifolds. We normalize constant period to X 0 = 1. The system (32) has 3 regular singular points: (i) the point z = 0 of maximal unipotent monodromy, ˜ = 0 corresponding to a nodal singularity (called the conifold point), (ii) the point (iii) the point z → ∞ (a Zn orbifold point). Because of (iii), a single cover variable ψ is sometimes more convenient. It is customary to introduce the latter as z=

(−1)n , (nψ)n

(33)

so the conifold is at ψ n = 1. More precisely we define the conifold divisor as = (1 − ψ n ) . Solutions to (32) can be obtained as k

∂ Xk = X 0 (z, ρ) 2πi∂ρ

(34)

,

(35)

ρ=0

12 One can also consider the elliptic fibration over Pn given by the hypersurface of degree 6n in the weighted projective space Pn+1 (1n , 2n, 3n), apply Batyrev’s reflexive polyhederal mirror construction, and take the large fiber limit on both sides.


643

where we define X (z, ρ) := 0

∞ k=0

z k+ρ . (−n(k + ρ) + 1)(k + ρ + 1)n

(36)

Specializing to n = 4, we find the compact part of the mirror geometry is related to the K3 given by the quartic in P3 obtained by setting n = 4 in the above equations. The meromorphic differential is given by d 1 , λ= 2πi γ0 p where the contour γ0 is around p = 0 and d is the canonical measure on P3 . The single logarithmic solution is 1 log(z) + 24 z + 1260 z 2 + 123200 z 3 + O(z 4 ) . X1 = (37) 2πi We define q = exp(2πit) and with t = X 1 / X 0 = X 1 we obtain by inverting the mirror map the series z = q − 24 q 2 − 396 q 3 − 39104 q 4 + O(q 5 ) .

(38)

The first term of (28) vanishes in the local case as X 0 = 1. The holomorphic limit of the Kähler potential term is trivial. We must determine the holomorphic ambiguity. As f is a modular invariant, f can be expressed in terms of ψ 4 . As there is a non-degenerate conformal field theory description at ψ = 0 given by the σ -model on the orbifold Cn /Zn , F1 cannot be singular at this point. On the other hand the CFT degenerates at ψ n = 1 and at ψ = ∞, and F1 is expected to be logarithmically divergent at the conifold and at the point of maximal unipotent monodromy. The former behavior can be argued by comparison with the 3-fold case while the latter follows directly from (29) and the leading behavior of (37). Therefore, we are left with the ansatz f = x log(), where x is unknown. We obtain

1 ∂ψ F1 = log − log(). (39) ∂t 24 The first term comes from the holomorphic limit of the Weil-Peterson metric. The x coefficient of the last term is matched to the first term in the localization calculation that can be done in the local case. The leading behavior at boundary divisors in the moduli space will only depend on the type of the singularities. We expect therefore that the 1 leading − 24 log() behavior will be universal at every conifold in 4-folds. The series F1 generates the genus 1 Gromov-Witten invariants of the local Calabi-Yau O(−4) → P4 , N1,d q d , F1 = C + d>0

where C is an integration constant. Our result for F1 agrees up to degree 6 with the calculation of Mayr [30]. Mayr uses the localization and Hodge integral formulas of [9,17] to calculate up to degree 6. The associated invariants n 0,d (L) and n 1,d are given in Table 1. We have checked integrality of the invariants up to degree d = 100.

644

A. Klemm, R. Pandharipande Table 1. Integer invariants n 0,d (L) and n 1,d for O(−4) → P3 g=0

g=1

−20 −820 −68060 −7486440 −965038900 −137569841980 −21025364147340 −3381701440136400 −565563880222390140 −97547208266548098900 −17249904137787210605980 −3113965536138337597215480

0 0 11200 3747900 963762432 225851278400 50819375678400 11209456846594400 2447078892879536000 531302247998293196352 115033243754049262028000 24874518281284024213236000

d 1 2 3 4 5 6 7 8 9 10 11 12

6. Compact Calabi-Yau 4-Folds The holomorphic anomaly equation will now be used to verify the integrality conjectures for several compact Calabi-Yau 4-folds. The compact cases have much more interesting geometry than the local models previously considered.

6.1. The sextic 4-fold. Batyrev’s mirror construction gives the 1-parameter complex mirror family for degree n hypersurfaces in Pn−1 as p=

n

xkn − nψ

k=1

n

xk = 0

(40)

k=1

in13 Pn−1/Zn−2 n . The holomorphic (n, 0)-form can be written as =

1 2πi

γ0

ψd , p

where the contour γ0 is around p = 0. We obtain the Picard-Fuchs operator for the period integrals (4) in this family parameterized by the variable z = (−nψ)−n as L = θ n−1 − n z

n−1

(n θ + k) .

(41)

k=1

˜ = 1 − n n z, and z = 0 is the point of maximal unipotent monodrThe discriminant is omy. Solutions as in (35) are obtained from X (z, ρ) := 0

∞ (n(k + ρ) + 1) k=0

(k + ρ + 1)n

z k+ρ .

(42)

13 The orbifold is essentially irrelevant for the B-model period calculation. It only changes the normalization 1 . of the periods by a factor n−2 n


645

Here, there is a non-trivial holomorphic solution X0 =

∞ (nk)! k=0

(k!)n

zk .

(43)

Let X 6 ⊂ P5 be the sextic 4-fold. For X 6 , the first few terms of the inverse mirror map are z = q − 6264 q 2 − 8627796 q 3 − 237290958144 q 4 + O(q 5 ) .

(44)

The nonvanishing Hodge numbers of X 6 up to the symmetries of the Hodge diamond are h 00 = h 4,0 = 1, h 11 = 1, h 31 = 426, h 22 = 1752. Further one has χ= c4 = 2610, c2 = 15H 2 , c3 = −70H 3 , H 4 = 6, X

(45)

X

where H is the hyperplane class. The holomorphic ambiguity can be fixed as follows. A simple analytic continuation argument shows that X 0 ∼ ψ at the orbifold point ψ = 0. As there is no singularity 0 in F1 at this point, only the combination Xψ can appear in F1 . Furthermore, we use the universal behavior at the conifold = (1 − ψ 6 ) = 0 and obtain, using (28), 423 log F1 = 4

X0 ψ

∂ψ + log ∂t

−

1 log(). 24

(46)

As a consistency check, Eq. (29) is fulfilled. A further consistency check is obtained from classical methods for degrees d ≤ 4. The vanishing of the integer invariants n 1,1 = n 1,2 = 0 is expected from geometrical considerations. The invariants n 1,3 and n 1,4 are enumerative. The number of elliptic cubics on a general sextic X 6 ⊂ P5 can easily be computed by Schubert calculus14 to be 2734099200. The number of elliptic quartics on X 6 was computed to be 387176346729900 by Ellingsrud and Stromme in [8]. Finally, the integrality of n 1,d , which we have checked to d = 100, is highly non-trivial. The values for the first few n 1,d are listed in Table 2. We also report a few of the meeting invariants as they have an interesting interpretation as BPS bound states at threshold in Table 3. 14 We used the Maple package Schubert written by S. Katz and S. Stromme.

646

A. Klemm, R. Pandharipande Table 2. Integer invariants n 0,d (H 2 ) and n 1,d for X 6

d

g=0

g=1

1 2 3 4 5 6 7 8

60480 440884080 6255156277440 117715791990353760 2591176156368821985600 63022367592536650014764880 1642558496795158117310144372160 45038918271966862868230872208340160

0 0 2734099200 387176346729900 26873294164654597632 1418722120880095142462400 65673027816957718149246220800 2828627118403192025358734275898400

Table 3. Meeting invariants for X 6 m d1 ,d2

d2 = 1

d1 = 1 2 3

15245496000 0 0

2

3

111118033656000 809911567810170000 0

1576410499948536000 11490828530432030136000 16302908356356789337413600

6.2. Quintic fibrations over P1 . Genus 0 Gromov-Witten invariants for multiparameter Calabi-Yau 4-folds have been calculated in [21,29]. We determine the genus 1 GromowWitten invariants and test the integer expansion of F1 for two such cases. We first consider the quintic fibration over P1 realized as the resolution of the degree 10 orbifold hypersurface X 10 ⊂ P5 (1, 1, 2, 2, 2, 2). The non-vanishing Hodge numbers are h 0,0 = h 4,0 = 1, h 11 = 2, h 31 = 1452 up to symmetries of the Hodge diamond. We introduce the divisor F associated to the linear system generated by monomials of degree 2. For example, a representative would be x3 = 0 yielding a degree 10 hypersurface in P5 (1, 1, 2, 2, 2). The dual curve to F lies as a degree 1 curve in the quintic fiber with size t1 . Another divisor B is associated to the linear system generated by monomials of degree 1. Since B lies in a linear pencil of quintic fibers, B 2 = 0. The dual curve is the base P1 with size t2 . We calculate the classical intersection data by toric geometry as follows F4

= 10,

F3 B

= 5,

c2 ∧ F = 110, c2 ∧ B F = 50, M c3 ∧ F = −200, c3 ∧ B = −410 . M

c4 = 2160, M

2

M

(47)

M

By Batyrev’s construction the mirror is given also as a degree 10 hypersurface in P(1, 1, 2, 2, 2, 2)/(Z10 Z45 ), 2 k=1

2(n+1)

xk

+

n+2 k=3

(n+1)

xk

− 2φ

2 i=1

xin+1 − ψ

n+2 k=1

xi

(48)


647

with15 n = 4. We derive the Picard-Fuchs operators as n+1 L1 = θ1n−1 (2θ2 − θ1 ) − (n + 2) k=1 ((n + 2)θ1 − k) z 1 , L2 = θ2 − (2θ2 − θ1 − 1)(2θ2 − θ1 − 2)z 2 ,

(49)

φ 1 where θi = z i dzd i with z 1 = (−(n+2)ψ) n+2 and z 2 = (2φ)2 . The system has one conifold discriminant con and a ‘strong coupling’ discriminant s at

con = 1 − (ψ n+2 − φ)2 ,

s = 1 − φ 2 .

(50) (1)

Let us now turn to the calculation of n 0,β (Si ) and n 0,β (c2 ). We denote by A1 = J F (1) and A2 = J B the harmonic (1, 1)-forms dual to F and B. We chose further a basis (2) 2 2,2 (M). Toric geometry A(2) 1 = J F and A2 = J B J F for the vertical subspace of H implies that the latter can be obtained from the leading θ -polynomials Lˆ i of the PicardFuchs operators. More precisely the subspace is spanned by the degree two elements of the graded multiplicative ring (1) (1) R = C[A1 , . . . , Ah 11 ]/Id(Lˆ i |θ →A(1) ) , i

(51)

i

where the Lˆ i are the formal limits Lˆ i = lim zi →0 Li of the Picard-Fuchs operators. Following [19,21,29], we calculate the genus 0 quantum cohomology intersection (1) (1) (1) Ci jα = Ai ∧ A j ∧ A(2) (52) α + instanton corrections, M

in the B model as follows. Using the flat coordinates ti =

Xi 1 log(z i ) + O(z) = X0 2πi

for the identification of the B-model structure at the point of maximal unipotent monodromy with the A-model structure [19], we find Ci(1) jα = ∂ti ∂t j

(2) α , X0

(53)

(2) where given an A(2) α the dual period α is specified by the leading quadratic behavior 16 in the logarithms as (2) log(z i ) log(z j ) α 1 (1) (1) = Ai ∧ A j ∧ A(2) + O(z) . (54) α × 0 X 2 (2πi)2 M ij

For example, using (47), the period c2 , whose expansion in qi yields the n 0,β (c2 ), is specified by the leading logarithmic behavior c2 1 = (55 log(z 1 )2 + 50 log(z 1 ) log(z 2 )). X0 (2πi)2 15 These formulas apply to n-dimensional degree 2(n + 1) hypersurfaces in P(1, 1, 2n ). 16 Note that admixtures of periods with lower leading logarithmic behavior does not affect C (1) due to the i jα

derivatives in (52).

648

A. Klemm, R. Pandharipande Table 4. Integer invariants for the resolution of X 10 ⊂ P5 (1, 1, 2, 2, 2)

n 10,β d1 = 0 1 2 3 4

d2 = 0

n 20,β d1 = 0 1 2 3 4

d2 = 0 * 2875 1218500 951619125 969870120000

2

3

4

1

2

3

4

5 0 0 0 9375 0 0 0 17669375 5243750 0 0 34150175000 50975575000 4766665625 0 66623314796875 253824223203750 125716582171875 5379339875000

d2 = 0

1

2

3

4

* 0 0 −2768250 −17325370250

0 0 0 7297250 90447173500

0 0 0 7297250 699252105750

0 0 0 −2768250 90447173500

0 0 0 0 −17325370250

n 1,β d1 = 0 1 2 3 4

1

* 0 0 0 0 12250 12250 0 0 0 6462250 35338750 6462250 0 0 5718284750 85125750000 85125750000 5718284750 0 6349209995000 192339896968750 507648446407500 192339896968750 6349209995000

Table 5. Meeting invariants for the resolution of X 10 ⊂ P5 (1, 1, 2, 2, 2) m β1 ,β2 (0, 1) (0, 2) (1, 0) (1, 1) (1, 2) (2, 0) (2, 1)

(0, 1)

(0, 2)

(1, 0)

(1, 1)

(1, 2)

(2, 0)

(2, 1)

−10

−10 10

6500 0 10781250

0 0 19237750 10768250

0 0 0 0 0

4025250 0 5310625000 10532668750 0 2555792968750

4025250 0 43309206500 43309206500 0 22836787744000 124882678630250

With this information, we calculate the invariants (2)

n i0,β = n 0,β (Ai ) as well as n 0,β (c2 ). Finally, for the genus 1 Gromov-Witten invariants, we obtain

F1 = 86 log

X0 ψ

+ log det

∂(ψ, φ) ∂(t1 , t2 )

−

1 7 log(con ) − log(s ). 24 24

(55)

The only difference from the calculation for the sextic is that the behavior at s must be determined. The latter determination is made by imposing (29) with M c3 ∧ B = −410. The second condition in (29) is a check. At s = 0 we have divisor collapsing, which is an P1 fibration over the degree 5 hypersurface in P3 . Integer as well as meeting invariants are listed in Tables 4 and 5. It is interesting to compare the above quintic fibration with a different quintic fibration given by the hypersurface of bidegree (2, 5), X 2,5 ∈ P1 × P4 .


649

The Hodge diamond of the second fibration is the same as the previous case. The divisors B and F correspond to the pull-backs of the hyperplane classes on P1 and P4 respec2 tively. We use the same basis for the vertical subspace of H 2,2 (M) as before A(2) 1 = JF (2) and A2 = J B J F . Due to the different fibration structure, the topological data differ from the previous case: c2 ∧ F 2 = 70, c2 ∧ BF = 50, F 4 = 2, F 3 B = 5, M M (56) c4 = 2160, c3 ∧ F = −200, c3 ∧ B = −330 . M

M

M

We derive the Picard-Fuchs equations in the standard large volume variables z 1 and z 2 [19] as L1 = θ13 (5θ1 − 2θ2 ) − 55 z 1 L2 = θ2 − z 2

2

4

(5θ1 + 2θ2 + k) + 4z 2 (5θ1 + 2θ2 + 1),

k=1

(57)

(5θ2 + 2θ1 + k) .

k=1

The system has only one conifold discriminant = (1 − x12 ) − 5x2 (1 + 4x1 ) + 10x22 (1 − x1 ) − x23 (10 − 5x2 + x22 ) .

(58)

We have introduced rescaled variables x1 = 55 z 1 and x2 = 22 z 2 . Here, we know no further regularity conditions in the interior of the moduli space. Therefore, we simply impose (29) with c3 ∧ B = −330 and c3 ∧ B = −200. M

M

That fixes the coefficients of the log(z 1 ) and log(z 2 ) terms in the most general ansatz of the ambiguity

1 51 ∂(z 1 , z 2 ) 22 − log() + log(z 1 ) + log(z 2 ). F1 = 86 log X 0 + log det ∂(t1 , t2 ) 24 4 3 (59) The integer invariants listed in Table 6 are compatible with the previous quintic fibration — we get the same invariants in the fiber direction, as expected. 6.3. Elliptically fibered Calabi-Yau 4-folds. A simple elliptic fibration over P3 compactifies the local model in Sect. 5. Consider the resolution of the degree 24 orbifold hypersurface X 24 ⊂ P5 (1, 1, 1, 1, 8, 12) in weighted projective space. The genus 0 invariants have be calculated in [21]. The resolution has the following non-vanishing Hodge numbers: h 0,0 = h 4,0 = 1, h 11 = 2, h 31 = 3878, h 22 = 15564 up to symmetries.

650

A. Klemm, R. Pandharipande Table 6. Integer invariants for X 2,5 ⊂ P1 × P4

n 10,β d1 = 0 1 2 3 4

d2 = 0

1

2

3

* 9950 5487450 4956989450 5573313899000

0 171750 533197250 1342522028500 3120681190272750

0 609500 9651689750 64483365881000 301443864603401500

0 609500 63917722000 1152361680367750 10812807897775185750

n 20,β d1 = 0 1 2 3 4

d2 = 0

1

2

3

* 2875 1218500 951619125 969870120000

125 195875 369229625 713334157250 1390949237651750

0 1248250 10980854250 53873269172000 205222409245164750

0 1799250 101591346500 1308427978728875 9819953566670512000

d2 = 0

1

2

3

* 0 0 −2768250 −17325370250

0 0 0 218986250 2510820252500

0 0 0 82508848750 1468762788741875

0 0 0 2759605738750 94873159058300000

n 1,β d1 = 0 1 2 3 4

We introduce the linear system B generated by linear polynomials in the four degree 1 variables. The linear system maps X 24 to P3 with fibers given by elliptic curves. The second linear system E is generated by polynomials of degree 4. The curve dual to E is a curve extending over the fiber E with size denoted by t1 . The curve dual to B is a degree one curve in P3 with size denoted by t2 . The intersections of the divisors are E 4 = 64,

E 3 B = 16,

E 2 B 2 = 4,

E B 3 = 1,

B4 = 0 .

Further topological data are c4 = 23328, c3 ∧ B = −960, c3 ∧ E = −3860, M M M c2 ∧ B 2 = 48, c2 ∧ B E = 182, c2 ∧ E 2 = 728. M

M

(60)

(61)

M

The mirror family is likewise given by an hypersurface of degree 24 in P(1, 1, 1, 1, 8, 12)/(Z324 ), n

2 3 x16n + xn+1 + xn+2 − nφ

n

xi2n − 6nψ

n+2

xi

(62)

L1 = θ1 (θ1 − nθ2 ) − 12(6 θ1 − 5)(6 θ1 − 1)z 1 , L2 = θ2n − nk=1 (n θ2 − θ1 − k)z 2 ,

(63)

k=1

k=1

i=1

with17 n = 4. We derive the Picard-Fuchs operators as

nφ (−1) where θi = z i dzd i with z 1 = (nψ) 6 and z 2 = (nφ)n . The system has two conifold discriminants n

1 = 1 − φ n ,

2 = 1 − φ˜ n ,

17 These formulas apply to n-dimensional degree 6n hypersurfaces in P(1n , 2n, 3n).

(64)


651

Table 7. Integer invariants for the resolution of X 24 n 10,β d1 = 0 1 2 3 4 n 20,β d1 = 0 1 2 3 4 n 1,β d1 = 0 1 2 3 4

d2 = 0

1

2

3

4

0 960 1920 2880 3840

0 5760 −1817280 421685760 2555202430080

0 181440 −98640000 29972448000 −6353500619520

0 13791360 −10715760000 4447212981120 −1273702762398720

0 1458000000 −1476352644480 783432258136320 −285239128072550400

d2 = 0

1

2

3

4

0 0 0 0 0

−20 7680 −1800000 278394880 623056099920

−820 491520 −159801600 35703398400 −6039828417600

−68060 56256000 −24602371200 7380433205760 −1683081588149760

−7486440 7943424000 −4394584496640 1662353371955200 −478655396625235200

d2 = 0

1

2

3

4

0 −20 0 0 0

0 −120 45720 −10662240 1638152760

0 −3780 2245680 −719326800 160844654520

11200 −7852120 2858334000 −719497580160 140278855296640

3747900 −3536410200 1724679193440 −573686979645680 145314212874711600

where we defined φ˜ = ψ 6 − φ. The solutions to the Picard-Fuchs equations can be obtained similarly as in Sect. 6.1 using the methods outlined in [21]. For example the holomorphic solution at the point of maximal unipotent monodromy is given by ∞

X = 0

k1 =0,k2 =0

(6k1 )!(nk2 )! z k1 z k2 . (2k1 )!(3k1 )!k1 !(k2 !)n 1 2

(65)

The considerations, which lead to the expression of F1 in the holomorphic limit, are very similar to those of Sect. 6.1,

F1 = 928 log

X0 ψ

∂(ψ, φ) + log det ∂(t1 , t2 )

A new feature here is

lim log det

ψ→0

∂(ψ, φ) ∂(t1 , t2 )

1 log(i ) . 24 2

+ 3 log(ψ) −

(66)

i=1

∼ ψ −3 ,

as is shown by simple analytic continuation of X 0 and the two logarithmic solutions to ψ = 0. To maintain the expected regularity at ψ = 0, we have to add the explicit 3 log(ψ) term to the holomorphic ambiguity. As a check of the result (66), we note again that (29) with (61) is fulfilled. (2) (2) 1 We chose further a basis A1 = 17 (4J E2 + J E J B ) and A2 = J B2 and calculate as before the genus 0 and genus 1 invariants. As a consistency check we note that scaling the size of the elliptic fiber t1 to infinity leaves us precisely with the O(−4) → P3 geometry. The corresponding invariants are listed in Table 7. Acknowledgements. The paper began in a conversation with I. Coskun about the classical enumerative geometry of canonical curves.

652


We thank M. Aganagic, J. Bryan, D. Maulik, S. Theisen, and especially C. Vafa for useful discussions. A. K. was partially supported by the DOE grant DE-FG02-95ER40986. R. P. was partially supported by the Packard foundation and the NSF grant DMS-0500187. The project was started during a visit of R. P. to MIT in the fall of 2006.

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29. Mayr, P.: Mirror symmetry, N = 1 superpotentials and tensionless strings on Calabi-Yau four-folds, Nucl. Phys. B 494, 489 (1997) 30. Mayr, P.: Summing up open string instantons and N=1 string amplitudes. http://arixiv.org/list/hepth/ 0203237, 2002 31. Pandharipande, R.: The canonical class of M 0,n (Pr , d) and enumerative geometry. IMRN 173–186 (1997) 32. Pandharipande, R.: Hodge integrals and degenerate contributions. Comm. Math. Phys. 208, 489–506 (1999) 33. Pandharipande, R.: Three questions in Gromov-Witten theory, Proceedings of the ICM (Beijing 2002), Vol II., Beijing: Higher Ed. Press, 2002, pp. 503–512 34. Pestun, V., Witten, E.: The Hitchin functionals and the topological B-model at one loop. Lett. Math. Phys. 74, 21–51 (2005) 35. Ray, D.B., Singer, I.M.: Analytic torsion for complex manifolds, Ann. of. Math. 98, 154 (1973) 36. Taubes, C.: SW ⇒ Gr: from the Seiberg-Witten equations to pseudo-holomorphic curves. J. AMS 9, 845–918 (1996) 37. Taubes, C.: Counting pseudo-holomorphic submanifolds in dimension 4. J. Diff. Geom. 44, 818–893 (1996) 38. Taubes, C.: Gr ⇒ SW: from pseudo-holomorphic curves to Seiberg-Witten solutions. J. Diff. Geom. 51, 203–334 (1999) 39. Taubes, C.: GR = SW: counting curves and connections. J. Diff. Geom. 52, 453–609 (1999) 40. Thomas, R.: A holomorphic Casson invariant for Calabi-Yau 3-folds and bundles on K3 fibrations. JDG 54, 367–438 (2000) 41. Vafa, C., Witten, E.: A one loop test of string duality, Nucl. Phys. B 447, 261 (1995) 42. Witten, E.: Mirror manifolds and topological field theory. http://arixiv.org/list/hepth/9112056, 1991 Communicated by N.A. Nekrasov


Communications in


Quasi-Linear Dynamics in Nonlinear Schrödinger Equation with Periodic Boundary Conditions M. Burak Erdo˘gan, Vadim Zharnitsky Department of Mathematics, University of Illinois, Urbana, IL 61801, USA. E-mail: [email protected]; [email protected] Received: 27 May 2007 / Accepted: 7 August 2007 Published online: 7 March 2008 – © Springer-Verlag 2008

Abstract: It is shown that a large subset of initial data with finite energy (L 2 norm) evolves nearly linearly in nonlinear Schrödinger equation with periodic boundary conditions. These new solutions are not perturbations of the known ones such as solitons, semiclassical or weakly linear solutions.

1. Introduction The nonlinear Schrödinger (NLS) equation iqt + q + |q|2 q = 0,

(1)

where q : Rt × Mx → C, frequently appears as the leading approximation of the envelope dynamics of a quasi-monochromatic plane wave propagating in a weakly nonlinear dispersive medium. It arises in a number of physical models in the description of nonlinear waves such as the propagation of a laser beam in a medium whose index of reflection is sensitive to the wave amplitude. NLS has been considered on various domains such as M = Rn , Tn , with periodic or Dirichlet boundary conditions. One dimensional cubic NLS is integrable [17] and the explicit (or approximately explicit) solutions can be obtained as solitons, cnoidal waves, and their perturbations. There have been also many interesting results on the long time asymptotics of solutions of integrable NLS in the limit of small dispersion, see e.g. the recent monograph [10,6,16,3] and references therein. Recent results in optical communication literature (see, e.g. [2,7,13], and the Appendix) suggest that for some initial data (highly localized pulses) the evolution is nearly linear. Based on these studies, we introduce a large class of solutions, which we call quasi-linear, for one dimensional cubic NLS with periodic boundary conditions. These The authors were partially supported by NSF grants DMS-0505216 (V. Z.) and DMS-0600101 (B. E.).

656

M. B. Erdo˘gan, V. Zharnitsky

solutions can be characterized by the magnitude of Fourier coefficients of the initial data. We prove that these solutions evolve nearly linearly using a normal form reduction and estimates on Fourier sums. Although we do not explicitly use integrability, we do rely on the integrability of the quartic normal form which is partially responsible for quasi-linear behavior. Therefore, similar results can be obtained for some nonlinear PDEs, such as iqt + qx x x x + |q|2 q = 0, for which there are no integrability results. We do not study long time asymptotics but rather the finite time dynamics in the limit of spectral broadening of initial data. This broadening forces q(x, 0) H s to grow to infinity, making the analysis rather nontrivial even for the finite time interval. While, we consider the focusing case, our result holds for the defocusing case as well. The reader will be able to see that our proof can be immediately adapted for the defocusing case, since nowhere our arguments rely on the nonlinearity sign. In many engineering and physics applications, nonlinearity is unavoidable while modeling and optimizing a linear behavior is much easier than a nonlinear one. Therefore, it is an important question whether a nonlinear system can be made to behave linearly. In applied mathematics and physics literature, such a behavior has been observed in e.g. [1,7,8,14,15]. We believe that our result gives a systematic way to analyze this behavior in nonlinear systems when the energy is distributed over many Fourier harmonics. 2. Main Results We consider the nonlinear Schrödinger equation with periodic boundary conditions, iqt + qx x + 2|q|2 q = 0, with initial data in q(0) ∈ L 2 (−π, π ). In [4], Bourgain proved the L 2 global wellposedness of this equation. The numerical simulations of quasi-linear regime for light wave communication systems suggest that the following statement should hold (see, e.g., [7,14]) Observation 1. Assume that initial data is a localized Gaussian 1 − x2 q(x, 0) = √ e ε2 h(x), ε where h(x) is a smooth cutoff near x = ±π/2. Then the initial data evolves quasi-linearly, q(x, t) − eit (+4P) q(x, 0)2 → 0,

(2)

as ε → 0 and for t ≤ T , where T is a fixed positive number, and P = q(·, 0)22 /2π . We will prove (2) for a large class of initial data (including the ones above) characterized by the magnitude of Fourier coefficients. We will use Fourier transform in the form q(x, t) = u(n, t)einx , n∈Z

u(m, t) =

1 2π

π

−π

q(x, t)e−imx d x,

Quasi-Linear Dynamics in NLS

657

so that the NLS equation takes the form i

du(m) − m 2 u(m) + 2 dt m

u(m 1 )u(m 2 )u(m ¯ 3 ) = 0.

(3)

1 +m 2 −m 3 =m

Our main result is the following theorem. Theorem 2.1. Let P > 0 and C > 0 be fixed. Assume that the Fourier sequence of the initial data u(n, 0) = q(·, 0)(n) satisfies 1

1

u(·, 0)1 ≤ Cε− 2 ,

u(·, 0)∞ ≤ Cε 2 ,

for sufficiently small ε ∈ (0, 1). Then, for each t > 0, q(·, t) − eit (+4P) q(·, 0) L 2 t ε1− , (4) √ where P = q(·, 0)22 /2π , t = 1 + t 2 and the implicit constant depends only on C. Remark 2.1. The initial data in the observation above satisfies the hypothesis of the theorem. In fact, if f is an H s function for some s > 1/2 with compact support on (−π, π ), then 1 f ε (x) = √ f (x/ε) ε satisfies the hypothesis of the theorem. By continuous dependence on initial data in L 2 , it suffices to prove (4) for any δ > 0 and for any initial data in the following subset of L 2 : ⎧ ⎫ ∞ 1/ p ⎨ ⎬ 1 1 − δ = f ∈ L 2 : fˆ p,δ := | fˆ(n)| p eδ|n| p ≤ Cε 2 p , p ∈ [1, ∞] . Bε,C ⎩ ⎭ n=−∞

δ ⊂ H 1 , we can introduce the Hamiltonian [11] Since Bε,C H (u) = i n 2 |u(n)|2 − i u(n 1 )u(n 2 )u(n ¯ 3 )u(n ¯ 4 ), n

l(n)=0

with conjugated variables {u(n), u(n)} ¯ n∈Z , where l(n) = n 1 + n 2 − n 3 − n 4 . The Hamiltonian flow is then given by u(n) ˙ =

∂H · ∂ u(n) ¯

Theorem 2.1 follows from the following by continuous dependence on initial data in L 2. Theorem 2.2. Let P > 0 and C > 0 be fixed. Assume that q(0)22 = 2π P, and δ q(·, 0) ∈ Bε,C for some δ > 0, and for sufficiently small ε ∈ (0, 1). Then, for each t > 0, q(·, t) − eit (+4P) q(·, 0)2 t ε1− , (5) where the implicit constant depends only on C.

658


The proof of Theorem 2.2 is based on the normal form transformations, see, e.g., [11,12] and [5]. In Sect. 3, we introduce a canonical transformation u = u(v) in the Fourier space which brings the equation into the form1 , see (15) and (16) below, v(n) ˙ = i(n 2 + 4P)v(n) + E(v)(n).

(6)

We prove that the transformation u = u(v) is near-identical in the following sense. δ or v ∈ B δ , then Proposition 2.1. If u ∈ Bε,C ε,C

u2 = v2 , and

3

u − v p,δ ε 2

− 1p −

for 1 ≤ p ≤ ∞, where the implicit constant depends on C and p. In particular, if ε is δ implies v ∈ B δ sufficiently small, then u ∈ Bε,C ε,2C and vice versa. Then, we estimate the error term E(v) as follows δ , then the error term E(v) in the transformed equation (6) Proposition 2.2. If v ∈ Bε,C satisfies 3

E(v) p,δ ε 2

− 1p −

,

for 1 ≤ p ≤ ∞, where the implicit constant depends on C and p. δ for some Propositions 2.1 and 2.2 imply Theorem 2.2. Indeed, assume that q(·, 0) ∈ Bε,C

δ > 0, C > 0, and for sufficiently small ε ∈ (0, 1). Multiplying (6) with e−i(n and integrating over t, we obtain t 2 −i(n 2 +4P)t − v(n, 0) = e−i(n +4P)τ E(v)dτ. v(n, t)e

2 +4P)t

0

This and Propositions 2.1 and 2.2 imply, for each p ∈ [1, ∞], that 3

v(t) − ei Lt v(0) p,δ = v(t)e−i Lt − v(0) p,δ t ε 2

− 1p −

,

(7)

where L(v)(n) = (n 2 + 4P)v(n). Finally, Proposition 2.1 and (7) imply, for p ∈ [1, ∞], that u(t) − ei Lt u(0) p,δ ≤ u(t) − v(t) p,δ + v(t) − ei Lt v(0) p,δ + ei Lt v(0) − ei Lt u(0) p,δ 3

t ε 2

− 1p −

,

where the implicit constant depends on C. In particular, this yields the assertion of Theorem 2.2 as follows q(t) − eit (+2P) q(0)2 = u(t) − ei Lt u(0)2 ≤ u(t) − ei Lt u(0)2,δ t ε1− .

Notation. We will frequently use convolution with 1/|n|, which √ will be denoted by ρ(n) =

1 |n| χZ\{0} (n).,

and we will also use the notation n =

1 + n2.

1 Similar quasi-linear behavior can be obtained for the nonintegrable NLS iq + q 2 t x x x x + |q| q = 0 with the leading behavior given by v(n) ˙ = i(n 4 + 4P)v(n).


659

We always assume by default that the summation index avoids the terms with vanishing denominators. To avoid using unimportant constants, we will use sign: A B means there is an absolute constant K such that A ≤ K B. In some cases the constant will depend on parameters such as p. A B(η−) means that for any γ > 0, A ≤ Cγ B(η − γ ). A B(η+) is defined similarly. 3. Normal Form Calculations Consider the change of variables u n → vn , generated by the time 1 flow of a purely imaginary Hamiltonian F. Namely, solve dw ∂F = , w|s=0 = v, ds ∂ w¯ thus producing a symplectic transformation u = u(v) := w|s=1 . Let X sF be the time s map of the flow of F. Using Taylor expansion [11,12], we have H ◦ X 1F (v) = H (v) + {H, F}(v) + · · · +

(8)

k 1

+ 0

where

1 {· · · {{H, F}, F}, . . . , F }(v) k!

(1 − s)k {. . . {{H, F}, F}, · · · , F } ◦ X sF (v) ds, k! k+1

∂A ∂B ∂A ∂B {A, B} = − ∂u(n) ∂ u(n) ¯ ∂ u(n) ¯ ∂u(n) n

(9)

is the Poisson bracket. Recall that H has a quadratic and a quartic part H = 2 + H4 , where

2 = i

m 2 |u(m)|2 .

(10)

(11)

We write H4 = H4nr + H4r , where the superscripts “nr” and “r” denotes the non-resonant and resonant terms: H4nr = i v(m 1 )v(m 2 )v(m ¯ 3 )v(m ¯ 4 ), l(m)=0, q(m)=0

H4r

=i

v(m 1 )v(m 2 )v(m ¯ 3 )v(m ¯ 4 ),

l(m)=0, q(m)=0

where l(m) = m 1 + m 2 − m 3 − m 4 and q(m) = m 21 + m 22 − m 23 − m 24 . As usual H4r is the part of the Hamiltonian that commutes with 2 . Note that we can further decompose H4r as H4r = −i |v(m)|4 + 2i |v(m 1 )|2 |v(m 2 )|2 := H4r1 + H4r2 . m

m 1 ,m 2

660


We sequentially apply two normal form transformations generated by F1 and F2 . We choose F1 so that the following cancellation property holds: { 2 , F1 } = −H4nr . We will prove that F1 commutes with with k = 2, we obtain

H4r2 .

(12)

Using these cancellation properties in (8)

1 1 H ◦ X 1F1 = 2 + H4r1 + H4r2 + {H4r1 , F1 } + {H4nr , F1 } + g 2F1 (H4 ) 2 2 1 (1 − s)2 3 + g F1 (H ) ◦ X sF1 ds, 2 0 where we used the notation g 0F (H ) = H,

k g k+1 F (H ) = {g F , F}, k = 0, 1, 2, . . . .

Now, we apply the second transformation2 generated by F2 to eliminate the non-resonant terms in 21 {H4nr , F1 }, i.e., 1 (13) { 2 , F2 } = − {H4nr , F1 }nr . 2 We will also prove that F2 commutes with H4r2 . Using these cancellation properties as above in (8) (with k = 1), we obtain H ◦ X 1F1 ◦ X 1F2 = 2 + H4r2 + R, where 1 R = H4r1 + {H4r1 , F1 } + {H4nr , F1 }r + {H4r1 , F2 } + {{H4r1 , F1 }, F2 } + K 2 1 1 + {{H4nr , F1 }, F2 } + {K , F2 } + (1 − s)g 2F2 (H ◦ X 1F1 ) ◦ X sF2 ds, 2 0 where

1 1 2 (1 − s)2 3 g F1 (H4 ) + g F1 (H ) ◦ X sF1 ds. 2 2 0 The transformed evolution equation is given by K =

∂(H ◦ X 1F1 ◦ X 1F2 )

. ∂ v¯ Note that the contribution of the “leading” terms, 2 + H4r2 , is given by ∂ |v(m 1 )|2 |v(m 2 )|2 = i(n 2 + 4P)v(n). i m 2 |v(m)|2 + 2i ∂ v(n) ¯ m ,m v(n) ˙ =

1

(14)

2

Therefore, we can rewrite (14) as v(n) ˙ = i(n 2 + 4P)v(n) + E(v)(n),

(15)

where E(v)(n) =

∂R . ∂ v(n) ¯

(16)

2 It turns out that the transform generated by F is not enough since the term {H nr , F } is present in the 1 1 4 Hamiltonian. The direct estimate of this term produces a finite order nonlinear effect (see Subsect. 4.4).


661

3.1. Calculation of F1 and F2 . To obtain (12), we take F1 of the form F1 = f (m 1 , m 2 , m 3 , m 4 )v(m 1 )v(m 2 )v(m ¯ 3 )v(m ¯ 4 ). l(m)=0

We have

∂ F1 ∂ F1 ¯ − v(m) m 2 v(m) ∂ v(m) ¯ ∂v(m) m =i (m 21 + m 22 − m 23 − m 24 ) f (m 1 , m 2 , m 3 , m 4 )v(m 1 )v(m 2 )v(m ¯ 3 )v(m ¯ 4 ).

{ 2 , F1 } = i

l(m)=0

Therefore, we let F1 =

v(m 1 )v(m 2 )v(m v(m 1 )v(m 2 )v(m ¯ 3 )v(m ¯ 4) ¯ 3 )v(m ¯ 4) . = 2 2 2 2 2(m 1 − m 3 )(m 2 − m 3 ) m1 + m2 − m3 − m4 l(m)=0 l(m)=0

(17)

Now, we calculate F2 . Using the Hamiltonian structure3 ∂H ∂H =− , ∂ v(n) ¯ ∂v(n)

∂ F2 ∂ F2 =− ∂ v(n) ¯ ∂v(n)

we obtain {H4nr , F1 }nr = 2i

v(m 1 )v(m 2 )v(m 3 )v(m ¯ 4 )v(m ¯ 5 )v(m ¯ 6) − c.c. (m 2 − m 6 )(m 3 − m 6 )

m 4 ,m 5 =m 1 , m 2 ,m 3 =m 6

l(m)=0, q(m)=0

Therefore, a calculation similar to the one for F1 yields F2 =

m 4 ,m 5 =m 1 , m 2 ,m 3 =m 6

v(m 1 )v(m 2 )v(m 3 )v(m ¯ 4 )v(m ¯ 5 )v(m ¯ 6) − c.c. q(m)(m 2 − m 6 )(m 6 − m 3 )

(18)

l(m)=0, q(m)=0

Here, l(m) = m 1 +m 2 +m 3 −m 4 −m 5 −m 6 , and q(m) = m 21 +m 22 +m 23 −m 24 −m 25 −m 26 . 3.2. Proof of Proposition 2.1. First we state a simple corollary of Young’s inequality. Recall that ρ(n) = 1/|n| for n = 0 and ρ(0) = 0. Lemma 3.1. For any p > 1, for any choices of ± signs, w p− . w(±n ± j)ρ(± j) p j n

With some abuse of notation, we denote each sum of the above form by w ∗ ρ. 3 These identities follow from the following easily checked ones: ¯ = 0 and ∂v H¯ (v, v) ¯ = ∂v¯ H (v, v). ¯ + ∂v H¯ (v, v) ¯

(H ) = 0, ∂v H (v, v)

662


Proof. Recall that by Young’s inequality, w∗ρ p wq ρr , where 1+ 1p = q1 + r1 . The lemma follows since ρ ∈ q for any q > 1. Proof of Proposition 2.1. First note that the equality of the 2 norms follows from Hamiltonian formalism. Indeed, it is straightforward to verify that {F, Q} = 0 (where Q(u) = u22 ), which implies 2 norm conservation. To prove the second statement, we should estimate the time 1 map of the flow of F1 and of F2 . We start with F1 , dw(n) ∂ F1 = = ds ∂ w(n) ¯

m 1 +m 2 −m 3 −n=0

w(m 1 )w(m 2 )w(m ¯ 3) . (m 1 − n)(m 2 − n)

(19)

δ ) Multiplying with eδ|n| , we estimate (assuming that w ∈ Bε,C

δ|n| dw(n) e ds ≤ m 1 +m 2 −m 3 −n=0

≤ w∞,δ

e−δ(|m 1 |+|m 2 |+|m 3 |−|n|) |w(m 1 )eδ|m 1 | w(m 2 )eδ|m 2 | w(m 3 )eδ|m 3 | | |m 1 − n||m 2 − n|

|w(m 1 )|eδ|m 1 | |w(m 2 )|eδ|m 2 | ≤ w∞,δ [|w|eδ|·| ∗ ρ]2 (n). |m − n||m − n| 1 2 m ,m 1

2

In the second line, we used the fact that |m 1 | + |m 2 | + |m 3 | − |n| ≥ 0. Therefore, by Lemma 3.1, we obtain dw δ|·| 2 2 ds ∞,δ ≤ w∞,δ |w| e ∗ ρ∞ ≤ w∞,δ wq,δ for any 1 ≤ q < ∞. Similarly, using Lemma 3.1, we obtain dw δ|·| 2 2 ds 1,δ ≤ w∞,δ |w| e ∗ ρ2 w∞,δ w2−,δ . δ (or w(1) ∈ B δ ) then The last two inequalities imply that if w(0) ∈ Bε,C ε,C 3

1

w(s) − w(0)∞,δ ε 2 − ,

w(s) − w(0)1,δ ε 2 − .

This completes the proof for F1 . In the proof for F2 , we omit some of the details, in particular the multiplication with eδ|n| argument above, since it works exactly the same way. To estimate the p norm of the right-hand side of dw(n) ∂ F2 = , ds ∂ w(n) ¯ we use duality: ∂ F2 ∂ w(n) ¯

p

=

sup

h

p

=1

∂ F2 h(n) . ∂ w(n) ¯

(20)


663

Note that the right-hand side of (20) can be estimated by the sum of six terms of the form

F˜2 (w1 , . . . , w6 ) :=

m 4 ,m 5 =m 1 , m 2 ,m 3 =m 6

w1 (m 1 ) · · · w6 (m 6 ) , |q(m)||m 2 − m 6 ||m 6 − m 3 |

(21)

l(m)=0, q(m)=0

where in the j th term w j = |h| and the others are |v|. The required estimates for these terms follow by applying Lemma 3.2 below with arbitrarily small η and with i = j if p = 1 and with k = j if p = ∞. Lemma 3.2. For any η > 0 and for any distinct i, k ∈ {1, 2, 3, 4, 5, 6}, there is a permutation (i 1 , i 2 , i 3 , i 4 ) of the remaining indices such that F˜2 (w1 , . . . , w6 ) wi 1 wk ∞ wi1 1

4

1

η

1+η wil 1+η ∞ wil 1 .

l=2

Proof. Fix η > 0, i, and k. By Holder’s inequality we have ⎤

⎡ ⎢ F˜2 ≤ ⎢ ⎣

m 4 ,m 5 =m 1 , m 2 ,m 3 =m 6

⎡ ×⎣

⎥ w1 (m 1 ) · · · w6 (m 6 ) ⎥ |q(m)|1+η |m 2 − m 6 |1+η |m 6 − m 3 |1+η ⎦

l(m)=0, q(m)=0

⎤

1 1+η

(22)

η 1+η

w1 (m 1 ) · · · w6 (m 6 )⎦

.

l(m)=0

The second line is bounded by η

wk 1+η ∞

6 l=1,l=k

η

wl 1+η 1 .

The required estimate for the sum in the first line follows from the following claim: For any permutation ( j1 , j2 , j3 ) of {1, 4, 5}, and for any permutation (n 1 , n 2 , n 3 ) of {2, 3, 6}, we have m 4 ,m 5 =m 1 , m 2 ,m 3 =m 6

w1 (m 1 ) · · · w6 (m 6 ) |q(m)|1+η |m 2 − m 6 |1+η |m 6 − m 3 |1+η

(23)

l(m)=0, q(m)=0

w j1 ∞ w j2 ∞ w j3 1 wn 1 ∞ wn 2 ∞ wn 3 1 . To prove this inequality, replace m j1 in the sum with a linear combination of other indices using the identity l(m) = 0. We claim that |q(m)|−1−η 1m

j2

1,

664


where the implicit constant is independent of the remaining indices. Indeed, it suffices to consider the cases j1 = 1, j2 = 4 and j1 = 4, j2 = 5 since m 4 and m 5 enter symmetrically. In the former case q(m) = (m 4 + m 5 + m 6 − m 2 − m 3 )2 + m 22 + m 23 − m 24 − m 25 − m 26 = C1 m 4 + C2 , where the integers C1 , C2 depend on m 2 , m 3 , m 5 , m 6 . Moreover, C1 = 0 since m 1 = m 4 . Therefore, sup

m 2 ,m 3 ,m 5 ,m 6

|q(m)|−1−η 1m 1. 4

In the latter case q(m) = C1 + C2 m 5 − 2m 25 , where the integers C1 , C2 depend on m 1 , m 2 , m 3 , m 6 . Since for any integers n, C1 , C2 , the equation n = C1 + C2 m 5 − 2m 25 has at most two solutions, we have sup

m 1 ,m 2 ,m 3 ,m 6

|q(m)|−1−η 1m 1. 5

Using this claim, we obtain

w2 (m 2 )w3 (m 3 )w6 (m 6 ) |m 2 − m 6 |1+η |m 6 − m 3 |1+η wn 3 (m n 3 ) ≤ w j1 ∞ w j2 ∞ w j3 1 wn 1 ∞ wn 2 ∞ |m 2 − m 6 |1+η |m 6 − m 3 |1+η w j1 ∞ w j2 ∞ w j3 1 wn 1 ∞ wn 2 ∞ wn 3 1 .

(23) w j1 ∞ w j2 ∞ w j3 1

3.3. Cancellation property of H4r2 . We claim that {H4r2 , F j } = 0, j = 1, 2. Indeed, by (17) and (18), both F1 and F2 have the phase invariant property F j (v) = F j (veiφ ), but the evolution induced by H4r2 is just uniform phase rotation, v(n, t) = ei2Pt v(n, 0). Thus, {H4r2 , F j } :=

d ) = 0, F j (X t=0 H4r2 dt

j = 1, 2.


665

4. Proof of Proposition 2.2 δ , we should prove that the p,δ norm of each of the summands Assuming that v ∈ Bε,C 3/2−1/ p− in (16) is ε for p = 1 and p = ∞. To simplify the exposition, we will do this only in the case δ = 0. The proof for the case δ > 0 is similar by using the simple multiplication by eδ|·| argument we used in the proof of Proposition 2.1. Note that it suffices to consider the ∂v(k) derivatives of the following terms ¯

H4r1 , {H4r1 , F1 }, {H4r1 , F2 }, {H4nr , F1 }r , g bF2 g aF1 (H4 ), a + b ≥ 2, and the terms involving integrals. We define

f 1 (v1 , v2 , v3 )(k) :=

m 1 ,m 2 =k

v1 (m 1 )v2 (m 2 )v3 (m 1 + m 2 − k) (m 1 − k)(m 2 − k)

¯ = ∂v(k) so that f 1 (v, v, v)(k) ¯ F1 . Similarly we define f 2 (v1 , v2 , v3 , v4 , v5 )(k) so that f 2 (v, v, v, v, ¯ v)(k) ¯ = ∂v(k) ¯ F2 . The following lemma will be used repeatedly: Lemma 4.1. I) For any q ∈ [1, ∞] and any permutation (i 1 , i 2 , i 3 ) of (1, 2, 3), we have f 1 (v1 , v2 , v3 )q vi1 q vi2 ∞− vi3 ∞− . II) For any q ∈ [1, ∞], for any η > 0, and for any i ∈ {1, 2, 3, 4, 5} there is a permutation (i 1 , i 2 , i 3 , i 4 ) of the set {1, 2, 3, 4, 5}\{i} such that f 2 (v1 , v2 , v3 , v4 , v5 )q vi q vi1 1

4

1

η

1+η vil 1+η ∞ vil 1 .

l=2

Proof. Part I can easily be verified following the proof of Proposition 2.1 with δ = 0. Part II follows from Lemma 3.2 and interpolation. r1 4.1. Estimate of ∂v(k) ¯ H4 . Recall that

H4r1 = i

|v(m)|4 ,

m

and hence ∂ H4r1 = 2i|v(k)|2 v(k). ∂ v(k) ¯ We estimate the contribution of this term as ∂ H r1 4 v 3 ∞ ε3/2 , ∂ v(·) ¯ ∞

and

∂ H r1 4 ∂ v(·) ¯

3

1

v 3 1 = v33 ε 2 −1 .

666


r1 r1 4.2. Estimates for ∂v(k) ¯ {H4 , F1 } and ∂v(k) ¯ {H4 , F2 }. Let

H˜ 4r1 (v1 , v2 , v3 , v4 ) :=

v1 (n)v2 (n)v3 (n)v4 (n).

n

We use duality as in (20). Note that the following two terms:

k

r1 |∂v(k) ¯ {H4 , F1 }||h(k)| is bounded by the sum of

H˜ 4r1 (| f 1 (|v|, |v|, |v|)|, |h|, |v|, |v|),

H˜ 4r1 (| f 1 (|h|, |v|, |v|)|, |v|, |v|, |v|),

and similar terms obtained by permuting the arguments. The following estimates (with p = 1 and p = ∞), which follow from the definition of H˜ 4r1 and Lemma 4.1, completes r1 the analysis of ∂v(k) ¯ {H4 , F1 }: H˜ 4r1 (| f 1 (|v|, |v|, |v|)|, |h|, |v|, |v|) h p v p v∞ f 1 (|v|, |v|, |v|)∞ h p v p v2∞ v2∞− ε 2

5

− 1p −

,

5

− 1p −

.

H˜ 4r1 (| f 1 (|h|, |v|, |v|)|, |v|, |v|, |v|) f 1 (|h|, |v|, |v|) p v p v2∞ h p v p v2∞ v2∞− ε 2

r1 We estimate ∂v(k) ¯ {H4 , F2 } similarly. The estimates below imply the required bound

H˜ 4r1 (| f 2 (|v|, . . . , |v|)|, |h|, |v|, |v|) h p v p v∞ f 2 (|v|, . . . , |v|)∞ 3

3η

1+η h p v p v2∞ v1 v1+η ∞ v 1 5

ε2

− 1p −

, r1 ˜ H4 (| f 2 (|h|, |v|, |v|, |v|, |v|)|, |v|, |v|, |v|) f 2 (|h|, . . . , |v|) p v p v2∞ 3

3η

1+η h p v1 v p v2∞ v1+η ∞ v 1 5

ε2

− 1p −

.

In both estimates, the last inequality is obtained by taking η sufficiently small. nr r 4.3. Estimate of ∂v(k) ¯ {H4 , F1 } . Based on the calculations in Sect. 3.1, we have

{H4nr , F1 }r = 2i

m 4 ,m 5 =m 1 , m 2 ,m 3 =m 6

v(m 1 )v(m 2 )v(m 3 )v(m ¯ 4 )v(m ¯ 5 )v(m ¯ 6) − c.c. (m 2 − m 6 )(m 3 − m 6 )

l(m)=0, q(m)=0

Using duality as above we need to estimate 6 terms of the form m 4 ,m 5 =m 1 , m 2 ,m 3 =m 6

l(m)=0, q(m)=0

v1 (m 1 )v2 (m 2 )v3 (m 3 )v4 (m 4 )v5 (m 5 )v6 (m 6 ) , |m 2 − m 6 ||m 3 − m 6 |

(24)


667

where v j = |h| and others are |v|. The required estimates follow from the following claim: For any η > 0, for any permutation ( j1 , j2 , j3 ) of {1, 4, 5}, and for any permutation (n 1 , n 2 , n 3 ) of {2, 3, 6}, we have 1 η (24) v j1 ∞ v j3 1 vn 1 ∞ vn 3 1 v j2 ∞ vn 2 ∞ 1+η v j2 1 vn 2 1 1+η . As in the proof of Lemma 3.2, see (22), the claim follows from an estimate for the following sum: m 4 ,m 5 =m 1 , m 2 ,m 3 =m 6

v1 (m 1 )2 v(m 2 )v3 (m 3 )v4 (m 4 )v5 (m 5 )v6 (m 6 ) . |m 2 − m 6 |1+η |m 3 − m 6 |1+η

(25)

l(m)=0, q(m)=0

First we replace j1 in the equation q(m) = 0 using l(m) = 0. By symmetry it suffices to consider two cases j1 = 1, j1 = 4. In the former case we have 0 = (m 2 + m 3 − j2 − j3 − m 6 )2 + m 22 + m 23 − j22 − j32 − m 26 = −2 j2 (m 2 + m 3 − j3 − m 6 ) + (m 2 + m 3 − j3 − m 6 )2 + m 22 + m 23 − j32 − m 26 . Moreover, m 2 + m 3 − j3 − m 6 = 0 since m 1 = m 4 , m 5 . Therefore, both j1 and j2 are determined by the remaining indices. This implies that (25) v j1 ∞ v j2 ∞ v j3 1

m 2 ,m 3 =m 6

v(m 2 )v3 (m 3 )v6 (m 6 ) |m 2 − m 6 |1+η |m 3 − m 6 |1+η

v j1 ∞ v j2 ∞ v j3 1 vn 1 ∞ vn 2 ∞ vn 3 1 , which leads to the desired estimate as in the previous sections. The case j1 = 4 is similar, the only difference is that j2 is determined as roots of a quadratic polynomial instead of a linear one. 4.4. Estimate of ∂v(k) g bF2 g aF1 (H4 ).. The bounds for ∂v(k) g bF2 g aF1 (H4 ) will be obtained ¯ ¯ inductively. Although we only need to consider the cases when a + b ≥ 2, we start with the case a = 1, b = 0 for clarity. Note that g 1F1 (H4 ) is a sum of terms of the form H4 (v1 , v2 , v3 , v4 ) = v1 (n 1 )v2 (n 2 )v3 (n 3 )v4 (n 4 ), n 1 −n 2 +n 3 −n 4 =0

1 ¯ To estimate ∂v(k) where one of vi ’s is f 1 or f¯1 and the others are v or v. ¯ g F1 (H4 ) p , we use duality as before: ∂ ∂ 1 1 (26) g F1 (H4 ) ≤ sup g F1 (H4 ) |h(k)|. ∂ v(k) ¯ ∂ v(k) ¯ h p =1 p k

Note that the sum in the right-hand side of (26) is bounded by the sum of the following two terms: H4 (| f 1 (|v|, |v|, |v|)|, |h|, |v|, |v|),

H4 (| f 1 (|h|, |v|, |v|)|, |v|, |v|, |v|),

and similar terms obtained by permuting the arguments. The following lemma will be used to estimate these terms and the ones appearing in the higher order commutators.

668


Lemma 4.2. For any q ∈ [1, ∞] and any permutation (i 1 , i 2 , i 3 , i 4 ) of (1, 2, 3, 4), we have |H4 (v1 , v2 , v3 , v4 )| ≤ vi1 q vi2 q vi3 1 vi4 1 . Proof. Note that for any permutation we can write H4 (v1 , v2 , v3 , v4 ) = vi1 ( j) vi2 ∗ vi3 ∗ vi4 ( j). j

The statement follows from Hölder’s and Young’s inequalities. Using Lemma 4.2 and Lemma 4.1, we obtain H4 (| f 1 (|v|, |v|, |v|)|, |h|, |v|, |v|) h p v p v1 f 1 (|v|, |v|, |v|)1 h p v p v1 v1 v2∞− 1

h p ε 2

− 1p −

.

Similarly, we have H4 (| f 1 (|h|, |v|, |v|)|, |v|, |v|, |v|) f 1 (|h|, |v|, |v|) p v p v21 h p v2∞− v p v21 1

h p ε 2

− 1p −

.

Similar bounds follow for the terms obtained by permuting the arguments. Therefore we have ∂ 1 1 1 2 − p −. g F1 (H4 ) ∂ v(k) p ε ¯

Note that this gives an error of order 1 when p = 2. This explains why we consider higher order commutators and a second normal form transform (see footnote 2). This proof motivates the following generalization: Lemma 4.3. Consider H4 (|v|, |v|, |v|, |v|). Repeatedly (a times) replace one of the v’s with f 1 (|v|, |v|, |v|). Then repeatedly (b times) replace one of the v’s with f 2 (|v|, |v|, |v|, |v|, |v|). Finally, replace one of the v’s with h. We denote any such function by H4,a,b ( f 1 , f 2 , h, v). Then, for p = 1 and p = ∞, we have |H4,a,b ( f 1 , f 2 , h, v)| h p ε

a+b− 21 − 1p −

.

Proof. First by using Lemma 4.1 repeatedly (with sufficiently small η, we see that any composition of f 1 ’s and f 2 ’s satisfy b 3− 1+ , (27) · q vq v2a ∞− v1 v∞ where a is the number of f 1 ’s and b is the number of f 2 ’s appearing in the composition. Now, note that H4 has four arguments. Let a j (resp. b j ) be the number of f 1 ’s (resp. f 2 ’s) appearing in the j th argument. Only one of the arguments contains h, say the first one. Using Lemma 4.2, we have |H4 (v1 , v2 , v3 , v4 )| v1 p v2 p v3 1 v4 1 .


669

Using (27), we have 2(a +a3 +a4 )

v2 p v3 1 v4 1 v p v1 v1 v∞−2

v1+ v3− ∞ 1

b2 +b3 +b4

.

Next, note that v1 is either |h| (in which case we stop) or f 1 (v1,1 , v1,2 , v1,3 ) or f 2 (v1,1 , . . . , v1,5 ). In the latter cases, without loss of generality, v1,1 contains |h|. We estimate, using (27) and a simple induction, b1 1 v1+ v1 p h p v2a v3− . ∞ ∞− 1 Combining these estimates we obtain 3− 1+ |H4,a,b ( f 1 , f 2 , h, v)| h p v p v1 v1 v2a ∞− v1 v∞ h p ε

a+b− 21 − 1p −

b

.

b a Using duality as above we see that the right-hand side of (26) for ∂v(k) ¯ g F2 g F1 (H4 ) can be bounded by a finite sum of functions H4,a,b ( f 1 , f 2 , h, v). Therefore, Lemma 4.3 implies that ∂g b g a (H4 ) 3 1 F2 F1 a+b− 21 − 1p − − − ε 2 p , if a + b ≥ 2. ε p ∂ v(k) ¯

4.5. Remainder Estimates. It remains to estimate the terms involving integrals. Note that it suffices to prove the inequalities ∂ 3 1 − − 3 s sup g F1 (H ) ◦ X F1 ε 2 p , ∂ v(k) ¯ s∈[0,1] p ∂ 3 1 − − 3 s sup {g F1 (H ) ◦ X F1 , F2 } ε2 p , ¯ p s∈[0,1] ∂ v(k) ∂ 3 1 − − 2 1 s sup g F2 (H ◦ X F1 ) ◦ X F2 ε2 p ∂ v(k) ¯ p s∈[0,1] 1

−1

for p = 1, ∞ assuming that v p ε 2 p , p ∈ [1, ∞]. Since we have to estimate the composite function derivative, we first study the bounds on the derivatives of X sF j (v), j = 1, 2, s ∈ [0, 1], more precisely, let w(m) = [X sF j (v)](m), which is the solution at t = s of the system ∂ Fj dw(m) = , dt ∂ w(m) ¯

w|t=0 = v.

Differentiating this equation with d respect to initial condition w(n)|t=0 = v(n) and using the notation Dn , we see that dt Dn w(m) is bounded by a sum of terms of the form f 1 (v1 , v2 , v3 )(m), for j = 1, f 2 (v1 , v2 , v3 , v4 , v5 )(m), for j = 2,

670


where one of the vk ’s is |Dn w| and the others are |w|. Without loss of generality we can d Dn w. ¯ Note that at s = 0, we assume that v1 = |Dn w|. We have a similar formula for ds have Dn w(m)∞ 1 = Dn w(m)∞ 1 = 1. m n n m 1 We will prove that both of these norms remain bounded for s ∈ [0, 1]. Taking the ∞ m n norm of f j we obtain

d Dn w(m) dt

1 ∞ m n

f j (|Dn w|, . . . , |w|)(m)∞ 1 m n

≤ f j (Dn w1n , . . . , |w|)(m)

∞ m

1− Dn w(m)∞ . 1ε m n

In the last line, we used Lemma 4.1 (for sufficiently small η). This implies that (with w(m) = [X sF j (v)](m), j = 1, 2) sup Dn w(m)∞ 1 1. m n

(28)

sup Dn w(m)∞ 1 1. n m

(29)

0≤s≤1

Similarly, we obtain 0≤s≤1

We also need the following estimates for the higher order derivatives of w = X sF1 (v) with respect to the initial conditions: 1

D j Dn w(k)∞ 1 ε 2 − , j,n k

D j Dm Dn w(k)∞

1 j,m,n k

1,

1

D j Dn w(k)∞ 1 ε 2 − , k,n j

D j Dm Dn w(k)∞

1,

1 k,m,n j

which can be obtained using Lemma 4.1 as in the proof of (28), (29). Remark 4.1. For δ > 0, a similar argument implies δ|n−m| δ|n−m| Dn w(m) ∞ 1 1, Dn w(m) e e

1 ∞ n m

m n

and for higher order derivatives of w = X sF1 (v) we have δ| j1 +···+ jk −m| D j1 . . . D jk w(m) ∞ 1, e j ,..., j 1m 1 k δ| j1 +···+ jk −m| D j1 . . . D jk w(m) ∞ 1. e 1 m, j

2 ,..., jk j1

The rest of the argument follows as in other sections.

1,


671

s 3 nr 4.5.1. Estimation of ∂v(k) ¯ g F1 (H ) ◦ X F1 (v). Since g F1 ( 2 ) = −H4 , it suffices to estimate ∂ a s sup g F1 (H4 ) ◦ X F1 (v) p , a = 2, 3. ¯ s∈[0,1] ∂ v(k)

When a = 2, we estimate this expression rather than the one containing H4nr (as we should have) because it simplifies the notation and still implies the estimate for the required expression. Note that ∂g a (H4 ) ∂ w(i) ∂ ¯ F1 a s g F1 (H4 ) ◦ X F1 (v) ≤ + c.c.1 ∂ v(k) k ¯ ∂ w(i) ¯ ∂ v(k) ¯ 1 1 k

k

i

a ∂g F1 (H4 ) Dk w(i) ≤ ¯ i∞ 1k + c.c.1k 1 ∂ w(i) ¯ i

ε

1 2−

,

for a ≥ 2 by (28) and the estimates we obtained in Subsect. 4.4. Similarly, using (29), we obtain ∂ 3 a s g F1 (H4 ) ◦ X F1 (v) ε 2 − , for a ≥ 2. ∂ v(k) ¯ ∞ k

s s 3 2 1 The estimates for ∂v(k) ¯ {g F1 (H ) ◦ X F1 , F2 } and ∂v(k) ¯ g F2 (H ◦ X F1 ) ◦ X F2 are similar. The only difference is that we also require the higher order derivative estimates of w = X sF1 (v) listed above. We omit the details.

Appendix A. Nonlinear Fiber Optics Application One of the most important applications of NLS concerns light-wave communication systems, where optical pulses in a retarded time frame evolve according to the one dimensional NLS i A z + Sd(z)Aτ τ + g(z)|A|2 A = 0.

(30)

Here z is the rescaled distance, τ is the rescaled retarded time, A is the amplitude of the optical wave envelope, d(z) is the group velocity dispersion, which is usually piecewise constant, and S is the dispersion strength parameter. Finally, g(z) > 0 is the nonlinear coefficient which accounts for the losses and amplifications. For the derivation of NLS from Maxwell’s equations, one can consult many references, e.g. [9]. It is a standard assumption that d(z) and g(z) are periodic. In general, in light-wave communication, the information is transmitted with localized pulses (with Gaussian or exponential tails) in allocated time slots. The presence of pulse corresponds to “1” and the absence of pulse corresponds to “0” in binary format. Naturally, it is preferable that the incoming waveform would appear undistorted at the end of the transmission line. It can be achieved by optimizing an individual pulse, so it would propagate without distortion, and sending such pulses together, keeping them sufficiently far apart (i.e. taking time slots sufficiently large), so they would not interact. Such regime is usually called “soliton regime” in the optical communication literature,

672


where the word “soliton” does not usually mean that the equation is integrable. The pulses could be, for example, dispersion managed solitons, which are approximately periodic localized solutions of the above equation. In other words, the main feature of the soliton regime is that the pulses do not interact (or rather pulse to pulse interaction is weak compared to the pulse self-interaction) during the propagation through the transmission line. An alternative regime (often called the quasi-linear regime) has been found when the pulses strongly overlap during the transmission, see e.g. the survey paper [7]. Surprisingly, it was observed that up to a linear transformation of the transmitted waveform, the pulses appeared undistorted. Note that even though the pulses spread over many time slots, the average optical energy (L 2 norm square) per bit does not change and therefore nonlinear effects remain strong. It is usually implicitly assumed in the engineering literature that “nonlinearity gets averaged out” due to the high frequency of the initial data. In this article, we rigorously explain the quasi-linear phenomenon for a model problem when d(z) and g(z) are constant and all bits are occupied by 1 s, in the limit of vanishing pulse width. This case (of all identical 1’s) leads to the formulation with periodic boundary conditions. Although, this is a special case, we hope that our proof can be extended to the more general case: pseudo-random sequence of 1 s and 0 s. Note that constant d(z) and g(z) assumption is not restrictive since if the evolution is quasi-linear on each interval where d(z), g(z) are constant, then the evolution is quasi-linear on their union. There has been previous work on the quasi-linear regime. In [15], the limit of the short pulse width for dispersion managed NLS on the real line is considered. An effective evolution equation was derived which turned out to be integrable and weakly nonlinear. The equation was later improved in [1]. On the real line the energy disperses to infinity and therefore nonlinearity becomes small. This leaves an open question: what will happen if the energy does not disperse to infinity or in other words, there is an infinite bit stream. The problem considered in this paper models this situation: nonlinearity remains strong which is due to the periodic boundary conditions. Finally, we note that on R, the dispersion strength S and pulse width ε can be combined into a single effective parameter S/ε2 by scaling τ . This implies that the limits S → ∞ and ε → 0 are equivalent. This is not the case in our model since the characteristic τ -scale, bit size, is already present. Therefore, the two parameter problem in S, ε should be considered. However, the limit S → ∞ is insufficient to achieve quasi-linear evolution and must be supplemented with ε → 0. On the other hand, the limit ε → 0 does produce quasi-linear evolution with S being fixed but arbitrary. This motivated us to consider only this case. We put S = 1 in order not to obscure the exposition. Acknowledgement. The authors would like to thank Eugene Wayne for many helpful discussions.

References 1. Ablowitz, M., Hirooka, T., Biondini, G.: Quasi-linear optical pulses in strongly dispersion-managed transmission systems. Optics Lett. 26, 459–462 (2000) 2. Bergano, N.S. et al.: Dig. Optical Fiber Communications Conf., 1998, Postdeadline Paper 12, Washington, DC: Opt. Soc of Amer., 1998 3. Biondini, G., Kodama, Y.: On the Whitham equations for the defocusing nonlinear Schrödinger equation with step initial data. J. Nonlinear Sci. 16(5), 435–481 (2006)


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4. Bourgain, J.: Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Part I: Schrödinger equations. GAFA 3(2), 107–156 (1993) 5. Bourgain, J.: A remark on normal forms and the “I -method” for periodic NLS. J. Anal. Math. 94, 125–157 (2004) 6. Deift, P., Zhou, X.: Perturbation theory for infinite-dimensional integrable systems on the line. A Case Study. Acta Math. 188(2), 1871–2509 (2002) 7. Essiambre, R.-J., Raybon, G., Mikkelsen, B.: Pseudo-linear transmission of high-speed TDM signals: 40 and 160 Gb/s. Optical Fiber Communications IV, I. Kaminow, T. Li, ed. San Diego, CA: Academic, 2002, pp. 232–304 8. Gershgorin, B., Lvov, Y., Cai, D.: Renormalized waves and discrete breathers in β-Fermi-Pasta-Ulam chains. Phys. Rev. Lett. 95, 264302 (2005) 9. Hasegawa, A., Kodama, Y.: Solitons in Optical Communication. New York: Oxford University Press, 1995 10. Kamvissis, S., McLaughlin, K., Miller, P.: Semiclassical soliton ensembles for the focusing nonlinear Schrödinger equation. Annals of Mathematical Studies 154, Princeton, NJ: Princeton University Press, 2003 11. Kuksin, S.: Analysis of Hamiltonian PDEs. New York: Oxford University Press, 2000 12. Kuksin, S., Poschel, J.: Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation. Ann. Math. 142, 149–179 (1995) 13. Mamyshev, P.V., Mamysheva, N.A.: Pulse-overlapped dispersion-managed data transmission and inrachannel four-wave mixing. Opt. Lett. 24, 1454–1456 (1999) 14. Mikkelsen, B. et al.: 320-Gb/s Single-Channel pseudolinear transmission over 200 km of nonzero-dispersion fiber. IEEE Photon. Technol. Lett. 12, 1400–1402 (2000) 15. Manakov, S.V., Zakharov, V.E.: On propagation of short pulses in strong dispersion managed optical lines. Sov. Phys. JETP Lett. 70, 578–582 (1999) 16. Tovbis, A., Venakides, S., Zhou, X.: On the long-time limit of semiclassical (zero dispersion limit) solutions of the focusing nonlinear Schrödinger equation: pure radiation case. Comm. Pure Appl. Math. 59, 1379–1432 (2006) 17. Zakharov, V.E., Shabat, A.B.: Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP Lett. 37, 823–828 (1973) Communicated by P. Constantin


Communications in


Height Fluctuations in the Honeycomb Dimer Model Richard Kenyon Department of Mathematics, Brown University, Providence, RI, USA. E-mail: [email protected] Received: 29 May 2007 / Accepted: 18 January 2008 Published online: 29 May 2008 – © Springer-Verlag 2008

Abstract: We study a model of random surfaces arising in the dimer model on the honeycomb lattice. For a fixed “wire frame” boundary condition, as the lattice spacing → 0, Cohn, Kenyon and Propp [3] showed the almost sure convergence of a random surface to a non-random limit shape 0 . In [12], Okounkov and the author showed how to parametrize the limit shapes in terms of analytic functions, in particular constructing a natural conformal structure on them. We show here that when 0 has no facets, for a family of boundary conditions approximating the wire frame, the large-scale surface fluctuations (height fluctuations) about 0 converge as → 0 to a Gaussian free field for the above conformal structure. We also show that the local statistics of the fluctuations near a given point x are, as conjectured in [3], given by the unique ergodic Gibbs measure (on plane configurations) whose slope is the slope of the tangent plane of 0 at x. Contents 1.

2.

Introduction . . . . . . . . 1.1 Dimers and surfaces . . 1.2 Results . . . . . . . . . 1.2.1 Limit shape. . . . . 1.2.2 Local statistics. . . 1.2.3 Fluctuations. . . . 1.3 The Gaussian free field 1.4 Beltrami coefficient . . 1.5 Examples . . . . . . . 1.6 Proof outline . . . . . . Definitions . . . . . . . . . 2.1 Graphs . . . . . . . . . 2.1.1 Dual graph. . . . .

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2.1.2 Forms. . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Heights and asymptotics . . . . . . . . . . . . . . . . 2.3 Measures and gauge equivalence . . . . . . . . . . . . 2.4 Kasteleyn matrices . . . . . . . . . . . . . . . . . . . 2.5 Measures in infinite volume . . . . . . . . . . . . . . . 2.6 T -graphs . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Definition. . . . . . . . . . . . . . . . . . . . . . 2.6.2 Associated dimer graph and Kasteleyn matrix. . . . 2.6.3 Harmonic functions and discrete analytic functions. 2.6.4 Green’s function and K G−1 . . . . . . . . . . . . . . D Constant-Slope Case . . . . . . . . . . . . . . . . . . . . . 3.1 T -graph construction . . . . . . . . . . . . . . . . . . 3.2 Boundary behavior . . . . . . . . . . . . . . . . . . . 3.2.1 Flows and dimer configurations . . . . . . . . . . 3.2.2 Canonical flow. . . . . . . . . . . . . . . . . . . . 3.2.3 Boundary height. . . . . . . . . . . . . . . . . . . 3.3 Continuous and discrete harmonic functions . . . . . . 3.3.1 Discrete and continuous Green’s functions. . . . . 3.3.2 Smooth functions. . . . . . . . . . . . . . . . . . 3.4 K G−1 in constant-slope case . . . . . . . . . . . . . . . D 3.4.1 Values near the diagonal. . . . . . . . . . . . . . . General Boundary Conditions . . . . . . . . . . . . . . . . 4.1 The complex height function . . . . . . . . . . . . . . 4.2 Gauge transformation . . . . . . . . . . . . . . . . . . 4.3 Embedding . . . . . . . . . . . . . . . . . . . . . . . 4.4 Boundary . . . . . . . . . . . . . . . . . . . . . . . . Continuity of K −1 . . . . . . . . . . . . . . . . . . . . . . Asymptotic Coupling Function . . . . . . . . . . . . . . . Free Field Moments . . . . . . . . . . . . . . . . . . . . . Boxed Plane Partition Example . . . . . . . . . . . . . . .

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1. Introduction 1.1. Dimers and surfaces. A dimer covering, or perfect matching, of a finite graph is a set of edges covering all the vertices exactly once. The dimer model is the study of random dimer coverings of a graph. Here we shall for the most part deal with the uniform measure on dimer coverings. A good reference is [11]. In this paper we study the dimer model on the honeycomb lattice (the periodic planar graph whose faces are regular hexagons), or rather, on large pieces of it. This model, and more generally dimer models on other periodic bipartite planar graphs, are statistical mechanical models for discrete random interfaces. Part of their interest lies in the conformal invariance properties of their scaling limits [8,9]. Dimer coverings of the honeycomb graph are dual to tilings with 60◦ rhombi, also known as lozenges, see Fig. 1. Lozenge tilings can in turn be viewed as orthogonal projections onto the plane P111 = {x + y + z = 0} of stepped surfaces which are polygonal surfaces in R3 whose faces are squares in the 2-skeleton of Z3 (the stepped surfaces are monotone in the sense that the projection is injective), see Figs. 1,3. Each stepped surface is the graph of a function, the normalized height function, on the √ underlying tiling, which is linear on each tile. This function is defined simply as 3 times the

Height Fluctuations in the Honeycomb Dimer Model

677

Fig. 1. Honeycomb dimers (solid) and the corresponding “lozenge” tiling (green)

√ distance from the surface to the plane P111 . (The scaling factor 3 is just to make the function integer-valued on Z3 . The value of the height function at a vertex is the sum of its coordinates.) 1.2. Results. We are interested in studying the scaling limit of the honeycomb dimer model, that is, the limiting behavior of a uniform random dimer covering of a fixed plane region U when the lattice spacing goes to zero. Equivalently, we take stepped surfaces in Z3 and let → 0. As boundary conditions we are interested in stepped surfaces spanning a “wire frame” which is a simple closed polygonal path γ in Z3 . We take γ converging as → 0 to a smooth path γ which projects to ∂U . 1.2.1. Limit shape. Let U be a domain in P111 , and γ be a smooth closed curve in R3 , projecting orthogonally to ∂U . For each > 0 let γ be a nearest-neighbor path in Z3 approximating γ (in the Hausdorff metric) and which can be spanned by a monotone stepped surface , monotone in the sense that it projects injectively to P111 , or in other words it is the graph of a function on P111 . See for example Fig. 3 (although there the boundary is only piecewise smooth). The existence of such an approximating sequence imposes constraints on γ , see [3,5], as follows: the curve γ can be spanned by a surface , which is the graph of a continuous function on U , and such that the normal ν to the surface points into the positive orthant R3≥0 . Conversely, any such curve γ can be approximated by γ as above, see [3]. The condition of positivity of the normal to can be stated in terms of the gradient of the function h whose graph is : this gradient must lie in a certain triangle. The formulation in terms of the normal is more symmetric, however. For a given γ there are, typically, many spanning surfaces and we study the limiting properties of the uniform measure on the set of as → 0. For a surface spanning γ , let h : P111 → R be the normalized height function, defined on the region enclosed by U := π111 (γ ), whose graph is . Under the above hypotheses Cohn, Kenyon and Propp proved the existence of a limit shape: Theorem 1.1 (Cohn, Kenyon, Propp [3]). The distribution of h converges as → 0 a.s. to a nonrandom function h¯ : U → R. The function h¯ is the unique function h which minimizes the “surface tension” functional

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σ (∇h) d x d y,

min h

U

where, in terms of the normal vector ( pa , pb , pc ) ∈ R3 to the graph of h scaled so that pa + pb + pc = 1, we have σ ( pa , pb , pc ) = − π1 (L(π pa ) + L(π pb ) + L(π pc )) and x L(x) = − 0 log(2 sin t) dt is the Lobachevsky function. Here the minimum is over Lipschitz functions whose graph has normal with nonnegative coordinates. Equivalently, these are functions whose gradient lies in a certain triangle. In the above formula the surface tension σ is the negative of the exponential growth rate of the number of discrete surfaces of average slope ∇h. The function h¯ is called the asymptotic height function. Its graph is a surface 0 spanning γ . 1.2.2. Local statistics. Suppose that the gradient of h¯ is not maximal at any point in U¯ , i.e. the normal to 0 has nonzero coordinates at every point of U¯ . In this paper we show that, if the precise local behavior of the approximating curves γ is chosen in a particular way, then both the local statistics and the global height fluctuations of can be determined. Here is the result on the local statistics. Theorem 1.2. Suppose that the gradient of h¯ is not maximal at any point in U¯ , that is, the normal vector to the surface has nonzero coordinates at every point. Under appropriate hypotheses on the local structure of the approximating curves γ , the local statistics of near a given point are given by the unique Gibbs measure1 of slope equal to the slope of h¯ at that point. For the precise statement see Theorem 6.1. In particular the hypotheses on γ are explained in Sects. 2.6 and 3.2. Suppose that ( pa , pb , pc ) is a normal vector to the surface at a point. Recall that pa , pb , pc > 0. If we rescale so that pa + pb + pc = 1, then one consequence of Theorem 6.1 is that the quantities pa , pb , pc are the densities of the three orientations of lozenges near the corresponding point on the surface. 1.2.3. Fluctuations. The fluctuations are the image of the Gaussian free field under a certain diffeomorphism from the unit disk D to U . To describe the fluctuations, we first describe the relevant conformal structure on U . It is a function of the normal to the ¯ and is defined as follows. Let ( pa , pb , pc ) be the normal to h, ¯ scaled so graph of h, that pa + pb + pc = 1. Let θa = π pa , θb = π pb , θc = π pc . Let a, b, c be the edges of a Euclidean triangle with angles θa , θb , θc . Let z = −e−iθc and w = −eiθb so that a + bz + cw = 0. See Fig. 2; here the triangle on the left has edges a, bz, cw when these edges are oriented counterclockwise. We define = −cw/a. All of these quantities are functions on U , although a, b, c are only defined up to scale. Let x, ˆ yˆ , zˆ be the unit vectors in P111 in the directions of the projections of the standard basis vectors in R3 . 1 Sheffield [19] showed that there is a 2-parameter family of Gibbs measures on dimer covers of the honeycomb lattice, one for each possible average slope of the height function; see Sect. 2.5 below.


679

Φ θa

θa

b

c

θc

θb a

θb

θc

0

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Fig. 2. The triangle and a scaled copy with vertices 0, 1,

Theorem 1.3 ([12]). The function satisfies the complex Burgers equation xˆ + yˆ = 0,

(1)

where xˆ , yˆ are directional derivatives of in directions x, ˆ yˆ respectively. The function : U → C can be used to define a conformal structure on U , as follows. A function g : U → C is defined to be analytic in this conformal structure on U if it satisfies gxˆ + g yˆ = 0, where gxˆ , g yˆ are the directional derivatives of g in the directions x, ˆ yˆ respectively. By the Alhfors-Bers theorem there is a diffeomorphism f : U → D satisfying f xˆ + f yˆ = 0; the conformal structure on U is the pull-back of the standard conformal structure on D under f . The conformal structure on U can be described by a Beltrami coefficient ξ (see below) which in the current case is ξ = ( − eiπ/3 )/( − e−iπ/3 ). In the special cases that is constant (which correspond to the cases where γ is contained in a plane), this means that the conformal structure is just a linear image of the standard conformal structure. For example, note that in the standard conformal structure on P111 , a function g is analytic if gxˆ + eiπ/3 g yˆ = 0. So the case = eiπ/3 , which corresponds to the case a = b = c, gives the standard conformal structure (recall that the vectors xˆ and yˆ are 120◦ apart). Theorem 1.4. Suppose that the gradient of h¯ is not maximal at any point in U¯ . Under the same hypotheses on γ as in Theorem 1.2, the fluctuations of the unnormalized height ¯ have a weak limit as → 0 which is the Gaussian free field in the function, 1 (h − h), complex structure defined by , that is, the pull-back under the map f : U → D above, of the Gaussian free field on the unit disk D. For the definition of the Gaussian free field see below. As mentioned above, we require that the normal to the graph of h¯ be strictly inside the positive orthant, so that we have a positive lower bound on the values of pa , pb , pc , whereas the results of [3] and [12] do not require this restriction. Indeed, in many of the simplest cases the surface 0 will have facets, which are regions on which ( pa , pb , pc ) = (1, 0, 0), (0, 1, 0) or (0, 0, 1). Our results do not apply to these situations. This work builds on work of [9,13,12]. Previously Theorem 1.4 was proved for the dimer model on Z2 in a special case which in our context corresponds to the wire frame γ lying in the plane P111 , see [9]. In that case the conformal structure is the standard conformal structure on U . In the present case we still require special boundary conditions, which generalize the “Temperleyan” boundary conditions of [8,9]. It remains an open question whether the result holds for all boundary conditions. The fluctuations in the presence of facets are also unknown, and the current techniques do not seem to immediately extend to this more general setting.

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1.3. The Gaussian free field. The Gaussian free field X on D [18] is a random object in the space of distributions on D, defined on smooth test functions as follows. For any smooth test function ψ on D, D ψ(x)X (x)|d x|2 is a real Gaussian random variable of mean zero and variance given by D D

ψ(x1 )ψ(x2 )G(x1 , x2 )|d x1 |2 |d x2 |2 ,

where the kernel G is the Dirichlet Green’s function on D: x1 − x2 1 . log G(x1 , x2 ) = − 2π 1 − x¯1 x2 A similar definition holds (for the standard conformal structure) on any bounded domain in C, only the expression for the Green’s function is different. An alternative description of the Gaussian free field is that it is the unique Gaussian process which satisfies E[X (x1 )X (x2 )] = G(x1 , x2 ). Higher moments of Gaussian processes can always be written in terms of the moments of order 2; for the Gaussian free field we have E[X (x1 ) . . . X (xn )] = 0 if n is odd, and E[X (x1 ) . . . X (x2k )] =

G(xσ (1) , xσ (2) ) . . . G(xσ (2k−1) , xσ (2k) ),

(2)

pairings

where the sum is over all (2k − 1)!! pairings of the indices. Any process whose moments satisfy (2) is the Gaussian free field [9].

1.4. Beltrami coefficient. A conformal structure on U can be defined as an equivalence class of diffeomorphisms φ : U → D, where mappings φ1 , φ2 are equivalent if the composition φ1 ◦ φ2−1 is a conformal self-map of D. The Beltrami differential ξ(z) ddzz¯ of φ is defined by the formula ξ(z)

φz¯ d z¯ d z¯ = . dz φz dz

The Beltrami differential is invariant under post-composition of φ with a conformal map, so it is a function only of the conformal structure (and in fact defines the conformal structure as well). It is not hard to show that |ξ(z)| < 1; note that ξ(z) = 0 if and only if the map is conformal. The Ahlfors-Bers uniformization theorem [1] says that any smooth function (even any measurable function) ξ(z) satisfying |ξ(z)| < 1 defines a conformal structure.


681

Fig. 3. Boxed plane partition

1.5. Examples. The simplest case is when the wire frame γ is contained in a plane {(x, y, z) ∈ R3 | pa x + pb y + pc z = const}. In this case the limit surface 0 is linear. The normal ( pa , pb , pc ) is constant, and the conformal structure is a linear image of the standard conformal structure. That is, the map f : U → D is a linear map L composed with a conformal map. For a more interesting case, consider the boxed plane partition (BPP) shown in Fig. 3, which is a random lozenge tiling of a regular hexagon. In [4] it was shown that, for a random tiling of the hexagon, the asymptotic height function h¯ is linear outside of the inscribed circle and analytic inside (with an explicit but somewhat complicated formula). Although our theorem does not apply to this case because of the facets outside the inscribed circle, if we choose boundary conditions inside the inscribed circle, and boundary values equal to the graph of the function there, our results apply.√ Suppose that the hexagon has sides of √ length 1, so that the inscribed circle has radius 3/2. Let U be a disk of radius r < 3/2 concentric with it. Suppose that the normalized height function on the boundary of U is chosen to agree with the asymptotic height function of the corresponding region in the BPP, so that the asymptotic height function of U equals the asymptotic height function of the BPP restricted to U . Then the fluctuations on U can be computed using Theorem 7.1. In fact in this setting, the conformal structure can be explicitly computed: take the standard conformal structure on a hemisphere in R3 , and project it orthogonally onto the plane containing its equator. Identifying the equator with the inscribed circle in the BPP gives the relevant conformal structure in U . Remarkably, in this example the Beltrami coefficient is rotationally invariant, even though h¯ itself is not. See Sect. 8. √ We conjecture that the fluctuations for the BPP are given by the limit r → 3/2 of this construction (it is known [4] that fluctuations in the “frozen” regions outside the circle are exponentially small in 1/).

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Fig. 4. The honeycomb graph

1.6. Proof outline. The fundamental tool in the study of the dimer model is the Kasteleyn matrix (defined below). Minors of the inverse Kasteleyn matrix compute (multiple-) edge probabilities in the model. The main goal of the paper is to obtain an asymptotic expansion of the inverse Kasteleyn matrix. This is complicated by the fact that it grows exponentially in the distance between vertices (except in the special case when the boundary height function is horizontal). However by pre- and post-composition with an appropriate diagonal matrix, we can remove the exponential growth and relate K −1 to the standard Green’s function with Dirichlet boundary conditions on a related graph GT . Here is a sketch of the main ideas. 1. We construct a discrete version of the map f of Theorem 1.4. For each we define a directed graph GT embedded in the upper half plane H, and a geometric map φ from U to GT , such that the Laplacian on GT is related (via the construction of [14,15]) with the Kasteleyn matrix on U . The existence of such a graph GT follows from [15]. This is done in Sect. 3.1 in the “constant slope” case and Sect. 4.3 in the general case. 2. Standard techniques for discrete harmonic functions yield an asymptotic expansion for the Green’s function on GT . This is done in Sect. 2.6.4. 3. The asymptotic expansion of the inverse Kasteleyn matrix on U is obtained from the derivative of the Green’s function on GT , pulled back under the mapping φ. See Sects. 2.6.4 and 6. 4. Asymptotic expansions of the moments of the height fluctuations are computed via integrals of the asymptotic inverse Kasteleyn matrix. These moments are the moments of the Gaussian free field on H pulled back under φ. 2. Definitions 2.1. Graphs. Let π111 be the orthogonal projection of R3 onto P111 . Let x, ˆ yˆ , zˆ be the π111 -projections of the unit basis vectors. Define e1 = 13 (xˆ − zˆ ), e2 = 13 ( yˆ − x), ˆ 1 e3 = 3 (ˆz − yˆ ), so that xˆ = e1 − e2 , yˆ = e2 − e3 , zˆ = e3 − e1 . Let H be the honeycomb lattice in P111 : vertices of H are L ∪ (L + e1 ), where L is the lattice L = Z(e1 − e2 ) + Z(e2 − e3 ) = Zxˆ + Z yˆ , and edges connect nearest neighbors. Vertices in L are colored white, those in L + e1 are black. See Fig. 4. 2.1.1. Dual graph. Let U be a Jordan domain in P111 with smooth boundary. In H, take a simple closed polygonal path with approximates ∂U in a reasonable way, for example the polygonal curve is locally monotone in the same direction as the curve ∂U .


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Fig. 5. The “dual” graph G ∗ (solid lines) of the graph G of Fig. 4

Let G be the subgraph of H bounded by this polygonal path. We define a special kind of dual graph G ∗ as follows. Let H∗ be the usual planar dual of H. For each white vertex of G take the corresponding triangular face of H∗ ; the union of the edges forming these triangles, along with the corresponding vertices, forms G ∗ . In other words, G ∗ has a face for each white vertex of G, as well as for black vertices which have all three neighbors in G. See Fig. 5. Throughout the paper the graph G and its related graphs will be scaled by a factor over the corresponding graphs H, and so the G graphs have edge lengths of order , and the graph H and its related graphs have edge lengths of order 1. 2.1.2. Forms. For an edge in G joining vertices b and w, we denote by (bw)∗ the dual edge in G ∗ , which we orient at +90◦ from the edge bw (when this edge is oriented from b to w). A 1-form ω on a graph is a function on directed edges which is antisymmetric with respect to reversing the orientation: ω(v1 v2 ) = −ω(v2 v1 ). A 1-form is also called a flow. If the graph is planar one can similarly define a 1-form on the dual graph. If ω is a 1-form, ω∗ is the dual 1-form, defined by ω∗ ((v1 v2 )∗ ) := ω(v1 v2 ). On a planar graph with a 1-form ω, dω is a function on oriented faces defined by dω( f ) = e ω(e), where the sum is over the edges on a path around the face going counterclockwise. This is also known as the curl of the flow ω. The form dω∗ is a function on vertices (faces of the dual graph), defined by dω∗ (v) = v ∼v ω(v, v ). In other words it is the divergence of the flow ω. A 1-form ω is closed if dω = 0, that is, the sum of ω along any cycle is zero (in the language of flows, the flow has zero curl). It is exact if ω = d f for some function f on the vertices, that is ω(v1 v2 ) = f (v1 ) − f (v2 ). A 1-form is co-closed if its dual form is closed. The corresponding flow is divergence-free. If dω∗ = 0, the integral of ω∗ between two faces of G (i.e. on a path in the dual graph) is the flux, or total flow, between those faces.

2.2. Heights and asymptotics. The unnormalized height function, or just height function, of a tiling is the integer-valued function on the vertices of the lozenges (faces of G) which is the sum of the coordinates of the corresponding point in Z3 . It changes by

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±1 along each edge of a tile. When we scale the lattice by , so that we are discussing surfaces in Z3 , the height function is defined as 1/ times the coordinate sum, so that it is still integer-valued. The normalized height √ function is times the height function, and is the function which, when scaled by 3, has graph which is the surface in Z3 . Let u : ∂U → R be a continuous function with the property that u can be extended to a Lipschitz function u˜ on the interior of U having the property that the normal to the graph of u˜ has nonnegative coordinates, that is, the normal points into the positive orthant R3≥0 . In other words, the graph of u is a wire frame γ of the type discussed before. Let h¯ : U → R be the asymptotic height function with boundary values u, from Theorem 1.1. It is smooth assuming the hypothesis of Theorem 1.4. ¯ scaled so that pa + pb + Let ν = ( pa , pb , pc ) be the normal vector to the graph of h, pc = 1. The directional derivatives of h¯ in the directions x, ˆ yˆ , zˆ are respectively 3 pc − 1, 3 pa − 1, 3 pb − 1.

(3)

2.3. Measures and gauge equivalence. We let µ = µ(G) be the uniform measure on dimer configurations on a finite graph G. If edges of G are given positive real weights, we can define a new probability measure, the Boltzmann measure, giving a configuration a probability proportional to the product of its edge weights. Certain edge-weight functions lead to the same Boltzmann measure: in particular if we multiply by a constant the weights of all the edges in G having a fixed vertex, the Boltzmann measure does not change, since exactly one of these weights is used in every configuration. More generally, two weight functions ν1 , ν2 are said to be gauge equivalent if ν1 /ν2 is a product of such operations, that is, if there are functions F1 on white vertices and F2 on black vertices so that for each edge wb, ν1 (wb)/ν2 (wb) = F1 (w)F2 (b). Gauge equivalent weights define the same Boltzmann measure. It is not hard to show that for planar graphs, two edge-weight functions are gauge equivalent if and only if they have the same face weights, where the weight of a face is defined to be the alternating product of the edge weights around the face (that is, the first, divided by the second, times the third, and so on), see e.g. [13]. In this paper we will only consider weights which are gauge equivalent to constant weights (or nearly so), so the Boltzmann measure will always be (nearly) the uniform measure.

2.4. Kasteleyn matrices. Kasteleyn showed that one can count dimer configurations on planar graph with the determinant of the certain matrix, the “Kasteleyn matrix” [6]. In the current case, when the underlying graph is part of the honeycomb graph, the Kasteleyn matrix K is just the adjacency matrix from white vertices to black vertices. For more general bipartite planar graphs, and when the edges have weights, the matrix is a signed, weighted version of the adjacency matrix [17], whose determinant is the sum of the weights of dimer coverings. Each entry K (w, b) is a complex number with modulus given by the corresponding edge weight (or zero if the vertices are not adjacent), and an argument which must be chosen in such a way that around each face the alternating product of the entries (the first, divided by the second, times the third, and so on) is positive if the face has 2 mod 4 edges and negative if the face has 0 mod 4


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edges (since we are assuming the graph is bipartite, each face has an even number of edges. For nonbipartite graphs, a more complicated condition is necessary). The Kasteleyn matrix is unique up to gauge transformations, which consist of preand post-multiplication by diagonal matrices (with, in general, complex entries). If the weights are real then we can choose a gauge in which K is real, although in certain cases it is convenient to allow complex numbers (we will below). Probabilities of individual edges occurring in a random tiling can likewise be computed using the minors of the inverse Kasteleyn matrix: Theorem 2.1 ([7]). The probability of edges {(b1 , w1 ), . . . , (bk , wk )} occurring in a random dimer covering is k

K (wi , bi ) det{K −1 (bi , w j )}1≤i, j≤k .

i=1

On an infinite graph K is defined similarly but K −1 is not unique in general. This is related to the fact that there are potentially many different measures which could be obtained as limits of Boltzmann measures on sequences of finite graphs filling out the infinite graph. The edge probabilities for these measures can all be described as in the theorem above, but where the matrix “K −1 ” now depends on the measure; see the next section for examples.

2.5. Measures in infinite volume. On the infinite honeycomb graph H there is a twoparameter family of natural translation-invariant and ergodic probability measures on dimer configurations, which restrict to the uniform measure on finite regions (i.e. when conditioned on the complement of the finite region: we say they are conditionally uniform). Such measures are also known as ergodic Gibbs measures. They are classified in the following theorem due to Sheffield. Theorem 2.2 ([19]). For each ν = ( pa , pb , pc ) with pa , pb , pc ≥ 0 and scaled so that pa + pb + pc = 1 there is a unique translation-invariant ergodic Gibbs measure µν on the set of dimer coverings of H, for which the height function has average normal ν. This measure can be obtained as the limit as n → ∞ of the uniform measure on the set of those dimer coverings of Hn = H/n L whose proportion of dimers in the three orientations is ( pa : pb : pc ), up to errors tending to zero as n → ∞. Moreover every ergodic Gibbs measure on H is of the above type for some ν. The unicity in the above statement is a deep and important result. Associated to µν is an infinite matrix, the inverse Kasteleyn matrix of µν , K ν−1 = −1 (K ν (b, w)) whose rows index the black vertices and columns index the white vertices, and whose minors give local statistics for µν , just as in Theorem 2.1. From [13] there is an explicit formula for K ν−1 : let w = m 1 xˆ +n 1 yˆ and b = e1 +m 2 xˆ +n 2 yˆ = w +e1 +m xˆ +n yˆ , where m = m 2 − m 1 , n = n 2 − n 1 . Then a b −1 K ν−1 (b, w) = a( )m ( )n K abc (b, w), b c

(4)

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where a, b, c, z, w are as defined in Sect. (1.2.3) and z 1−m+n w1−n dz 1 dw1 1 −1 K abc (b, w) = (2πi)2 |z|=|w|=1 a + bz 1 + cw1 z 1 w1

−m+n −n

1 1 w z . +O = Im π cwm + an m 2 + n2

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(5) (6)

−1 This formula for K abc and its asymptotics were derived in [13]: they are obtained from the limit n → ∞ of the inverse Kasteleyn matrix on the torus H/n L with edge weights a, b, c according to direction. It is not hard to check from (5) that K K −1 = Id. −1 , that is, obtained by From (4), the matrix K ν−1 is just a gauge transformation of K abc pre- and post-composing with diagonal matrices. Defining F(w) = (bz/a)m 1 (cw/bz)n 1 and F(b) = a(bz/a)−m 2 (cw/bz)−n 2 we can write, using (6),

F(w)F(b) 1 1 + |F(b)F(w)|O( 2 K ν−1 (b, w) = Im ). (7) π cwm + an m + n2 We’ll use this function F below. As a sample calculation, the µν -probability of a single horizontal edge, from w = 0 to b = e1 , being present in a random dimer covering is (see Theorem 2.1) 1 dz 1 dw1 a θa −1 K ν−1 (b, w) = a K abc (b, w) = = , (2πi)2 T2 a + bz 1 + cw1 z 1 w1 π where θa is, as before, the angle opposite side a in a triangle with sides a, b, c. This is consistent with (3).

2.6. T -graphs. 2.6.1. Definition. T -graphs were defined and studied in [15]. A pairwise disjoint coln L is lection L 1 , L 2 , . . . , L n of open line segments in R2 forms a T-graph in R2 if ∪i=1 i connected and contains all of its limit points except for some finite set R = {r1 , . . . , rm }, where each ri lies on the boundary of the infinite component of R2 minus the closure of n L . See Fig. 6 for an example where the outer boundary is a polygon. Elements in R ∪i=1 i are called root vertices and are labeled in cyclic order; the L i are called complete edges. We only consider the case that the outer boundary of the T -graph is a simple polygon, and the root vertices are the convex corners of this polygon. (An example where the outer boundary is not a polygon is a “T” formed from two edges, one ending in the interior of the other.) Associated to a T -graph is a Markov chain GT , whose vertices are the points which are endpoints of some L i . Each non-root vertex is in the interior of a unique L j (because the L j are disjoint); there is a transition from that vertex to its adjacent vertices along L j , and the transition probabilities are proportional to the inverses of the Euclidean distances. Root vertices are sinks of the Markov chain. See Fig. 7. Note that the coordinate functions on GT are harmonic functions on GT \ R. More generally, any function f on GT which is harmonic on GT \ R (we refer to such functions as harmonic functions on GT ) has the property that it is linear along edges, that is, if v1 , v2 , v3 are vertices on the same complete edge then f (v2 ) − f (v3 ) f (v1 ) − f (v2 ) = . (8) v1 − v2 v2 − v3


r

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4

r

r

3

r2

5

r1

r

6

Fig. 6. A T-graph (solid) and associated graph G D (dotted)

Fig. 7. The Markov chain associated to the T -graph of Fig. 6. Note that root vertices are sinks of the Markov chain

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Fig. 8. A face of G D (dotted)

2.6.2. Associated dimer graph and Kasteleyn matrix. Associated to a T -graph is a weighted bipartite planar graph G D constructed as follows, see Fig. 6. Black vertices of G D are the L j . White vertices are the bounded complementary regions, as well as one white vertex for each boundary path joining consecutive root vertices r j and r j+1 (but not for the path from rm to r1 ). The complementary regions are called faces; the paths between adjacent root vertices are called outer faces. Edges connect the L i to each face it borders along a positive-length subsegment. The edge weights are equal to the Euclidean length of the bounding segment. To G D there is a canonically associated Kasteleyn matrix of G D : this is the n × n matrix K G D = (K G D (w, b)) with rows indexing the white vertices and columns indexing the black vertices of G D . We have K G D (w, b) = 0 if w and b are not adjacent, and otherwise K G D (w, b) is the complex number equal to the edge vector corresponding to the edge of the region w along complete edge b (taken in the counterclockwise direction around w). In particular |K G D (w, b)| is the length of the corresponding edge of w. Lemma 2.3. K G D is a Kasteleyn matrix for G D , that is, the alternating product of the matrix entries for edges around a bounded face is positive real or negative real according to whether the face has 2 mod 4 or 0 mod 4 edges, respectively. By alternating product we mean the first, divided by the second, times the third, etc. Proof. Let f be a bounded face of G D (we mean not one of the outer faces). It corresponds to a meeting point of two or more complete edges; this meeting point is in the interior of exactly one of these complete edges, L. See Fig. 8. In G D , for each other black vertex on that face the two edges of the T -graph to neighboring white vertices have opposite orientations. The two edges parallel to L (horizontal in the figure) have the same orientation, so their ratio is positive. This implies the result. Although we won’t need this fact, in [15] it is shown that the set of in-directed spanning forests of GT (rooted at the root vertices and weighted by the product of the transition probabilities) is in measure-preserving (up to a global constant) bijection with the set of dimer coverings of G D . 2.6.3. Harmonic functions and discrete analytic functions. To a harmonic function f on a T -graph GT we associate a derivative d f which is a function on black vertices of


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G D as follows. Let v1 and v2 be two distinct points on complete edge b, considered as complex numbers. We define d f (b) =

f (v2 ) − f (v1 ) . v2 − v1

(9)

Since f is linear along any complete edge (Eq. (8)), d f is independent of the choice of v1 and v2 . Lemma 2.4. If f is harmonic on a T -graph GT and K = K G D is the associated Kasteleny matrix, then b∈B K (w, b)d f (b) = 0 for any interior white vertex w. Proof. Let b1 , . . . , bk be the neighbors of w in cyclic order. To each neighbor bi is associated a segment of a complete edge L i . Let vi and vi+1 be the endpoints of that segment, and wi and wi be the endpoints of L i . Then f (wi ) − f (wi ) f (vi+1 ) − f (vi ) = vi+1 − vi wi − wi since the harmonic function is linear along L i . In particular K G D (w, bi )d f (bi ) = (vi+1 − vi )

f (wi ) − f (wi ) = f (vi+1 ) − f (vi ). wi − wi

Summing over i (with cyclic indices) yields the result. Note that at a boundary white vertex w, B K G D (w, b)d f (b) is the difference in f -values at the adjacent root vertices of GT . We will refer to a function g on black vertices of G D satisfying b∈B K G D (w, b)g(b) = 0 for all interior white vertices w as a discrete analytic function. The construction in the above lemma can be reversed, starting from a discrete analytic function d f (on black vertices of G D ) and integrating to get a harmonic function f on GT : define f arbitrarily at a vertex of GT and then extend to neighboring vertices (on a same complete edge) using (9). The extension is well-defined by discrete analyticity. . We can relate K G−1 to the conjugate Green’s function 2.6.4. Green’s function and K G−1 D D on GT using the construction of the previous section, as follows. Let w be an interior face of GT , and a path from a point in w to the outer boundary of GT which misses all the vertices of GT . For vertices v of GT , define the conjugate Green’s function G ∗ (w, v) to be the expected algebraic number of crossings of by the random walk started at v and stopped at the boundary. This is the unique function with zero boundary values which is harmonic everywhere except for a jump discontinuity of −1 across when going counterclockwise around w. (If there were two such functions, their difference would be harmonic everywhere with zero boundary values.) See Fig. 9 for an example. Let K G−1 (b, w) be the discrete analytic function of b defined from G ∗ (w, v) as in the D (b, w) using two points on b previous section; on an edge which crosses , define K G−1 D = I by Lemma 2.4, and on the same side of . This function clearly satisfies K G D K G−1 D therefore is independent of the choice of .

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Fig. 9. Conjugate Green’s function example. Here the sides of the triangle are bisected in ratio 2 : 3 and interior complete edges 1 : 1

Note that K G−1 also has the following probabilistic interpretation: take two particles, D started simultaneously at two different points v1 , v2 of the same complete edge, and couple their random walks so that they start independently, take simultaneous steps, and when they meet they stick together for all future times. Then the difference in their winding numbers around w is determined by their crossings of before they meet. That is, K G−1 (b, w)(v1 − v2 ) is the expected difference in crossings before the particles meet D or until they hit the boundary, whichever comes first. 3. Constant-Slope Case (Theorem 3.7) in the speIn this section we compute the asymptotic expansion of K G−1 D cial case is when γ is planar. In this case the normalized asymptotic height function h¯ is linear, and its normal ν is constant. This case already contains most of the complexity of the general case, which is treated in Sect. 4. ¯ scaled as usual so that pa + pb + pc = 1. Let ν = ( pa , pb , pc ) be the normal to h, The angles θa , θb , θc are constant, and we choose a, b, c as before to be constant as well. Define a function F(w) = (bz/a)m (cw/bz)n at a white vertex w = m xˆ + n yˆ and F(b) = a(bz/a)−m (cw/bz)−n at a black vertex b = e1 + m xˆ + n yˆ . These functions are defined on all of the honeycomb graph H. Let K H be the adjacency matrix of H (which, as we mentioned earlier, is a Kasteleyn matrix for H). Lemma 3.1. We have b

K H (w, b)F(b) = 0 =

w

F(w)K H (w, b),

(10)


691

where K H is the adjacency matrix of H. Proof. This follows from the equation a + bz + cw = 0.

3.1. T -graph construction. Define a 1-form on edges of H by (wb) = −(bw) = 2Re(F(w))F(b). ∗

(11)

∗ ((wb)∗ )

By (10) the dual form (defined by = (wb)) is closed (the integral around any closed cycle is zero) and therefore ∗ = d for a complex-valued function on H∗ . Here H∗ , the dual of the honeycomb, is the graph of the equilateral triangulation of the plane. Extend linearly over the edges of H∗ . This defines a mapping from H∗ to C with the property that the images of the white faces are triangles similar to the a, b, c-triangle (via orientation-preserving similarities), and the images of black faces are segments. This follows immediately from the definitions: if b1 , b2 , b3 are the three neighbors of a white vertex w of H, the edges wbi have values 2Re(F(w))F(bi ) which are proportional to F(b1 ) : F(b2 ) : F(b3 ), which in turn are proportional to a : bz : cw by (10). If w1 , w2 , w3 are the three neighbors of a black vertex then the corresponding edge values are 2Re(F(wi ))F(b) which are proportional to Re(F(w1 )) : Re(F(w2 )) : Re(F(w3 )), that is, they all have the same slope and sum to zero. It is not hard to see that the images of the white triangles are in fact non-overlapping (see [15], Sect. 5 for the proof, or look at Fig. 10). It may be that ReF(w) = 0 for some w; in this case choose a generic modulus-1 complex number λ and replace F(w) by λF(w) and F(b) by λF(b). So we can assume that each white triangle is similar to the a, b, c-triangle. In fact, this operation will be important later; note that by varying λ the size of an individual triangle varies; by an appropriate choice we can make any particular triangle have maximal size (side lengths a, b, c). Lemma 3.2. The mapping is almost linear, that is, it is a linear map φ(m, n) = cwm + an plus a bounded function. Proof. Consider for example a vertical column of horizontal edges {w1 b1 , . . . , wk bk } of H connecting a face f 1 to face f k+1 . We have k

∗ ((wi bi )∗ ) =

i=1

k

(wi bi )

i=1

=

k (F(wi ) + F(wi ))F(bi ) i=1

= ka +

k

F(wi )F(bi )

i=1

= ka + a

k

z 2i w −2i

i=1

= φ( f k+1 ) − φ( f 1 ) + osc,

(12)

where osc is oscillating and in fact O(1) independently of k (by hypothesis z/w ∈ R). In the other two lattice directions the linear part of is again φ, so that is almost linear in all directions.

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Fig. 10. The -image of H∗ is a T -graph covering R2

This lemma shows that the image of H∗ under is an infinite T -graph HT covering all of R2 . The images of the black triangles are the complete edges and have lengths O(1). If we insert a λ as above and let λ vary over the unit circle, one sees all possible local structures of the T -graph, that is, the geometry of the T -graph HT = HT (λ) in a neighborhood of a triangle (w) only depends up to homothety on the argument of λF(w). Recall that G ∗ is a subgraph of H∗ approximating U . We can restrict to 1 G ∗ thought of as a subgraph of 1 H, and then multiply its image by . Thus we get a finite sub-T -graph GT of HT . Let = ◦ ◦ 1 so that acts on G ∗ . The union of the -images of the white triangles in G ∗ forms a polygon P. Define the “dimer” graphs H D associated to HT , and G D associated to GT as in Sect. 2.6.2. See Fig. 6 for the T -graph arising from the graph G ∗ of Fig. 5. Note that G D contains G (defined in Sect. 2.1.1) but has extra white vertices along the boundaries. These extra vertices make the height function on G D approximate the desired linear function (whose graph has normal ν), see the next section. From (11) we have Lemma 3.3. Edge weights of H D and G D are gauge equivalent to constant edge weights. Indeed, for w not on the boundary of G D we have K G D (w, b) = 2Re(F(w))F(b)K G (w, b)

(13)

and similarly for all w, K H D = 2Re(F(w))F(b)K H (w, b).

(14)


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Asymptotics of K G−1 are described below in Sect. 3.4. For K H D , from (7) and (14) D we have

F(w)F(b) 1 1 −1 Im + O( (b, w) = ). (15) KH D 2π Re(F(w))F(b) φ(b) − φ(w) |φ(b) − φ(w)|2 −1 Here K H is the inverse constructed from the conjugate Green’s function on the T -graph D

−1 (b, w) coincides with K ν−1 of (7) since HT . We use the fact that 2Re(F(w))F(b)K H D both satisfy the equation that d K −1 equals the conjugate Green’s function.

3.2. Boundary behavior. Recall that from our region U we constructed a graph G (Sect. 2.1.1). From the normalized height function u on ∂U (which is the restriction of a linear function to ∂U ) we constructed the T -graph GT and then the dimer graph G D . In this section we show that the normalized boundary height function of G D when G D ¯ This is proved in a roundabout way: we first show is chosen as above approximates h. that the boundary height does not depend (up to local fluctuations) on the exact choice of boundary conditions, as long as we construct GT from G ∗ as in Sect. 2.1. Then we compute the height change for “simple” boundary conditions. 3.2.1. Flows and dimer configurations Any dimer configuration m defines a flow (or 1-form) [m] with divergence 1 at each white vertex and divergence −1 at each black vertex: just flow by 1 along each edge in m and 0 on the other edges. The set 1 ⊂ [0, 1] E of unit white-to-black flows (i.e. flows with divergence 1 at each white vertex and −1 at each black vertex) and with capacity 1 on each edge is a convex polytope whose vertices are the dimer configurations [16]. On the graph H define the flow ω1/3 ∈ 1 to be the flow with value 1/3 on each edge wb from w to b. Up to a factor 1/3, this flow can be used to define the height function, in the sense that for any dimer configuration m, [m] − ω1/3 is a divergence-free flow and the integral of its dual (which is closed) is 1/3 times the height function of m. That is, the height difference between two points is three times the flux of [m] − ω1/3 between those points. This is easy to see: across an edge which contains a dimer the height changes by ±2 (depending on the orientation); if the edge does not contain a dimer the height changes by ∓1, for so that the height change can be written ±(3χ − 1) where χ is the indicator function of a dimer on that edge. For a finite subgraph of H the boundary height function can be obtained by integrating around the boundary (3 times) the dual of [m] − ω1/3 for some m. 3.2.2. Canonical flow. On H D there is a canonical flow ω ∈ 1 defined as illustrated in Fig. 11: let v1 v2 be the two vertices of GT along b adjacent to w. The flow from w to b is 1/(2π ) times the sum of the two angles that the complete edges through v1 and v2 make with b. One or both of these angles may be zero. Note that the total flow out of w is 1. For an edge b, the total flow into b is also 1 as illustrated on the right in Fig. 11. Now G D is a subgraph of H D with extra white vertices around its boundary. The canonical flow on H D restricts to a flow on G D except on edges connecting to these extra white vertices (we define the canonical flow there to be zero). This flow has divergence 1, −1 at white/black vertices, except at the black vertices of G D connected to the boundary white vertices, and the boundary white vertices themselves. If m is a dimer

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θ5

θ4

θ 3+ θ4

θ4 θ1

θ5+ θ 6 θ6

θ3

θ1 + θ 2 θ1

θ1

θ2 θ3 θ3 + θ 4

θ2

θ2

Fig. 11. Defining the canonical flow (divide the angles by 2π ).

covering of G D , the flow [m] − ω is now a divergence-free flow on G D except at these black and white boundary vertices. Lemma 3.4. Along the boundary of G D the divergence of [m] − ω for any dimer configuration m is the turning angle of the boundary of P. Proof. Consider a complete edge L corresponding to black vertex b. The canonical flow into b has a contribution from the two endpoints of L. The flow [m] can be considered to contribute −1/2 for each endpoint. Recall that each endpoint of L ends in the interior of another complete edge or at a convex vertex of the polygon P. If an endpoint of L ends in the interior of another complete edge, and there are white faces adjacent to the two sides of this endpoint (that is, the endpoint is not a concave vertex of P) then the contribution of the canonical flow is also −1/2, so the contribution of [m] − ω is zero. Suppose the endpoint is at a concave vertex v of P with exterior angle θ < π . The θ contribution from v to the flow of [m] − ω into L is −1/2 + 2π . The other complete edge at v does not end at v and so has no contribution. This quantity is 1/2π times the turning angle of the boundary. Suppose the endpoint is at a convex vertex v of P of interior angle θ < π . There is some complete edge L of HT , containing v in its interior, which is not in GT . The sum of the contributions of [m] − ω for the endpoints of the two complete edges L 1 , L 2 of GT meeting at v is −1/2 − θ/2π . This is the 1/2π times the turning angle at the convex vertex, minus 1. The contribution from [m] from the boundary white vertices is 1 per white boundary vertex, that is, one per convex vertex of P. This lemma proves in particular that the divergence of [m] − ω is bounded for any [m]. 3.2.3. Boundary height. Recall that the height function of a dimer covering m can be defined as the flux of ω1/3 − [m]. In particular, the flux of ω1/3 − ω defines the boundary height function up to O(1) (the turning of the boundary), since the flux of [m] − ω is the boundary turning angle. The flux of ω1/3 − ω between two faces can be computed along any path, and in fact because both ω1/3 and ω are locally defined from H D , we see that the flux does not depend on the choice of the nearby boundary. Let us compute this flux and show that it is linear along lattice directions, and therefore linear everywhere in H D . Take a vertical column of horizontal edges in G, and let us compute ω on this set of edges. The image of the dual of this column is a polygonal curve η whose j th edge is


695

Fig. 12. A column of triangles from GT

(using (12)) a constant times 1 + ( wz )2 j . The image of the triangles in the vertical column of G ∗ is as shown in Fig. 12. The j th edge is part of a complete edge corresponding to the j th black vertex in the column. By the argument of the previous section, the flux is equal to the number of convex corners of this polygonal curve η, that is, corners where the curve turns left. The curve η has a convex corner when

1 + (z/w)2 j+2 ≥ 0, Im 1 + (z/w)2 j that is, using z/w = −eiθa , when 2 jθa ∈ [π, π + 2θa ]. Assuming that θa is irrational, a this happens with frequency 2θ 2π , so the flux of ω1/3 − ω along a column of length n θa 1 is n( π − 3 ), and the average flux per edge is pa − 1/3. Therefore the average height change per horizontal edge is 3 pa − 1. If θa is rational, a continuity argument shows that 3 pa − 1 is still the average height change per horizontal edge. A similar result holds in the other directions, and (3) shows that the average height has normal ( pa , pb , pc ) as desired. 3.3. Continuous and discrete harmonic functions. To understand the asymptotic expansion of G ∗ , the conjugate Green’s function on GT , from which we can get K −1 , we need two ingredients. We need to understand the conjugate Green’s function on HT , and also the relation between continuous harmonic functions on domains in R2 and discrete harmonic functions on (domains in) HT . 3.3.1. Discrete and continuous Green’s functions. On HT , the discrete conjugate Green’s function G ∗ (w, v), for w ∈ H and v ∈ HT , can be obtained from integrating the exact formula for K ν−1 given in (4) as discussed in Sect. 2.6.4. As we shall see, this formula differs even in its leading term from the continuous conjugate Green’s function, due to the singularity at the diagonal. Basically, because our Markov chain is directed, a long random walk can have a nonzero expected winding number around the origin. This causes the conjugate Green’s function, which measures this winding, to have a component of Re log v as well as a part Im log v. The continuous conjugate Green’s function on the whole plane is g ∗ (v1 , v2 ) =

1 Im log(v2 − v1 ). 2π

(We use G ∗ to denote the discrete conjugate Green’s function and g ∗ the continuous version.) To compute the discrete conjugate Green’s function G ∗ , if we simply restrict g ∗ to the vertices of HT , it will be very nearly harmonic as a function of v2 for large |v2 −v1 |, but the discrete Laplacian of g ∗ at vertices v2 near v1 (within O() of v1 ) will be of constant order in general. We can correct for the non-harmonicity at a vertex v by adding an

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appropriate multiple of the actual (non-conjugate) discrete Green’s function G(v, v2 ). The large scale behavior of this correction term is a constant times Re log(v2 − v ). So we can expect the long-range behavior of the discrete conjugate Green’s function G ∗ (w, v), for |w − v| large, to be equal to g ∗ (w, v) plus a sum of terms involving the real part Re log(v − v ) for v s within O() of φ(w). These extra terms sum to a function of the form c log |v − φ(w)| + s(v) + O( 2 ), where s is a smooth function and c is a real constant, both c and s depending on the local structure of HT near φ(w). This form of G ∗ can be seen explicitly, of course, if we integrate the exact formula (15). We have Lemma 3.5. The discrete conjugate Green’s function G ∗ on the plane HT is asymptotically

Im(F(w)) 1 Im log(v − φ(w)) + Re log(v − φ(w)) G ∗ (w, v) = 2π Re(F(w)) 1 +s(w, v) + O( 2 ) + O( ) (16) |v − φ(w)| 1 Im (F(w) log(v − φ(w))) + s(w, v) = 2π Re(F(w)) 1 ), (17) +O( 2 ) + O( |v − φ(w)| where s(w, v) is a smooth function of v. Proof. From the argument of the previous paragraph, it suffices to compute the constant in front of the Re log(v − φ(w)) term. This can be computed by differentiating the above formula with respect to v and comparing with formula (15) for K −1 . The differential of s(w, v) is O(). Let v1 , v2 be two vertices of complete edge b, coming from adjacent faces of H, adjacent across an edge bw1 (so that v1 − v2 = ∗ (bw1 ) = 2Re(F(w1 ))F(b)). We have for w far from w1 , −1 (b, w) = KH D

=

G ∗ (w, v1 ) − G ∗ (w, v2 ) v − v2 1 F(w) v1 −v2 1 2π Im Re(F(w)) φ(b)−φ(w)

+ O() 2Re(F(w1 ))F(b)

1 F(w)F(b) = Im + O(). 2π Re(F(w))F(b) φ(b) − φ(w)

Note that in fact for any complex number λ of modulus 1, we get a discrete conjugate Green’s function G ∗λ on the graph HT (λ) (from Sect. 3.1) with similar asymptotics. 3.3.2. Smooth functions. Constant and linear functions on R2 , when restricted to HT , are exactly harmonic. More generally, if we take a continuous harmonic function f on R2 and evaluate it on the vertices of HT , the result will be close to a discrete harmonic function f , in the sense that the discrete Laplacian will be O( 2 ): if v is a vertex of HT


697

and v1 , v2 are its (forward) neighbors located at v1 = v − d1 eiθ and v2 = v + d2 eiθ then the Taylor expansion of f about v yields f (v) = f (v) −

d1 d2 f (v − d1 eiθ ) − f (v + d2 eiθ ) = O( 2 ). d1 + d2 d1 + d2

This situation is not as good as in the (more standard) case of a graph like Z2 , where if we evaluate a continuous harmonic function on the vertices, the Laplacian of the resulting discrete function is O( 4 ): f (v) −

1 ( f (v + ) + f (v + i) + f (v − ) + f (v − i)) = O( 4 ). 4

In the present case the principal error is due to the second derivatives of f . To get an error smaller than O( 2 ), we need to add to f a term which cancels out the 2 error. We can add to f a function which is 2 times a bounded function f 2 whose value at a point v depends only on the local structure of HT near v and on the second derivatives of f at v. Lemma 3.6. For a smooth harmonic function f on R2 whose second partial derivatives don’t all vanish at any point, there is a bounded function f on HT such that f (z) = f (z) + 2 f 2 (z) has discrete Laplacian of order O( 3 ), where f 2 (z) depends only the second derivatives of f at z and on the local structure of the graph HT at z. Proof. We have exact formulas for one discrete harmonic function, the conjugate Green’s function on HT , and we know its asymptotics (Lemma 3.5), Eq. (17), which are G ∗ (w, z 2 ) ≈ η(z 1 , z 2 ) =

1 Im(c log(z 2 − z 1 )) 2π

for a constant c depending on the local structure of the graph near w, and where z 1 is a point in φ(w). We’ll let w be the origin in HT and z 1 = 0; η(0, z 2 ) is a continuous harmonic function of z 2 . For z ∈ R2 consider the second derivatives of the function f , which by hypothesis are not all zero. There is a point z 2 = β(z) ∈ R2 at which η(0, z 2 ) has the same second derivatives. Indeed, f has three second partial derivatives, f x x , f x y , and f yy , but because f is harmonic f yy = − f x x . We have ∂z∂ 2 η(0, z 2 ) = 21pi Im zc2 , and the second partial derivatives of η are the real and imaginary parts of const/z 22 , which is surjective, in fact 2 to 1, as a mapping of R2 − {0} to itself. In particular there are two choices of z 2 for which f (z) and η(β(z)) have the same second derivatives. Since f and η are smooth, by taking a consistent choice, β can be chosen to be a smooth function as well. Consider the function f (z) = f (z) + G ∗λ (0, β(z)) − η(0, β(z)), where G ∗λ is the discrete conjugate Green’s function and λ is chosen so that HT (λ) (see Sect. 3.1) has local structure at β(z) identical to that of HT at z. We claim that the discrete Laplacian of f is O( 3 ). This is because G ∗λ (0, β(z)) is discrete harmonic, and f (z) − η(0, β(z)) has vanishing second derivatives. We also have that G ∗ (0, β(z)) − η(0, β(z)) = O( 2 ), see Lemma 3.5 above.

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3.4. K G−1 in constant-slope case. D Theorem 3.7. In the case of constant slope ν and a bounded domain U , let ξ be a conformal diffeomorphism from φ(U ) to H. When b and w are converging to different points as → 0 we have K G−1 (b, w) = D

1 2π Re(F(w))F(b) ξ (φ(b))F(w)F(b) ξ (φ(b))F(w)F(b) + + O(). (18) ×Im ξ(φ(b)) − ξ(φ(w)) ξ(φ(b)) − ξ(φ(w))

Proof. The function G ∗GT (w, v) is equal to the function (16) for the whole plane, plus a harmonic function on GT whose boundary values are the negative of the values of (16) on the boundary of GT . Since discrete harmonic functions on GT are close to continuous harmonic functions on U , we can work with the corresponding continuous functions. From (17) we have G ∗HT (w, v) =

F(w) 1 Im log(v − z 1 ) + s(z 1 , v) + O()2 , 2π Re(F(w))

(19)

where z 1 is a point in face φ(w). The continuous harmonic function of v on U whose values on ∂U are the negative of the values of (19) on ∂U is

F(w) 1 Im log(v − z 1 ) 2π ReF(w) F(w) F(w) 1 Im log(ξ(v) − ξ(z 1 ))+ log(ξ(v)−ξ(z 1 )) +s2 (z 1 , v)+ O()2, + 2π ReF(w) ReF(w)

−

(20) where s2 is smooth. The discrete Green’s function G ∗GT must be the sum of (19) and (20): 1 Im G ∗GT (w, v) = 2π

F(w) F(w) log(ξ(v) − ξ(z 1 )) + log(ξ(v) − ξ(z 1 )) ReF(w) ReF(w)

+s3 (z 1 , v) + O()2 . Differentiating gives the result (as in Lemma 3.5).

It is instructive to compare the discrete conjugate Green’s function in the above proof with the continuous conjugate Green’s function on U which is g ∗ (z 1 , z 2 ) =

1 1 Im log(ξ(z 1 ) − ξ(z 2 )) + Im log(ξ(z 1 ) − ξ(z 2 )). 2π 2π


699

3.4.1. Values near the diagonal. Note that when b is within O() of w, and neither is close to the boundary, the discrete Green’s function G ∗ (w, v) for v on b is equal to the discrete Green’s function on the plane G ∗HT (w, v) plus an error which is O(1) coming from the corrective term due to the boundary. The error is smooth plus oscillations of order O( 2 ), so that within O() of w the error is a linear function plus O(). Therefore when we take derivatives −1 K G−1 (b, w) = K H (b, w) + O(1), D D −1 is of order O( −1 ) when |b − w| = O(), implies that the local which, since K H D statistics are given by µν .

Theorem 3.8. In the case of constant slope ν, the local statistics at any point in the interior of U are given in the limit → 0 by µν , the ergodic Gibbs measure on tilings of the plane of slope ν. 4. General Boundary Conditions ¯ the normalized Here we consider the general setting: U is a smooth Jordan domain and h, asymptotic height function on U , is not necessarily linear.

4.1. The complex height function. The equation (1) implies that the form log( − 1)d xˆ − log(

1 − 1)d yˆ

is closed. Since U is simply connected it is d H for a function H : U → C which we call the complex height function. ¯ we have arg( − 1) = π − θc (Fig. 2) and The imaginary part of H is related to h: arg( 1 − 1) = θa − π , which gives Im d H = (π − θc )d xˆ + (π − θa )d yˆ . From (3) we have d h¯ = (3 pc − 1)d xˆ + (3 pa − 1)d yˆ , so 3 ¯ Im d H = 2(d xˆ + d yˆ ) − d h. π The real part of H is the logarithm of a special gauge function which we describe below. We have Hxˆ = log( − 1), 1 Hyˆ = − log( − 1), − yˆ xˆ = , Hxˆ xˆ = −1 −1 1 yˆ . Hyˆ yˆ = − −1

(21) (22) (23) (24)

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4.2. Gauge transformation. The mapping is a real analytic mapping from U to the upper half plane. It is an open mapping since Im( xˆ / yˆ ) = −Im = 0, but may have isolated critical points. The Ahlfors-Bers theorem gives us a diffeomorphism φ from U onto the upper half plane satisfying the Beltrami equation dφ d z¯ dφ dz

=

d d z¯ d dz

=

− eiπ/3 , − e−iπ/3

that is φxˆ = − φ yˆ . Such a φ exists by the Ahlfors-Bers theorem [1]. It follows that is of the form f (φ) for some holomorphic function f from H into H. Since ∂U is smooth, φ is smooth up to and including the boundary, and φxˆ , φ yˆ are both nonzero. For white vertices of G define

1 F(w) = e H (w) φ yˆ (w)(1 + M(w)), (25) where M(w) is any function which satisfies

φx2ˆ yˆ φxˆ yˆ yˆ H φ H x ˆ x ˆ x ˆ y ˆ x ˆ x ˆ x ˆ e−Hxˆ Mxˆ − e Hyˆ M yˆ = e−Hxˆ − − − 2+ 8 6 4φ yˆ 4φ yˆ 8φ yˆ 2 Hyˆ yˆ Hyˆ yˆ yˆ Hyˆ yˆ φ yˆ yˆ φ y2ˆ yˆ φ yˆ yˆ yˆ H yˆ + + +e − 2+ . 8 6 4φ yˆ 4φ yˆ 8φ yˆ Hxˆ2xˆ

(26)

The existence of such an M follows from the fact that the ratio of the coefficients of Mxˆ and M yˆ is −e Hxˆ +Hyˆ = , so (26) is of the form Mxˆ + M yˆ = J (x, y) for some smooth function J . This is the ∂¯ equation in coordinate φ. We don’t need to know M explicitly; the final result is independent of M. We just need its existence to get better estimates on the error terms in Lemma 4.1 below. For black vertices b define

1 F(b) = e− H (w) φ yˆ (w)(1 − M(w)), (27) where w is the vertex adjacent to and left of b and M(w) is as above. Lemma 4.1. For each black vertex b with three neighbors in G we have

F(w)K (w, b) = O( 3 )

w

and for each white vertex w we have b

K (w, b)F(b) = O( 3 ).


701

Proof. This is a calculation. Let w, w − x, ˆ w + yˆ be the three neighbors of b. Then, setting H = H (w), φ yˆ = φ yˆ (w), and M = M(w) we have 1 2 3 2 4 F(w − x) ˆ = e (H − Hxˆ + 2 Hxˆ xˆ − 6 Hxˆ xˆ xˆ +O( )) φ yˆ − φ yˆ xˆ + φ yx x + O( 3 ) 2 ×(1 + M − 2 Mxˆ + O( 3 )), F(w + yˆ ) = e

1 2 (H + H yˆ + 2

3

H yˆ yˆ + 6 H yˆ yˆ yˆ +O( 4 ))

φ yˆ + φ yˆ yˆ +

2 φ yˆ yˆ yˆ + O( 3 ) 2

×(1 + M + 2 M yˆ + O( 3 )). The sum of the leading order terms in F(w) + F(w − x) ˆ + F(w + yˆ ) is

1 1 1 + ) = 0. e H φ yˆ (1 + e−Hxˆ + e Hyˆ ) = e H φ(1 + −1 1− 1 The sum of the terms of order is 2 e H φ yˆ times φ yˆ xˆ φ yˆ yˆ + Hxˆ xˆ ) + e Hyˆ ( + Hyˆ yˆ ) φ yˆ φ yˆ

− yˆ yˆ φ yˆ yˆ φ yˆ yˆ 1 + = 0, + yˆ + − = −1 φ yˆ −1 1 − φ yˆ ( − 1)

e−Hxˆ (−

1 and the sum of the order- 2 terms is 2 e H φ yˆ times 2 φx2ˆ yˆ φxˆ yˆ yˆ H H φ H x ˆ x ˆ x ˆ y ˆ x ˆ x ˆ x ˆ xˆ xˆ − − − 2+ −e−Hxˆ Mxˆ + e Hyˆ M yˆ + e−Hxˆ 8 6 4φ yˆ 4φ yˆ 8φ yˆ 2 Hyˆ yˆ Hyˆ yˆ yˆ Hyˆ yˆ φ yˆ yˆ φ y2ˆ yˆ φ yˆ yˆ yˆ H yˆ + + +e − 2+ = 0. (28) 8 6 4φ yˆ 4φ yˆ 8φ yˆ

A similar calculation holds at a white vertex, and we get the same expression for the 2 contribution (changing the signs of M, H, d/d xˆ and d/d yˆ gives the same expression).

It is clear from this proof that the error in the statement can be improved to any order O( 3+k ) by replacing M(w) with M0 (w) + M1 (w) + · · · + k Mk (w), where each M j satisfies an equation of the form (26) except with a different right hand side—the right-hand side will depend on derivatives of H, φ and the Mi for i < j. For our proof below we need an error O( 4 ) and therefore the M0 and M1 terms, even though the final result will depend on neither M0 nor M1 . 4.3. Embedding. Define a 1-form (wb) = 2Re(F(w))F(b)K (w, b) on edges of G, where F is defined in (25,27) (since K (w, b) = is a constant, this is an unimportant factor for now, but in a moment we will perturb K ). By the comments

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after the proof of Lemma 4.1, the dual form ∗ on G ∗ can be chosen to be closed up to O( 4 ) and so there is a function φ˜ on G ∗ , defined up to an additive constant, satisfying d φ˜ = ∗ + O( 3 ). In fact up to the choice of the additive constant, φ˜ is equal to φ plus an oscillating function. This can be seen as follows. For a horizontal edge wb we have F(w)F(b) = φ yˆ (w) + O( 2 ). Thus on a vertical column {w1 b1 , . . . , wk bk } of horizontal edges we have k

∗ (wi bi ) =

i=1

k

(F(wi ) + F(wi ))F(bi )

i=1

=

φ yˆ (wi ) +

k

F(wi )F(bi ) + O( 3 ).

i=1

The first sum gives the change in φ from one endpoint of the column to the other, and the second sum is oscillating (F(wi ) and F(bi ) have the same argument which is in (0, π ) and which is a continuous function of the position) and so contributes O(). Similarly, in the other two lattice directions the sum is given by the change in φ plus an oscillating term. Therefore by choosing the additive constant appropriately, φ˜ maps G ∗ to a small ˆ which shrinks to H as → 0. neighborhood of H in the spherical metric on C, The map φ˜ has the following additional properties. The image of the three edges of a black face of G ∗ are nearly collinear, and the image of a white triangle is a triangle nearly similar to the a, b, c-triangle, and of the same orientation. Thus it is nearly a mapping onto a T -graph. In fact near a point where the relative weights are a, b, c (weights which are slowly varying on the scale of the lattice) the map is up to small errors the map of Sect. 3. We can adjust the mapping φ˜ by O( 3 ) so that the image of each black triangle is an exact line segment: this can be arranged by choosing for each black face a line such that ˜ the φ-image of the corresponding black face is within O( 3 ) of that line; the intersections of these lines can then be used to define a new mapping : G ∗ → C which is an exact T -graph mapping. The mapping will then correspond to the above 1-form but for a matrix K˜ with slightly different edge weights. Let us check how much the weights differ from the original weights (which are ). As long as the triangular faces are of size of order (which they are typically), the adjustment will change the edge lengths locally by O( 3 ) and therefore their relative lengths by 1 + O( 2 ). There will be some isolated triangular faces, however, which will be smaller—of order O( 2 ) because of the possibility that F(w) might be nearly pure imaginary. We can deal with these as follows. Once we have readjusted the “large” triangular faces we have an exact T -graph mapping on most of the graph. We can then multiply F(w) by λ = i and F(b) by λ = −i: the readjusted weights give (for most of the graph) a new exact T -graph mapping (because we now have K˜ F = F K˜ = 0 exactly for these weights), but now all faces which were too small before become O() in size and we can readjust their dimensions locally by a factor 1 + O( 2 ). In the end we have an exact mapping of G ∗ onto a T -graph GT and it distorts the edge weights of K by at most 1 + O( 2 ). We shall see in Sect. 5 that this is close enough to get a good approximation to K −1 .


703

In conclusion the Kasteleyn matrix for G D is equal to 2Re(F(w))F(b) K˜ (w, b) where K˜ has edge weights + O( 3 ).

4.4. Boundary. Along the boundary we claim that the normalized height function of G D follows u. Since G D arises from a T -graph, we can use its canonical flow (Sect. 3.2.2). Near any given point the canonical flow looks like the canonical flow in the constantweight case—since the weights vary continuously, they vary slowly at the scale , the scale of the graph. Since the canonical flow defines the slope of the normalized height function, we have pointwise convergence of the derivative of h¯ along the boundary to the derivative of u. Thus h¯ converges to u. In fact this argument shows that the normalized height function of the canonical flow converges to the asymptotic height function in the interior of U as well. 5. Continuity of K −1 In this section we show how K −1 changes under a small change in edge weights. Lemma 5.1. Suppose 0 < δ . If G is a graph identical to G D but with edge weights which differ by a factor 1 + O(δ), then K G−1 = K G−1 (1 + O(δ/)). D In particular since δ = 2 in our case this will be sufficient to approximate K −1 to within 1 + O(). Proof. For any matrix A we have ⎛

(K + δ A)−1

⎞ ∞ = K −1 ⎝1 + (−1) j (δ) j (AK −1 ) j ⎠ , j=1

as long as this sum converges. From Theorem 6.1 below we have that K G−1 (b, w) = O(1/|b − w|). When A represents a bounded, weighted adjacency matrix of G D , the matrix norm of AK −1 is then AK −1 ≤ max w

| A(w , b)K −1 (b, w)| ≤ 3a max w

b

w

w=w

|w

1 = O( −2 ), − w|

where a is the maximum entry of A. In particular when δ we have (K + δ A)−1 = K −1 (1 + O(δ/)).

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6. Asymptotic Coupling Function We define K G D as in (13) using the values (25,27) for F (and K G is the adjacency matrix of G). Theorem 6.1. If b, w are not within o(1) of the boundary of U , we have F(w)F(b) F(w)F(b) 1 −1 K G D (b, w) = + O(). (29) Im + 2π Re(F(w))F(b) φ(b) − φ(w) φ(b) − φ(w) When one or both of b, w are near the boundary, but they are not within o(1) of each other, then K G−1 (b, w) = O(1). D Proof. The proof is identical to the proof in the constant-slope case, see Theorem 3.7, except that ξ φ there is φ here. The ξ factors from (18) are here absorbed in the definition of the functions φ and F. As in the case of constant slope, when b and w are close to each other (within O()) and not within o(1) of the boundary, Theorem 3.8 applies to show that the local statistics are given by µν . 7. Free Field Moments As before let φ be a diffeomorphism from U to H satisfying φxˆ = − φ yˆ , where is defined from h¯ as in Sect. 4.1. Theorem 7.1. Let h¯ be the asymptotic √ height function on G D , h the normalized height ¯ Then hˆ converges weakly as → 0 function of a random tiling, and hˆ = 23π (h − h). to φ ∗ F, the pull-back under φ of F, the Gaussian free field on H. Here weak convergence means that for any smooth test function ψ on U , zero on the boundary, we have ˆ f) → 2 ψ( f )h( ψ(x)F(φ(x))|d x|2 , GD

U

where the sum on the left is over faces f of G D . Proof. We compute the moments of F. Let ψ1 , . . . , ψk be smooth functions on U , each zero on the boundary. We have ⎡⎛ ⎞ ⎛ ⎞⎤ ˆ f 1 )⎠ · · · ⎝ 2 ˆ f k )⎠ ⎦ E ⎣⎝ 2 ψ1 ( f 1 )h( ψk ( f k )h( f 1 ∈G D

=

2k

f k ∈G D

ˆ f 1 ) . . . h( ˆ f k )]. ψ1 ( f 1 ) · · · ψk ( f k )E[h(

f 1 ,..., f k

From Theorem 7.2 below the sum becomes = ··· E[F(φ(x1 )) . . . F(φ(xk ))] ψi (xi )|d xi |2 + o(1), U

U

that is, the moments of hˆ converge to the moments of the free field φ ∗ (F). Since the free field is a Gaussian process, it is determined by its moments. This completes the proof.


705

Theorem 7.2. Let s1 , . . . , sk ∈ U be distinct points in the interior of U . For each let f 1 , f 2 , . . . , f k be faces of G D , with f i converging to si as → 0. If k is odd we have ˆ f 1 ) . . . h( ˆ f k )] = 0, lim E[h(

→0

and if k is even we have ˆ f 1 ) . . . h( ˆ f k )] = lim E[h(

→0

k/2

G(φ(sσ (2 j−1) ), φ(sσ (2 j) )),

pairings σ j=1

where z − z 1 G(z, z ) = − log 2π z − z¯

is the Dirichlet Green’s function on H and the sum is over all pairings of the indices. If two or more of the si are equal, we have ˆ f 1 ) . . . h( ˆ f k )] = O( − ), E[h( where is the number of coincidences (i.e. k − is the number of distinct si ). Proof. We first deal with the case that the si are distinct. Let γ1 , . . . , γk be pairwise disjoint paths of faces from points si on the boundary to the f i . We assume that these paths are far apart from each other (that is, as → 0 they converge to disjoint paths). The height h ( f i ) can be measured as a sum along γi . We suppose without loss of generality that each γi is a polygonal path consisting of a bounded number of straight segments which are parallel to the lattice directions x, ˆ yˆ , zˆ . In this case, by additivity of the height change along γi and linearity of the moment in each index, we may as well assume that γi is in a single lattice direction. Now the change in hˆ along γi is given by the sum of ai j − E(ai j ) where ai j is the indicator function of the jth edge crossing γi (with a sign according to the direction of γi ). So the moment is ˆ f 1 ) . . . h( ˆ f k )] = E[h(

E[(a1 j1 − E[a1 j1 ]) . . . (ak jk − E[ak jk ])].

j1 ∈γ1 ,..., jk ∈γk

If wi ji , bi ji are the vertices of edge ai ji , this moment becomes (see [8]) 0 K −1 (b2 j , w1 j ) ... K −1 (bk j , w1 j ) 1 2 1 k ⎞ −1 K (b1 j , w2 j ) 0 k 1 2 . ⎝ K (wi j , bi j )⎠ . . . i i . . . j1 ,..., jk i=1 . . . −1 −1 K (b1 j , wk j ) 0 , wk j ) ... K (bk−1 j 1 k k k−1 ⎛

In particular, the effect of subtracting off the mean values of the ai ji is equivalent to cancelling the diagonal terms K −1 (bi ji , wi ji ) in the matrix.

706

R. Kenyon

Expand the determinant as a sum over the symmetric group. For a given permutation σ , which must be fixed-point free or else the term is zero, we expand out the corresponding product, and sum along the paths. For example if σ is the k-cycle σ = (12 . . . k), the corresponding term is k sgn(σ ) K (wi ji , bi ji ) K −1 (b1 j1 , w2 j2 ) j1 ,..., jk

×K

= −(

−1 k ) 4πi

⎛

⎝

j1 ,..., jk

−1

i=1

(b2 j2 , w3 j3 ) · · · K −1 (bk jk , w1 j1 ) =

⎞ F(b1 j )F(w2 j ) F(b1 j )F(w2 j ) F(b1 j )F(w2 j ) F(b1 j )F(w2 j ) 2 − 1 2 − 1 2 ⎠... 1 1 2 + φ(b1 j ) − φ(w2 j ) φ(b1 j ) − φ(w2 j ) φ(b1 j ) − φ(w2 j ) φ(b1 j ) − φ(w2 j ) 1 2 1 2 1 2 1 2 ⎞ ⎛ F(bk j )F(w1 j ) F(bk j )F(w1 j ) F(bk j )F(w1 j ) F(bk j )F(w1 j ) 1 1 − 1 ⎠ 1 − k k k k ...⎝ + φ(b1 j ) − φ(w2 j ) φ(bk j ) − φ(w1 j ) φ(bk j ) − φ(w1 j ) φ(bk j ) − φ(w1 j ) 1 2 1 k 1 1 k k

(30)

plus lower-order terms. Multiplying out this product, all 4k terms have an oscillating coefficient (as some ji varies) except for the terms in which the pairs F(bi ji ) and F(wi ji ) in the numerator are either both conjugated or both unconjugated for each i. That is, any term involving F(bi ji )F(wi ji ) or its conjugate will oscillate as ji varies and so contribute negligibly to the sum. There are only 2k terms which survive. Let z i denote the point z i = φ(bi ji ) ≈ φ(wi ji ). For a term with F(bi ji ) and F(wi ji ) both conjugated, the coefficient F(bi ji )F(wi ji ) is equal to dz i , otherwise it is equal to dz i , where dz i is the amount that φ changes when moving by one step along path γi , that is, when ji increases by 1. So the above term for the k-cycle σ becomes ⎛ ⎞ −1 k ⎝ εj⎠ −( ) 4πi ×

ε1 ,...,εk =±1

j

φ(γ1 )

...

(ε )

(ε )

dz 1 1 . . . dz k k

φ(γk )

(z 1(ε1 ) − z 2(ε2 ) )(z 2(ε2 ) − z 3(ε3 ) ) . . . (z k(εk ) − z 1(ε1 ) ) (1)

(−1)

plus an error of lower order, where z j = z j and z j = z¯ j , with similar expressions for other σ . When we now sum over all permutations σ , only the fixed-point free involutions do not cancel: Lemma 7.3 ([2]). For n > 2 let Cn be the set of n-cycles in the symmetric group Sn . Then n σ ∈Cn i=1

1 = 0, z σ (i) − z σ (i+1)

where the indices are taken cyclically.


707

Proof. This is true for n odd by antisymmetry (pair each cycle with its inverse). For n even, the left-hand side is a symmetric rational function whose denominator is the Van dermonde i< j (z i − z j ) and whose numerator is of lower degree than the denominator. Since the denominator is antisymmetric, the numerator must be as well. But the only antisymmetric polynomial of lower degree than the Vandermonde is 0. By the lemma, in the big determinant all terms cancel except those for which σ is a fixed-point free involution. It remains to evaluate what happens for a single transposition, since a general fixed-point free involution σ will be a disjoint product of these: 1 (4πi)2

φ(s ) φ(s ) φ(s ) φ(s ) φ(s ) φ(s ) φ(s1 ) φ(s2 ) dz dz d z¯ 1 dz 2 dz 1 d z¯ 2 d z¯ 1 d z¯ 2 1 2 1 2 1 2 1 2 − − + 2 2 2 2 φ(s1 ) φ(s2 ) (z 1 − z 2 ) φ(s1 ) φ(s2 ) (¯z 1 − z 2 ) φ(s1 ) φ(s2 ) (z 1 − z¯ 2 ) φ(s1 ) φ(s2 ) (¯z 1 − z¯ 2 )

1 = (4πi)2 =

φ(s1 ) φ(s2 )

φ(s1 )

φ(s2 )

dz 1 dz 2 (z 1 − z 2 )2

1 G(φ(s1 ), φ(s2 )), 2π

where we used φ(si ) ∈ R. Now suppose that some of the si coincide. We choose paths γi as before but √ suppose that the γi are close only at those endpoints where the si coincide. Let δ = . For pairs of edges ai ji , ai , ji on different paths, both within δ of such an endpoint we use 1 −1 the bound K (b, w) = O( ), so that the big determinant, multiplied by the prefactor i K (wi ji , bi ji ), is O(1). In the sum over paths the net contribution for each coincidence is then O(δ/)2 = O( −1 ), since this is the number of terms in which both edges of two different paths are near the endpoint. 8. Boxed Plane Partition Example For the boxed plane partition, whose hexagon has vertices {(1, 0), (1, 1), (0, 1), (−1, 0), (−1, −1), (0, −1)} in (x, ˆ yˆ ) coordinates, see Fig. 3, it is shown in [12] that satisfies (− x + y)2 = 1 − + 2 , or 1 − 2x y + 4(x 2 − x y + y 2 ) − 3 (x, y) = 2(1 − x 2 ) for (x, y) inside the circle x 2 − x y + y 2 ≤ 3/4. maps the region inside the inscribed circle with degree 2 onto the upper half-plane, with critical point (0, 0) mapping to z−eπi/3 eπi/3 . If we map the half plane to the unit disk with the mapping z → z−e −πi/3 , the composition is √ (x, y) − eπi/3 2r 2 − 3 + 9 − 12r 2 = (31) (x, y) − e−iπ/3 2(y − ωx) ¯ 2 (where r 2 = x 2 − x y + y 2 ) which maps circles concentric about the origin to circles concentric about the origin. To see this, note that z = i(y − ωx) defines the standard

708

R. Kenyon

conformal structure, and |z|2 = x 2 − x y + y 2 = r 2 , so that the right-hand side of Eq. (31) is f (r )/¯z 2 . Note also that the Beltrami differential of , which is µ(x, y) =

z¯ − eπi/3 = , z − e−πi/3

satisfies µ=

µz¯ , µz

that is, it is its own Beltrami differential! This is simply a restatement of the PDE (1) in terms of µ. The diffeomorphism φ from√the region inside the inscribed circle to the unit disk is also very simple, it is just φ = µ, or √ √ 3 − 3 − 4r 2 iθ iθ e . φ(r e ) = 2r The inverse of φ which maps D to U is even simpler: it is √ z 3 −1 φ (z) = . 1 + |z|2 This map can be viewed as the orthogonal projection of a hemisphere onto the plane through its equator, if we identify D conformally with the upper hemisphere sending 0 to the north pole. In conclusion if the domain U is the disk {(x, y) | x 2 −x y + y 2 ≤ r 2 }, where r 2 < 3/4, and the height function on the boundary of U is given by the height function h¯ of the BPP on ∂U , then the h¯ on U will equal the h¯ on BPP restricted to U , and the fluctuations of the height function are the pull-back of the Gaussian free field on the disk of radius √ √ 3− 3−4r 2 under φ. 2r Acknowledgements. Many ideas in this paper were inspired by conversations with Henry Cohn, Jim Propp, Jean-René Geoffroy, Scott Sheffield, Béatrice deTilière, Cédric Boutillier, and Andrei Okounkov. We thank the referees for useful comments. This paper was partially completed while the author was visiting Princeton University and IPAM, and the bulk of the paper was prepared while the author was at the University of British Columbia.

References 1. 2. 3. 4.

Ahlfors, L., Bers, L.: Riemann’s mapping theorem for variable metrics. Ann. Math 72, 385–404 (1960) Boutillier, C.: Thesis, Université Paris-Sud, 2004 Cohn, H., Kenyon, R., Propp, J.: A variational principle for domino tilings. J. AMS 14, 297–346 (2001) Cohn, H., Larsen, M., Propp, J.: The shape of a typical boxed plane partition. New York J. Math 4, 137–165 (1998) 5. Fournier, J.C.: Pavage des figures planes sans trous par des dominos: fondement graphique de l’algorithme de Thurston, parallélisation, unicité et décomposition. C. R. Acad. Sci. Paris Sér. I Math. 320(1), 107–112 (1995) 6. Kasteleyn, P.: Graph theory and crystal physics. In: Proc. 1967 Graph Theory and Theoretical Physics, London: Academic Press, pp. 43–110


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7. Kenyon, R.: Local statistics of lattice dimers. Ann. Inst. H. Poincar Probab. Statist. 33(5), 591–618 (1997) 8. Kenyon, R.: Conformal invariance of domino tiling. Ann. Probab. 28, 759–795 (2000) 9. Kenyon, R.: Dominos and the Gaussian free field. Ann. Probab. 29, 1128–1137 (2001) 10. Kenyon, R.: The Laplacian and Dirac operators on critical planar graphs. Invent. Math. 150, 409–439 (2002) 11. Kenyon, R.: Lectures on dimers. PCMI Lecture notes, to appear 12. Kenyon, R., Okounkov, A.: Limit shapes and the complex Burgers equation. Acta. Math. 199(2), 263–302 (2007) 13. Kenyon, R., Okounkov, A., Sheffield, S.: Dimers and amoebae. Ann. of Math. (2) 163(3), 1019–1056 (2006) 14. Kenyon, R., Propp, J., Wilson, D.: Trees and matchings. Electr. J. Combin. 7 (2000), research paper 25 15. Kenyon, R., Sheffield, S.: Dimers, tilings and trees. J. Combin. Theory Ser. B 92(2), 295–317 (2004) 16. Lovasz, L., Plummer, M.: Matching Theory, North-Holland Mathematics Studies, 121 Annals of Discrete Mathematics 29, Amsterdam: North-Holland Publishing Co., 1986 17. Percus, J.: One more technique for the dimer problem. J. Math. Phys. 10, 1881–1888 (1969) 18. Sheffield, S.: Gaussian free fields for mathematicians. Probab. Theory Related Fields 139(3–4), 521–541 (2007) 19. Sheffield, S.: Random surfaces. Astérisque No. 304 (2005) Communicated by H. Spohn


Communications in


Ergodic Theory of Parabolic Horseshoes Mariusz Urbanski ´ 1, , Christian Wolf2, 1 Department of Mathematics, University of North Texas, P.O. 311430, Denton,

TX 76203-1430, USA. E-mail: [email protected]

2 Department of Mathematics, Wichita State University, Wichita, KS 67260, USA.

E-mail: [email protected] Received: 10 June 2007 / Accepted: 18 November 2007 Published online: 15 May 2008 – © Springer-Verlag 2008

Abstract: In this paper we develop the ergodic theory for a horseshoe map f which is uniformly hyperbolic, except at one parabolic fixed point ω and possibly also on W s (ω). We call f a parabolic horseshoe map. In order to analyze dynamical and geometric properties of such horseshoes, by making use of induced maps, we establish, in the context of σ -finite measures, an appropriate version of the variational principle for continuous potentials with mild distortion defined on subshifts of finite type. Staying in this setting, we propose a concept of σ -finite equilibrium states (each classical probability equilibrium state is a σ -finite equilibrium state). We then study the unstable pressure function t → P(−t log |D f |E u |), the corresponding finite and σ -finite equilibrium states and their associated conditional measures. The main idea is to relate the pressure function to the pressure of an embedded parabolic iterated function system and to apply the developed theory of the symbolic σ -finite thermodynamic formalism. We prove, in particular, an appropriate form of the Bowen-Ruelle-Manning-McCluskey formula, the existence of exactly two σ -finite ergodic conservative equilibrium states for the potential −t u log |D f |E u | (where t u denotes the unstable dimension), one of which is the Dirac δ-measure supported at the parabolic fixed point and the other being nonatomic. We also show that the conditional measures of this non-atomic equilibrium state on unstable manifolds, are equivalent to (finite and positive) packing measures, whereas the Hausdorff measures vanish. As an application of our results we obtain a classification for the existence of a generalized physical measure, as well as a criteria implying the non-existence of an ergodic measure of maximal dimension.

The research of the first author was supported in part by the NSF Grant DMS 0400481. The research of the second author was supported in part by the National Science Foundation under Grant

No. EPS-0236913 and matching support from the State of Kansas through Kansas Technology Enterprise Corporation.

712

M. Urbański, C. Wolf

Contents 1.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Statement of the main results . . . . . . . . . . . . . . . . . . . . . . . 2. Symbol Space and the Shift Map . . . . . . . . . . . . . . . . . . . . . . . 3. Variational Principle and σ -Finite Equilibrium States . . . . . . . . . . . . . 4. One-Dimensional Parabolic Iterated Function Systems . . . . . . . . . . . . 5. Parabolic Smale’s Horseshoes . . . . . . . . . . . . . . . . . . . . . . . . . 6. Horseshoe and the Associated Parabolic Iterated Function System . . . . . . 7. Topological Pressures, Unstable Dimension, Hausdorff and Packing Measures 8. Equilibrium States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Conditional Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. Dimension of the Horseshoe . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Generalized Physical Measures . . . . . . . . . . . . . . . . . . . . . . . . 12. Measures of Maximal Dimension . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

712 712 712 714 716 725 727 730 733 737 740 742 744 746 750

1. Introduction 1.1. Motivation. Even mild parabolic features in one dimensional non-invertible dynamics have a profound impact on the dynamical and geometric character of the reference dynamical systems. Our goal in this paper is to understand whether in the case of higher dimensional systems the presence of, even weakly, parabolic points deeply affects the dynamics and, especially the geometry of the corresponding invariant set. We test this issue on one of the simplest, at least in our opinion, examples of parabolic higher dimensional systems. Namely, we introduce the concept of a parabolic horseshoe, and investigate it in detail throughout the paper. Even though parabolic horseshoes can be derived by slightly perturbing a hyperbolic horseshoe at one of its fixed points (see Example 1 in Sect. 5), the phenomena we discover differentiate promptly our parabolic system from hyperbolic ones. The main results are stated below in Subsect. 1.2. In order to perform our analysis of a parabolic horseshoe, we develop an appropriate form of the thermodynamic formalism of σ -finite measures, we borrow from the theory of parabolic iterated function systems and develop the “parabolic” approach to study generalized physical measures and measures of maximal dimension, existing up to our knowledge, so far only in hyperbolic contexts. 1.2. Statement of the main results. Let S ⊂ R2 be a closed topological disk whose boundary is smooth except at finitely many (possibly none) points. Let f : S → R2 be a parabolic horseshoe map of smooth type and let ω ∈ S be its parabolic fixed point (see Sect. 5 for the definition and details). We call the set = {x ∈ S : f n (x) ∈ S for all n ∈ Z} the parabolic horseshoe of f . Consider the potential φu : → R defined by φu (x) = log |D f (x)| E xu |. We define the unstable pressure function by P u (t) = P( f | , −tφu ) : R → R, where P( f | , .) denotes the topological pressure with respect to the dynamical system f |. Our first result is the following (Proposition 7.3 and Theorem 7.4) version of a Bowen-Ruelle-Manning-McCluskey type of formula for the unstable dimension of .

Ergodic Theory of Parabolic Horseshoes

713

Theorem 1.1. Let f : S → R2 be a parabolic horseshoe map of smooth type. Then the unstable dimension t u = dim H W u (x) ∩ is independent of x ∈ . Moreover, t u is the smallest zero of the unstable pressure function t → P u (t) and 0 < t u < 1. Let Ht (A) and Pt (A) denote the t-dimensional Hausdorff respectively packing measure of a set A. In the case of hyperbolic horseshoes and more generally for uniformly u (x) ∩ has positive and finite hyperbolic sets on surfaces it is well-known that Wloc u t -dimensional Hausdorff measure. The next result (see Theorem 7.5 in the text) shows that alone the occurrence of one parabolic fixed point can cause a drastic change on this phenomenon. Theorem 1.2. Let f : S → R2 be a parabolic horseshoe map of smooth type and let u (x) ∩ ) = 0 and 0 < P u (W u (x) ∩ ) < ∞. x ∈ . Then Ht u (Wloc t loc Next, we discuss results concerning the equilibrium states of the potential −t u φu . In particular, we consider finite as well as σ -finite equilibrium states. Let µω denote the Dirac δ-measure supported on the parabolic fixed point ω which is clearly an equilibrium state of the potential −t u φu . We show in Theorem 8.3 that there exists a unique (up to a multiplicative constant) ergodic conservative σ -finite equilibrium state µt u of the potential −t u φu which is distinct from µω . It turns out that the question whether µt u is finite is closely related to the behavior of f near ω. Let β be defined as in Eq. (5.1). Roughly speaking, the exponent β determines the rate at which orbits starting close to ω escape from ω. The following theorem compiles results from Theorems 8.1 and 8.3 in the text. Theorem 1.3. Let f : S → R2 be a parabolic horseshoe map of smooth type. Then the following are equivalent: (i) µt u is finite, in which case P u is not differentiable at t u ; (ii) t u > 2β/(β + 1). Since t u < 1, the measure µt u being finite implies that β < 1. On the other hand, by Eq. (5.1), 1 + β is an upper bound for the maximal possible regularity of f , i.e., f is at most of class C 1+β . Therefore, if f is a C 2 -diffeomorphism then µt u is always an infinite measure. In order to reasonably speak about σ -finite equilibrium states, we develop in Sect. 3 an appropriate form of a thermodynamic formalism: variational principle (Theorem 3.7) and equilibrium states (Definition 3.8) for σ -finite measures on subshifts of finite type (keep in mind that our parabolic horseshoe is topologically conjugate to the full shift on two elements). We now discuss two applications of our results concerning the existence of certain natural invariant measures of f . Recall that an ergodic invariant probability measure µ is called a generalized physical measure if its basin B(µ) has the same Hausdorff dimension as the stable set of (see Sect. 11 and [Wo] for more details). Applying Theorem 1.3 we are able to prove that the finiteness of the equilibrium state µt u is equivalent to the existence of a generalized physical measure (see Theorem 11.2). In particular, there are parabolic horseshoes having no generalized physical measure (see Corollary 11.3). This contrasts the case of hyperbolic surface diffeomorphisms which always have a unique generalized physical measure (see [Wo]). Another application concerns the existence of ergodic measures of maximal dimension. Given an invariant probability measure µ, we denote by dim H µ the Hausdorff

714


dimension of µ (see (12.1) for the definition). Assume now that µ is ergodic. Following [BW1] we say that µ is an ergodic measure of maximal dimension if dim H µ = sup dim H ν, ν

where the supremum is taken over all ergodic invariant probability measures ν. These measures have been intensively studied in the context of hyperbolic diffeomorphism in [BW1]. It turns out that for hyperbolic sets on surfaces there always exists an ergodic measure of maximal dimension. Moreover, this measure is, in general, not unique (see the example in [Ra]). In contrast to these results, we show in Theorem 12.4 that for certain parabolic horseshoe maps there exists no ergodic measure of maximal dimension. 2. Symbol Space and the Shift Map In this section we recall some notions from symbolic dynamics. We will discuss simultaneously one-sided and two-sided shift maps. We denote by Z the set of all integers and by N = {0, 1, 2, 3, . . .} the set of all non-negative integers. Given a countable, either finite or infinite set E and a function A : E × E → {0, 1}, called and incidence matrix, we define E +A = (ωn )+∞ ∈ E N : Aωn ωn+1 = 1 ∀ n ∈ N and 0 +∞ Z E +− = (ω ) ∈ E : A = 1 ∀ n ∈ Z . n ω ω n n+1 −∞ A We refer to either of these sets as a shift or a symbol space. In the case when we do not want to specify the shift space or also when it is clear from the context which shift space is meant, we write E A instead of E +A or E +− A . Given any s ∈ (0, 1), the space E A can be endowed with the metric ρ = ρs defined by ρ (ωn )n , (τn )n = s min{n≥0: ωn =τn or ω−n =τ−n } . Here we use the common convention that s +∞ = 0. All the metrics ρs , s ∈ (0, 1) are Hölder continuously equivalent and induce (the same) Tychonoff topology on E A . It is well-known that E A endowed with the product (Tychonov) topology is compact in the case when E is a finite set. The (left) shift map σ : E A → E A is defined by the formula σ (xn )n ) = (xn+1 )n . Occasionally, in order to avoid confusion, we write σ+ for σ : E +A → E +A and σ+− for +− +− σ : E +− A → E A . Note that in the case of E A , the shift map is injective and in the case of E +A , the shift map performs cutting off the zeroth coordinate. Let m ≤ n and let ω = (ωm , ωm+1 , · · · , ωn ) ∈ E n−m+1 . We call [ω] = {τ ∈ E A : τ j = ω j for all m ≤ j ≤ n}, the cylinder generated by ω. If ω ∈ E A and m ≤ n, we define ω|nm = (ωm , ωm+1 , . . . , ωn ), and, if m = 0, we frequently write ω|n instead of ω|n0 . For every n ≥ 1 an n-tuple τ of elements in E is said to be A-admissible provided that Aab = 1 for all pairs


715

of consecutive elements ab in τ . The number n is then called the length of τ and is denoted by |τ |. We denote by E nA the set of A-admissible tuples of length n. We also put +− +− n + + E ∗A = ∞ n=1 E A . Denote by : E A → E A the projection from E A to E A defined by +∞ (ωn )+∞ −∞ = (ωn )0 . If D ⊂ E then we write D A for D A| D×D . Let g : E A → R be a function. Given an integer n ∈ N we define the n th partition function Z n (D, g) by Z n (D, g) = exp sup Sn g(τ ) , ω∈D nA

τ ∈[ω]∩D A

where Sn g :=

n−1

g ◦ σ j.

j=0

In the case when D = E, then we simply write Z n (g) instead of Z n (E, g). A straightforward argument shows that the sequence (log Z n (D, f ))n∈N is subadditive. Therefore, we can define the topological pressure of g with respect to the shift map σ : D A → D A by

1 1 log Z n (D, g) : n ∈ N . (2.1) PD (g) = lim log Z n (D, g) = inf n→∞ n n If D = E, then we simply write P(g) instead of PE (g). Recall from [MU2] that a function g : E +A → R is said to be acceptable if it is uniformly continuous and sup sup g|[e] − inf g|[e] ) : e ∈ E < ∞. The following fact, which will be needed in the next section, was proven in [MU2]. Theorem 2.1. If g : E +A → R is acceptable, then

P(g) = sup hµ (σ ) + gdµ = sup{PD (g)}, where the first supremum is taken over all Borel probability (ergodic) σ -invariant measures µ on E such that the function −g is µ-integrable and the second supremum is taken over all finite subsets D of E. In the case when E = {0, 1, 2, . . . , d − 1}, where d ≥ 2, and the incidence matrix A consists of 1s only, we rather use the notation d+ := {0, 1, 2, . . . , d − 1}+A = {0, 1, 2, . . . , d − 1}N and Z d+− := {0, 1, 2, . . . , d − 1}+− A = {0, 1, 2, . . . , d − 1} .

Similarly as above, in the case when we do not want to specify the shift space or also when it is clear from the context which shift space is meant, we write d instead of d+ or d+− .

716


3. Variational Principle and σ -Finite Equilibrium States Let (X, A, µ) be a measure space, where µ is a σ -finite measure. Moreover, let T : X → X be a measurable map which is µ-invariant (that is, µ ◦ T −1 = µ), ergodic and conservative (see [A] for the definitions). Consider a fixed set F ∈ A with µ(F) > 0. Then the first return time τ := τ F : F → {1, 2, . . .} ∪ {∞} given by the formula τ (x) = min{k ≥ 1 : T k (x) ∈ F} is finite in the complement (in F) of a set of measure zero. Therefore, the first return map TF : F → F given by the formula TF (x) = T τ F (x) (x) is well-defined µ-a.e. on F. If φ : X → R is a measurable function, we set φ F (x) =

τ (x)−1

φ ◦ T j (x), x ∈ F.

j=0

The function φ F is A F -measurable, where A F is the σ -algebra of all subsets of F that belong to A. If in addition µ(F) < ∞, then it is well-known (see [A] for example) that the conditional measure µ F on F defined by µ F (B) = µ(B)/µ(F), B ∈ A F , is TF -invariant. Also if φ is µ-integrable, that is |φ|dµ < +∞, then φ F is µ F -integrable and

φdµ . (3.1) φ F dµ F = µ(F) F However, the converse is not in general true, i.e. µ F integrability of φ F does not imply µ-integrability of φ, even if the measure µ is finite. We now shall provide a short proof of the following, essentially immediate, consequence of Abramov’s formula, which is due to Krengel ([K]). Theorem 3.1. Let µ be a σ -finite measure on X , and let T : X → X be an ergodic, conservative and µ-invariant map. If E and F are measurable sets with 0 < µ(E), µ(F) < ∞, then µ(E)hµ E (TE ) = µ(F)hµ F (TF ). Proof. Put D = E ∪ F. Obviously 0 < µ(D) < +∞. Let TED and TFD be the first return maps respectively to E and F induced by the map TD : D → D. It follows from Abramov’s formula that µ D (E)h(µ D ) E (TED ) = hµ D (TD ) = µ D (F)h(µ D ) F (TFD ).

(3.2)

Since µ D (E) = µ(E)/µ(D), µ D (F) = µ(F)/µ(D), TE = TED , TF = TFD (as E, F ⊂ D), and (µ D ) E = µ E , (µ D ) F = µ F , we may conclude that µ(E)hµ E (TE ) = µ(F)hµ F (TF ). Following Krengel we denote this common value µ(F)hµ F (TF ) by hµ (T ) and call it the entropy of the transformation T with respect to the invariant measure µ. Assume now that T : X → X is a continuous self-map of a compact metric space X . Given a Borel


717

measure µ on X let X µ (∞) be the set of all points x ∈ X such that µ(U ) = +∞ for every open set U containing x. Following [U3] we call X µ (∞) the set of points of infinite condensation of µ. Notice that X µ (∞) is a closed subset of X and X µ (∞) = ∅ if and only if the measure µ is finite. Denote by M∞ T the family of all Borel ergodic conservative T invariant measures µ for which µ(X \ X µ (∞)) > 0. Note that if µ ∈ M∞ T , then X µ (∞) is forward invariant, i.e. T (X µ (∞)) ⊂ X µ (∞). The following simple proposition shows that X µ (∞) is measurably negligable and, in the transitive case, it is also topologically negligable. Proposition 3.2. If T : X → X is a continuous self-map of a compact metric space X and µ ∈ M∞ T , then µ(X µ (∞)) = 0. If, in addition, T : X → X is topologically transitive, then X µ (∞) is a nowhere dense subset of X . Proof. Since T(X µ (∞)) since, by ergodicity and conservativity of µ, ⊂ nX µ (∞) and the measure µ X \ ∞ T (X (∞)) = 0 whenever µ(X µ (∞)) = 0, the condiµ n=0 tion µ(X \ X µ (∞)) > 0 implies that µ(X µ (∞)) = 0. Suppose now that T is topologically transitive and assume X µ (∞) is not nowhere dense. Then it follows that ∞ ∞ n n n=0 T (X µ (∞)) is a dense subset of X . Since however n=0 T (X µ (∞)) ⊂ X µ (∞) and since X \ X µ (∞) = ∅ (as µ(X \ X µ (∞)) > 0), we get a contradiction which completes the proof. Proposition 3.3. If T : X → X is a continuous self-map of a compact metric space X and µ ∈ M∞ T , then µ is σ -finite. Proof. By the definition of X µ (∞), for every point x ∈ X \ X µ (∞) there exists an open ball Bx ⊂ X \ X µ (∞) such that x ∈ Bx and µ(Bx ) < +∞. Since X \ X µ (∞) is a separable metric space, there exists a countable set Y ⊂ X such that y∈Y B y = X \ X µ (∞). Thus {X µ (∞)} ∪ {B y } y∈Y forms a countable cover of X by Borel sets of finite measure (µ(X µ (∞)) = 0) by Proposition 3.2. Note that if F is a closed (compact) subset of X \ X µ (∞), then µ(F) < +∞. Motivated by the remark after formula (3.1), we say that a Borel function φ : X → R is dynamically integrable with respect to µ ∈ M∞ T on X provided that F |φ F |dµ F < +∞ for every closed set F ⊂ X \ X µ (∞) with µ(F) > 0 (closed sets can be equivalently repla ced by those whose closure is disjoint from X µ (∞)). Then the number µ(F) F φ F dµ F is independent of F, its common value is denoted by

ˆ

φdµ,

and it is referred to as the dynamical integral of φ against the measure µ. Thus, we have

ˆ

φ F dµ F

φdµ = µ(F)

(3.3)

F

whenever F is a closed subset of X \ X µ (∞) with positive measure µ. Of course, each integrable function is dynamically integrable, but, as has been said, not conversly. Keep T : X → X as a continuous self-map of a compact metric space X . A closed forward invariant subset Q of X is called ergodically finite provided T | Q : Q → Q admits no conservative ergodic infinite σ -finite invariant measure. Fix a continuous

718


function φ : X → R. Then it is a consequence of the well-known variational principle that

(3.4) sup hµ (T ) + (φ − P(φ))dµ = 0, where P(φ) denotes the topological pressure of the potential φ, and the supremum is taken over all ergodic T -invariant Borel probability measures on X . Denote by M∞ φ the family of all measures µ ∈ M∞ for which the function φ − P(φ) is dynamically T integrable with respect to the measure µ. The potential φ : X → R is called VPA (Variational Principle Admissible) provided that

ˆ ∞ sup hµ (T ) + (φ − P(φ))dµ : µ ∈ Mφ = 0. (3.5) One of the most common ergodic theory constructions is the concept of Rokhlin’s natural extension. The next lemma and the corollaries following establish close canonical relations between the classes M∞ and M∞ as well as M∞ and M∞ , where T˜ : X˜ → X˜ T

φ

T˜

φ◦

is Rokhlin’s natural extension of T : X → X and : X˜ → X is the projection onto the 0th coordinate. It will be the basic tool in Sect. 7 to establish links between parabolic horseshoes and parabolic iterated function systems. Lemma 3.4. Suppose that T : X → X is a continuous map of a compact metric space X . Let T˜ : X˜ → X˜ be Rokhlin’s natural extension of T . Then the map µ → µ ◦ −1 is ∞ a bijection between M∞ ˜ and MT . In addition, T

hµ (T˜ ) = hµ◦−1 (T ),

(3.6)

, the function φ and, if φ : X → R is a Borel function, then for every measure µ ∈ M∞ T˜ is dynamically integrable against µ ◦ −1 if and only if φ ◦ is dynamically integrable against the measure µ. Furthermore, if one of these integrabilities holds, then

ˆ

ˆ φ ◦ dµ = φdµ ◦ −1 . X˜

X

Proof. Recall that X˜ = {(xn )+∞ n=0 : T (x n+1 ) = x n for all n ≥ 0}, and +∞ T˜ ((xn )+∞ n=0 ) = (T (x n ))n=0 .

Recall that T˜ : X˜ → X˜ is a homeomorphism and its inverse is the left shift map (cutting off the 0th coordinate). For every n ≥ 0 let n : X˜ → X be the projection onto the n th coordinate. We put := 0 . Fix µ ∈ M∞ . Since T˜ lim dist T˜ n (−1 (( X˜ µ (∞)))), X˜ µ (∞) = 0, n→∞

it follows from conservativity of the measure µ that µ −1 (( X˜ µ (∞))) \ X µ (∞) = 0. Therefore,


719

µ ◦ −1 (( X˜ µ (∞))) = µ(−1 (( X˜ µ (∞)))) = µ(( X˜ µ (∞) ∪ (−1 (( X˜ µ (∞))) \ X˜ µ (∞))) = µ(( X˜ µ (∞)) + µ(−1 (( X˜ µ (∞))) \ X˜ µ (∞)) = 0 + 0 = 0. (3.7) Now, if ω ∈ X \ ( X˜ µ (∞)), then there exists an open neighbourhood G ⊂ X of ω such that G ∩ ( X˜ µ (∞)) = ∅. Then −1 (G) ∩ X˜ µ (∞) = ∅, and, as −1 (G) is compact, µ(−1 (G)) < +∞. Hence, µ ◦ −1 (G) < ∞, so ω ∈ / X µ◦−1 (∞) and X µ◦−1 (∞) ⊂ ( X˜ µ (∞)), and, by virtue of (3.7), µ ◦ −1 (X µ◦−1 (∞)) = 0. On the other hand, if µ is T˜ -invariant and µ ◦ −1 ∈ M∞ T , then X˜ µ (∞) ⊂ −1 (X µ◦−1 (∞)),

(3.8)

µ( X˜ µ (∞)) ≤ µ ◦ −1 (X µ◦−1 (∞)) = 0.

(3.9)

( X˜ µ (∞)) = X µ◦−1 (∞).

(3.10)

and (therefore),

We also claim that

Indeed, the inclusion ( X˜ µ (∞)) ⊂ X µ◦−1 (∞) follows directly from (3.8). Suppose now for a contradiction that there exists a point x ∈ X µ◦−1 (∞)\( X˜ µ (∞)). Then there exists a closed neighborhood U of x such that U ∩ ( X˜ µ (∞)) = ∅. But then −1 (U ) is a compact set disjoint from X˜ µ (∞)). Hence µ ◦ −1 (U ) = µ(−1 (U )) < +∞, contrary to the assumption that x ∈ X µ◦−1 (∞). Thus, identity (3.10) is proven. Now, it is obvious that if µ ∈ M∞ (it suffices to know that µ is shift-invariant and T˜ ergodic conservative), then µ ◦ −1 is ergodic conservative. On the other hand, suppose that µ is a Borel T˜ -invariant measure on X˜ such that µ ◦ −1 ∈ M∞ T . We want to show that µ is ergodic conservative. Indeed, if µ has atoms, then µ ◦ −1 has them as well (images of atoms of µ under ), and since µ◦−1 is ergodic, the measure µ◦−1 must be finite and supported on a single periodic orbit of T . Therefore, µ must be also finite and supported on a single periodic orbit of T˜ . Thus, µ is ergodic conservative. If however µ has no atoms, then in virtue of Proposition 1.2.1 from [A] (saying that for invertible nonsingular transformations with respect to a non-atomic measure, ergodicity implies conservativity), it suffices to show that µ is ergodic. Since µ ◦ −1 ∈ M∞ T , there exists a compact set F ⊂ X \ X µ◦−1 (∞) such that µ◦−1 (F) > 0 (then also µ(F) < +∞). By Proposition 1.5.2(1) in [A], the first return map TF : F → F is ergodic with respect to the conditional measure µ◦−1 | F on F. Using the uniqueness property of Rokhlin’s natural extension, it is easy to see that the authomorphism T˜−1 (F) : −1 (F) → −1 (F) with the invariant measure µ−1 (F) is canonically isomorphic to Rokhlin’s natural extension T˜F : F˜ → F˜ of TF : F → F. But, by ergodicity of the latter, the former map (T˜F : F˜ →

720


˜ is also ergodic (see [KFS], Theorem 1 in Sect. 10.4). Hence, T˜−1 (F) : −1 (F) → F) −1 is ergodic conservative for T : X → X , −1 (F) is ergodic. Since the measure µ ◦ ∞ −1 −n ˜ −n (π −1 (F))) = 0. At this point X \ n=0 T (F) = 0. Hence, µ X˜ \ ∞ µ◦ n=0 T Proposition 1.5.2(2) in [A] applies, to yield that the map T˜ : X˜ → X˜ is ergodic with respect to the measure µ. Since µ is non-atomic, it now follows from Proposition 1.2.1 in [A] that µ is conservative. So, µ is ergodic conservative, and by using (3.7), we may conclude that µ ∈ M∞ . The above considerations with the set F provide also the T˜ −1 = ν ◦ −1 , injectivity of the map µ → µ ◦ −1 . Indeed, if µ, ν ∈ M∞ ˜ and µ ◦ T

−1 then µ◦−1 F = ν ◦ F . So, from injectivity of Rokhlin’s natural extension construction on probability measures, and because of the canonical isomorphism established above, we conclude that µ−1 (F) = ν−1 (F) . Thus, using also the fact that µ(−1 (F)) = ν(−1 (F)), we may conclude that µ = ν. Summarizing what we have done so far, we have shown that the map µ → µ ◦ −1 is an injection from M∞ to M∞ T , and that in T˜ order to prove this map to be bijective, it suffices to demonstrate that if m ∈ M∞ T , then −1 ˜ ˜ there exists a Borel T -invariant measure µ on X such that m = µ ◦ . And indeed, fix again a compact set F ⊂ X \ X µ◦−1 (∞) such that µ ◦ −1 (F) > 0 (then also µ(F) < +∞). Since the homeomorphism T˜−1 (F) : −1 (F) → −1 (F) is canonically isomorphic (topologically conjugate) with (topological) natural extension T˜F : F˜ → F˜ of TF : F → F, there exists a finite Borel T˜−1 (F) -invariant measure m˜F on −1 (F) such that m F = m˜F ◦ −1 . Then there exists a finite Borel T˜ -invariant measure µ on X˜ such that µ−1 (F) = m˜F . Hence, (µ◦−1 ) F = µ−1 (F) ◦−1 = m˜F ◦−1 = m F . So, m = µ ◦ −1 , and bijectivity of the map ν → ν ◦ −1 from M∞ to M∞ T is established. T˜ In order to prove the entropy formula (3.6), we first notice that Rokhlin’s natural extension of probability invariant measures preserves entropies. Fix now µ ∈ M∞ and T˜ −1 a Borel set F ⊂ X such that 0 < µ ◦ (F) < +∞. Then

hµ◦−1 (T ) = µ ◦ −1 (F)h(µ◦−1 ) F (TF ) = µ(−1 (F))hµ−1 (F) (T˜−1 (F) ) = hµ (T˜ ). Passing to the proof of the last claim of our lemma, fix µ ∈ M∞ and a Borel function T˜ φ : X → R. Suppose that φ and φ ◦ are dynamically integrable respectively against measures µ ◦ −1 and µ. Fix a compact set F ⊂ X \ X µ (∞) such that µ ◦ −1 (F) > 0. Then

ˆ

φdµ ◦ −1 = µ ◦ −1 (F) φ F d(µ ◦ −1 ) F X

F = µ(−1 (F)) φ F dµ−1 (F) ◦ −1

F −1 = µ( (F)) φ F ◦ dµ−1 (F) −1 (F)

= µ(−1 (F)) (φ F ◦ )−1 (F) dµ−1 (F) =

−1 (F)

ˆ X˜

φ ◦ dµ.


721

Since dynamical integrability of φ ◦ against µ clearly implies dynamical integrability of φ against µ ◦ −1 , we are therefore left to show that dynamical integrability of φ against µ ◦ −1 implies dynamical integrability of φ ◦ against µ. In order to do this, fix a compact set F ⊂ X˜ \ X˜ µ (∞) such that µ(F) > 0. Since the sets of the form −1 n (G), where G is an open subset of X and n is an integer ≥ 0, form a basis of topology for X˜ , there exist finitely many integers 0 ≤ n 1 < n 2 < . . . < n k and open sets G 1 , G 2 , . . . , G k ⊂ X such that F⊂

k

−1 −1 ˜ ˜ −1 n j (G j ) and n 1 (G 1 ) ∪ . . . ∪ n k (G k ) ⊂ X \ X µ (∞).

(3.11)

j=1

Since n j = T n k −n j ◦ n k for all j = 1, 2, . . . , k, we get that F ⊂ −1 n k (), where =

k j=1

(3.12)

T −(n k −n j ) (G j ). Set l = n k . By (3.12), we get T˜ −l (F) ⊂ (l ◦ T˜ l )−1 () = −1 ()

(3.13)

and, l−1 () =

k

k k l−1 T −(n k −n j ) (G j ) = (T (n k −n j ) ◦ l )−1 (G j ) = −1 n j (G j ).

j=1

j=1

j=1

Hence, −1 () ∩ X˜ µ = (T l ◦ l )−1 () ∩ T˜ −l ( X˜ µ ) = (l ◦ T˜ l )−1 () ∩ T˜ −l ( X˜ µ ) = T˜ −l (−1 ()) ∩ T˜ −l ( X˜ µ ) = T˜ −l (−1 ()) ∩ X˜ µ ) l

l

⊂ T˜ −l (∅) = ∅. Thus, using also (3.10), we get ∩ X µ◦−1 = ◦ ( X˜ µ ) = ∅. Since is compact and φ is dynamically integrable against µ ◦ −1 , we conclude that φ is integrable against (µ ◦ −1 ) , or equivalently, φ ◦ −1 () is integrable against µ−1 () . We are now going to show that φT˜ l (−1 ()) is integrable against µT˜ l (−1 ()) . Indeed, since T˜ ˜ l −1 ◦ T˜ l = T˜ l ◦ T˜−1 () and since the measure µ is T˜ l T (

())

invariant, the map T˜ l : −1 () → T˜ l (−1 iso ()) establishes a measure-preserving morphism between the dynamical systems T˜−1 () ,−1 (),µ−1 () and T˜T˜ l (−1 ()) ,

722


T˜ l (−1 ()), µT˜ l (−1 ()) . Therefore,

T˜ l (−1 ())

|φ ◦ T˜ l (−1 ()) |dµT˜ l (−1 ())

= |φ ◦ T˜ l (−1 ()) |dµ−1 () ◦ T˜ −l l −1 ˜ T ( ())

= |φ ◦ T˜ l (−1 ()) ◦ T˜ l |dµ−1 () −1 ()

= |(φ ◦ ◦ T˜ l )−1 () |dµ−1 () −1 ()

=

−1 ()

−

l−1

τ−1 () (x)+l−1 τ−1 () (x)−1 ˜ j (x)) + φ ◦ ( T φ ◦ (T˜ j (x)) j=0

j ˜ φ ◦ (T (x))dµ−1 ()

j=τ−1 () (x)

j=0

|φ ◦ )−1 () | + 2l||φ||∞ dµ−1 ()

= 2l||φ||∞ + |φ ◦ )−1 () |dµ−1 () < +∞. ≤

−1 ()

−1 ()

Thus φ ◦ T˜ l (−1 ()) is integrable against µT˜ l (−1 ()) . By (3.13), F ⊂ T˜ l (−1 ()), and therefore, (φ ◦ ) F = (φ ◦ )T˜ l (−1 ()) F is integrable against µ F . We are done. We will need the following. Lemma 3.5. Suppose that T : X → X is a continuous self-map of a compact metric space X . Suppose φ, ψ : X → R are two continuous functions cohomologous in the class of continuous functions. Then for every µ ∈ M∞ T , the function ψ is dynamically ˆ integrable against µ if and only if φ is, and ψdµ = ˆ φdµ. Furthermore, ψ satisfies the generalized variational principle if and only if φ does, and E S ∞ (ψ) = E S ∞ (φ), the sets of all (σ -finite) equilibrium states of ψ and φ respectively. Proof. By our hypothesis there exists a continuous function u : X → R such that ψ − φ = u − u ◦ T . Fix F, a compact subset of X \ X µ (∞). Then ψ F − φ F = u − u ◦ TF . Since u is bounded, ψ F is thus integrable against µ F if and only if φ F is, and ψ dµ = φ F dµ f . Therefore ψ is dynamically integrable against µ if and only F f F ˆ F if φ is, and ψdµ = ˆ φdµ. The rest of the lemma follows now from the observation that P(ψ) = P(φ) and (ψ − P(ψ)) − (φ − P(φ)) = u − u ◦ T . We remark that this lemma fails if dynamical integrability against µ is replaced by (sheer) integrability against µ. In order to prove the main results of this section, a sufficient condition for a potential to be VPA, our function φ : X → R needs to satisfy some further conditions. Let d be


723

a metric on X compatible with the topology of X . Given n ≥ 1 the metric dn on X is defined as follows: dn (x, y) = max{d(T j (x), T j (y)) : 0 ≤ j ≤ n − 1}. Each metric dn is equivalent to d (they induce the same topology) and the set Bn (x, δ) = Bdn (x, δ), an open ball with respect to the metric dn , is called the Bowen ball centered at x with radius δ. Considering again a function φ : X → R, we need the following definition. Definition 3.6. A continuous function φ : X → R is said to be of mild distortion provided that there exists an ergodically finite set Q ⊂ X such that ∀ε > 0 ∃δ > 0 s.th. ∀x ∈ X ∀n ≥ 1 with T n (x) ∈ / B(Q, ε) we have |Sn φ(y) − Sn φ(x)| ≤ ε

(3.14)

whenever y ∈ Bn (x, δ). If we need to be more specific about the set Q, we say that the function φ is of mild distortion modulo Q. In the context of subshifts of finite type and potentials of mild distortion we obtain the following stronger version of the variational principle (3.4) which now also takes σ -finite measures into account. Theorem 3.7. Suppose that E = ∅ is a finite set and let A : E × E → {0, 1} be an irreducible incidence matrix. If φ : E +A → R is a continuous function with mild distortion, then φ is VPA, i.e.

ˆ ∞ sup hµ (σ ) + (φ − P(φ))dµ : µ ∈ Mφ = 0. Proof. In view of (3.3), (3.4) and Theorem 3.1 it is sufficient to demonstrate that

hµ F (σ F ) + (φ − P(φ)) F dµ F ≤ 0 (3.15) F

+ + for every infinite measure µ ∈ M∞ φ and some set F ⊂ F ⊂ E A \ E A,µ (∞) (depending on µ) with µ(F) > 0. Consider such an infinite measure µ ∈ M∞ φ . It follows from + Proposition 3.2 and the ergodic finiteness of Q that µ(E A,µ (∞) ∪ Q) = 0. Hence, there exists an A-admissible finite word ω such that [ω] ⊂ E A \ (E +A,µ (∞) ∪ Q) and µ([ω]) ∈ (0, ∞). Let F be the set of all those ρ ∈ E A that return to [ω] infinitely often under the forward iteration of the shift map σ : E +A → E +A . Since µ is conservative it follows that µ(F) = µ([ω]) ∈ (0, ∞). Let

E ω = {ρ|τ F (ρ)−1 : ρ ∈ F}. Obviously E ω is a countable set (as a subset of E ∗A ). Define an incidence matrix Aω : E ω × E ω → {0, 1} by declaring that Aω (α, β) = 1 if and only if αβ ∈ E ∗A . Clearly Aω is irreducible and F coincides with the set E ω∞ of all infinite concatenations of elements from E ω admissible by Aω . Furthermore, the map I : F → E ω∞ which ascribes to each element ρ ∈ F its unique representation as an infinite word over the alphabet E ω , is a

724


homeomorphism, and I establishes topological conjugacy between σ F : F → F and the full shift map σ : E ω∞ → E ω∞ . That is, the following diagram commutes, F ⏐ ⏐ I E ω∞

σF −→ σ −→

F ⏐ ⏐ I E ω∞

or equivalently I ◦ σ F = σ ◦ I . For every n ≥ 1 let Dn be the (finite) set of all those elements in E ω whose length (when treated as elements in E ∞ A ) is bounded above by n. Then I −1 (Dn∞ ) is a compact subset of E ∞ , the time of the first return from I −1 (Dn∞ ) to A −1 ∞ −1 ∞ I (Dn ) of every element x ∈ I (Dn ) is equal to τ F (x) ≤ n, and the first return map σn : I −1 (Dn∞ ) → I −1 (Dn∞ ) is continuous. In what follows, subtracting P(φ) from φ, we may assume without loss of generality that P(φ) = 0 and we may treat, via the conjugacy I , the function ψ = φ F as defined on the shift space E ω∞ . Since [ω] ∩ Q = ∅ and both Q and [ω] are compact sets, we have ε = dist(Q, [ω]) > 0. Let δ > 0 be associated to ε according to Definition 3.6. Replacing [ω] by its sufficiently long subcylinder, we may assume without loss of generality that diam([ω]) < δ. Then σ |ω| ([ρ|τ F (ρ)−1 ω]) ⊂ Bτ F (ρ)−|ω| (σ |ω| (ρ), δ), and it therefore follows from Definition 3.6 (note also that σ τ F (ρ)−|ω| (σ |ω| (ρ)) = σ τ F (ρ) (ρ) ∈ / Q) that the function φ F is acceptable. Thus, in view of Theorem 2.1 and the classical variational principle, for every θ > 0 there exists n ≥ 1 and a Borel probability σ F -invariant measure m supported on I −1 (Dn ) such that

hµ F (σ F ) + (φ) F dµ F ≤ hm (σ F ) + φ F dm + θ. (3.16) F

F

Since m is supported on I −1 (Dn ) (τ F ≤ n!), there exists a Borel probability σ -invariant measure mˆ on E +A such that mˆ F = m. Hence, by (3.4), hmˆ (σ )+ E + φd mˆ ≤ 0. Therefore, A by (3.1) and Theorem 3.1, we get that hm (σ F ) + F φdm ≤ 0. Inserting this-into (3.16) gives hµ F (σ F )+ F (φ) F dµ F ≤ θ . Letting finally θ 0 yields hµ F (σ F )+ F (φ) F dµ F ≤ 0. We are done. It is clear that every Hölder continuous potential has mild distortion with respect to Q = ∅. In the forthcoming sections we will provide further classes of potentials with mild distortion. Passing to σ -finite equilibrium states, we start with the following. Definition 3.8. Let T : X → X be a continuous self-map of a compact metric space and let φ : X → R be a continuous VPA potential. A σ -finite measure µ ∈ M∞ φ is said to be an equilibrium state of φ provided that

ˆ (φ − P(φ))dµ = 0. hµ (T ) + X

The space of all (σ -finite) equilibrium states of φ is denoted by E S ∞ (φ).


725

+− Since the two-sided shift map σ+− : E +− A → E A is canonically topologically conjugate to Rokhlin’s natural extension of the one-side shift σ+ : E +A → E +A , we now can easily prove the following.

Proposition 3.9. If ψ : E +A → R has mild distortion and φ : E +− A → R is a continuous function such that ψ ◦ and φ are cohomologous in the class of continuous functions, −1 is a bijection between the sets of then φ : E +− A → R is VPA, the map µ → µ ◦ equilibrium states of φ and ψ respectively, P(φ) = P(ψ), and

ˆ

ˆ hµ (σ+− ) = hµ◦−1 (σ+ ) and φdµ = ψdµ ◦ −1 . E +− A

E +A

Proof. It follows from Lemma 3.4 and the classical Variational principle that P(φ) = P(ψ ◦ ) = P(ψ). Our proposition is then an immediate consequence of Theorem 3.7, Lemma 3.4 and Lemma 3.5. 4. One-Dimensional Parabolic Iterated Function Systems The concept of a parabolic Cantor set was introduced in [U2]. The concept of a parabolic iterated function system was formally introduced in [MU1]. Both concepts largely overlap and the object dealt with in this and subsequent sections belongs to this overlap. We briefly summarize here the definition of a parabolic iterated function system following [MU1] and partially adopting it to the much more concrete setting we will need in the sequel for our applications. Let be a compact line segment. Suppose that we have at least two and at most finitely many C 1+ε maps φi : → , i ∈ I , satisfying the following conditions: (1) Open Set Condition: φi (int()) ∩ φ j (int()) = ∅ for all i = j. (2) |φi (x)| < 1 everywhere except for finitely many pairs (i, xi ), i ∈ I , for which xi is the unique fixed point of φi and |φi (xi )| = 1. Such pairs and indices i will be called parabolic and the set of parabolic indices will be denoted by . All other indices will be called hyperbolic. (3) ∀n ≥ 1 ∀ω = (ω1 , ω2 , . . . , ωn ) ∈ I n if ωn is a hyperbolic index or ωn−1 = ωn , then φω = φω1 ◦ φω2 . . . φωn extends in a C 1+ε manner to an open line segment V and maps V into itself. (4) If i is a parabolic index, then n≥0 φi n () = {xi } and so the diameters of the sets φi n () converge to 0. Here i n denotes the n-tuple whose entries are all i. (5) ∃s < 1 ∀n ≥ 1 ∀ω ∈ I n if ωn is a hyperbolic index or ωn−1 = ωn , then ||φω || ≤ s. (6) Bounded Distortion Property: ∃K ≥ 1 ∀n ≥ 1 ∀ω = (ω1 , . . . , ωn ) ∈ I n ∀x, y ∈ V if ωn is a hyperbolic index or ωn−1 = ωn , then |φω (y)| ≤ K. |φω (x)| We call such a system of maps F = {φi : i ∈ I } a 1-dimensional subparabolic iterated function system. If = ∅, we call the system F = {φi : i ∈ I } a 1-dimensional parabolic iterated function system. From now on throughout the entire section we assume the system F to be parabolic.

726


The Bounded Distortion Property (6) is not obvious in the 1-dimensional case. But there is a natural easily verifiable sufficient condition for this property to hold. Indeed, it was proved in [U1] and [U2] that condition (6) (Bounded Distortion Property) follows from all conditions (1)-(5) enlarged by the requirement that if i is a parabolic element, then the map φi has the following representation in a neighborhood of the the parabolic point xi : φi (x) = x ± a|x − xi |βi +1 + o(|x − xi |βi +1 )

(4.1)

with some βi > 0 and with the sign “−” if x − xi ≥ 0 and the sign “+” if x − xi ≤ 0. It was also proven in [U1] and [U2] that for every parabolic element i and every n ≥ 0, |φin (x)| (n + 1)

−

βi +1 βi

and |φin (x) − xi | (n + 1)

− β1

i

outside every fixed open neighborhood of xi . In particular |φin j (x)| (n + 1)

−

βi +1 βi

and |φin j (x) − xi | (n + 1)

− β1

i

for every parabolic element i and every j ∈ I \{i} and all n ≥ 0. Recall that the elements of the set I \ are called hyperbolic (see Condition 2). We extend this name to all the words appearing in Conditions (5) and (6). By I ∗ we denote the set of all finite words with alphabet I and by I ∞ all infinite sequences with elements in I . It follows from (3) that for every hyperbolic word ω, φω (V ) ⊂ V . Note that our conditions insure that φi (x) = 0, for all i and x ∈ V . It has been proven in [MU1] that for all ω = (ωn )n≥0 ∈ I ∞ the intersection n≥0 φωn () is a singleton. Furthermore, lim sup{diam(φω ()) : ω ∈ I ∗ , |ω| = n)} = 0.

n→∞

Thus we can define the coding map π : I ∞ → , defining π(ω) to be the only element of the intersection n≥0 φωn (), and this map is uniformly continuous. The limit set J = JF of the system F = {φi }i∈I is defined to be π(I ∞ ). It turns out that J is compact and satisfies the following invariance property: J = ∪i∈I φi (J ). Consider now the system F ∗ generated by I ∗ defined by F ∗ = {φi n j : n ≥ 1, i ∈ , i = j} ∪ {φk : k ∈ I \ }. It immediately follows from our assumptions that the following holds. Theorem 4.1. The system F ∗ is a hyperbolic (though with the infinite alphabet I ∗ ) conformal iterated function system, i.e. F ∗ has no parabolic elements. F ∗ is called the hyperbolic iterated function system associated to the parabolic system F. The limit set generated by the system F ∗ is denoted by J ∗ . A proof of the following lemma can be found in [MU1]. Lemma 4.2. The limit sets J and J ∗ of the systems F and F ∗ respectively differ only by a countable set. More precisely, J ∗ ⊂ J and J \ J ∗ is countable. In this paper we will only be interested in the special case when I = {0, 1}, φ0 () ∩ φ1 () = ∅, and 0 is the only parabolic index with the parabolic point x0 . The following result is an immediate consequence of the Bounded Distortion Property (6).


727

Proposition 4.3. The function ω → − log |φω 1 (π(σ (ω))|, ω ∈ 2+ , has mild distortion modulo Q = {0∞ }. Note that now the projection π : 2+ → JF is a homeomorphism, there is a welldefined map F : JF → JF , where F(x) = φi−1 (x) if x ∈ φi (), which has a C 1+ε extension φi−1 to each interval φi (V ), and the projection π : 2+ → JF establishes a canonical conjugacy between the shift map σ : 2+ → 2+ and the map F : JF → JF . Consequently, the entire discussion from Sect. 2 about symbol spaces and shift maps applies to the map F : JF → JF . We will frequently invoke the results proven in [MU1] and [U2] concerning the dynamics of the map F and the geometry of the limit set JF called a parabolic Cantor set in [U2]. 5. Parabolic Smale’s Horseshoes In this section we describe the main object of interest in this paper, a parabolic horseshoe of smooth type. Let S ⊂ R2 be a closed topological disk whose boundary is smooth except at finitely many (possibly none) points. We consider a C 1+ε -diffeomorphism f : S → R2 onto its image having the following properties: (a) The intersection f (S) ∩ S consists of two disjoint closed topological disks R0 and R1 . (b) The closure f (S) \ S consists of three mutually disjoint topological disks R2 , R3 , R4 . Furthermore, it is required that there exist two 1-dimensional mutually transversal foliations W u and W s of S ∪ f (S) consisting of smooth connected leaves with the following properties: (c) If x ∈ Ri , i = 0, 1, 2, 3, 4, then the sets W u (x) ∩ Ri and W s (x) ∩ Ri are connected. (d) For all points x, y ∈ Ri , i = 0, 1, 2, 3, 4, W u (x) ∩ W s (y) ∩ Ri is a singleton denoted by [x, y]. (e) For every point x ∈ Ri , i = 0, 1, 2, 3, 4, the map [·, ·] : W s (x)∩Ri ×W u (x)∩Ri → Ri , (y, z) → [y, z], is a homeomorphism. (f) For every point x ∈ Ri , i = 0, 1, f W u (x) ∩ Ri = W u ( f (x)). If f (x) ∈ R j , j = 0, 1, 2, 3, 4, then f −1 W s ( f (x)) ∩ R j = W s (x). u/s

For every point x ∈ S we denote by E x the tangent space of W u/s (x) at x. We use the notation Du/s f (x) for the derivative of the map f : W u/s (x) ∩ S → W u/s ( f (x)); u/s hence Du/s f (x) = D f (x)|E x for all x ∈ S. Let ω be the fixed point of f lying in R0 . We require that the following conditions hold: (g) (h) (i) (j)

There exists 0 < γ < 1 such that |Ds f (x)| ≤ γ for all x ∈ S. If x ∈ S then |Du f (x)| ≥ 1. Moreover, if x ∈ S \ W s (ω), then |Du f (x)| > 1. Du f (ω) = 1. In the oriented arc-length parametrization of W u (ω) starting at ω, we have f −1 (x) = x ± a|x|β+1 + o(|x|β+1 )

(5.1)

in a sufficiently small neighborhood of 0 with some positive constants a and β and with the sign “−” if x ≥ 0 and the sign “+” if x ≤ 0. We call a map f : S → R2 satisfying properties (a) - (j) a parabolic horseshoe map.

728


Remark. We note that property (j) restricts the regularity of f , namely by (5.1) the map f is at most of class C 1+β . In particular, if β < 1, then f is not twice differentiable. It is easy to see that for every τ ∈ 2+− , the intersection +∞

f −n (Rτn )

n=−∞ +− is a singleton, which +− we denote by (τ ). The map : 2 → S is a homeomorphism on its image 2 , which we denote by . Hence,

= {x ∈ S : f ±n (x) ∈ S for all n ∈ N}.

(5.2)

Obviously, is a compact f -invariant set, i.e. f () = = f −1 (). Moreover, the map : 2+− → establishes a topological conjugacy between the shift map σ : 2+− → 2+− and f : → , i.e., the following diagram commutes: 2+− ⏐ ⏐

σ −→ f −→

2+− ⏐ ⏐

In the sequel we will frequently identify the cylinders on the symbol space 2+− and their images under the homeomorphism . Given a point x ∈ Ri , i = 0, 1, 2, 3, 4, we put s Wloc (x) = W s (x) ∩ Ri .

These sets are called local stable manifolds. Given a point x ∈ Ri , i = 0, 1, 2, 3, 4, we define Wiu (x) = W u (x) ∩ Ri . For i ∈ {0, 1} we define u (x) = W u (x) ∩ Ri⊕1 (x), Wi⊕1

where the symbol “⊕” denotes the addition mod(2) in the group {0, 1}. Given two points x, y ∈ f (S) we denote by u Hx,y : W u (x) → W u (y)

the holonomy map along local stable manifolds. Moreover, we denote by H u : f (S) → W u (ω) the holonomy map from f (S) to W u (ω) along local stable manifolds, that is, u . H u (x) = Hx,ω


729

We define ω0 = ω and pick ω1 ∈ arbitrary having the property that f (W1u (ω)) = W u (ω1 ). We say that the parabolic horseshoe f : S → R2 is of smooth type if the u : W u (x) → holonomy map H u : f (S) → W u (ω) and all the holonomy maps Hx,y W u (y) are C 1+ε , and in addition, the derivative along unstable manifolds of the map f −1 ◦ Hωu, ω1 : W u (ω) → W u (ω) has norm less than 1 at every point of W u (ω) except possibly at ω. Given two points x, y ∈ f (S) we denote by s s s : Wloc (x) → Wloc (y) Hx,y

the holonomy map along unstable manifolds. We say that the parabolic horseshoe f : s : S → R2 has a Lipschitz continuous unstable foliation if all the holonomy maps Hx,y s s Wloc (x) → Wloc (y) are Lipschitz continuous with a uniform Lipschitz constant. The following example provides a parabolic horseshoe map f : S → R2 of smooth type having a Lipschitz continuous unstable foliation. In particular, the unstable and stable manifolds in R0 and R1 are vertical and horizontal lines respectively. Example 1. An Almost Linear Parabolic Horseshoe Map. Let S ⊂ R2 be a unit square and let h : S → R2 be a linear horseshoe map with constant contraction rate λs < 21 and constant expansion rate λu > 2, see Fig. 1. Let ω = (ω1 , ω2 ) be the fixed point of h contained in R0 . Let δ > 0 such that (ω1 , ω2 + δ), (ω1 , ω2 − δ) ∈ intS. Fix constants β > 0, a > 0 and η > 1. Let ϕ : [−δ, δ] → [−δ, δ] be a C 1+ -diffeomorphism satisfying the following properties: (i) ϕ −1 (t) = λu t ± λu a|t|β+1 for t sufficiently close to 0, and with the sign "−" if t ≥ 0 and the sign "+" if t ≤ 0; (ii) λ−1 u < ϕ (t) ≤ η for all t ∈ [−δ, δ] \ {0}; (iii) ϕ (−δ) = ϕ (δ) = 1. Let Aδ ⊂ h(S) be defined as in Fig. 1, that is, Aδ = [a1 , a2 ] × [b1 , b2 ], where b1 = ω2 − δ, b2 = ω2 + δ and a2 − a1 = λs . We define the map g : h(S) → h(S) by ⎧ ⎨ (x1 , ω2 + ϕ(x2 − ω2 )) if x ∈ Aδ g(x) = (5.3) ⎩x if x ∈ h(S) \ Aδ for all x = (x1 , x2 ) ∈ h(S). Clearly, g is a C 1+ -diffeomorphism which preserves vertical and horizontal lines in R0 ∪ R1 = S ∩ h(S). We now define the map f = g ◦ h. It follows immediately from the construction that f is a parabolic horseshoe map. In particular, property (5.1) holds. Indeed (5.1) is a consequence of the facts that the foliations W u |R0 ∪ R1 and W s |R0 ∪ R1 of f are given by vertical and horizontal lines −1 respectively and that pr 2 (h −1 (ω1 , x2 )) = ω2 + λ−1 u (x 2 − ω2 ) and pr 2 (g (ω1 , x 2 )) = ω2 + ϕ −1 (x2 − ω2 ) if x2 is sufficiently close to ω2 . Here pr 2 denotes the projection in R2 on the 2nd coordinate. Moreover, it is easy to see that f is of smooth type and has a Lipschitz continuous unstable foliation. A simple calculation shows that |Ds f (x)| = λs and

1 ≤ |Du f (x)| ≤ ηλu for all x ∈ .

(5.4)

Remark. (i) We note that the diffeomorphism f is uniquely determined on S ∩ h(S) once we have chosen the constants a, β, η, δ, λu , λs , the map ϕ and h. In particular, the contraction rate λs ∈ (0, 1/2) can be chosen independently of a, β, η, δ, λu , and ϕ. (ii) The method of construction of an almost linear parabolic horseshoe map in Example 1

730


Fig. 1. An Almost Linear Parabolic Horseshoe Map f = g ◦ h

can be generalized. Namely, let h be a hyperbolic (not necessarily linear) C 1+ -horseshoe map which has the property that h as well as h −1 is of smooth type. Then we can construct analogously to Example 1 a C 1+ -diffeomorphism g : h(S) → h(S) such that f = g ◦ h is a parabolic horseshoe map of smooth type with a Lipschitz (even C 1+α ) unstable foliation. 6. Horseshoe and the Associated Parabolic Iterated Function System In this section we introduce for a parabolic horseshoe map an associated parabolic iterated function system on the unstable manifold of the parabolic fixed point. The goal is to do this in such a way that the parabolic horseshoe and the parabolic iterated function system share many ergodic-theoretical features. Let f be a parabolic horseshoe map of smooth type. We define ω0 = ω and pick ω1 ∈ arbitrary having the property that f (W1u (ω)) = W u (ω1 ). We introduce the 1-dimensional iterated function system on W u (ω) defined by the following two maps: u φi = f −1 ◦ Hω,ω : W u (ω) → Wiu (ω) ⊂ W u (ω), i = 0, 1. i

(6.1)

It is easy to check that the iterated function system = {φ0 , φ1 } satisfies all the requirements (in particular (4.1)) of a 1-dimensional parabolic iterated function system introduced in Sect. 4 except possibly item (2) (it may happen that |φ1 (ω)| = 1). We shall now demonstrate that after a C ∞ change of the Riemannian metric on W u (ω), Condition (2) is also satisfied and = {φ0 , φ1 } becomes a parabolic iterated function system. The following formula immediately follows from the definition of a smooth parabolic horseshoe: |φi (x)| < 1 for all x ∈ W u (ω) \ {ω} and |φi (ω)| ≤ 1, i = 0, 1.

(6.2)

Since φ1 (φ1 (ω)), φ0 (φ1 (ω)) = φ1 (ω), there exists a closed segment T ⊂ W u (ω) with the following properties: (a) T ⊂ W1u (ω), (b) φ1 (ω) ∈ intW1u (ω) T , (c) φ1 (T ) ∩ T = ∅. Define M = sup{max{|φ0 (x)|, |φ1 (x)| : x ∈ T }.

(6.3)


731

In view of (6.2), item (a) above, and the continuity of the functions x → |φ0 (x)|, |φ1 (x)|, we conclude that M < 1.

(6.4)

Therefore there exists a C ∞ function ρ : W u (ω) → (0, 1] such that M < ρ(x) < 1

for all x ∈ intT

(6.5)

and ρ(x) = 1

for all x ∈ W u (ω) \ T.

(6.6)

Consider the Riemannian metric ρ(x)d x on W u (ω) and let |g (x)|ρ = ρ(g(x))|g (x)|/ ρ(x) be the norm of the derivative of a differentiable function g : W u (ω) → W u (ω) calculated with respect to the Riemannian metric ρ(x)d x. We shall prove the following. Lemma 6.1. We have |φi (x)|ρ < 1 for all x ∈ W u (ω) \ {ω}, i = 0, 1, |φ0 (ω)|ρ = 1 and |φ1 (ω)|ρ < 1. Proof. The equality |φ0 (ω)|ρ = 1 is immediate. Consider an arbitrary point x ∈ intT . Then both φ0 (x), φ1 (x) ∈ / T by (a) and (c) respectively. Thus, using (6.2)-(6.3), we obtain for i = 0, 1 that |φi (x)|ρ = ρ(φi (x))|φi (x)|/ρ(x) = |φi (x)|/ρ(x) < M −1 |φi (x)| ≤ 1. Next we assume that x ∈ W u (ω) \ intT . Then |φi (x)|ρ = ρ(φi (x))|φi (x)| ≤ |φi (x)|.

(6.7)

So, if x = ω, then it follows from (6.2) that |φi (x)|ρ ≤ |φi (x)| < 1. Hence, we are left to consider the case when x = ω and i = 1. It then follows from (b), (6.5) and (6.2) that |φ1 (ω)|ρ = ρ(φ1 (ω))|φ1 (ω)|/ρ(φ1 (ω)) < 1. Therefore, as long as we are dealing with the iterated function system itself, we may assume without loss of generality that is a parabolic iterated function system and, in particular, all the considerations from Sect. 4 apply. Similarly as in the case of the horseshoe we will frequently identify in the sequel the cylinders on the symbol space 2+ and their images under the homeomorphism π : 2+ → J . Note that u φi−1 = (Hω,ω )−1 ◦ f = Hωui ,ω ◦ f : Wiu (ω) → W u (ω), i = 0, 1 i

(6.8)

Hωui ,ω = H u |W u (ωi ) .

(6.9)

and

We first prove a preliminary result. Lemma 6.2. For all x ∈ S and all n ≥ 0 we have that (H u ◦ f )k (x) ∈ S for all k = 0, 1, . . . , n if and only if f k (x) ∈ S for all k = 0, 1, . . . , n. In this case we have u s s (H ◦ f )n (x) = Wloc ( f n (x)). Wloc

732


Proof. We shall prove this lemma by induction with respect to n ≥ 0. For n = 0 there is nothing to prove. So, take n ≥ 1 and suppose that our lemma is true for all non-negative integers less than n. Assume that f k (x) ∈ S for all k = 0, 1, . . . , n. Then s ( f n−1 (x)) (H u ◦ f )n (x) = H u ◦ f (H u ◦ f )n−1 (x) ∈ H u ◦ f Wloc s s ( f n (x)) ⊂ Wloc ( f n (x)), ⊂ H u (Wloc and, in particular, (H u ◦ f )n (x) ∈ S. We now assume that (H u ◦ f )k (x) ∈ S for all k = 0, 1, . . . , n. Thus, s u (H ◦ f )n−1 (x) f n (x) = f ( f n−1 (x)) ∈ f Wloc u s s ⊂ Wloc f ◦ (H u ◦ f )n−1 (x) = Wloc H ◦ f ) ◦ (H u ◦ f )n−1 (x) s ((H u ◦ f )n (x)). = Wloc

This implies that f n (x) ∈ S. The desired result now follows from the induction assumption. For all integers n ≥ 0 and also for n = +∞ we define Sn =

n

f −k (S)

k=0

and S−n =

n

f k (S).

k=0

Next, we prove the following. s (x) ⊂ S . Lemma 6.3. If n ≥ 0 and x ∈ Sn , then Wloc n s (z) ⊂ S = S for all z ∈ S. So, suppose Proof. For n = 0 this is obvious because Wloc 0 that our lemma is true for some n ≥ 0 and fix a point x ∈ Sn+1 . Then x ∈ Sn , and in s (x) ⊂ S . Hence, f n+1 W s (x) is well-defined view of our inductive hypothesis, Wloc n s n+1 loc s and, as f n+1 (x) ∈ S, we get that f n+1 Wloc f (x) ⊂ S. This completes (x) ⊂ Wloc the proof.

We recall that J is the limit set of the iterated function system . The relation between this limit set and the horseshoe is given by the following. Lemma 6.4. J = ∩ W u (ω). Proof. We shall show first by induction that J ⊂ Sn for all n ≥ 0. Indeed, if z ∈ J , then z ∈ φ0 (W u (ω)) ∪ φ1 (W u (ω)) = W0u (ω) ∪ W1u (ω) ⊂ S = S0 and our inclusion is proved for n = 0. So, suppose that n ≥ 1 and that J ⊂ Sk for all k = 0, 1, . . . , n − 1. Let us consider an arbitrary point z ∈ J . We write z = π(τ ), where τ ∈ I ∞ . Then z ∈ J ⊂ Sn−1 and z = φτ0 (π(σ (τ ))). Hence, u u (π(σ (τ ))) = f n−1 Hω,ω (π(σ (τ ))) . f n (z) = f n ◦ φτ0 (π(σ (τ ))) = f n ◦ f −1 ◦ Hω,ω τ τ 0

0


733

s (π(σ (τ ))) and since π(σ (τ )) ∈ J ⊂ S u Since Hω,ω (π(σ (τ ))) ∈ Wloc n−1 , we may τ0 u n conclude from Lemma 6.3 that Hω,ωτ (π(σ (τ ))) ∈ Sn−1 . Thus, f (z) ∈ S, which 0 implies that z ∈ Sn . Therefore, the inductive proof is complete, and we obtain that J ⊂ S+∞ . Since also J ⊂ W0u (ω) ∪ W1u (ω) ⊂ S−∞ , we therefore obtain that

J ⊂ W u (ω) ∩ S+∞ ∩ S−∞ = W u (ω) ∩ .

(6.10)

In order to prove the opposite inclusion we consider an arbitrary point z ∈ ∩ W u (ω). We shall prove by induction that there exists an infinite word τ = ∅τ1 τ2 . . . ∈ {∅} × I ∞ , such that for every n ≥ 0, z = φτ |n (x) for some x ∈ W u (ω). Indeed, for n = 0 we have z = φ∅ (z) and z ∈ W u (ω). So, suppose that for some n ≥ 1, the word τ |n = ∅τ1 τ2 . . . τn has been constructed. This means that z = φτ |n (x) with some x ∈ W u (ω). As z ∈ , we have f n (z) ∈ S, and, in view of Lemma 6.2, (H ◦ f )n (z) ∈ S. Using (6.8) and (6.9), u n we therefore obtain that x = φτ−1 |n (z) = (H ◦ f ) (z) ∈ S. And applying the last part of Lemma 6.2 along with the fact that f n+1 (z) ∈ S, we get that s s ( f n (z)) ⊂ Wloc ( f n+1 (z)) ⊂ S. f (x) = f ◦ (H u ◦ f )n (z) ∈ f Wloc Moreover, f (x) ∈ f (S ∩ W u (ω)) = f W0u (ω) ∪ W1u (ω) = f W0u (ω) ∪ f W1u (ω) = W u (ω) ∪ W u (ω1 ), u (y) with some i ∈ {0, 1} and y ∈ W u (ω). Consequently, and thus f (x) = Hω,ω i u (y) = φ (y). Thus z = φ (φ (y)) = φ (y), and the inductive proof x = f −1 ◦ Hω,ω i τ i τi i is complete by putting τn+1 = i. Note that z = π(τ ) ∈ J . This gives the inclusion W u (ω) ∩ ⊂ J . Combining this with (6.10) completes the proof.

7. Topological Pressures, Unstable Dimension, Hausdorff and Packing Measures Let f be a parabolic horseshoe map of smooth type. Since the holonomy maps along local stable manifolds are smooth, it follows that def

t u = dim H W u (ω) ∩ = dim H W u (x) ∩ is independent of x ∈ . We call the quantity t u the unstable dimension of the set . The main goal of this section is to establish a Bowen-Ruelle-Manning-McCluskey type of formula for t u . In order to do this we will make use of Lemma 6.4, will introduce the topological pressure of several dynamical systems and potentials, and will apply results from the thermodynamic formalism of parabolic iterated function systems derived in ˆ [MU1] and [U2]. First we consider different pressure functions. For all t ≥ 0 let P(t) denote the pressure function associated with the iterated function system as defined in Sect. 4 (see [MU1,U2] for details). Moreover, we define P u (t) = P( f, −t log |Du f |), where P( f, −t log |Du f |) is the ordinary topological pressure of the potential −t log |Du f | : → R with respect to the dynamical system f |. We claim that the following diagram commutes:

734


⏐ ⏐ Hu J

f −→ ◦ f −→

Hu

⏐ ⏐ u H

(7.1)

J

that is, H u ◦ f = (H u ◦ f ) ◦ H u .

(7.2)

s (x) = W s (H u (x)). This implies that W s ( f (x)) = Indeed, if x ∈ , then Wloc loc loc s u Wloc ( f (H (x))), and therefore H u ( f (x)) = H u ( f (H u (x))), which proves the claim. We define ˜ = P H u ◦ f, −t log |D(H u ◦ f )| and P(t) = P f, −t log |D(H u ◦ f )| ◦ H u . P(t)

We also set, φu = log |Du f | : → R, and φ˜ u = log |D(H u ◦ f )| : J → R. Since H u ◦ f : J → J is the dynamical system generated by the inverses φ0−1 and φ1−1 , we immediately obtain that ˜ ˆ P(t) = P(t), t ≥ 0.

(7.3)

Differentiating both sides of Eq. (7.2) along the unstable manifolds, we get (Du H u ◦ f ) · Du f = D(H u ◦ f ) ◦ H u · Du H u . This implies the following. Lemma 7.1. log |D(H u ◦ f ) ◦ H u | − log |Du f | = log |Du H u ◦ f | − log |Du H u |, and consequently the potentials −t log |D(H u ◦ f ) ◦ H u | and −t log |Du f | are, for all t ≥ 0, cohomologous in the class of continuous functions. As an immediate consequence of this lemma along with Proposition 3.9 and Proposition 4.3, we get the following. Lemma 7.2. For every t ≥ 0, we have that ¯ ˆ P u (t) = P(t) = P(t). In addition, the potentials −tφu : → R and −t φ˜ u : J → R, are VPA and the map µ → µ ◦ (H u )−1 is a bijection between E S ∞ (−tφu ) and E S ∞ (−t φ˜ u ), the sets of (σ -finite) equilibrium states for −tφu and −t φ˜ u respectively. Combining Lemma 7.2 and Bowen’s formula proven in [U2] (compare [MU1]) provides the following.


735

Proposition 7.3. The unstable dimension t u of is the smallest zero of the pressure ˆ function t → P(t) = P(t), t ≥ 0. Now, let us take more fruits of the results proven in [U2]. First, Theorem 6.5 from [U2] and Lemma 6.4 give the following. Theorem 7.4. The unstable dimension t u ∈ (0, 1). Remark. It follows from work in [DV] that if f is a C 2 -diffeomorphism (which in particular implies that β ≥ 1), then t u ≥ 21 . Let us denote by Ht (A) respectively Pt (A) the t-dimensional Hausdorff respectively packing measure of the set A. Combining Theorem 7.4, Lemma 6.4, and Theorem 6.4 of [U2] provides the following. u (z) ∩ ) = 0 and 0 < P u (W u (z) ∩ ) < ∞ for Theorem 7.5. We have that Ht u (Wloc t loc all z ∈ .

We end this section with the following result. Theorem 7.6. The unstable pressure function t → P u (t) is real-analytic on (0, t u ). Proof. Consider the hyperbolic iterated function system ∗ = {φ0n 1 }∞ n=0 associated to the system as in Sect. 4. Consider the two-parameter family G t,s of the functions 0 1 (z) = t log |φ0 n 1 (z)| − s(n + 1) : t, s ∈ R, n ≥ 0, z ∈ W u (ω). gt,s n

With the terminology of Sect. 3 of [MU2] (see also [U4] and [HMU], where these were introduced) we shall prove the following. Lemma 7.7. For all t, s ∈ R the family G t,s is Hölder continuous. For all (t, s) ∈ R × (0, +∞) the family G t,s is summable. Proof. The fact that the family G t,s is Hölder follows immediately from the sentence located just beneath the proof of Theorem 8.4.2 in [MU2]. Since ||φ0 n 1 || (n + 1) (see Lemma 2.3 in [U2]), we see that if t ∈ R and s > 0, then 0n 1 = exp sup gt,s exp sup t log |φ0 n 1 (z)| − s(n + 1) : z ∈ W u (ω) n≥0

n≥0

=

e−s(n+1) ||φ0 n 1 ||t

n≥0

(n + 1)

− β+1 β

− β+1 β t −s(n+1)

e

n≥0

< +∞. This precisely means that our family G t,s is summable, and we are done. Let gt,s : {0n 1 : n ≥ 0}N → R be the amalgamated function (see [MU2], comp. [U4] 0n 1 }∞ . This function is given by the formula and [HMU]) of the family {gt,s n=0 τ0 gt,s (τ ) = gt,s (π∗ (σ∗ (τ ))) = t log |φτ 0 | − s|τ0 |,

(7.4)

where σ∗ : {0n 1 : n ≥ 0}N → {0n 1 : n ≥ 0}N is the shift map associated with the iterated function system ∗ and π∗ : {0n 1 : n ≥ 0}N → J∗ is the corresponding canonical projection. Summability of the family G t,s proven in Lemma 7.7 precisely means

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summability of the amalgamated function gt,s . Hölder continuity of this amalgamated function follows from Lemma 7.7 and Lemma 3.1.3 from [MU2]. The following lemma is now an immediate consequence of Theorem 2.6.12 and Proposition 3.1.4, both from [MU2]. Lemma 7.8. The function (t, s) → P(gt,s ), (t, s) ∈ R × (0, +∞) is real-analytic in both variables t and s, where the topological pressure P(gt,s ) is defined in Sect. 3.1 of [MU2]. We now prove the conclusion of Theorem 7.6. Since the dynamical system H u ◦ f : J → J is expansive, it follows from Theorem 3.12 in [DU] that for every t ≥ 0 there exists a Borel probability measure m t supported on J and such that

˜ m t (H u ◦ f (A)) = e P(t) |(H u ◦ f ) |t dm t (7.5) A

for all Borel sets A ⊂ J having the property H u ◦ f | A is one-to-one. Hence,

˜ m t (φi (E)) = e− P(t) |φi |t dm t E

for i = 0, 1 and E, any Borel subset of J . In addition, m t φ0 (W u (ω))∩φ1 (W u (ω)) = 0 as these sets φ0 (W u (ω)) and φ1 (W u (ω)) are disjoint. We therefore get by a straightforward induction that

0n 1 m t φ0n 1 (E) = dm t exp gt,P(t) E

and m t φ0n 1 (W u (ω)) ∩ φ0k 1 (W u (ω)) = 0 ˜ by P u (t) due to for all t ≥ 0 and all n, k ≥ 0 with n = k. We have replaced here P(t) u u Lemma 7.2 and (7.3). If now t ∈ (0, t ), then P (t) > 0 and the family G t P(t) is Hölder and summable due to Lemma 7.7. So, looking at the definition of G t P(t) -conformal measures, i.e. formulas (3.5) and (3.6) from [MU2], we see that m t is a unique G t P(t) conformal measure and that P gt,s = P G t P(t) = 0, t ∈ (0, t u ). (7.6) Looking at Theorem 3.2.3, Corollary 2.7.5 and Proposition 2.6.13 from [MU2] and at the formula (7.4), we conclude that for all t ∈ (0, t u ),

∂ P gt,s |(t,P(t)) = −|τ0 |d µ˜ t (τ ) = 0, (7.7) ∂s where µ˜ t = µ˜ gt,P(t) is the σ∗ -invariant Gibbs state proved to exist by Corollary 2.7.5 of [MU2]. Hence, applying formula (7.7) (also using (7.6)), the proof follows by applying Lemma 7.8 and the Implicit Function Theorem.


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8. Equilibrium States In this section we provide a complete description of all ergodic σ -finite equilibrium states of the potential −t u φu , where, we recall, φu = log |Du f | : → R, with respect to the dynamical system f : → . We start our analysis with the potential −t u φ˜ u , where, we also recall, φ˜ u = log |D(H u ◦ f )| : J → R with respect to the dynamical system H u ◦ f : J → J . Let µ˜ ω be the Dirac δ-measure supported on ω. Let m˜ t u be the t u -conformal measure established in (7.5) with t = t u . ˜ u ) = 0. It follows from Theorem 7.2 in [U2] that there exists a unique We note that P(t (up to a multiplicative constant) ergodic σ -finite H u ◦ f -invariant measure µ˜ t u on J equivalent to m˜ t u . The measure µ˜ t u is ergodic and conservative. The following necessary and sufficient condition for the measure µ˜ t u to be finite was established in [U2]. Theorem 8.1. The measure µ˜ t u is finite if and only if tu > 2

β . β +1

We shall prove the following. Theorem 8.2. The measures µ˜ t u and µ˜ ω are the only ergodic equilibrium states of the potential −t u φ˜ u with respect to the dynamical system H u ◦ f : J → J . u˜ ˜ Proof. u Since by Lemma 3.3 and Theorem 4.3 in [U2], P(−t φu ) = 0 and since −t φ˜ u d µ˜ ω = 0, it follows that the Dirac δ-measure µ˜ ω is an equilibrium state of the potential −t u φ˜ u . Next, we demonstrate that µ˜ t u is also an equilibrium state. We define

ρ=

d µ˜ t u . d m˜ t u

(8.1)

It follows from Theorem 7.2 in [U2] that ρ|[0n 1|n0 ] n+1. Since in addition m˜ t u [0n 1|n0 ] (n + 1)

u − β+1 β t

and since |(H u ◦ f ) (z) − 1| (β + 1)z β (n + 1)−1

(8.2)

for all z ∈ [0n 1|n0 ] (so log |(H u ◦ f ) (z)| (n + 1)−1 ), we may conclude that

0 ≤ −t u φ˜ u d µ˜ t u < ∞.

(8.3)

Now notice that 0 < µ˜ t u ([1|00 ]) < ∞. In order to see that µ˜ t u is an equilibrium state for −t u φ˜ u let us induce the map H u ◦ f on the cylinder [1|00 ]. Denote µ˜ t u [1|0 ] by µ˜ 1 . Consider 0

the partition α of [1|00 ] generated by the first return time. Obviously α is a generating partition for the measure µ˜ 1 and the first return map on [1|00 ]. Since for µ˜ 1 -a.e. point τ ∈ [1|00 ] there is an infinite sequence {kn }∞ n=1 of positive integers such that τkn = 1, and

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since µ˜ 1 is equivalent to m˜ t u |[1|0 ] with the Radon-Nikodym derivative bounded away 0 from zero and infinity, we get that µ˜ 1 [τ |k0n ] m˜ t u [τ |k0n ] exp −t u Skn φ˜ u (τ ) . But [τ |k0n ] = α n (τ ) and Skn −t u φ˜ u (τ ) = Sn1 −t u φ˜ u1 (τ ), where α n is the n th refined partition with respect to the first return map on [1|00 ], S 1j is the j th ergodic sum with respect to the first return map on [1|00 ], and −t u φ˜ u1 = −t u φ˜ u |[1|0 ] . So, µ˜ 1 (β n (τ )) = 0 exp −t u Sn1 φ˜ u1 (τ ) . Applying now the Shannon-McMillan-Breiman Theorem and Biru khoff’s Ergodic Theorem (along with the observation that µ˜ 1 is ergodic because m˜ t is), we get that hµ˜ 1 + (−t u φ˜ u1 )d µ˜ 1 = 0, the equality that demonstrates that µ˜ t u is an equilibrium state for −t u φ˜ u . In order to prove that µ˜ ω and µ˜ t u are the only ergodic conservative equilibrium states for −t u φ˜ u , suppose that µ˜ is an ergodic conservative equilibrium state for −t u φ˜ u different from µ˜ ω . Suppose that µ˜ has an atom. Because of ergodicity and conservativity of µ, ˜ this measure must be supported on a periodic orbit of σ containing this atom. But then µ˜ is (up to a multiplicative constant) a probability measure, hµ˜ = 0 and φ˜ u d µ˜ > 0 since µ˜ = µ˜ ω . Consequently, hµ˜ + (−t u φ˜ u )d µ˜ < 0 contrary to the fact that µ˜ is an ergodic conservative equilibrium state for −t u φ˜ u . So, µ˜ is atomless, and similarly as in the proof of Theorem 3.7, there exists an initial cylinder F, with the first coordinate equal to 1, such that µ(F) ˜ ∈ (0, ∞). Let α be the countable partition of F induced by the first return time. We shall prove the following. Claim 1. hµ˜ F (α) < ∞. Proof. Suppose on the contrary that hµ˜ F (α) = ∞. It then immediately follows from Shannon-McMillan-Breiman Theorem that − log µ˜ F (α n (ω)) =∞ (8.4) lim n→∞ n for µ˜ F -a.e. ω ∈ F, say ω ∈ F1 . Since µ˜ ∈ M−t u φ˜u , the function |φ˜ u | = −φ˜ u is µ-integrable, ˜ and therefore 0 < χ := |φ˜ u |d µ˜ F < ∞. Thus, by Birkhoff’s Ergodic Theorem, lim

n→∞

1 F F S φ˜ (ω) = χ n n u

(8.5)

for µ-a.e. ˜ ω ∈ F, say ω ∈ F2 . Put F3 = F1 ∩ F2 . Then µ˜ F (F3 ) = 1. Since φ1 is a hyperbolic element of our parabolic iterated function system , it follows from item (b) of Sect. 4 and the definition of measure µ˜ t u that for every ω ∈ F and every n ≥ 1, µ˜ t u F (α n (ω)) exp −t u SnF φ˜ u (ω) . Therefore, using (8.5), we get for all ω ∈ F3 and all n ≥ 1 large enough, that µ˜ t u F (α n (ω)) ≥ exp −t u (χ + 1)n . Combining this and (8.4), we see that for all ω ∈ F3 , µ˜ F (α n (ω)) = 0. n→∞ µ ˜ t u F (α n (ω)) lim

Thus µ˜ F (F3 ) = 0 and this contradiction finishes the proof of Claim 1.


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Let Jµ−1 ˜ F with ˜ F : F → [0, 1] be the inverse of the (weak) Jacobian of the measure µ respect to the first return map σ F : F → F, i.e

−1 µ˜ F (B) = Jµ−1 ˜F ˜ F ◦ (σ F | B ) d µ σ F (B)

for every Borel set B ⊂ F such that σ F | B is injective. Let Lµ˜ F : L 1 (µ˜ F ) → L 1 (µ˜ F ) be the Perron-Frobenius operator associated to the measure µ˜ F . The operator Lµ˜ F is given by the formula Jµ−1 Lµ˜ F g(z) = ˜ F (x)g(x). x∈σ F−1 (z)

Notice that for the measure µ˜ t u we have Jµ−1 ˜ u (z) = t F

ρ(z) u |(H u ◦ f )F (z)|−t ρ(σ F (z))

(8.6)

and this Jacobian is a continuous function. Furthermore, Lµ˜ t u F g(z) =

x∈σ F−1 (z)

ρ(x) u |(H u ◦ f )F (z)|−t g(x) ρ(z)

(8.7)

and Lµ˜ t u F g : F → R is continuous for every continuous function g : F → R. In particular, Lµ˜ t u F ρ = ρ, and this equality holds throughout the whole set F. We now shall prove the following: −1 ˜ F -a.e. z ∈ F. Claim 2. Jµ−1 ˜ F (z) = Jµ˜ u (z) for µ t F

Proof. Since Lµ˜ F (11) = 11 and since Lµ˜ t u F (11) = 11, applying (8.7), we get

−1 −1 1 = 11d µ˜ F = Lµ˜ t u F (11)d µ˜ F Jµ−1 Jµ˜ u (x)Jµ−1 ˜ F (z) ˜ F (x) ˜ F (x)d µ

x∈σ F−1 (z)

−1 −1 Lµ˜ F = Lµ˜ F Jµ−1 Jµ˜ u d µ˜ F ˜ F (x) t F

−1 −1 −1 ≥ 1 + log Jµ˜ F (x) Jµ˜ u d µ˜ F ≥ 1 t F

u + log Jµ˜ F ρ · (ρ ◦ σ F )−1 |(H u ◦ f )F |−t d µ˜ F

= 1 + log(Jµ˜ F )d µ˜ F + log ρd µ˜ F − t u log(ρ ◦ σ F )

u −t log |(H u ◦ f )F |d µ˜ F

= 1 + log(Jµ˜ F )d µ˜ F − t u φ˜ uF d µ˜ F . =

−1 −1 Jµ˜ u d µ˜ F Jµ−1 ˜ F (x) t F

t F

(8.8)

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Applying Claim 1 yields log(Jµ˜ F )d µ˜ F = hµ˜ F (σ F ). Therefore, since P(−t u φ˜ u ) = 0 and since µ˜ is an equilibrium state of −t u φ˜ u , we conclude that the most right-hand sided formula in (8.8) is equal to 1. Hence, the signs "≥" appearing in (8.8) must be all equal −1 ˜ F a.e. The proof of Claim 2 is to the "=" signs. As a consequence, Jµ−1 ˜ F = Jµ˜ t u F -µ complete. Since σ Fn (A) = F for all n ≥ 1 and all A ∈ α n , it immediately follows from Claim 2, (8.6), (8.1), and the Bounded Distortion Property (item (6) in Sect. 4), that if µ˜ F (A) > 0, then µ˜ F (A) µ˜ t u F (A). Thus, the measure µ˜ F is absolutely continuous with respect to the measure µ˜ t u F (A), and since the latter is ergodic, µ˜ F = µ˜ t u F . Hence, µ˜ = µ˜ t u , and we are done. Let µω be the Dirac δ-measure on supported on ω. The following, main result of this section, provides a complete description of the structure of equilibrium states of the potential −t u φu : → R, where φu is given by the formula φu (x) = log |Du f (x)|,

(8.9)

with respect to the dynamical system f : → . As an immediate consequence of Theorem 8.2 and Lemma 7.2, we get the following. Theorem 8.3. There are exactly two (up to a multiplicative constant) ergodic equilibrium states for the potential −t u φu . Namely, µω and µt u , the latter uniquely determined by the condition that µt u ◦ (H u )−1 = µ˜ t u . 9. Conditional Measures Suppose that (X, A, ν) is a σ -finite measure space. Suppose also that A is a sub-σ algebra of A. It easily follows from the probabilistic case that for every ν-integrable function g : X → R there exists E(g|A) : X → R, a unique expected value of g with respect to the σ -algebra A, i.e. an A-measurable function for which

E(g|A)dν = gdν A

A

for every set A ∈ A. Let P = PA be the measurable partition generated by the σ -algebra A. The canonical system {ν x }x∈X of ν conditional measures with respect to the σ -algebra A (and partition P) is given by the following formula: ν x (B ∩ P(x)) = E(11 B |A)(x)

(9.1)

for every set B ∈ A. We note that for ν-a.e. x ∈ X the value ν x (B ∩P(x)) is independent of the choice of a set B ∈ A with the property that B ∩ P(x) = B ∩ P(x). Since ν is σ -finite, it easily follows from Martingale’s Convergence Theorem that if {An }∞ n=0 is an ascending sequence of sub-σ -algebras of A, generating A, then (9.2) E(11 B |A)(x) = lim E 11 B |An (x) n→∞

for ν-a.e. x ∈ X . We now consider the σ -finite measure space (, B, µt u ), the partition P u of into unstable manifolds Wiu (x), x ∈ , i = 0, 1, and the σ -algebra B u


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generated by the partition P u . In the language of the symbol space 2+− the unstable manifolds take on the form W0u (γ ) = [γ |0−∞ ]. Without confusion we will frequently use either of the two languages: symbolic or "differentiable". For every n ≥ 0 let Bnu be the finite σ -algebra generated by the cylinders [γ |0−n ], γ ∈ 2+− . Obviously, {Bnu }∞ n=0 is an ascending sequence of sub-σ -algebras generating B u . Applying (9.1) and (9.2), we see that for every γ ∈ 2+− and every τ ∈ {0, 1}q , we have γ γ q q µt u [γ |0−∞ τ |−∞ ] = µt u [τ |1 ] ∩ [γ |0−∞ ] = E 11|[τ |q ] |B u (γ ) 1 u = lim E 11|[τ |q ] |Bn (γ ) 1 n→∞ q q µt u [τ |1 ] ∩ [γ |0−n ] µt u [γ |0−n τ |−n ] = lim = lim n→∞ n→∞ µt u ([γ |0−n ]) µt u ([γ |0−n ]) 0 n+q n+q µt u [γ |−n τ |0 ] µ˜ t u [γ |0−n τ |0 ]+ = lim = lim , n→∞ µt u ([γ |0 |n ]) n→∞ µ ˜ t u ([γ |0−n |n0 ]+ ) −n 0 where we could have written the second last equality sign since the measure µt u is shiftinvariant and we wrote the second equality sign because of Theorem 8.3. We recall that µ˜ t u u ˜ u ρ = ddm ˜ t u . Using the fact that P(t ) = P(t ) = 0, we further can write by making use of (7.5) that u u |φγ |0 τ |t ρ ◦ φγ |0 τ d µ˜ t u |φγ |0 τ |t d µ˜ t u J J −n γ q −n −n lim , µt u [γ |0−∞ τ |−∞ ] = lim tu ρ ◦ φ 0 dµ tu ˜ u n→∞ n→∞ u |φ | ˜ |φ t t 0 γ | J γ | J γ |0 | d µ −n

−n

−n

where we wrote the comparability sign using (8.3). We now assume that τ = 0q and γ |0−∞ = 0∞ |0−∞ . Let i ≥ 0 be the least integer such that γ−i = 1. Then the distortion property allows us to continue as follows: γ q µt u [γ |0−∞ τ |−∞ ]

K i,(3)j

||φγ |0

−n τ

||t

u

u ||φγ |0 ||t −n

K i, j ||φρ 0 τ ||h (3)

||φγ |0 ||t ||φτ ||t u

K i,(2)j

−n

u ||φγ |0 ||t −n q K i, j m˜ h [ρ0 τ |0 ] ,

u

= K i,(2)j ||φτ ||t

u

(2)

where the constants K i, j , K i, j , K i, j and K i, j depend on i and j = q − l, where l is the last position of the letter 1 in the word τ = τ1 τ2 . . . τq . Since the conformal measure µ˜ t u is a constant multiple of packing measure Pt u | J (see Theorem 6.4 in [U2]), since the holonomy maps between unstable manifolds are uniformly Lipschitz, and since q q H [γ |0−∞ τ |−∞ ] = [γ0 τ |0 ], we obtain that γ q q µt u [γ |0−∞ τ |−∞ ] K i, j Pt u [γ |0−∞ τ |−∞ ] . We now may conclude that γ

µt u |[γ |0

n −∞ ]∩[1|n ]

K i,0 Pt u |[γ |0

n −∞ ]∩[1|n ]

γ for all n ≥ 0. Since in addition µt u [γ |0−∞ 0∞ ] = Pt u [γ |0−∞ 0∞ ] = 0 (also using that 0 n µt u is non-atomic), and since [γ |−∞ ] = [γ |0−∞ 0∞ ] ∪ ∞ n=0 [1|n ], we therefore have proven the following main result of this section.

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Theorem 9.1. For every i = 0, 1 and all x ∈ ∩ Ri the conditional measure µtxu on the unstable manifold Wiu (x) is equivalent to the packing measure Pt u restricted to the manifold Wiu (x). 10. Dimension of the Horseshoe In this section we establish formulas for the stable dimension and the dimension of the parabolic horseshoe. In this and in the subsequent sections we consider probability measures rather than σ -finite measures. In particular, the notion of equilibrium states will be from this point on exclusively used in the context of probability measures. Let f be a parabolic horseshoe map. We denote by M the space of all Borel f -invariant probability measures on endowed with weak* topology. This makes M a compact convex space. Moreover, we denote by M E ⊂ M the subset of ergodic measures. Let µ ∈ M. We define

log |Du f | dµ and λs (µ) = log |Ds f | dµ. (10.1) λu (µ) =

Note that λu (µ) and λs (µ) coincide with the µ-average of the pointwise Lyapunov exponents of f . It follows from properties (g), (h) and (i) of the parabolic horseshoe (see Sect. 5) that λu (µ) ≥ 0 and λs (µ) ≤ log γ < 0

(10.2)

for all µ ∈ M, where γ < 1 is the constant in property (g) of the parabolic horseshoe. We say that µ ∈ M is a hyperbolic measure if λu (µ) > 0. In this case we refer to λu/s (µ) as the positive/negative Lyapunov exponent of µ. Recall that µω denotes the Dirac-δ measure supported on the parabolic fixed point ω. We begin with a preliminary result. Lemma 10.1. Let µ ∈ M. Then µ is a hyperbolic measure if and only if µ = µω . Proof. We first consider the case when µ is ergodic. Obviously, if µ = µω then λu (µ) = 0, and µ is not hyperbolic. Assume now that µ = µω . Therefore, Birkhoff’s Ergodic Theorem implies that µ(W s (ω)) < 1, and since µ is ergodic we obtain µ(W s (ω)) = 0. Note that |Du f (x)| > 1 for all x ∈ \ W s (ω). Therefore, it follows from the definition of the Lyapunov exponent and the fact that x → Du f (x) is continuous that λu (µ) > 0. By using that λs (ν) < 0 for all ν ∈ M we obtain that µ is hyperbolic. Finally, the case when µ is not ergodic follows from the ergodic case by using an ergodic decomposition of µ. We define the stable pressure function P s : R → R by P s (t) = P( f |, tφs ), where P s ( f |, .) is the ordinary topological pressure of the dynamical system f | and φs = log |Ds f | : → R. Next we establish a Bowen-Ruelle-Manning-McCluskey type of formula for the stable dimension of . Theorem 10.2. Let f be a parabolic horseshoe map having a Lipschitz continuous def s (x) ∩ does not depend on x ∈ . Moreover, unstable foliation. Then t s = dim H Wloc s t is given by the unique solution of P s (t) = 0, and 0
0 are analogous to the case of uniformly hyperbolic sets on surfaces (see [MM]). Remark. (i) Similar to the case of uniformly hyperbolic surface diffeomorphisms one can show that the potential tφs has a unique equilibrium state for all t ≥ 0. (ii) We note that Theorem 10.2 in particular applies to almost linear parabolic horseshoe maps (see Example 1) as well as to the more general case of small perturbations of hyperbolic horseshoes of smooth type which were discussed in part (ii) of the remark after Example 1. Theorem 10.3. Let f be a parabolic horseshoe map of smooth type having a Lipschitz continuous unstable foliation. Then dim H = t u + t s . Proof. Let i = 0, 1. We consider the set i = ∩ Ri . Given x ∈ i it follows from property (e) of the parabolic horseshoe (see Sect. 5) that [·, ·] : W u (x) ∩ i × W s (x) ∩ i → i is a homeomorphism. Moreover, since f is of smooth type and since the unstable foliation of f is Lipschitz continuous, it follows that [·, ·] as well as [·, ·]−1 are Lipschitz continuous and therefore preserve Hausdorff dimension. Recall that dim B A denotes the upper box dimension of the set A. Similar, as in the case of hyperbolic sets on surfaces one can show that dim H W s (x) ∩ i = dim B W s (x) ∩ i . Using the formula for the Hausdorff dimension of products (see for example [F]) we obtain dim H W u (x)∩i ×W s (x)∩i = dim H W s (x) ∩ i + dim H W u (x) ∩ i . This completes the proof. We define the stable set of by W s () = x ∈ S : f n (x) ∈ S for all n ∈ N, lim dist ( f n (x), ) = 0 . n→∞

(10.4)

Similarly we define the unstable set W u () of as the stable set of with respect to f −1 . It follows immediately from the properties of the parabolic horseshoe that W s/u () = W s/u (x). (10.5) x∈

Note that (10.5) is also a consequence of the Shadowing Lemma. Theorem 10.4. Let f be a parabolic horseshoe map of smooth type having a Lipschitz continuous unstable foliation. Then dim H W s/u () = t u/s + 1 < 2.

744


Proof. We only prove the formula for the dimension of the stable set. The proof for the unstable set is analogous. Note that by Theorem 7.4, t u < 1, which gives the right-hand side inequality. Combining that has a local product structure with (10.5) implies that it suffices to prove that ⎞ ⎛ s dim H ⎝ Wloc (y)⎠ = t u + 1 (10.6) u (x)∩ y∈Wloc

u ( p) ∩ . Since has a local product structure, for all x ∈ . Let x ∈ . Set A x = Wloc there exists a homeomorphism s Wloc (y), (10.7) H : A x × (−1, 1) → y∈A x s (y) for all y ∈ A . Moreover, since f with the property that H (y × (−1, 1)) = Wloc x is of smooth type and since f has a Lipschitz continuous unstable foliation it follows that H as well as H −1 are Lipschitz continuous. Applying Theorem 10.2 completes the proof.

11. Generalized Physical Measures Let f be a parabolic horseshoe map of smooth type having a Lipschitz continuous unstable foliation. In this section provide a classification for f having a generalized physical measure. Given µ ∈ M we define the basin of µ by

n−1 1 B(µ) = x ∈ S : f (x) ∈ S for all n ∈ N, lim δ f i (x) = µ . n→∞ n n

(11.1)

i=0

Here δ f i (x) denotes the Dirac-δ measure on f i (x). The basin of µ is sometimes also called the set of future generic points of µ, see [DGS] and [Ma]. A measure µ ∈ M E is called a physical measure if B(µ) has positive Lebesgue measure. Obviously, B(µ) ⊂ W s ().

(11.2)

Therefore, by Theorem 10.4 the map f can not have a physical measure. Following [Wo] we say that µ ∈ M E is a generalized physical measure if it is non-atomic and B(µ) is as large as possible in the sense that dim H B(µ) = dim H W s ().

(11.3)

We now prove a formula for the Hausdorff dimension of the basin of a hyperbolic measure. Proposition 11.1. Let µ ∈ M E \ {µω }. Then dim H B(µ) =

hµ( f ) + 1. λu (µ)

(11.4)


745

Proof. Note that by Lemma 10.1, λu (µ) > 0; hence the right-hand side of (11.4) is well-defined. It is easy to see that if y ∈ f (S) then s y ∈ B(µ) if and only if ∃x ∈ ∩ B(µ) with y ∈ Wloc (x).

(11.5)

Combining (11.5) with property (e) of the parabolic horseshoe (see Sect. 5) it follows that it is sufficient to show that for each x ∈ , ⎛ ⎞ hν ( f ) s Wloc (y)⎠ = dim H ⎝ + 1, (11.6) λu (ν) y∈A x

u (x) ∩ B(µ). Pick x ∈ . The same methods used by Manning [Ma] where A x = Wloc in the context of hyperbolic surface diffeomorphisms (see [Me] for the analogous result in the non-uniformly hyperbolic setting) can be used to show that

dim H A x =

hµ( f ) . λu (µ)

(11.7)

Therefore, (11.6) can be shown analogously as Theorem 10.4 by using the bi-Lipschitz continuous homeomorphism H . Recall that φu = log |Du f | : → R. We now present our main result about generalized physical measures. Theorem 11.2. Let f be a parabolic horseshoe of smooth type having a Lipschitz continuous unstable foliation. Then the following are equivalent: (i) (ii) (iii) (iv) (v)

f admits a generalized physical measure; The potential −t u φu has more than one (finite) equilibrium state; The potential −t u φu has precisely two ergodic (finite) equilibrium states; P u is not differentiable at t u ; β . t u > 2 β+1

Proof. (i)⇒(ii): Let µ be a generalized physical measure of f . It follows from Theorem 10.4 and Proposition 11.1 that t u = h µ ( f )/λu (µ). Thus, µ is an ergodic equilibrium state of the potential −t u φu . Using that µ = µω implies (ii). The implication (ii)⇒(iii) is a consequence of Theorem 8.3. Moreover, (iii)⇒(iv) follows from [J, Cor. 1]. The implication (iv)⇒(v) follows from [J, Cor. 1] and Theorem 8.1. Finally, if (v) holds, then by Theorem 8.1 there exists a probability ergodic equilibrium state µ of the potential −t u φu with µ = µω . Hence, t u = h µ ( f )/λu (µ), and we may conclude from Theorem 10.4 and Proposition 11.1 that µ is a generalized physical measure for f . As an application of Theorem 11.2 we construct parabolic horseshoe maps with as well as without generalized physical measures. Corollary 11.3. There exists a parabolic horseshoe map having a generalized physical measure as well as one having no generalized physical measure. Proof. We first construct an example of a parabolic horseshoe map having a generalized physical measure. Pick 0 < β < 1, η > 1 and λu > 2 such that β log 2 . >2 log η + log λu β +1

(11.8)

746


Let f be a parabolic horseshoe map as defined in Example 1 with the corresponding constants β, η and λu . In particular, f is of smooth type and has a Lipschitz continuous unstable foliation. It follows from (5.4) that λu (ν) ≤ log λu + log η

(11.9)

for all ν ∈ M. Let µ denote the measure of maximal entropy of f , i.e. the unique measure satisfying h µ ( f ) = log 2. Therefore, (11.9) and Theorem 1 in [Me] imply that β t u > 2 β+1 . We now may conclude from Theorem 11.2 that f has a generalized physical measure. The existence of a parabolic horseshoe map having no generalized physical measure can be easily seen. Just pick any parabolic horseshoe map f of smooth type having a Lipschitz continuous unstable foliation (for instance a map as in Example 1) with β ≥ 1. Recall that t u < 1 (see Theorem 7.4); hence t u < 2β/(β + 1), and therefore, Theorem 11.2 implies that f does not have a generalized physical measure. Corollary 11.4. The existence of a generalized physical measure is not a (topological) conjugacy invariant. Proof. By Corollary 11.3 there exist parabolic horseshoe maps f k , k = 1, 2 such that f 1 has a generalized physical measure and f 2 does not have a generalized physical measure. Since f 1 |1 and f |2 are both topological conjugate to the shift map σ : 2+− → 2+− , the result follows. 12. Measures of Maximal Dimension In this section we discuss the existence of ergodic measures of maximal dimension for a particular subclass of parabolic horseshoe maps. In particular, we provide a criterion which guarantees that no ergodic measure of maximal dimension exists. Recall that in this section the notion of equilibrium states is meant in the space of probability measures. Let f : S → R2 be a parabolic horseshoe map. We say that f has constant contraction rate if there is 0 < c < 1 such that |Ds f (x)| = c for all x ∈ . For example the almost linear parabolic horseshoe maps in Example 1 have constant contraction rate c < 1/2. Given µ ∈ M we define the Hausdorff dimension of µ by dim H µ = inf{dim H A : µ(A) = 1}.

(12.1)

Following [BW1] we say that µ ∈ M E is an ergodic measure of maximal dimension if def

dim H µ = sup{dim H ν : ν ∈ M E } = δ( f ).

(12.2)

It follows from work in [BW2] that the definition of δ( f ) in (12.2) is the same when the supremum is taken over all (not necessarily ergodic) measures in M. Let µ ∈ M \ {µω }. It follows from Young’s formula [Y] (also using Lemma 10.1) that if µ is ergodic, then dim H µ = d(µ), where def

d(µ) = h µ ( f )

! 1 1 − . λu (µ) λs (µ)

(12.3)


747

We now define a one-parameter family of measures (νt )t∈[0,t u ] which will be crucial for the analysis of measures of maximal dimension. For t ∈ [0, t u ) we define νt to be the unique equilibrium state of the potential −tφu . Note that νt is well-defined. Indeed, P u is differentiable in [0, t u ). This is a consequence of Theorem 7.6 and the fact that f | has a unique measure of maximal entropy. Thus, by [J, Cor. 1] the potential −tφu has a unique equilibrium state. Next, we define νt u . In the case when the potential −t u φu has more than one equilibrium state we define νt u to be the unique hyperbolic ergodic equilibrium state of −t u φu . Otherwise, we define νt u = µω . The results of this section will be based on a careful analysis of the dimension of the measures νt . For simplicity we write h(t) = h νt ( f ) and λu (t) = λu (νt ) for all t ∈ [0, t u ]. Thus, P u (t) = h(t) − tλu (t).

(12.4)

It follows from standard properties of the topological pressure (see for example [J]) that if P u is differentiable at t0 then d P u (t0 ) = −λu (t0 ). dt

(12.5)

We first prove a preliminary result. Lemma 12.1. We have the following: (i) If νt u = µω then P u ∈ C 1 ([0, t u ]); (ii) If νt u = µω then P u ∈ C 1 ([0, t u )). Proof. We already know that P u is real-analytic on (0, t u ) (see Theorem 7.6). Therefore, we only have to consider t = 0 and t = t u . Since ν0 is the unique measure of maximal entropy it follows from [J] that P u is differentiable at 0. Similarly, if νt u = µω then P u is differentiable at t u . We claim that if νt u = µω then P u is C 1 in a left neighborhood of t u . Let tn ≤ t u with tn → t u for n → ∞. By (12.5) is suffices to show that λu (tn ) → λu (t u ) for n → ∞. By convexity of P u and (12.5), λu is decreasing. Thus, a = limn→∞ λu (tn ) exists. By compactness of M there exists µ ∈ M such that µ is a weak* cluster point of the measures νtn . Since λu is continuous on M we may conclude that λu (µ) = a. Using that the entropy map ν → h ν ( f ) is upper semi-continuous on M (also using (12.4)) we obtain that µ is an equilibrium state of the potential −t u φu ; hence µ = νt u which proves the claim. The proof of the statement that P u is C 1 in a right neighborhood of 0 is entirely analogous. Since P u is real-analytic in (0, t u ), (12.5) implies that λu is also real-analytic in (0, t u ). Hence, by (12.4), h is also real-analytic in (0, t u ). Moreover, t0 dλu (t0 ) dh(t0 ) = dt dt

(12.6)

for all t0 ∈ (0, t u ). We now prove another preliminary result. Lemma 12.2. The functions λu and h are continuous in [0, t u ]. Moreover, {λu (t) : t ∈ [0, t u ]} = [λu (t u ), λu (0)].

(12.7)

748


Proof. We first consider the case νt u = µω . In this case (12.4), (12.5) and Lemma 12.1 imply that the functions λu and h are continuous in [0, t u ]. Moreover, by (12.5), {λu (t) : t ∈ [0, t u ]} is a compact interval; thus, (12.7) follows from the fact that t → λu (t) is decreasing in [0, t u ]. Next, we consider the case νt u = µω . Similarly as above, we can show that λu is continuous in [0, t u ). We now prove the continuity of λu at t u . Let E S(t u ) ⊂ M be the set of all equilibrium states of the potential −t u φu . It is well-known that E S(t u ) is a compact convex set whose extreme points are the ergodic measures in E S(t u ), see e.g. [J]. Therefore, Theorem 8.3 implies that E S(t u ) = {sµω + (1 − s)νt u : s ∈ [0, 1]}.

(12.8)

Since λu is decreasing in [0, t u ) it follows that limt→t u − λu (t) exists. Since M is compact, there exist a sequence (tn )n∈N converging to t u from the left and a measure µ ∈ M such that µ is a weak* limit of the sequence of measures (νtn )n∈N . Since the entropy map is upper semi-continuous, we may conclude from the continuity of λu that µ is an equilibrium state of the potential −t u φu . Thus, µ = sµω +(1−s)νt u for some s ∈ [0, 1]. We claim that s = 0. Indeed, otherwise there would exist > 0 and δ > 0 such that λu (t) < λu (t u )− for all t < t u with t u −t < δ. But this contradicts the convexity of P u and the fact that νt u is an equilibrium state of −t u φu . We conclude that λu is continuous at t0 . Finally, the continuity of h and identity (12.7) can be shown analogously as in the case νt u = µω (see above). The following result shows that each ergodic measure of maximal dimension must be contained in the family of measures (νt )t∈[0,t u ] . Theorem 12.3. Let f be a parabolic horseshoe map of smooth type with constant contraction rate. Then δ( f ) = sup{dim H νt : t ∈ [0, t u ]}.

(12.9)

Moreover, if µmax is an ergodic measure of maximal dimension for f then there is t ∈ [0, t u ] such that µ = νt . Proof. Since f has constant contraction rate, there exists 0 < c < 1 such that λs (ν) = log c

(12.10)

for all ν ∈ M. Let µ ∈ M E . We claim that there exists t ∈ [0, t u ] such that dim H µ ≤ dim H νt . Note that the claim clearly implies (12.9). To prove the claim we first consider the case λu (µ) > λu (ν0 ). Using that ν0 is the unique measure of maximal entropy, we conclude from (12.3) and (12.10) that d(µ) < d(ν0 ). Next we assume that λu (µ) ∈ [λu (ν0 ), λu (νt u )]. It follows from (12.7) that there exists t ∈ [0, t u ] with λu (νt ) = λu (µ). We now may conclude from the definition of νt and (12.4) that h νt ( f ) ≥ h µ ( f ). As before, (12.3) and (12.10) gives d(µ) ≤ d(νt ). Finally, we consider the case λu (µ) < λu (νt u ). Clearly, this is only possible if νt u = µω . Hence, νtu is the unique ergodic hyperbolic equilibrium state of the potential −t u φu . We now may conclude from (12.3), (12.10) and the definition of t u that dim H νt u = t u −

h νt u ( f ) . log c

(12.11)


749

On the other hand, by (12.4), h νt u ( f ) ≥ h µ ( f ). We conclude that d(νt u ) > d(µ). This completes the proof of the claim. Assume now that µmax is an ergodic measure of maximal dimension for f . Applying the same arguments as before to µmax instead of µ implies that there exists tmax ∈ [0, t u ] such that λu (µmax ) = λu (νtmax ) and h νtmax ( f ) ≥ h µmax ( f ). On the other hand, the fact that µmax is an ergodic measure of maximal dimension and (12.3) imply h µmax ( f ) ≥ h νtmax ( f ); hence h µmax ( f ) = h νtmax ( f ). We conclude that µmax is an ergodic equilibrium state of the potential −tmax φu . Using that µmax = µω implies µmax = νtmax . This completes the proof. We now present the main result of this section. Theorem 12.4. Let f be a parabolic horseshoe map of smooth type with constant contraction rate c. Suppose that µω is the unique equilibrium state of the potential −t u φu and that u

P (t) u η = inf : t ∈ [0, t ) > 0. (12.12) λu (νt )2 Then there exists c0 = c0 (η) > 0 such that if 0 < c < c0 , then f has no ergodic measure of maximal dimension. Proof. Note that in order to show that f has no ergodic measure of maximal dimension it suffices to prove that t → d(νt )

(12.13)

is strictly increasing on [0, t u ). This follows from Theorem 12.3 and the fact that dim H νt u = 0. We claim that P u is not affine on [0, t u ). Indeed, because otherwise (12.5) would imply that the measure of maximal entropy ν0 is an equilibrium state of the potential −t u φu , in contradiction to the hypothesis. Since P u a real-analytic convex function on (0, t u ), we may conclude that d 2 P u (t0 ) ≥ 0, dt 2

(12.14)

where all the zeros of d 2 P u /d 2 t in (0, t u ) are isolated. Moreover, νt0 = µω for all t0 ∈ [0, t u ). Thus νt0 is a hyperbolic measure (see Lemma 10.1). Let now t0 ∈ (0, t u ). It follows from (12.3), (12.5), (12.6) and an elementary calculation that ! h(t0 ) h(t0 ) − λu (t0 ) log c u (t ) P t0 λu (t0 ) 0 − = −λu (t0 ) 2 λu (t0 ) log c ! 2 u u t0 d P (t0 ) P (t0 ) + = dt 2 λu (t0 )2 log c

d d d(νt0 ) = dt dt

def

=

d 2 P u (t0 ) A(t0 ). dt 2

(12.15)

750


We define c0 = exp(−t u /η). It follows from an easy computation that if 0 < c < c0 then A(t0 ) > 0 for all t0 ∈ (0, t u ). Therefore, (12.15) implies that d d(νt0 ) ≥ 0, dt

(12.16)

d d(νt0 ) in (0, t u ) are isolated. We conclude that d(νt ) is strictly where all the zeros of dt u increasing in [0, t ). This completes the proof.

Remark. (i) We note that by L’Hospital’s rule the hypotheses (12.12) holds for instance in the case when P u has uniformly bounded second derivatives on [0, t u ). (ii) A strategy to construct a parabolic horseshoe map having no ergodic measure of maximal dimension is as follows: Consider an almost linear parabolic horseshoe map f as given in Example 1 such that −t u log |Du f | has only one ergodic equilibrium state (for example one can choose β ≥ 1) and that (12.12) holds. In particular, f is of smooth type and has constant contraction rate; hence Theorem 12.4 applies. As mentioned in Remark (i) after Example 1, we may decrease the contraction rate c without changing Du f on W u (ω), insuring that c < c0 . Therefore, Lemma 7.2 guarantees that P u (t), t ∈ [0, t u ) is not affected by this procedure, and therefore, (12.12) still holds. Finally, applying Theorem 12.4 shows that f does not have an ergodic measure of maximal dimension.

References [A] [BW1] [BW2] [DU] [DV] [DGS] [F] [HMU] [J] [KFS] [K] [Ma] [MM] [MU1] [MU2] [Me] [PU]

Aaronson, J.: An introduction to infinite ergodic theory. Mathematical Surveys and Monographs, 50. Providence, RI: Amer. Math. Soc., 1997 Barreira, L, Wolf, C.: Measures of maximal dimension for hyperbolic diffeomorphisms. Commun. Math. Phys. 239, 93–113 (2003) Barreira, L., Wolf, C.: Pointwise dimension and ergodic decompositions. Erg. Theory Dyn. Syst. 26(3), 653–671 (2006) Denker, M., Urbański, M.: On the existence of conformal measures. Trans. A.M.S. 328, 563–587 (1991) Diaz, L., Viana, M.: Discontinuity of hausdorff dimension and limit capacity on arcs of diffeomorphisms. Erg. Theory Dyn. Syst. 9(3), 403–425 (1989) Denker, M., Grillenberger, C., Sigmund, K.: Ergodic theory on compact spaces, Lecture Notes in Math. 527. Berlin: Springer-Verlag, 1976 Falconer, K.: Fractal Geometry: Mathematical Foundations and applications. New York: Wiley, 2003 Hanus, P., Mauldin, D., Urbański, M.: Thermodynamic formalism and multifractal analysis of conformal infinite iterated function systems. Acta Math. Hungarica 96, 27–98 (2002) Jenkinson, O.: Rotation, Entropy, and Equilibrium States. Trans. Amer. Math. Soc. 353, 3713–3739 (2001) Kornfeld, P., Fomin, S.V., Sinai, Y.G.: Ergodic Theory. Berlin-Heidelberg-New York: Springer, 1982 Krengel, U.: Entropy of conservative transformations. Z. Wahr. Verw. Geb. 7, 161–181 (1967) Manning, A.: A relation between Lyapunov exponents, Hausdorff dimension and entropy. Ergodic Theory Dy. Syst. 1(4), 451–459 (1982) Manning, A., McCluskey, H.: Hausdorff dimension for Horseshoes. Erg. Theory Dyn. Syst. 3, 251–260 (1983) Mauldin, D., Urbański, M.: Parabolic iterated function systems. Erg. Th. & Dyna. Syst. 20, 1423–1447 (2000) Mauldin, D., Urbański, M.: Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets. Cambridge: Cambridge Univ. Press, 2003 Mendoza, L.: The entropy of C 2 surface diffeomorphisms in terms of Hausdorff dimension and a Lyapunov exponent. Erg. Theory Dyn. Syst. 5(2), 273–283 (1985) Przytycki, F., Urbański, M.: Fractals in the Plane - Ergodic Theory Methods. To appear Cambridge Univ. Press, available on Urbański’s webpage, http://www.math.unt.edu/~urbanski/


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Rams, M.: Measures of maximal dimension for linear horseshoes. Real Analysis Exchange 31, 55–62 (2006) Ruelle, D.: Thermodynamic formalism. Reading, MA: Addison-Wesley, 1978 Urbański, M.: Parabolic Cantor sets. Preprint 1995, available on Urbański’s webpage, http://www. math.unt.edu/~urbanski/ Urbański, M.: Parabolic cantor sets. Fund. Math. 151, 241–277 (1996) Urbański, M.: Geometry and ergodic theory of conformal nonrecurrent dynamics. Erg. Th. Dyn. Syst. 17, 1449–1476 (1997) Urbański, M.: Hausdorff measures versus equilibrium states of conformal infinite iterated function systems. Periodica Math. Hung. 37, 153–205 (1998) Walters, P.: An introduction to ergodic theory. Graduate Texts in Mathematics 79, Berlin-Heidelberg-New York: Springer, 1981 Wolf, C.: Generalized physical and SRB measures for hyperbolic diffeomorphisms. J. Stat. Phys. 122(6), 1111–1138 (2006) Young, L.-S.: Dimension, entropy and Lyapunov exponents. Erg. Theory Dyn. Syst. 2, 109–124 (1982)

Communicated by G. Gallavotti


Communications in


Entanglement of Positive Definite Functions on Compact Groups J. K. Korbicz1,2 , J. Wehr2,3 , M. Lewenstein2 1 Dept. d’Estructura i Constituents de la Matèria, Universitat de Barcelona, 647 Diagonal,

08028 Barcelona, Spain. E-mail: [email protected]

2 ICREA and ICFO–Institut de Ciències Fotòniques, Mediterranean Technology Park,

08860 Castelldefels (Barcelona), Spain

3 Department of Mathematics, University of Arizona, 617 N. Santa Rita Ave.,

Tucson, AZ 85721-0089, USA Received: 13 June 2007 / Accepted: 17 October 2007 Published online: 15 May 2008 – © Springer-Verlag 2008

Abstract: We define and study entanglement of continuous positive definite functions on products of compact groups. We formulate and prove an infinite-dimensional analog of the Horodecki Theorem, giving a necessary and sufficient criterion for separability of such functions. The resulting characterisation is given in terms of mappings of the space of continuous functions, preserving positive definiteness. A relation between the developed group-theoretical formalism and the conventional one, given in terms of density matrices, is established through the non-commutative Fourier analysis. It shows that the presented method plays the role of a “generating function” formalism for the theory of entanglement. 1. Introduction Entanglement is a property of states of composite quantum mechanical systems. This concept lies at the very heart of quantum mechanics, and it concerns all of the important aspects of quantum theory: from philosophical aspects [1,2], through physical [3] and mathematical [4,5] fundamentals1 , to applications in quantum information and metrology [7]. The importance of entangled states for the understanding of quantum theory was recognized quite early, mainly thanks to Einstein (e.g. in the famous EPR paper by Einstein, Podolsky, and Rosen [8]). Only with the advent of new experimental techniques in recent years, it became clear that entanglement may in fact also be used as a resource for transmission and processing of (quantum) information, e.g. for quantum cryptography or quantum computing (for a recent review, see Ref. [4] ; see also Ref. [7]). In the present work we develop a novel framework for studying quantum entanglement, based on analysis of continuous functions on compact groups. With respect to the standard formalism of entanglement theory, our approach plays a role analogous 1 For a description of positive maps from a physical point of view, see e.g. Ref. [6].

754

J. K. Korbicz, J. Wehr, M. Lewenstein

to that of a “generating function” method—various group-theoretical objects serve as “generating functions” for the corresponding families of operator-algebraic objects (like density matrices, positive maps, etc), operating in different dimensions. This allows one to formulate and address the questions of entanglement theory in a unified, dimensionwise, way. Before we proceed with the group-theoretical formalism, let us first recall some basic facts and define the notion of entanglement precisely. A quantum system is associated with a Hilbert space H, which we will assume to have a countable basis. A state of the system is then represented by a positive, trace-class operator (a density matrix), satisfying normalization condition tr = 1. If the system under consideration is composite, i.e. it can be thought of being composed of two subsystems A and B, each of which is treated as an independent individual, then, according to the postulates of quantum theory, the Hilbert space of the system is H = HA ⊗ HB . The following definition thus makes sense [9]: Definition 1.1. A state on HA ⊗ HB is called separable if it can be approximated in the trace norm by convex combinations of the form: K

pm |xm xm | ⊗ |ym ym |, where xm ∈ HA , ym ∈ HB ,

m=1

pm 0,

K

pm = 1. (1)

m=1

Otherwise is called entangled. This definition can be easily generalized to multipartite systems with more than two parties involved. In the light of Definition 1.1 a natural question arises, known as the separability problem: Given a state decide if it is separable or not. The problem turns out to be computationally very hard: although efficient algorithms employing positive definite programming methods exist in lower dimensions [10] (for a specific formulation of semi-definite approach for 2 ⊗ N systems see Ref. [11]), it has been proven that the problem belongs to the N P complexity class as dimensions of the Hilbert spaces involved grow [12]. In terms of operational entanglement criteria up to date there are only partial answers known, in both finite and infinite dimensions. We briefly quote below few basic results, referring the reader to Ref. [4] for a complete overview. One astonishingly powerful, given its simplicity, necessary criterion for separability follows immediately from the definition of separable states [13,14]: Theorem 1.1 (Positivity of Partial Transpose (PPT)). If a state on HA ⊗ HB is separable then the partially transposed operator TB := (1A ⊗ T ) is positive, where T is a transposition map and 1A is the identity operator on HA . In the lowest non-trivial dimensions dimHA = dimHB = 2 and dimHA = 2, dimHA = 3 PPT criterion provides both necessary and sufficient conditions for separability (see Ref. [14], Theorem 3). However, in higher dimensions there exist states, called PPT or bound entangled, which satisfy the PPT criterion, but are nevertheless entangled. The first examples of such states were constructed in Ref. [15] (although in a different context of, so called, indecomposable maps) and in Ref. [16]. In infinite dimension, a complete solution to the separability problem exists only for a special family of states—so called Gaussian states [17]. As mentioned above the separability problem is connected to other open mathematical problems. In their fundamental work [14] Horodecki et al. established an important link

Entanglement of Positive Definite Functions on Compact Groups

755

between this problem and the problem of characterization of positive maps on finitedimensional matrix algebras (cf. Ref. [14], Theorem 2; see also Refs. [5,15,18,19] ): Theorem 1.2 (M., P., and R. Horodecki). Let L(H) denote the space of linear operators on H and let ∈ L(HA ⊗HB ) be a density matrix on a finite dimensional Hilbert space HA ⊗HB . Matrix is separable if and only if for all linear maps : L(HB ) → L(HA ) preserving positive operators (such maps are called positive), (1A ⊗ ) 0 as an operator on HA ⊗ HA . The starting point for the present work is the non-commutative Fourier analysis on a compact group, which we proposed to employ for studying entanglement in Ref. [20] (cf. Ref. [21] where the same method was used for a rigorous derivation of the classical limit of quantum state space). Namely, one can pass from operators A ∈ L(HA ⊗ HB ) to their non-commutative Fourier transforms2 in two steps: i) identify the spaces HA , HB with representation spaces of unitary, irreducible representations πα , τβ of some compact groups G 1 and G 2 respectively (there are no a priori restrictions on G 1 , G 2 apart from possessing representations in suitable dimensions); ii) pass from A to a function ϕ A : G 1 × G 2 → C, the non-commutative Fourier transform of A, through: A → ϕ A (g1 , g2 ) := tr Aπα (g1 ) ⊗ τβ (g2 ) .

(2)

The above transform is called non-commutative, since apart from the trivial case dimHA = dimHB = 1, groups G 1 and G 2 are necessarily non-Abelian. In case A = is a quantum state, the corresponding function ϕ is called a non-commutative characteristic function of . The transformation (2) is invertible—one can recover A from ϕ A . Hence, one expects that for density matrices their non-commutative characteristic functions should encode entanglement in some way [20]. This is indeed the case and in what follows we define and study the notion of separability for suitably generalized noncommutative characteristic functions (general continuous positive definite functions on G 1 × G 2 ; cf. Definition 2.1). We then prove an analog of the Horodecki Theorem 1.2 for such functions, which constitutes the main result of the paper. Since the framework we work in is countably infinite-dimensional (unless both G 1 , G 2 are finite) our result can be viewed as a generalization of the Horodecki Theorem to an infinite-dimensional setting. The usual quantum-mechanical formalism, given by density matrices, and the presented group-theoretical one are then shown to be related through non-commutative harmonic analysis. In particular, by employing the non-commutative Fourier transform we demonstrate how our approach turns out to be a “generating function” method for the theory of entanglement. Let us finally remark that the formalism of non-commutative Fourier transform (2) is closely related to that of generalized coherent states [23]. The difference is that in the coherent state formalism one assigns to an operator A a function (called P-representation of A), which is defined not on the whole group G, but on a homogeneous space G/H , where H is an isotropy subgroup of a fixed vector. However, unlike the non-commutative Fourier transform ϕ A , P-representation is generally non-unique (e.g. in the SU (2) case) and does not encode positivity of a density matrix in a simple manner. For some applications of generalized coherent states to the study of entanglement see e.g. Refs. [24]. 2 We note that in Ref. [22] the term “noncommutative Fourier transform” is used in a slightly different— though very closely related—sense.

756


2. Preliminary Notions In the main part of the work G 1 , G 2 will be compact groups. The principal object of our study are continuous positive definite functions on the product group G 1 × G 2 . But first we recall some basic definitions and facts, valid for any locally compact G (see e.g. Refs. [25–27] for a complete exposition). Definition 2.1. A continuous complex function ϕ on a group G with the Haar measure dg is called positive definite if it is bounded and satisfies: dgdh f (g)ϕ(g −1 h) f (h) 0 (3) (bar denotes complex conjugation) for any continuous function f with compact support. We will denote by P(G) the set of positive definite functions on G and by P1 (G) its subset consisting of the functions which satisfy the normalization ϕ(e) = 1, where e is the neutral element of G. P(G) is a closed convex cone in C(G) — the space of continuous complex-valued functions on G equipped with the topology of uniform convergence on compact sets, called compact convergence in the sequel. The structure of P(G) is described by the following deep, fundamental result of representation theory, often referred to as the GNS construction (see e.g. Ref. [25], Theorem 3.20; Ref. [26], Theorem 13.4.5): Theorem 2.1 (Gel’fand, Naimark, Segal). With every ϕ ∈ P(G) we can associate a Hilbert space Hϕ , a unitary representation πϕ of G in Hϕ and a vector vϕ , cyclic for πϕ , such that: ϕ(g) = vϕ |πϕ (g)vϕ .

(4)

The representation πϕ is unique up to a unitary equivalence. The above result provides a tool for a systematic study of P(G) in terms of representations of G (and conversely). In the sequel we will need some basic properties of positive-definite functions. While they all follow from the definition by standard, elementary arguments, we find the proofs based on the GNS representation particularly transparent. A function ϕ will be called pure if πϕ is irreducible. Pure normalized functions are the extreme points of P1 (G) (cf. Ref. [25], Theorem 3.25); we denote their set by E1 (G). Every ϕ ∈ P1 (G) is a limit, in the topology of compact convergence, of convex combinations of extreme points of P1 (G) (cf. Ref. [26], Theorem 13.6.4): g →

N

pm εm (g), where εm ∈ E1 (G), pm 0,

m=1

N

pm = 1.

(5)

m=1

There is also an integral representation (provided G is separable as a topological space), sometimes called the generalized Bochner Theorem. Namely, for any ϕ ∈ P1 (G) there exists a probability measure µϕ concentrated on E1 (G) such that (cf. Ref. [26], Propo. 13.6.8): ϕ(g) = dµϕ (ε) ε(g) for any g ∈ G. (6) E1 (G)


757

From this point on, we assume that G 1 , G 2 are compact and consider positive definite functions on G 1 × G 2 . Let us introduce the algebraic tensor product C(G 1 ) ⊗ C(G 2 ) as the space of finite (complex) linear combinations of product functions f ⊗ξ : (g1 , g2 ) → f (g1 )ξ(g2 ). Then C(G 1 )⊗C(G 2 ) is uniformly dense in C(G 1 × G 2 ). This standard fact follows, for example, from the Stone-Weierstrass Theorem (see e.g. Ref. [28]). Every product φ ⊗ ψ, where φ ∈ P1 (G 1 ), ψ ∈ P1 (G 2 ), is positive definite on G 1 × G 2 , since, by the GNS Theorem 2.1, φ(g1 )ψ(g2 ) = vφ ⊗ vψ |πφ (g1 ) ⊗ πψ (g2 )vφ ⊗ vψ which is of the form (4) on G 1 × G 2 . It follows that convex combinations of such products are positive definite and hence so are uniform limits of such convex combinations. The resulting class of positive definite functions, introduced formally in the next definition, is our fundamental object of study (compare Definition 1.1). Definition 2.2. We define Sep0 as the set of all functions ϕ ∈ P1 (G 1 × G 2 ) which can be represented as finite convex combinations ϕ(g1 , g2 ) =

K

pm εm (g1 )ηm (g2 ), where εm ∈ E1 (G 1 ), ηm ∈ E1 (G 2 ).

(7)

m=1

A function ϕ ∈ P1 (G 1 × G 2 ) is called separable if it is a uniform limit of elements of Sep0 . The set of separable functions is denoted by Sep. Functions which are not separable are called entangled. The definitions of separable and entangled functions generalize without any change to arbitrary (i.e. not necessarily normalized) positive definite functions. This includes our main result, Theorem 3.2, together with its proof (since for a nonzero positive definite function ϕ(e1 , e2 ) = ||ϕ||∞ > 0, we can replace ϕ by ϕ/ϕ(e1 , e2 ) and reduce the proof to the normalized case). The normalization is, however, natural from the physical point of view. Geometrically E1 (G 1 ) × E1 (G 2 ) is embedded into E1 (G 1 × G 2 ) through the map (ε, η) → ε ⊗ η. Then Sep is a closed convex hull of E1 (G 1 ) × E1 (G 2 ). We note that every ϕ ∈ Sep admits an integral representation: ϕ(g1 , g2 ) = dµϕ (ε, η) ε(g1 )η(g2 ) for any (g1 , g2 ) ∈ G 1 × G 2 , (8) E1 (G 1 )×E1 (G 2 )

but we will not use this fact.

3. Necessary and Sufficient Criterion for Separability of Positive Definite Functions The main problem we would like to address is that of finding an intrinsic characterization of separable functions ϕ ∈ Sep. This is known as the generalized separability problem [20]. By Eq. (4) for every positive definite function φ, φ(g −1 ) = φ(g) and this function is again positive definite. It now follows immediately from Definition 2.2 and from uniform closedness of P(G 1 × G 2 ) in C(G 1 × G 2 ) that: Theorem 3.1. If ϕ ∈ Sep then the function (g1 , g2 ) → ϕ(g1 , g2−1 ) is positive definite.

758


The above simple criterion is only a necessary condition—there are functions satisfying it which are nevertheless entangled. This can be seen by noting that Theorem 3.1 is a group-theoretical analog of the PPT criterion, given by Theorem 1.1, (we will show it in Sect. 5; see also Ref. [20], Theorem 2) and, as we mentioned in the Introduction, there exist PPT entangled (or, equivalently, bound entangled) quantum states [16]. A natural question arises whether one obtains a complete characterization of separable functions when in place of the inverse g → g −1 one considers all possible linear maps of functions, preserving positive definiteness. The affirmative answer is the main result of our work: Theorem 3.2. A function ϕ ∈ P1 (G 1 × G 2 ) is separable if and only if for every bounded linear map : C(G 2 ) → C(G 1 ), such that P(G 2 ) ⊂ P(G 1 ), function (id ⊗ )ϕ is positive definite on G 1 × G 1 . Tensor product id ⊗ : C(G 1 × G 2 ) → C(G 1 × G 1 ) is defined in the natural way: we first define it on the algebraic product C(G 1 ) ⊗ C(G 2 ) and then extend by continuity to all of C(G 1 × G 2 ). The above theorem is a group-theoretical analog of the Horodecki Theorem 1.2. In fact, we derive a version of the Horodecki result as a corollary in Sect. 4 (cf. Theorem 4.2). We will adopt the standard terminology of entanglement theory and say that an entangled function ϕ is detected by a map if the function (id ⊗ )ϕ is not positive definite. As mentioned earlier, the above theorem, as well as the following proof, hold for arbitrary positive definite functions, but in order to use more natural concepts from the physical point of view we state and prove it for normalized ones. The proof in one K direction is immediate and follows directly from Definition 2.2: (id⊗ ) m=1 pm εm ⊗ K ηm = m=1 pm εm ⊗ ηm ∈ P(G 1 × G 1 ) and since P(G 1 × G 1 ) is uniformly closed in C(G 1 × G 1 ) this holds on all of Sep. For the proof in the other direction, let C(G 1 × G 2 ) denote the space of continuous linear functionals on C(G 1 × G 2 ) (the space dual to C(G 1 × G 2 )). Since Sep is a closed convex set, it follows from the Hahn-Banach Theorem (see e.g. Ref. [29], Theorem V.4) that for every ϕ ∈ / Sep there exists a functional l ∈ C(G 1 × G 2 ) and a real number γ , such that: Rel(ϕ) < γ Rel(σ ) for any σ ∈ Sep,

(9)

where Rel denotes the real part of the functional l: Rel(ϕ) := Re[l(ϕ)]. From the Riesz Representation Theorem (see e.g Ref. [29], Theorem IV.17) we know that each linear functional l on C(G 1 × G 2 ) can be uniquely represented by a complex measure µl with finite total variation |µl | on G 1 × G 2 . Denoting the space of such measures by M(G 1 × G 2 ) we have: C(G 1 × G 2 ) = M(G 1 × G 2 ). We will interchangeably treat elements of C(G 1 × G 2 ) as either linear functionals or as the corresponding measures. To work with the normalized functions ϕ which we are interested in, it is convenient to introduce a modification of the functional l in (9): L := l − γ δ(e1 ,e2 ) , where δ(e1 ,e2 ) is the Dirac delta (point mass), concentrated at the neutral element (e1 , e2 ) of G 1 × G 2 . Hence, for every entangled ϕ ∈ P1 (G 1 × G 2 ) there exists L ∈ C(G 1 × G 2 ) such that: ReL(ϕ) < 0 ReL(σ ) for any σ ∈ Sep.

(10)

As an easy consequence of the above condition we obtain the following lemma, crucial for the rest of the proof:


759

Lemma 3.1. A function ϕ is separable if and only if for every functional L ∈ C(G 1 × G 2 ) , satisfying ReL(ψ1 ⊗ ψ2 ) 0 for every ψ1 ∈ P1 (G 1 ), ψ2 ∈ P1 (G 2 ), we have ReL(ϕ) 0. K K Indeed, for every Sep0 ϕ = m=1 pm εm ⊗ηm , L(ϕ) = m=1 pm L(εm ⊗ηm ) 0 and by continuity of L this extends to all of Sep. Conversely, assume that for every L satisfying the condition in the statement of the lemma, ReL(ϕ) 0 but ϕ ∈ / Sep. Then from the Hahn-Banach Theorem (see (10)) we know that there exists a functional L 0 such that ReL 0 (σ ) 0 for every separable σ —in particular for every function of the form ψ1 ⊗ ψ2 —and ReL 0 (ϕ) < 0, which contradicts our assumption. In order to pass from linear functionals on C(G 1 × G 2 ) to linear maps from C(G 2 ) to C(G 1 ), we first employ an algebraic isomorphism between functionals from C(G 1 ×G 2 ) : C(G 2 ) → C(G 1 ) . For each L ∈ C(G 1 × G 2 ) we define and bounded linear maps the corresponding map L by: L ξ( f ) := L( f ⊗ ξ );

(11)

L is bounded:

L || = || =

L ξ || ∞ = sup ||

||ξ ||∞ =1

sup

sup

sup |L( f ⊗ ξ )|,

||ξ ||∞ =1 || f ||∞ =1

L ξ( f )| sup |

||ξ ||∞ =1 || f ||∞ =1

(12)

where || · || ∞ is the norm on C(G 1 ) induced by the supremum norm || · ||∞ . Conversely, : C(G 2 ) → C(G 1 ) , Eq. (11) defines a functional L for any bounded map on C(G 1 ) ⊗ C(G 2 ), which is bounded by Eq. (12), and uniquely defined, since if L ( f ⊗ ≡ 0 (note that L ξ ) = 0 for all f ∈ C(G 1 ), ξ ∈ C(G 2 ), then by Eq. (11) depends linearly on ) . As C(G 1 ) ⊗ C(G 2 ) is uniformly dense in C(G 1 × G 2 ), L can be uniquely extended to a continuous functional on all of C(G 1 × G 2 ). This establishes the L . claimed isomorphism L ↔ L , analogous to the one given by Next, we establish a positivity criterion Jamiołkowski in Ref. [30] for operators on finite dimensional Hilbert spaces: Lemma 3.2. A functional L ∈ C(G 1 × G 2 ) satisfies ReL(ψ1 ⊗ ψ2 ) 0 for all L maps positive definite functions from ψ1 ∈ P1 (G 1 ), ψ2 ∈ P1 (G 2 ) if and only if Re P(G 2 ) to positive definite measures from M(G 1 ). Positive definite measure on G is a measure µ satisfying a generalization of the condition (3)3 : (13) dµ ( f ∗ ∗ f ) 0 for any f ∈ C(G), where f ∗ (g) := f (g −1 ) is the involution and ( f ∗ ξ )(h) := dg f (g)ξ(g −1 h) is the L on f is defined as the real part of the functional convolution. The action of the map Re L f . The condition ReL(ψ1 ⊗ ψ2 ) 0 for all normalized ψ1 ∈ P1 (G 1 ), ψ2 ∈ P1 (G 2 )

3 On an arbitrary locally compact G positive definite measures are defined by requiring that condition (13) hold for all continuous functions with compact support (cf. Ref. [26], Def. 13.7.1).

760


is equivalent to ReL(ψ1 ⊗ ψ2 ) 0 for all ψ1 ∈ P(G 1 ), ψ2 ∈ P(G 2 ), since we can replace ψ1 , ψ2 = 0 by ψ1 /ψ1 (e) and ψ2 /ψ2 (e). From definition (11) it follows that: L ψ2 (ψ1 )] = Re L ψ2 (ψ1 ). ReL(ψ1 ⊗ ψ2 ) = Re[

(14)

A theorem by Godement (cf. Ref. [31], Theorem 17; Ref. [26], Theorem 13.8.6) states that every positive definite function can be uniformly approximated by functions of the form f ∗ ∗ f , where f is continuous with compact support. Hence, since G 1 , G 2 are compact, ReL(ψ1 ⊗ ψ2 ) 0 for every ψ1 ∈ P(G 1 ), ψ2 ∈ P(G 2 ) if and only if L ψ2 ( f ∗ ∗ f ) 0 for every f ∈ C(G 1 ) and ψ2 ∈ P(G 2 ). ReL ( f ∗ ∗ f ) ⊗ ψ2 = Re L ψ2 ∈ M(G 1 ) being But by Eqs. (13) and (14) this is equivalent to the measure Re positive definite for every ψ2 ∈ P(G 2 ). . Note that for any µ ∈ M(G 1 ) and As the next step we will regularize maps f ∈ C(G 1 ) the convolution: µ ∗ f (h) = dµ(g) f (g −1 h) (15) is a continuous function on G 1 . Let {ψU }, where U ⊂ G 1 runs through a neighbourhood base of the neutral element e1 ∈ G 1 , be an approximate identity in C(G 1 ). That is, for every U we have: o) ψU ∈ C(G 1 ), i) suppψU is a compact subset of U, ψU = 1. ii) ψU 0, iii) ψU (g −1 ) = ψU (g), iv)

(16) (17)

Using definition (15), let us define for every f ∈ C(G 1 ) function, f ∗ ψU .

U f :=

(18)

f as U → {e1 }. To see this, let us Then U f converges in weak-∗ topology to calculate dg U f (g)ξ(g) for an arbitrary ξ ∈ C(G 1 ): f ∗ ψU )(g)ξ(g) = dg d( f )(h)ψU (h −1 g)ξ(g) dg ( = d( f )(h) dgψU (g −1 h)ξ(g) f (ξ ∗ ψU ), =

(19)

where in the second step we used the symmetry of ψU : ψU (g −1 ) = ψU (g). But from the properties (16), (17) of {ψU } it follows that ξ ∗ ψU → ξ uniformly as U → {e1 } (cf. Ref. [25], Theorem 2.42), which proves the desired weak-∗ convergence. Thus any : C(G 2 ) → M(G 1 ) can be weakly-∗ approximated by bounded maps bounded map

U : C(G 2 ) → C(G 1 ) (boundedness of U for every U ⊂ G 1 follows immediately from the definitions (15) and (18)). ’s, introduced in Lemma 3.2, we In order to preserve the positivity property of the choose regularizing functions ψU in a special way. Namely, for every neighbourhood U of e1 ∈ G 1 we can find such open V e1 that: (cf. Ref. [25], Lemma 5.24): o) V ⊂ U, i) V −1 = V, ii) gV g −1 = V for every g ∈ G 1 .

(20)


761

Let us define the functions: 1 χV , |V | ψU := κV ∗ κV , κV :=

(21) (22)

where χV is the indicator function of V and |V | = dgχV (g) is the Haar measure of V . Then one easily shows that {ψU } form an approximate identity in C(G 1 ). Moreover, κV , and hence ψU , are central functions, i.e. for every g and h, κV (gh) = κV (hg), which follows from the property (ii) of the sets V . Using functions (22) to regularize , we find that: an arbitrary map f ∗ ψU = κV ∗ f ∗ κV = κV∗ ∗ f ∗ κV ,

U f =

(23)

where in the second step we used the fact that κV are central and hence µ ∗ κV = κV ∗ µ for any µ. In the last step we used the symmetry condition (i) and χV = χV . Now, from the special form of the regularization (23) we obtain: : C(G 2 ) → M(G 1 ) maps positive definite functions into positive Lemma 3.3. If definite measures, then for every neighbourhood U of e1 ∈ G 1 the regularized maps

U , defined by Eqs. (21–23), map P(G 2 ) into P(G 1 ). To prove Lemma 3.3, note that for an arbitrary φ ∈ P(G 2 ) and f ∈ C(G 1 ) one has:

φ ∗ κV (g −1 h) f (h) dgdh f (g) κV∗ ∗ φ)(b) κV (b−1 a −1 g −1 h) f (h) = dgdhda f (g) κV (a −1 ) d( = d( φ)(b) da dg f (g) κV (ag) dh f (h)κV (b−1 ah)

φ)(b) da f ∗ κˇ V ∗ (a) f ∗ κˇ V (a −1 b) = d(

φ ( f ∗ κV ∗ ∗ ( f ∗ κV ) , =

(24)

φ is a positive where κˇ V (g) := κV (g −1 ) = κV (g) by property (i) in Eq. (20). Hence, if definite measure from M(G 1 ) then for every U , U φ is a positive definite function on G 1 (note that f ∗ κV is continuous since f is). Let us introduce some terminology, analogous to that used in the theory of linear mappings of operators on finite-dimensional Hilbert spaces (see e.g. Refs. [32,19,18]): Definition 3.1. Let : C(G 2 ) → C(G 1 ) be a bounded linear map. Then is called: • positive definite (PD) if it preserves positive definite functions, i.e. if P(G 2 ) ⊂ P(G 1 ); • H -positive definite (H -PD), where H is a compact group, if id ⊗ : C(H × G 2 ) → C(H × G 1 ) is positive definite, i.e. if id ⊗ P(H × G 2 ) ⊂ P(H × G 1 ); • completely positive definite (CPD) if it is H -positive definite for any compact H .

762


Thus, rephrased in the terms introduced above, Lemma 3.3 states that every bounded : C(G 2 ) → M(G 1 ), mapping P(G 2 ) into positive definite measures, can linear map be weakly-∗ approximated by positive definite maps from C(G 2 ) to C(G 1 ). After we have established almost all the necessary facts, we return to the main Lemma 3.1. First, rewrite Eq. (11) using the regularization (3.3) in order to be able to write down explicitly the right-hand side of Eq. (11) for arbitrary functions, not only product ones. For product functions we have: L ξ( f ). ReL( f ⊗ ξ ) = Re

(25)

L , denoting the regularized operators by RU , We apply the regularization (23) to Re

RU : C(G 2 ) → C(G 1 ), rather than by Re L U , to obtain:

dg1 RU ξ (g1 ) f (g1 ) ReL( f ⊗ ξ ) = lim RU ξ( f ) = lim U →{e1 }

= lim

U →{e1 } G1

U →{e1 } G1

dg1 id ⊗ RU f ⊗ ξ (g1 , g1 ).

(26)

Formula (26) immediately extends to all of C(G 1 × G 2 ), so in particular for any ϕ ∈ P1 (G 1 × G 2 ) we have:

ReL(ϕ) = lim dg1 id ⊗ RU ϕ (g1 , g1 ). (27) U →{e1 } G1

Summarizing, by application of Lemmas 3.2 and 3.3 we obtain that for an arbitrary functional L ∈ C(G 1 × G 2 ) , such that ReL(ψ1 ⊗ ψ2 ) 0 for every ψ1 , ψ2 ∈ P1 (G), ReL(ϕ) is given by Eq. (27), where for every neighbourhood U ⊂ G 1 the maps RU : C(G 2 ) → C(G 1 ) are bounded and positive definite. We need two more simple facts. First, we note that if ϕ is a positive definite function on the product of two copies of G 1 , i.e. ϕ ∈ P(G 1 × G 1 ), then its restriction to the diagonal ϕ (g) := ϕ(g, g) is a positive definite function on G 1 . A particularly direct proof of this fact follows from the GNS construction (cf. Theorem 2.1): −1 dgdh f (g)vϕ |πϕ (g −1 , g −1 )πϕ (h, h) vϕ f (h) dgdh f (g)ϕ (g h) f (h) = (28) = dg f (g)πϕ (g, g)vϕ dh f (h)πϕ (h, h)vϕ 0. Second, we note that since G 1 is compact, dg φ(g) 0 for any φ ∈ P(G 1 ) (cf. Ref. [27], Theorem 34.8). Indeed, if G 1 is compact then the constant function 1 is a function support with compact and we can use it in the condition (3), which then implies that dgdh φ(g −1 h) = dh φ(h) 0, where we changed variables h → gh and used the normalization of dg. To finish the proof of the main Theorem 3.2, let us assume that for every bounded and positive definite map : C(G 2 ) → C(G 1 ), the function (id ⊗ )ϕ is positive definite. It then follows from the above discussion and from Eq. (27) that ReL(ϕ) 0 for every L, such that ReL(ψ1 ⊗ ψ2 ) 0 for every ψ1 ∈ P1 (G 1 ), ψ2 ∈ P1 (G 2 ). But then Lemma 3.1 implies that ϕ ∈ Sep.


763

4. Fourier Transforms and “Generating Function” Formalism In this section we establish a connection between the formalism of positive definite functions and standard notions of entanglement theory, thus ascribing a “physical meaning” to the former. Namely, as advertised in the Introduction, we show that various group-theoretical objects studied in the previous sections turn out to be “generating functions” for the corresponding operator-algebraic objects. For example a positive definite function generates a family of (subnormalized) density matrices, a positive definite map (cf. Definition 3.1) generates a family of positive maps, etc. We also derive here a weaker version of the Horodecki Theorem (cf. Theorem 1.2) from Theorem 3.2 and prove a number of other useful results. Our main tool will be non-commutative Fourier analysis of continuous functions on compact groups. Below we recall some basic notions and methods, which we will need (see e.g. Refs. [25,26] for more). The goal which we have in mind is to construct uniformly convergent Fourier series for continuous functions. By (a part of) the fundamental theorem of the theory—the Peter-Weyl Theorem (see e.g. Ref. [25], Theorem 5.12), any continuous function on a compact group can be uniformly approximated by linear combinations of matrix elements of irreducible representations, taken in some fixed orthonormal bases of the corresponding representation spaces. Here, we are primarily interested in product groups G 1 × G 2 , so we first recall 1 (G 2 ) the set of equivalence classes their representation structure. Let us denote by G of irreducible, strongly continuous, unitary representations (irreps) of G 1 (G 2 ). Since 1 and G 2 are discrete and (the classes of equivalent) irreducible G 1 , G 2 are compact, G representations can be labelled by discrete indices. We will denote irreps of G 1 and G 2 β . It can then by πα and τβ respectively and the spaces where they act by Hα and H be shown that for a large family of groups, including compact ones, every irrep of the product G 1 × G 2 can be chosen in the form πα ⊗ τβ , where: πα ⊗ τβ (g1 , g2 ) := πα (g1 ) ⊗ τβ (g2 )

(29)

β (cf. Ref. [25], Theorem 7.25; Ref. [26], Prop. 13.1.8). In acts in the space Hα ⊗ H 1 × G 2 . other words, G 1 × G 2 can be identified with G Next, we need matrix elements of the irreps πα ⊗ τβ . It is natural here to take them β . Thus, for each pair with respect to product bases of the representation spaces Hα ⊗ H of the representation indices α and β we fix an orthonormal base {ei }i=1,...,dimHα of β (we do not indicate explicitly the Hα and an orthonormal base {e˜k }k=1,...,dimH β of H dependence of {ei }, {e˜k } on the representation indices α, β in order not to complicate the notation). The corresponding matrix elements of πα ⊗ τβ are then simply given by products of the matrix elements of πα and τβ —that is, they are given by the functions β πiαj ⊗ τkl , where: β

πiαj (g1 ) := ei |πα (g1 )e j , τkl (g2 ) := e˜k |τβ (g2 )e˜l .

(30)

Now, for a given f ∈ C(G 1 × G 2 ) we can formally write the Fourier series: i jkl i jkl β β f = f αβ πiαj ⊗ τkl , f αβ := n α m β dg1 dg2 πiαj (g1 ) τkl (g2 ) f (g1 , g2 ), α,β i,...,l

G1 G2

(31)

764


where n α := dimHα , m β := dimHβ . However, for a generic function f ∈ C(G 1 × G 2 ) the Fourier series (31) converges only in the L 2 norm (since G 1 × G 2 is compact C(G 1 × G 2 ) ⊂ L 2 (G 1 × G 2 )) and not uniformly. The standard way around this difficulty is the following: i) regularize f so that the Fourier series of the regularized function converges uniformly; ii) perform the desired manipulations with the series; iii) at the end uniformly remove the regularization. For the regularization the same technique as in Sect. 3 (cf. Eqs. (16–18) and Eqs. (20–23)) is used. Thus, for a given f we consider a function f U := f ∗ ψU , where the regularizing functions ψU ∈ C(G 1 × G 2 ) are defined in an analogous way as in Eqs. (20–22), but this time on the product G 1 × G 2 . The sets U ⊂ G 1 × G 2 now run through a neighborhood base of the neutral element {e1 , e2 } ∈ G 1 × G 2 . Since, by construction, the ψU ’s are central on G 1 × G 2 (cf. property (ii) in Eq. (20)), a simple calculation shows that the Fourier series of f ∗ ψU takes the following form: αβ i jkl β cU f αβ πiαj ⊗ τkl , f ∗ ψU = αβ

α,β i,...,l

cU :=

dg1 dg2 ψU (g1 , g2 )χα (g1 ) χβ (g2 ),

(32)

G1 G2

where χα (g) := trπα (g) is the character of the representation πα and, analogously, χβ is the character of τβ . It can then be shown that the series (32) converges uniformly αβ for every U (cf. Ref. [25], p. 137). Thus, the role of the constants cU is to enhance ∗ convergence of the Fourier series (31). Note that since f ∗ ψU = κV ∗ f ∗ κV , where the sets V are defined as in Eq. (20) but on G 1 × G 2 , the regularization preserves positive definiteness (cf. Eq. (24) where we proved it for measures on a single group G 1 ). In fact, it preserves separability as well, as we will show later (see Lemma 4.1). Finally, the initial function f can be recovered from f ∗ ψU by letting U → {e1 , e2 } as then f ∗ ψU → f uniformly (cf. Ref. [25], Theorem 2.42). β ) by: Let us define operators fˆαβ ∈ L(Hα ⊗ H jilk fˆαβ := f αβ |ei e j | ⊗ |e˜k e˜l | i,...,l

= nα m β

dg1 dg2 f (g1 , g2 )πα (g1 )† ⊗ τβ (g2 )† ,

(33)

G1 G2

(note the change of the order of indices). Operators fˆαβ are inverse Fourier transforms of f : fˆαβ ≡ fˆ(πα ⊗ τβ ) [25,20] and Fourier series (31) and (32) can be rewritten as: αβ tr fˆαβ πα ⊗ τβ , f ∗ ψU = cU tr fˆαβ πα ⊗ τβ . f = (34) α,β

α,β

The last equation again explicitly shows the role of regularization in enhancing convergence of the Fourier series (31). Having recalled the technicalities of the Fourier analysis, we proceed to relate the group-theoretical formalism to the standard one. We begin by quoting a standard fact, which we will extensively use in what follows (see e.g. Ref. [27], Theorem 34.10): 1 and [τβ ] ∈ G 2 . Theorem 4.1. ϕ ∈ P(G 1 × G 2 ) if and only if ϕˆαβ 0 for all [πα ] ∈ G


765

This is a non-commutative analog of the fact that positive definiteness corresponds under (usual) Fourier transform to positivity. We present a proof of the above theorem just for the sake of completeness. Let us first introduce an abbreviation g := (g1 , g2 ) ∈ G 1 × G 2 . From Eq. (33) we then obtain β : for any v ∈ Hα ⊗ H v|ϕˆαβ v = n α m β dg1 dg2 ϕ(g1 , g2 ) v πα (g1 )† ⊗ τβ (g2 )† v = nα m β d hdgϕ(h−1 g) πα (h 1 )† ⊗ τβ (h 2 )† v πα (g1 )† ⊗ τβ (g2 )† v

n α ,m β

=

nα m β

αβ

αβ

d hdg vik (h) ϕ(h−1 g)vik (g) 0,

(35)

i,k=1

αβ where vik (h) := ei ⊗ e˜k πα (h 1 )† ⊗ τβ (h 2 )† v . In the second step above we inserted 1 = G 1 ×G 2 dh 1 dh 2 and then changed the variables g → h−1 g. Then we inserted the β unit matrix 1α ⊗ 1β , decomposed with respect to the fixed bases {ei }, {e˜k } of Hα , H and used positive definiteness of ϕ. Note that we do not need uniform convergence of the Fourier series here, and hence we used the L 2 -convergent series (31) of ϕ. The same applies to the proof in the other direction. Let us now assume that ϕˆαβ 0 for all α, β. Then from Eq. (31) we obtain: dgd h f (g)ϕ(g −1 h) f (h) i jkl β = ϕαβ dgd h f (g1 , g2 )πiαj (g1−1 h 1 )τkl (g2−1 h 2 ) f (h 1 , h 2 ) α,β i,...,l

=

α,β i,...,l,r,s

i jkl ϕαβ

β

dg1 dg2 f (g1 , g2 ) πriα (g1 ) τsk (g2 )

β × dh 1 dh 2 f (h 1 , h 2 )πrαj (h 1 )τsl (h 2 ) = cr s |ϕˆ αβ cr s 0

(36)

α,β r,s

β dg1 dg2 f (g1 , g2 )πriα (g1 )τsk (g2 ) ei ⊗ e˜k . for any f ∈ C(G 1 × G 2 ), where cr s := ik Thus, by the above theorem, a positive definite function generates a family of subnormalized states ϕˆαβ . The operators ϕˆαβ are not normalized (except in the trivial case when sum (34) consists of one term only) even if ϕ is, since from Eq. (34) we obtain that α,β tr(ϕˆ αβ ) = ϕ(e1 , e2 ) = 1, so tr(ϕˆ αβ ) 1. However, we can still speak of separability of the operators ϕˆαβ in the sense that they are decomposable into convex combinations of products of positive operators (cf. the corresponding remark after Definition 2.2). The following result holds (cf. Ref. [20] where a weaker version was proven): Lemma 4.1. A function ϕ ∈ P1 (G 1 × G 2 ) is separable if and only if the operators β ) are separable for all [πα ] ∈ G 1 and [τβ ] ∈ G 2 . ϕˆαβ ∈ L(Hα ⊗ H

766


K Indeed, from Eq. (33) it follows that if Sep0 ϕ = m=1 pm εm ⊗ ηm , then K (m) (m) (m) (m) ϕˆ αβ = m=1 pm εˆ α ⊗ηˆ β , where all the operators εˆ α , ηˆ β are positive by Lemma 4.1, since εm ∈ E1 (G 1 ), ηm ∈ E1 (G 2 ) for all m. This extends to all of Sep by the continuity for all α, β of the inverse Fourier transform (33) ϕ → ϕˆαβ and the fact that positive β ) [16]. separable matrices σ with trσ 1 form a compact convex subset of L(Hα ⊗ H The latter follows from the facts that i) the set of extreme points of the latter subset can be identified with CP n α × CP m β ∪ {0} and ii) the convex hull of a compact subset of R N is compact. K αβ αβ α α β β Conversely, assume that for every α, β, ϕˆαβ = m=1 pm |x m x m | ⊗ |ym ym |, β β . We will prove that ϕ is then a uniform limit of separawhere xmα ∈ Hα , ym ∈ H ble functions and hence is itself separable. First, we pass to the regularized function ϕU := ϕ ∗ ψU , according to the procedure we described above (cf. Eq. (32) and the surrounding paragraph). As we mentioned, ϕU is positive definite and hence ϕU (e1 , e2 ) = ||ϕU ||∞ > 0 (except in the trivial case ϕ ≡ 0), which allows us to pass to the normalized function ϕU /||ϕU ||∞ . Then Eq. (34) implies that: K αβ 1 1 αβ αβ ϕU (g1 , g2 ) = cU pm xmα |πα (g1 )xmα ymβ |τβ (g2 )ymβ ||ϕU ||∞ ||ϕU ||∞ α,β m=1

=

K αβ α,β m=1

×

1 αβ c p αβ ||x α ||2 ||ymβ ||2 ||ϕU ||∞ U m m

β β xα xmα ym ym m π τ , (g ) (g ) α 1 β 2 β ||xmα || ||xmα || ||ymβ || ||ym ||

(37)

and the series converges uniformly. The functions given by scalar products belong to E1 (G 1 ) and E1 (G 2 ) respectively, since πα and τβ are irreducible. From their definition in αβ Eq. (32) and the definition of ψU (22) it also follows that the factors cU are non-negative: αβ −1 cU = dgd hκV (h)κV (h g)χαβ (g) = dgd hκV (h−1 )κV (g)χαβ (hg) 1 = dgd hκV (h)κV (g)χαβ (h−1 g) = 2 2 tr κ (38) κV αβ )† 0, V αβ ( nα m β where we used definition (33) and the fact that κV is symmetric [cf. property (i) in Eq. (20)] and real. Evaluating ϕU /||ϕU ||∞ at the neutral element we see that the sum in Eq. (37) is in fact a convex combination of pure product functions (cf. the definition of a pure function in Sect. 2), since: K αβ α,β m=1

and

1 ϕU (e1 , e2 ) αβ αβ cU pm ||xmα ||2 ||ymβ ||2 = =1 ||ϕU ||∞ ||ϕU ||∞

1 αβ c p αβ ||x α ||2 ||ymβ ||2 0. ||ϕU ||∞ U m m

(39)

Thus, ϕU /||ϕU ||∞ is separable, as a uniform limit of separable functions. Since ϕU −−−−−−→ ϕ uniformly, ϕ ∈ Sep. U →{e1 ,e2 }


767

From Lemma 4.1 it follows that the problem of describing separable functions on G 1 × G 2 generates a family of separability problems in all pairs of dimensions where G 1 and G 2 have irreducible representations. In other words, it plays a role of a “generating function” for this family. Conversely, from the form of the Fourier transformation (2) β are in one-to-one and its inverse (33) it follows that density matrices on Hα ⊗ H correspondence with those functions ϕ from P1 (G 1 × G 2 ), which belong to the (finiteβ dimensional) linear span of πiαj ⊗ τkl , where πα , τβ are fixed. Moreover, since for an β ) its Fourier transform ϕ (cf. Eq. (2)) satisfies: arbitrary density matrix ∈ L(Hα ⊗ H (ϕ )γ ν = δαγ δβν ,

(40)

Lemma 4.1 implies that (cf. Ref. [20], Theorem 1): Corollary 4.1. A state is separable if and only if ϕ ∈ Sep . Next we examine bounded linear maps : C(G 2 ) → C(G 1 ). For an arbitrary function f ∈ C(G 2 ), we consider a function fU ∈ C(G 1 ), where fU := f ∗ ψU is the regularization of f . Now, the regularizing functions ψU ∈ C(G 2 ) are the single-group functions defined in Eq. (22), but now on the group G 2 , and the sets U run through a neighborhood base of e2 ∈ G 2 . Note, however, that unlike in Sect. 3 here we are regularizing the argument of and not its value. Calculating the Fourier transform of

fU from the single-group version of the definition (33) we obtain:

fU α = n α dg1 f ∗ ψU (g1 )πα (g1 )† G1

= nα

dg1 G1

=

β

β cU

β

cU

β

f βkl

β f βkl τkl (g1 )πα (g1 )†

k,l

nα

k,l

β dg1 τkl (g1 )πα (g1 )† ,

(41)

G1

where we used the uniform convergence of Fourier series for fU and the fact that is β continuous in the uniform norm. The regularizing constants cU are defined analogously as in Eq. (32), i.e. β (42) cU := dg2 ψU (g2 )χβ (g2 ). G2

β ) → L(Hα ) through an analog of Eq. ˆ βα : L(H We will find it useful to define maps (33): jiβ

kl |, i jβ := n α dg1 π α (g1 ) τ β (g1 ) ˆ βα :=

αlk |E i j H S E (43)

ij kl αkl i,...,l

G1

kl := |e˜k e˜l |, E i j := |ei e j | are the (note the change of the order of indices), where E bases of L(Hβ ) and L(Hα ) respectively, and A|B H S = tr(A† B). We can then rewrite Eq. (41) as follows: β

ˆ βα fˆβ .

fU α = cU (44) β

768


ˆ βα fˆβ ∈ L(Hα ) are finite-dimensional and Note that in the above series all operators hence the convergence can be understood in any of the equivalent norms on L(Hα ). β ) → L(Hα ) we can Fourier transform Conversely, given an arbitrary map : L(H it and assign to it a map : C(G 2 ) → C(G 1 ) through the following formula (compare with Eq. (2) where Fourier transform of operators was defined):

f (g1 ) := tr fˆβ πα (g1 ) for every f ∈ C(G 2 ). (45) It is obvious that f ∈ C(G 1 ), because we consider only continuous representations. Moreover from the definition of fˆβ we have:

† || f ||∞ = m β sup dg2 f (g2 )tr τβ (g2 ) πα (g1 ) g1 ∈G 1

G2

m β || f ||∞ sup

g1 ∈G 1

† dg2 tr τβ (g2 ) πα (g1 ) .

(46)

G2

The last supremum is finite, as the integrand is continuous and G is compact, and independent of f , so that is bounded. The transformation (45) is an inverse of the ˆ βα given by Eq. (43), since by an easy direct calculation one finds that mapping → (compare Eq. (40)): )µ ( ν = δνα δµβ .

(47)

By analogy with Theorem 4.1, which characterizes positive definite functions in terms of their inverse Fourier transforms, one would expect a corresponding characterization of positive definite (PD) maps (cf. Definition 3.1) in terms of their inverse Fourier ˆ βα . The next lemma provides such a characterization: transforms Lemma 4.2. A bounded linear map : C(G 2 ) → C(G 1 ) is positive definite, that is β ) → L(Hα ) are positive for all ˆ βα : L(H

P(G 2 ) ⊂ P(G 1 ), if and only if the maps [πα ] ∈ G 1 , [τβ ] ∈ G 2 . To prove it, let us take an arbitrary φ ∈ P(G 2 ), which by Theorem 4.1 is equivalent to ˆβ ˆ ˆ βα are positive, then cβ φˆ β 0 for all β. We employ Eq. (44). If all maps U α φβ 0 for β all β, since cU 0 (cf. Eq. (38) where we proved it for G 1 ×G 2 ). Since positive operators β β ˆ α φˆ β converges to a positive operator form a closed cone in L(Hα ), the series β cU

and hence φU α 0 for all α. Theorem 4.1 implies then that (φ ∗ ψU ) ∈ P(G 1 ). Taking the limit U → {e2 }, φ ∗ ψU converges to φ uniformly. Thus, using continuity of and uniform closedness of P(G 1 ) ⊂ C(G 1 ), we obtain that φ ∈ P(G 1 ) for any φ ∈ P(G 2 ). 2 and take an Conversely, assume to be positive definite. Let us fix [τβ ] ∈ G arbitrary positive operator ∈ L(Hβ ). Applying Fourier transform (2) to we obtain its characteristic function: kl β φ (g2 ) := tr τβ (g2 ) = τlk (g2 ), (48) k,l


769

which by Theorem 4.1 is positive definite, since (φ )γ = δγβ 0. Hence, φ is positive definite too. Then from Eq. (44), where we can neglect the regularization and β put cU = 1 since the Fourier series of φ contains only one term, and from Theorem 4.1

ˆ βα 0 for all α. Since τβ and were arbitrary, the result we obtain that

φ α = follows. Thus, from the above lemma and Eq. (47) it follows that (compare Corollary 4.1): β ) → L(Hα ) is positive if and only if the map Corollary 4.2. A map : L(H

: C(G 2 ) → C(G 1 ), defined in Eq. (45), is positive definite. Next we present a characterization of completely positive definite maps (cf. Definition 3.1) in terms of their Fourier transforms. We first prove the following fact: Lemma 4.3. A bounded linear map : C(G 2 ) → C(G 1 ) is G 2 -positive definite, i.e. β ) → L(Hα ) are ˆ βα : L(H (id ⊗ )P(G 2 × G 2 ) ⊂ P(G 2 × G 1 ), if and only if maps 1 , [τβ ] ∈ G 2 . completely positive for all [πα ] ∈ G 2 . Then the Let ϕ ∈ P(G 2 × G 2 ), so by Theorem 4.1 ϕˆβγ 0 for all [τβ ], [τγ ] ∈ G obvious generalization of Eq. (44) to G 2 × G 2 , implies that:

δγ

βγ δγ ˆ γα )ϕˆβγ , ⊗ βα ϕˆδγ = = cU id cU (1β ⊗ (49) (id ⊗ )ϕU βα

γ

δ,γ

where now U ⊂ G 2 × G 2 runs through a neighborhood base of {e2 , e2 } ∈ G 2 × G 2 . ˆ γα are completely positive, then cβγ (1β ⊗ ˆ γα )ϕˆ βγ 0 as operators from If all maps U α ), since cβγ 0 for all β, γ (cf. Eq. (38)). From closedness of the cone β ⊗ H L(H U of positive operators in L( Hβ ⊗ Hα ), the series in Eq. (49) converges to a positive operator as well. Hence (id ⊗ )ϕU 0 for all α, β and from Theorem 4.1 it βα

follows that (id ⊗ )(ϕ ∗ ψU ) ∈ P(G 2 × G 1 ). We remove the regularization by letting U → {e2 , e2 } so that ϕ ∗ ψU → ϕ uniformly. Then from the continuity of id ⊗ and uniform closedness of P(G 2 × G 1 ) it follows that (id ⊗ )ϕ ∈ P(G 2 × G 1 ) for any ϕ ∈ P(G 2 × G 2 ). For the proof in the other direction, we proceed along the same lines as in the proof of β ⊗ H γ ), 2 and arbitrary 0 ∈ L(H the previous lemma—for arbitrary [τβ ], [τγ ] ∈ G we consider the Fourier transform of , ϕ , defined in Eq. (2). Then positive definiteness µν of id⊗ , Theorem 4.1, and Eq.(44) generalized to G 2 ×G 2 (with cU = 1 as the Fourier

ˆ γα ) 0 series (31) of ϕ contains only one term) imply that (id ⊗ )ϕ = (1β ⊗ βα

for all γ . Since [τβ ], [τγ ], and are arbitrary, G 2 -positive definiteness of implies that 2 . Thus, in particular, every ˆ γα ˆ γα is (dimτ )-positive for all possible [τ ] ∈ G every map is m γ -positive, m γ = dimHγ . But then by the Choi Theorem (cf. Ref. [18], Theorem 2) ˆ γα being completely positive. this is equivalent to As a by-product we obtain an analog of the Choi Theorem (cf. Ref. [18], Theorem 2) for PD maps: Corollary 4.3. A map : C(G 2 ) → C(G 1 ) is completely positive definite if and only if it is G 2 -positive definite.

770


The proof in one direction follows immediately from Definition 3.1. For the opposite implication, let us assume that is G 2 -PD. Let H be an arbitrary compact group and let Latin indices a, b, . . . enumerate irreps of H . From the complete positivity of Fourier ˆ βα , guaranteed by Lemma 4.3, Theorem 4.1, Eq. (38) applied to H × G 2 , transforms and closedness of the cone of positive operators, we obtain that:

bβ ˆ βα )ϕˆbβ 0, (id ⊗ )ϕU = cU (1b ⊗ (50) bα

β

for every b and α and every ϕ ∈ P(H × G 2 ). Thus from Theorem 4.1, (id ⊗ )ϕU ∈ P(H × G 1 ). Letting U → {e H , e2 } ∈ H × G 2 , so that ϕU converges to ϕ uniformly, and using continuity of id ⊗ and uniform closedness of P(H × G 1 ), we obtain that (id ⊗ )ϕ ∈ P(H × G 1 ) for every compact H and every ϕ ∈ P(H × G 2 ). From Lemma 4.3 and Corollary 4.3 we finally obtain: Lemma 4.4. A bounded linear map : C(G 2 ) → C(G 1 ) is completely positive defβ ) → L(Hα ) are completely positive for all ˆ βα : L(H inite if and only if the maps 1 , [τβ ] ∈ G 2 . [πα ] ∈ G Comparison of Theorem 3.2 with the Horodecki Theorem 1.2 shows that positive definite mappings of continuous functions on compact groups play an analogous role to that of positive mappings of density matrices in the standard theory of entanglement [14]. The harmonic-analytical formalism described above makes this observation, as well as the “generating function” analogy, formal. Namely, Lemma 4.2 implies that every PD ˆ βα , acting between algebras of operators map generates a family of positive maps on representation spaces of G 1 and G 2 . Conversely, Fourier transform (45), together with property (47) and Lemma 4.2 allows one to assign a unique PD map to every suitable (cf. definition (45)) positive map. Analogously, Lemma 4.4 shows that each CPD map gives rise to a family of completely positive maps, and by Fourier transform (45) every (suitable) completely positive map defines a CPD map. Thus, PD and CPD maps between groups play a role of “generating functions” of families of positive and completely positive maps respectively. In order to compare Theorem 3.2 with the Horodecki Theorem 1.2, we first choose G 1 and G 2 so that they possess irreps in dimensions dimHA and dimHB , so that β for some [πα ] ∈ G 1 and [τβ ] ∈ G 2 we may identify HA ∼ = Hα and HB ∼ = H [20]. Apart from that there are no further restrictions on G 1 , G 2 . Finding such a group for a given finite-dimensional HA , HB is always possible—for example we can take G 1 = G 2 = SU (2), which possesses irreps in all finite dimensions. Having made the above identification, we obtain from Theorem 3.2 that: β ) is separable if and only if for all Theorem 4.2. A density matrix ∈ L(Hα ⊗ H β ) → L(Hγ ), (1α ⊗ γ ) 0 as an 1 and all positive maps γ : L(H [πγ ] ∈ G operator on Hα ⊗ Hγ . To prove it, we first Fourier transform , passing to its characteristic function ϕ = tr(πα ⊗ τβ ) ∈ P1 (G 1 × G 2 ). From Lemma 4.1 is separable if and only if ϕ ∈ Sep. Applying Theorem 3.2 to ϕ we obtain that is separable if and only if for all positive definite maps : C(G 2 ) → C(G 1 ), (id ⊗ )ϕ ∈ P(G 1 × G 1 ). From the δγ generalization of Eq. (44) to G 1 × G 2 with all cU = 1 (no regularization of ϕ is needed


771

because the Fourier series of ϕ contains only one non-zero term; cf. Eq. (40)) we obtain that:

ˆ βγ ). = (1α ⊗ (51) (id ⊗ )ϕ αγ

Then from Theorem 4.1 and Lemma 4.2 it follows that (id ⊗ )ϕ is a positive definite β ) → ˆ βγ ) 0 for every γ , where every map ˆ βγ : L(H function if and only if (1α ⊗ L(Hγ ) is positive. Thus, when applied to finite dimension, Theorem 3.2 turns out to be weaker than Theorem 1.2, since generically one has to check positive maps operating between the β ) and the whole family of spaces L(Hγ ), [πγ ] ∈ G 1 , and not only fixed space L(H β ) and L(Hα ) as in Theorem 1.2. Note, however, that the number of between L(H spaces Hγ to check need not be infinite, since it may be possible to find discrete G 1 [20]. This is particularly easy for low-dimensional initial spaces HA , HB . 5. Examples of Positive Definite Maps for G 1 = G 2 From the point of view of classification of separable functions using Theorem 3.2 only those positive definite maps which are not completely positive definite (cf. Definition 3.1) are interesting: if is a CPD map then all the functions (id⊗ )ϕ, ϕ ∈ P1 (G 1 ×G 2 ), are positive definite, whether ϕ is separable or not. Hence we encounter a similar problem as in the finite-dimensional linear algebra [19]: classify all positive definite but not completely positive definite maps from C(G 2 ) to C(G 1 ). In this section we give some examples of PD and CPD maps for the case G 1 = G 2 ≡ G. The first example of PD but not CPD map was already encountered in Theorem 3.1— the inversion map θ : θ f (g) := f (g −1 ).

(52)

To show that θ is not CPD (positive definiteness will be proven below in a more general setting), observe that θ corresponds through the Fourier transform to the transposition map T , T := T , acting on each representation space Hα of G: f (g −1 ) = f αi j πiαj (g −1 ) = f αji πiαj (g), and hence α

i, j

θf α¯ = T fˆα = fˆαT .

α

i, j

(53)

Here index α¯ denotes the complex conjugate π α of representation πα : π α (g) := πα (g). Equation (53) establishes the connection between the PPT criterion (Theorem 1.1) and Theorem 3.1 (see Ref. [20] for more details). Now, let ∈ L(Hα ⊗ Hβ ) be any positive operator such that its partial transpose (1α ⊗ T ) is not positive. Then, by Theorem 4.1, the Fourier transform ϕ of is positive definite, but (id ⊗ θ )ϕ is not. We propose to use the same terminology as in quantum information theory and call entangled functions ϕ not detected by θ (see the remark after Theorem 3.2) bound entangled. Let us consider more general substitutions of the argument in the tested function. Let α be an arbitrary automorphism of G and β an arbitrary anti-automorphism of G, i.e.: α(gh) = α(g)α(h), β(gh) = β(h)β(g).

(54) (55)

772


We define the corresponding maps from C(G) to C(G):

α f (g) := f (α(g)), β f (g) := f (β(g)).

(56)

Both α and β are positive definite, which follows most directly from the GNS construction (cf. Theorem 2.1):

dgdh f (g) α φ (g −1 h) f (h)

= dg f (g)πφ α(g) vφ dh f (h)πφ α(h) vφ 0,

dgdh f (g) β φ (g −1 h) f (h)

†

† = dh f (h)πφ β(h) vφ dg f (g)πφ β(h) vφ 0, where φ ∈ P(G) and we used the fact that α(g −1 ) = α(g)−1 and β(g −1 ) = β(g)−1 . Moreover, maps arising from automorphisms are completely positive definite. Indeed, from Corollary 4.3 it is enough to check the extension of to C(G × G). But then we obtain that (with the boldface characters denoting elements of G × G):

dgd h f (g) id ⊗ α ϕ (g −1 h) f ( h)

dh 1 dh 2 f (h 1 , h 2 )πϕ h 1 , α (h 2 ) vϕ 0, = dg1 dg2 f (g1 , g2 )πϕ g1 , α (g2 ) vϕ (57)

for an arbitrary f ∈ C(G × G). The maps arising from anti-automorphisms are not necessarily CPD—the above calculation leading to the inequality (57) cannot be repeated. −1 However, since every anti-automorphism can be written in the form β(g) = α(g) , where α is an automorphism, every map β arising from an anti-automorphism is of the form:

β = C P D ◦ θ,

(58)

where C P D is some CPD map. Hence, the use of a general anti-homomorphism β in Theorem 3.1 gives no improvement, since the function (id ⊗ β )ϕ is positive definite if (id ⊗ θ )ϕ is. In other words, PD maps of the type (58) cannot detect bound entangled functions. This is in close analogy to what one encounters in the study of standard separability problems. Indeed, from Lemma 4.3, Corollary 4.3, and Eq. (53) PD maps of the form (58) generate positive maps of the type C P ◦ T , where C P is a completely positive map. Clearly, by Theorem 1.2, such maps cannot detect bound entangled states , for which (1 ⊗ T ) 0. We can give a more general example of a CPD map, motivated by the Kraus decomposition of a completely positive map (cf. Ref. [32] and Ref. [18], Theorem 1). For an arbitrary measure µ from M(G) we define a map:

µ f (g) := µ∗ ∗ f ∗ µ (g) = dµ(a)dµ(b) f (agb−1 ), (59) G G


773

where the adjoint µ∗ is defined as µ∗ () := µ(−1 ) for any Borel set ⊂ G (cf. the corresponding definition for functions after Eq. (13)) and the convolution is defined through Eq. (15). Obviously, µ maps C(G) to C(G) and is a generalization of the regularization formula (23). The map µ is bounded on C(G), since −1 dµ(a)dµ(b) f (agb ) = |µ|2 (G). sup sup g∈G || f ||∞ =1

It is also completely positive definite, which can be easily proven using the GNS Theorem 2.1): dµ(a)dµ(b) vϕ πϕ (g1 , ag2 b−1 )vϕ (id ⊗ µ )ϕ(g1 , g2 ) = = dµ(a)dµ(b) πϕ (e, a)† vϕ πϕ (g1 , g2 )πϕ (e, b)† vϕ = πϕ (e, µ)† vϕ πϕ (g1 , g2 )πϕ (e, µ)† vϕ , (60) where πϕ (e, µ) := dµ(g)πϕ (e, g). Thus, (id ⊗ µ )ϕ is positive definite. Map (59) can be further generalized:

M f (g) := dM(µ) µ∗ ∗ f ∗ µ (g), (61) M(G)

where M is a positive measure on M(G) with a finite total variation, i.e. M ∈ M M(G) . We conjecture that any CPD map from C(G) to C(G) is of this form for some measure M, so that Eq. (61) is an analog of the Kraus decomposition of a completely positive map. 6. Conclusions The main conclusions are twofold. On one hand, this paper is directed to the audience of mathematicians working in the area of harmonic analysis. We have formulated here the separability problem in purely abstract terms of positive definite functions on compact groups. To our knowledge this is a new problem in harmonic analysis. One may hope that some well established methods of harmonic analysis will help to get more insight into the problem, solving it, at least partially. Several generalizations call for immediate attention: applications to quantum groups being perhaps one of the most fascinating ones. We hope that studies of entanglement within the harmonic analysis framework will open new avenues here. One the other hand, we expect that the harmonic analysis methods will help to study the physics of separability and entanglement. We expect to find new entanglement criteria, and get a better understanding of the whole problem, by looking at concrete examples and applications of our theoretical results. In particular, finite groups (which have finitely many irreducible representations) of various types (nilpotent, solvable) seem a rich source of interesting examples. As seen above, passing from linear-algebraic entanglement criteria for quantum states to the theory of positive-definite functions on compact groups is somewhat involved and requires technical work. We expect, on the other hand, that, going in the opposite direction, the above general results, when specialized to concrete groups (e.g. finite groups or SU (2)) will directly yield physically interesting results.

774


Acknowledgements. We would like to thank M. Bo˙zejko, P. Horodecki, M. Marciniak, P. Sołtan, and S. L. Woronowicz for discussions. We gratefully acknowledge the financial support of EU IP Programme “SCALA”, ESF PESC Programme “QUDEDIS”, Spanish MEC grants (FIS 2005-04627, Conslider Ingenio 2010 “QOIT”), and Trup Cualitat Generalitat de Catalunya. JW was partially supported by the NSF grant DMS 9706915.

References 1. Bell, J.S.: Speakable and Unspeakable in Quantum Mechamics. Cambridge: Cambridge University Press, 2004 2. Norsen, T.: Found. Phys. Lett. 19, 633 (2006); Found. Phys. 37, 311 (2007) 3. Peres, A.: Quantum Theory: Concepts and Mehtods. Dordrecht: Kluwer Academic Publishers, 1993 4. Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Rev. Mod. Phys., in press, available at http://arxiv.org/list/quant-ph/0702225v2, 2007 5. Strømer, E.: Acta Math. 110, 233 (1963) 6. Terhal, B.M.: Lin. Alg. Appl. 323, 61 (2001); Lewenstein, M., Kraus, B., Horodecki, P., Cirac, J.I.: Phys. Rev. A 63, 044304 (2001); Breuer, H.-P.: Phys. Rev. Lett. 97, 080501 (2006) 7. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge: Cambridge University Press, 2000 8. Einstein, A., Podolsky, B., Rosen, N.: Phys. Rev. 47, 777 (1935) 9. Werner, R.F.: Phys. Rev. A 40, 4277 (1989) 10. Doherty, A.C., Parrilo, P.A., Spedalieri, F.M.: Phys. Rev. Lett. 88, 187904 (2002) 11. Woerdmann, H.J.: Phys. Rev. A 67, 010303 (2003) 12. Gurvits, L.: J. Comp. Sys. Sci. 69, 448 (2004) 13. Peres, A.: Phys. Rev. Lett. 77, 1413 (1996) 14. Horodecki, M., Horodecki, P., Horodecki, R.: Phys. Lett. A 223, 1 (1996) 15. Choi, M.-D.: Linear Algebra Appl. 12, 95 (1975) 16. Horodecki, P.: Phys. Lett. A 232, 333 (1997) 17. Giedke, G., Kraus, B., Lewenstein, M., Cirac, J.I.: Phys. Rev. Lett. 87, 167904 (2001) 18. Choi, M.-D.: Linear Algebra Appl. 10, 285 (1975) 19. Woronowicz, S.L.: Rep. Math. Phys. 10, 165 (1976); Commun. Math. Phys. 51, 243 (1976); Kruszyński, P., Woronowicz, S.L.: Lett. Math. Phys. 3, 317 (1979) 20. Korbicz, J.K., Lewenstein, M.: Phys. Rev. A 74, 022318 (2006) 21. Korbicz, J.K., Lewenstein, M.: Found. Phys. 37, 879 (2007) 22. Holevo, A.S.: Probabilistic and statistical aspects of quantum theory. Amsterdam, North Holland, 1982 23. Perelomov, A.: Generalized Coherent States and Their Applications. Berlin, Springer, 1986 24. Barnum, H., Knill, E., Ortiz, G., Viola, L.: Phys. Rev. A 68, 032308 (2003); Barnum, H., Knill, E., Ortiz, G., Somma, R., Viola, L.: Phys. Rev. Lett. 92, 107902 (2004); Braunstein, S.L., Caves, C.M., Jozsa, R., ˙ Linden, N., Popescu, S., Schack, R.: Phys. Rev. Lett 83, 1054 (1999); Mintert, F., Zyczkowski, K.: Phys. Rev. A 69, 022317 (2004) 25. Folland, G.: A Course in Abstract Harmonic Analysis. Boca Raton: CRC Press, 1995 26. Dixmier, J.: C ∗ -Algebras. Amsterdam: North Holland, 1977 27. Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis. Vol. II, Berlin: Springer-Verlag, 1963 28. Folland, G.: Real Analysis. Modern Techniques and Their Applications. New York: Wiley, 1999 29. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Vol. I, San Diego: Academic Press, 1980 30. Jamiołkowski, A.: Rep. Math. Phys. 3, 275 (1972) 31. Godement, R.: Trans. Amer. Math. Soc. 63, 1 (1948) 32. Kraus, K.: States, Effects, and Operators: Fundamental Notions of Quantum Theory. Berlin: Springer, 1983 Communicated by M.B. Ruskai


Communications in


Strange Attractors in Periodically-Kicked Degenerate Hopf Bifurcations William Ott Courant Institute of Mathematical Sciences, New York, New York 10012, USA. E-mail: [email protected] Received: 18 June 2007 / Accepted: 18 November 2007 Published online: 15 May 2008 – © Springer-Verlag 2008

Abstract: We prove that spiral sinks (stable foci of vector fields) can be transformed into strange attractors exhibiting sustained, observable chaos if subjected to periodic pulsatile forcing. We show that this phenomenon occurs in the context of periodicallykicked degenerate supercritical Hopf bifurcations. The results and their proofs make use of a k-parameter version of the theory of rank one maps. 1. Introduction This paper aims to support the idea that shear and twist are natural mechanisms for the production of sustained, observable chaos in forced dynamical systems. Consider a weakly stable dynamical structure such as an equilibrium point or a limit cycle. If shear or twist is present, then forcing of various types can transform the weakly stable structure into a strange attractor. The nature of the forcing is not essential. Admissible types of forcing include periodic pulsatile drives, deterministic continuous-time signals, and random signals generated by stochastic processes. The strange attractors possess many of the dynamical, statistical, and geometrical properties commonly associated with chaotic dynamics. We study the simplest weakly stable dynamical structure. This is the spiral sink (or stable focus), an equilibrium point of a vector field with the property that the linearization of the field at the equilibrium point has a pair of complex conjugate eigenvalues α ± iβ satisfying α < 0 and β = 0. We consider the degenerate supercritical Hopf bifurcation in two dimensions. When a generic supercritical Hopf bifurcation occurs, the spiral sink becomes unstable and a limit cycle is born. In the degenerate case, the spiral sink loses its stability but no limit cycle is born. We prove that in the case of the degenerate supercritical Hopf bifurcation, periodic pulsatile drives transform the spiral sink into a strange attractor. The analysis is certainly not limited to Hopf bifurcations. We work in this context because the origin of the shear is transparent in the defining differential equations.

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The analysis is based on the beautiful dynamical theory of rank one maps formulated by Wang and Young (8,7). Speaking impressionistically, rank one maps are strongly dissipative maps exhibiting a single direction of instability. Rank one theory provides checkable conditions that imply the existence of strange attractors for a positive-measure set of parameters within a given parametrized family of rank one maps. The conditions appear within the following scheme. (1) Let dissipation go to infinity. This procedure produces the singular limit, a parametrized family of one-dimensional maps. (2) Check that the singular limit includes a map with strong expanding properties (a map of Misiurewicz type). (3) Verify a parameter transversality condition. (4) Verify a nondegeneracy condition. This allows information about the singular limit to be passed to the maps with finite dissipation. Steps (3) and (4) cumulatively require verifying that only finitely many quantities do not vanish. For good parameters, parameters corresponding to maps admitting strange attractors, rank one theory provides a reasonably complete dynamical description of the map. The attractor supports a positive, finite number of ergodic SRB measures. The orbit of Lebesgue almost-every point in the basin of attraction has a positive Lyapunov exponent and is asymptotically distributed according to one of the ergodic SRB measures. Each SRB measure satisfies the central limit theorem and exhibits exponential decay of correlations. A symbolic coding exists for orbits on the attractor. This symbolic coding implies the existence of equilibrium states and a measure of maximal entropy. Summarizing, the map has a nonuniformly hyperbolic character and exhibits sustained, observable chaos. Of the four steps in the rank one scheme, step (2) is the most fundamental and typically requires the most work. A Misiurewicz map has the property that the positive orbit of every critical point remains bounded away from the critical set. Existing papers on rank one theory view the singular limit as a one-parameter family { f a } of one-dimensional maps. This view makes locating Misiurewicz parameters difficult if the maps have multiple critical points. If each map f a has exactly one critical point c(a), then locating Misiurevicz parameters is relatively easy. Assuming that f a (c(a)) moves reasonably quickly as one varies a, simply locate an invariant set (a) that is disjoint from the critical set and then choose a ∗ such that f a ∗ (c(a ∗ )) ∈ (a ∗ ). The set (a) could be a periodic orbit or a Cantor set. If the singular limit consists of maps with multiple critical points, then one must locate a parameter a ∗ for which all of the critical orbits of f a ∗ are contained in good invariant sets. This is a serious challenge because good invariant sets such as periodic orbits and Cantor sets typically have Lebesgue measure zero. Wang and Young (9) overcome this challenge. However, their results assume that the maps in the singular limit possess an extremely large amount of expansion. We prove that significantly less expansion is needed if the singular limit is viewed as an m-parameter family for m sufficiently large. Assume that the singular limit consists of maps with k critical points. We prove that if the singular limit is viewed as a k-parameter family, then it contains Misiurewicz points assuming the maps are mildly expanding and assuming the parameters are independent in a sense to be made precise. This result widens the scope of rank one theory. We view this work as an element of a growing list of applications of rank one theory. The theory has been rigorously applied to simple mechanical systems (9), periodicallykicked limit cycles and Hopf bifurcations (10), and the Chua circuit (6). Guckenheimer,

Strange Attractors in Periodically-Kicked Degenerate Hopf Bifurcations

777

Wechselberger, and Young (1) connect rank one theory and geometric singular perturbation theory by formulating a general technique for proving the existence of chaotic attractors for three-dimensional vector fields with two time scales. Lin (2) demonstrates how rank one theory can be combined with sophisticated computational techniques to analyze the response of concrete nonlinear oscillators of interest in biological applications to periodic pulsatile drives. Lin and Young (3) study shear-induced chaos numerically in situations beyond the reach of current analytical tools. In particular, they consider stochastic forcing. This work supports the belief that shear-induced chaos is both widespread and robust. We organize the presentation of ideas as follows. In Sect. 2, we present the main results for periodically-kicked degenerate Hopf bifurcations. In Sect. 3, we prove the result concerning the existence of Misiurewicz points in k-parameter families of onedimensional maps and we present a two-parameter example. Section 4 presents rank one theory viewing the singular limit as a k-parameter family. Finally, in Sect. 5 we prove the results presented in Sect. 2.

2. Periodically-Kicked Degenerate Hopf Bifurcations The normal form for the supercritical Hopf bifurcation in two spatial dimensions is given in polar coordinates by ⎧ ⎨ r˙ = (µ − αµr 2 )r + r 5 gµ (r, θ ) ⎩˙ θ = ω + γµ µ + βµr 2 + r 4 h µ (r, θ ). Here µ is the bifurcation parameter and ω is a constant. The multipliers αµ , γµ , and βµ depend smoothly on µ. The functions gµ and h µ depend smoothly on µ and they are of class C 4 with respect to r and θ . The normal form for the degenerate Hopf bifurcation in two spatial dimensions is obtained by setting αµ = 0 for all µ and replacing µ with −µ, yielding ⎧ ⎨ r˙ = −µr + r 5 gµ (r, θ ) (2.1) ⎩˙ θ = ω + γµ µ + βµr 2 + r 4 h µ (r, θ ). For µ > 0, the origin is an asymptotically stable equilibrium point (a sink). We study (2.1) t denote the flow generated by (2.1). We perturb the flow F t with in this µ-range. Let F a ‘kick’ map κ defined as follows. Let L > 0 and let ρ2 > 0. The map κ = κµ,L ,ρ2 is given in rectangular coordinates by κ

r cos(θ ) r sin(θ )

=

r cos(θ )

r sin(θ ) + Lµρ2

.

t ◦ κ may be thought of as a perturbation followed by a period of The composition F t ◦ κ. Let A denote the annulus relaxation. We define an annulus map associated with F defined by A = {(r, θ ) : K 4−1 µρ1 r K 4 µρ1 },

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where K 4 > 1 and 0 < ρ2 < ρ1 . Let r˜ denote the distance from κ(A) to the origin. We have r˜ = Lµρ2 − K 4 µρ1 . Define the relaxation time τ (µ) by r˜ e−µτ (µ) = µρ1 . τ (µ) ◦ κ maps A into A. For µ sufficiently large, F τ (µ) ◦κ The following theorem states that under certain conditions, the annulus map F admits a strange attractor for a positive-measure set of values of µ. We make the crucial assumption that the twist factor β0 is nonzero. A nonzero twist factor implies the existence of an angular-velocity gradient in the radial direction for values of µ in a neighborhood of the bifurcation parameter µ = 0. This angular-velocity gradient allows the flow to stretch and fold the phase space, thereby producing chaos. The chaos in this setting is sustained in time and observable. The strange attractors possess many of the geometric and dynamical properties normally associated with chaotic systems. These properties include the existence of a positive Lyapunov exponent (SA1), the existence of SRB measures and basin property (SA2), and statistical properties such as exponential decay of correlations and the central limit theorem for dynamical observations (SA3). In addition, if |βL0 | is sufficiently large, then the annulus map admits a unique SRB measure (SA4). Properties (SA1)–(SA4) are described in detail in Sect. 4. Theorem 2.1. Assume β0 = 0. Let ρ1 and ρ2 satisfy ρ2 ∈ ( 38 , 21 ) and ρ1 + ρ2 = 1. M0 , then there exist L ∗ ∈ [L , L + |βπ0 | ] (1) There exists M0 > 0 such that if L |β 0| and µ0 > 0 satisfying the following. The parameter interval (0, µ0 ] contains a set τ (µ) ◦ κ admits = (L ∗ ) of positive measure such that for µ ∈ , the map F a strange attractor with properties (SA1), (SA2), and (SA3). The set intersects every interval of the form (0, µ] ˜ in a set of positive measure. M1 (2) There exists M1 M0 such that for all L |β , there exists a set = (L) with 0| the properties described in (1). (3) If L is sufficiently large and µ ∈ (L), then (SA4) holds as well.

3. Locating Misiurewicz Points Let I denote an interval or the circle S 1 . Let F : I × [a1 , a2 ] × [b1 , b2 ] → I be a C 2 map. The map F defines a two-parameter family F = { f a,b : a ∈ [a1 , a2 ], b ∈ [b1 , b2 ]} via f a,b (x) = F(x, a, b). Set A = [a1 , a2 ] and B = [b1 , b2 ]. We assume that for each (a, b) ∈ A × B, f a,b has two critical points. We label these critical points c(1) (a, b) and c(2) (a, b). Let C = C(a, b) = {c(1) (a, b), c(2) (a, b)}. For δ > 0, let Cδ denote the δ-neighborhood of C in I . We seek to identify conditions under which F contains strongly expanding (Misiurewicz) maps. We now introduce this class. Definition 3.1. We say that f ∈ C 2 (I, I ) is a Misiurewicz map and we write f ∈ M if the following hold for some neighborhood V of C. (A) (Outside of V ) There exist λ0 > 0, M0 ∈ Z+ , and 0 < d0 1 such that (1) for all n M0 , if f k (x) ∈ / V for 0 k n − 1, then |( f n ) (x)| eλ0 n , + k (2) for any n ∈ Z , if f (x) ∈ / V for 0 k n − 1 and f n (x) ∈ V , then n λ n 0 |( f ) (x)| d0 e .


779

(B) (Critical orbits) For all c ∈ C and n > 0, f n (c) ∈ / V. (C) (Inside V ) (1) We have f (x) = 0 for all x ∈ V , and (2) for all x ∈ V \C, there exists p0 (x) > 0 such that f j (x) ∈ / V for all j < p0 (x) 1

and |( f p0 (x) ) (x)| d0−1 e 3 λ0 p0 (x) . We first formulate hypotheses that imply the existence of maps in F that satisfy Definition 3.1(B). 3.1. The general result. We formulate the result for two-parameter families consisting of maps with two critical points. The result generalizes in a natural way for k-parameter families consisting of maps with k critical points. The first hypothesis is formulated in terms of the evolutions n (a, b) → γn(i) (a, b) where γn(i) (a, b) = f a,b (c(i) (a, b)). (i)

The evolutions {γn : n ∈ N} generate critical curve dynamics. Define n : A × B → (1) (2) I × I by n = (γn , γn ). We now present the general hypotheses. For J ⊂ I and ε > 0, let J ε denote the ε-neighborhood of J . Suppose there exist subintervals I1 and I2 of I , subintervals 1 ⊂ A and 2 ⊂ B, δ1 > 0, and ε1 > 0 such that the following hold. (H1) (Finite Misiurevicz condition) There exists n 0 ∈ Z+ such that n 0 (1 × 2 ) ⊃ I1 × I2 and for i ∈ {1, 2}, (a, b) ∈ 1 × 2 , and n < n 0 , we have (i) γn (a, b) ∈ I \Cδ1 (a, b). (H2) There exist fixed parameters aˆ ∈ 1 and bˆ ∈ 2 satisfying f ˆ (I1 ) × a, ˆ b

ε1 ε1 f a, ˆ bˆ (I2 ) ⊃ I1 × I2 . (H3) For all (a, b) ∈ 1 × 2 , we have I1 × I2 ⊂ I \Cδ1 (a, b) × I \Cδ1 (a, b).

Proposition 3.2. Suppose F satisfies (H1)–(H3). If √ 2 2 max{ ∂a F C 0 , ∂b F C 0 } · max{|1 |, |2 |} < ε1 ,

(3.1)

(a ∗ , b∗ )

then there exists ∈ 1 × 2 such that for i ∈ {1, 2} and for every n ∈ N, (i) γn (a ∗ , b∗ ) ∈ I \ Cδ1 (a ∗ , b∗ ). Proof of Proposition 3.2. Define G = ( f a, ˆ bˆ , f a, ˆ bˆ ). Applying (H1) and (H2), we have ε1 ε1 G(n 0 (1 × 2 )) ⊃ I1 × I2 . For every (a, b) ∈ 1 × 2 , we have n 0 +1 (a, b) − G(n 0 (a, b)) < ε1 , since ε1 satisfies (3.1). Therefore, n 0 +1 (1 ×2 ) ⊃ I1 ×I2 . Inductively, n (1 ×2 ) ⊃ I1 × I2 for all n n 0 . Define n 0 = (1 × 2 ) ∩ n−1 (I1 × I2 ). 0 For n > n 0 , define

−1 n+1 = n ∩ n+1 (I1 × I2 ).

Let = and choose (a ∗ , b∗ ) ∈ .

∞ k=n 0

k

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3.2. Verifying (H1). We present a two-step procedure for the verification of hypothesis (H1). First, we assume that 1 is a diffeomorphism on 1 × 2 . This implies that the image of 1 ×2 contains a rectangle in I × I . Second, if we assume that each map f a,b is expanding on I \Cδ1 , then the evolutions γ (1) and γ (2) will enlarge the rectangle to macroscopic size. The required time for this enlargement depends upon the magnitude of the expansion. Therefore, greater expansion results in a smaller value of n 0 . We now make these ideas precise. Suppose that 1 is a diffeomorphism on 1 × 2 such that for i ∈ {1, 2} and for (i) every (a, b) ∈ 1 × 2 , we have γ1 (a, b) ∈ I \Cδ1 . Define (1) ∂ γ (a, b) J (a, b) = a 1(2) ∂a γ1 (a, b)

∂b γ1(1) (a, b) . (2) ∂b γ1 (a, b)

Assume that there exists k0 > 0 such that |J | k0 on 1 × 2 . This implies that 1 (1 × 2 ) contains a box with side length bounded below by k02 min{|1 |, |2 |}, λM where λM =

sup

(a,b)∈1 ×2

sup{|λ| : λ is an eigenvalue of D1∗ D1 }.

| K > 1 on I\C . Lower Now suppose that for every (a, b) ∈ 1 × 2 we have | f a,b δ1 bounds on K will be given as the discussion proceeds. (1) (2) We choose K based on the magnitudes of the partial derivatives of γ1 and γ1 . Assume there exists ρ > 0 such that on 1 × 2 we have

(1) |∂a γ1(1) | ρ or |∂b γ1(1) | ρ, and (2) (2) (2) |∂a γ1 | ρ or |∂b γ1 | ρ. Suppose for the sake of definiteness that (1) and (2) hold with respect to the operator ∂a . We now relate spatial and parametric derivatives. The equation ∂ (i) ∂ (i) ∂ F (i) γn+1 (a, b) = f a,b γn (a, b) + (γ (a, b), a, b) (γn(i) (a, b)) · ∂a ∂a ∂a n

(3.2)

implies that parametric derivatives grow exponentially provided that spatial derivatives grow exponentially. If K satisfies 3 K , and 4 ∞ 1 ∂a F C 0 K−j , 4

Kρ − ∂a F C 0

j=2

then (3.2) implies that |∂a γn(i) (a, b)|

1 n K 2

(3.3)


781

(i)

provided γ j (a, b) ∈ I \Cδ1 for j < n. Hypothesis (H1) may be verified as follows. (i)

Look for a time n 0 such that for i ∈ {1, 2}, γn 0 (1 × {b}) ⊃ Ii for every b ∈ 2 and γk(i) (a, b) ∈ I \Cδ1 for all k < n 0 and (a, b) ∈ 1 × 2 . By (3.3), we have K n0 ≈

2λ M max{|I1 |, |I2 |} . k02 min{|1 |, |2 |}

3.3. A two-parameter example. Let S 1 = R/2π Z. Let : S 1 → R be a C 3 function with two nondegenerate critical points c(1) and c(2) . We assume that (c(1) ) = (c(2) ). Fix ζ ∈ S 1 . Consider the two-parameter family of circle maps F = { f a,L : a ∈ S 1 , L ∈ R+ } defined by f a,L (θ ) = ζ + L(θ ) + a. Small perturbations of this family frequently arise as singular limits of rank one families. Definition 3.3. We say that (a, L) is a Misiurewicz pair if f a,L ∈ M. The goal of this subsection is to prove the following result. Theorem 3.4. There exists L 0 > 0 such that if L L 0 , then there exists a Misiurewicz pair (a ∗ , L ∗ ) with L ∗ ∈ [L , L + 2π/|(c(2) ) − (c(1) )|] and a ∗ ∈ [0, 2π ). Remark 3.5. Misiurewicz points occur with greater frequency as L increases. Wang and Young (9) prove that there exists L 1 L 0 such that if L L 1 , then f a,L ∈ M for a O(1/L)-dense subset of parameters a ∈ [0, 2π ). Proof of Theorem 3.4. We prove Theorem 3.4 in two steps. We first show that for L sufficiently large, if f a,L satisfies Definition 3.1(B), then f a,L ∈ M. We then prove the existence of parameters for which f a,L satisfies Definition 3.1(B). Set f = f a,L for the sake of simplicity. Let k1 = 21 min{ (c(1) ), (c(2) )}. There exists δ2 = δ2 () such that |c(2) −c(1) | > 2δ2 and | | > k1 on Cδ2 . Notice that | f | > k1 L on Cδ2 . At this point we introduce the auxiliary constant K . This constant will be used to bound the derivative of f from below away from the critical set. Lower bounds on K will be given as the proof develops. We choose K before we choose L. Let σ = 2k1−1 L −1 K 3 and assume σ2 < δ2 . For x ∈ Cδ2 \C 1 σ we have | f (x)| K 3 . Choose L sufficiently large so that | f | K 3 2 outside C 1 σ . Summarizing, the map f has the following properties. 2

(P1) | f | > k1 L on Cδ2 , (P2) | f | K 3 outside C 1 σ . 2

The following recovery lemma asserts that if an orbit visits a small neighborhood of a critical point, then the derivative along this orbit regains a definite amount of exponential growth as this orbit tracks the orbit of the critical point for a period of time. Set K 2 = C 2 . Let V = {x ∈ S 1 : | f (x)| K } and note that V ⊂ C 1 σ . Together 2 with (P1) and (P2), Lemma 3.6 implies that if f satisfies Definition 3.1(B), then f ∈ M.

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Lemma 3.6 (Recovery estimate). Let c ∈ C be such that f n (c) ∈ / Cσ for all n ∈ N. For x ∈ V , let n(x) be the smallest value of n such that | f n (x) − f n (c)| > 4K1 2 K 3 L −1 . We have n(x) > 1 and |( f n(x) ) (x)| k3 K n(x) for some k3 = k3 (k1 , K 2 ). The proof of Lemma 3.6 uses the following distortion estimate. Sublemma 3.7 (Local distortion estimate). Let x, y ∈ S 1 . For i ∈ Z+ , let ωi denote the segment between f i (x) and f i (y). If n ∈ Z+ is such that |ωi | 4K1 2 K 3 L −1 and d(ωi , C) 21 σ for all 0 i < n, then

( f n ) (x) ( f n ) (y)

2.

Proof of Sublemma 3.7. We have

n

i n−1 ( f ) (x) f ( f (x)) log = log ( f n ) (y) f ( f i (y)) i=0

n−1 | f ( f i (x)) − f ( f i (y))| | f ( f i (y))| i=0

n−1 L K 2 | f i (x) − f i (y)| K3 i=0 n−1 L K2 1 | f n−1 (x) − f n−1 (y)| < log(2) K3 K 3i i=0

provided K is sufficiently large. Proof of Lemma 3.6. We first show that n(x) > 1. Since x ∈ V , we have K | f (x)| = | f (γ1 )| · |x − c| and therefore | f (x) − f (c)| =

| f (γ2 )| 2 C 0 2 1 | f (γ2 )| · |x − c|2 K . K 2 2 2| f (γ1 )| 2k1 L

We may assume the final quantity is less than ma 3.7 implies

1 4K 2

K 3 L −1 . If n(x) = 2, then Sublem-

1 K 3 L −1 < | f 2 (x) − f 2 (c)| = |( f 2 ) (γ3 )| · |x − c| 2|( f 2 ) (x)| · |x − c|. 4K 2 This inequality coupled with the estimate |x − c| |( f 2 ) (x)| >

K L C 0

implies

C 0 2 K . 8K 2

Now assume n = n(x) 3. Applying Sublemma 3.7 to estimate | f n−1 (x) − f n−1 (c)| and | f n (x) − f n (c)|, we have 1 1 1 | f (γ2 )| · |x − c|2 · |( f n−2 ) ( f (c))| K 3 L −1 , 2 2 4K 2 1 1 K 3 L −1 . | f (γ2 )| · |x − c|2 · 2|( f n−1 ) ( f (c))| > 2 4K 2

(3.4) (3.5)


783

The recovery estimate follows from the lower bound |( f n ) (x)|

1 | f (γ1 )| · |x − c| · |( f n−1 ) ( f (c))|. 2

Replacing |( f n−1 ) ( f (c))| with the lower bound provided by (3.5) and then replacing |x − c|−1 with the lower bound provided by (3.4) yields |( f n ) (x)|

3 3 k1 k1 n K 2 n− 2 K . 8K 2 8K 2

We have shown that for L sufficiently large, if f satisfies Definition 3.1(B), then f ∈ M with V = {x ∈ S 1 : | f (x)| K }. We now find a ∗ ∈ [0, 2π ) and L ∗ ∈ [L , L + 2π/|(c(2) ) − (c(1) )|) such that f a ∗ ,L ∗ satisfies Definition 3.1(B) by applying Proposition 3.2. Additional lower bounds on K will be given as the need arises. Hypotheses (H1)–(H3) are verified as follows. Let z ∈ S 1 be such that d(z, C) d(y, C) for all y ∈ S 1 . We have d(z, C 1 σ ) π2 − 21 σ . There exists a˜ ∈ [0, 2π ) and 2

(1) (2) ˜ L) = γ1 (a, ˜ L) = z. This is so L ∈ [L , L + 2π/|(c(2) ) − (c(1) )|) such that γ1 (a, because (2) (1) |∂ L γ1 (a, L) − ∂ L γ1 (a, L)| = |(c(2) ) − (c(1) )|.

Referring to the setting of Subsects. 3.1 and 3.2, we have J (a, L) = (c(2) ) − (c(1) ) and we therefore set k0 = |(c(2) ) − (c(1) )|. Let 1 be a parameter interval in a-space of length λkM2 K −3 centered at a˜ and let 2 be a parameter interval in L-space 0

of the same length centered at L. We assume K is sufficiently large so that 2 ⊂ [L , L + 2π/|(c(2) ) − (c(1) )|). The image 1 (1 × 2 ) contains a box such that the length of each of the sides is equal to K −3 . Let I1 and I2 be the vertical and horizontal projections of this box onto I , respectively. Since |γ1(i) (a, L) − z| max{1, |(c(1) )|, |(c(2) )|} ·

1 λ M −3 π K < − σ 2 2 2k02

for i ∈ {1, 2} and for all (a, L) ∈ 1 × 2 provided K is sufficiently large, we have I1 ⊂ S 1 \C 1 σ and I2 ⊂ S 1 \C 1 σ . 2 2 By construction, (H1) is satisfied with n 0 = 1 and the intervals I1 and I2 satisfy (H3). | K 3 on S 1\C Setting ε1 = 1, (H2) is satisfied because | f a,L 1 for all (a, L) ∈ 1 ×2 . 2σ If K is large enough so that (3.1) holds, then the application of Proposition 3.2 with δ1 = 21 σ produces a Misiurewicz pair (a ∗ , L ∗ ) ∈ 1 × 2 . 4. Theory of Rank One Attractors Let D denote the closed unit disk in Rn−1 and let M = S 1 × D. We consider a family of maps Ta,b : M → M, where a = (a1 , . . . , ak ) ⊂ is a vector of parameters and b ∈ B0 is a scalar parameter. Here = 1 × · · · × k ⊂ Rk is a product of intervals and B0 ⊂ R\{0} is a subset of R with an accumulation point at 0. Points in M are denoted by (x, y) with x ∈ S 1 and y ∈ D. Rank one theory postulates the following.

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(G1) Regularity conditions. (a) For each b ∈ B0 , the function (x, y, a) → Ta,b (x, y) is C 3 . (b) Each map Ta,b is an embedding of M into itself. (c) There exists K D > 0 independent of a and b such that for all a ∈ , b ∈ B0 , and z, z ∈ M, we have | det DTa,b (z)| K D. | det DTa,b (z )| (G2) Existence of a singular limit. For a ∈ , there exists a map Ta,0 : M → S 1 ×{0} such that the following holds. We select a special index j ∈ {1, . . . , k}. For every fixed set {ai ∈ i : i = j}, the maps (x, y, a j ) → Ta,b (x, y) converge in the C 3 topology to (x, y, a j ) → Ta,0 (x, y). Identifying S 1 × {0} with S 1 , we refer to Ta,0 and the restriction f a : S 1 → S 1 defined by f a (x) = Ta,0 (x, 0) as the singular limit of Ta,b . (G3) Existence of a sufficiently expanding map within the singular limit. There exists a∗ = (a1∗ , . . . , ak∗ ) ∈ such that f a∗ ∈ M. (G4) Parameter transversality. Let Ca∗ denote the critical set of f a∗ . Define a˜ j = (a1∗ , . . . , a ∗j−1 , a j , a ∗j+1 , . . . , ak∗ ). We say that the family { f a } satisfies the parameter transversality condition with respect to parameter a j if the following holds. For each x ∈ Ca∗ , let p = f (x) and let x(ã j ) and p(ã j ) denote the continuations of x and p, respectively, as the parameter a j varies around a ∗j . The point p(ã j ) is the unique point such that p(ã j ) and p have identical itineraries under f a˜ j and f a∗ , respectively. We have d d f a˜ j (x(ã j )) = p(ã j ) . da j da j a j =a ∗ a j =a ∗ j

j

(G5) Nondegeneracy at ‘turns’. For each x ∈ Ca∗ , there exists 1 n − 1 such that ∂ Ta∗ ,0 (x, 0) = 0. ∂ y (G6) Conditions for mixing. 1

(a) We have e 3 λ0 > 2, where λ0 is defined within Definition 3.1. (b) Let J1 , . . . , Jr be the intervals of monotonicity of f a∗ . Let Q = (qi j ) be the matrix defined by

1, if f a∗ (Ji ) ⊃ J j , qi j = 0, otherwise. There exists N > 0 such that Q N > 0. The following lemma often facilitates the verification of (G4). Lemma 4.1 ((5,4)). Let f = f a∗ . Suppose that for all x ∈ Ca∗ , we have ∞

1

k=0

|( f k ) ( f (x))|

< ∞.


785

Then for each x ∈ Ca∗ , ∞ [(∂ f )( f k (x))] a j a˜ j a j =a ∗j k=0

( f k ) ( f (x))

=

d d f a˜ (x(ã j )) − p(ã j ) da j j da j

a j =a ∗j

.

(4.1)

Rank one theory states that given a family {Ta,b } satisfying (G1)–(G5), a measuretheoretically significant subset of this family consists of maps admitting attractors with strong chaotic and stochastic properties. We formulate the precise results and we then describe the properties that the attractors possess. Theorem 4.2 ((8,7)). Suppose the family {Ta,b } satisfies (G1)–(G3) and (G5). For all 1 j k such that the parameter a j satisfies (G4) and for all sufficiently small b ∈ B0 , there exists a subset A j ⊂ j of positive Lebesgue measure such that for a j ∈ A j , Ta˜ j ,b admits a strange attractor with properties (SA1), (SA2), and (SA3). Theorem 4.3 ((8,9,7)). In the sense of Theorem 4.2, (G1)–(G6) =⇒ (SA1)–(SA4). (SA1) Positive Lyapunov exponent. Let U denote the basin of attraction of the attractor . For almost every (x, y) ∈ U with respect to Lebesgue measure, the orbit of (x, y) has a positive Lyapunov exponent. That is, lim

n→∞

1 log DT n (x, y) > 0. n

(SA2) Existence of SRB measures and basin property. (a) The map T admits at least one and at most finitely many ergodic SRB measures all of which have no zero Lyapunov exponents. Let ν1 , · · · , νr denote these measures. (b) For Lebesgue-a.e. (x, y) ∈ U , there exists j (x) ∈ {1, . . . , r } such that for every continuous function ϕ : U → R, n−1 1 ϕ(T i (x, y)) → ϕ dν j (x) . n i=0

(SA3) Statistical properties of dynamical observations. (a) For every ergodic SRB measure ν and every Hölder continuous function i + ϕ : → R, the sequence {ϕ ◦ T : i ∈ Z } obeys a central limit theorem. That is, if ϕ dν = 0, then the sequence n−1 1 ϕ ◦ Ti √ n i=0

converges in distribution to the normal distribution. The variance of the limiting normal distribution is strictly positive unless ϕ ◦ T = ψ ◦ T − ψ for some ψ.

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(b) Suppose that for some N 1, T N has an SRB measure ν that is mixing. Then given a Hölder exponent η, there exists τ = τ (η) < 1 such that for all Hölder ϕ, ψ : → R with Hölder exponent η, there exists L = L(ϕ, ψ) such that for all n ∈ N, (ϕ ◦ T n N )ψ dν − ϕ dν ψ dν L(ϕ, ψ)τ n . (SA4) Uniqueness of SRB measures and ergodic properties. (a) The map T admits a unique (and therefore ergodic) SRB measure ν, and (b) the dynamical system (T, ν) is mixing, or, equivalently, isomorphic to a Bernoulli shift. 5. Proof of Theorem 2.1 5.1. Degenerate Hopf bifurcation: the reduced equations. We study the two-dimensional system

r˙ = −µr (5.1) θ˙ = ω + γµ µ + βµr 2 . System (5.1) is obtained from (2.1) by setting gµ = h µ = 0. Let Ft denote the flow of (5.1). For µ sufficiently large, Fτ (µ) ◦ κ maps A into A. The study of this annulus map is the central goal of this subsection. We introduce a new coordinate system in order to standardize the position and size of A. Let r = µρ1 z. Written in terms of z and θ , system (5.1) becomes

z˙ = −µz (5.2) θ˙ = ω + γµ µ + µ2ρ1 βµ z 2 . Let G t denote the flow associated with (5.2). The kick map κ is now given in rectangular coordinates by

z cos(θ ) z cos(θ ) = . κ z sin(θ ) z sin(θ ) + Lµρ2 −ρ1 We have A = {(z, θ ) : K 4−1 z K 4 }. The relaxation time τ (µ) is given by z˜ e−µτ (µ) = 1,

(5.3)

where z˜ = Lµρ2 −ρ1 − K 4 . Let µ = G τ (µ) ◦ κ. For µ sufficiently large, µ maps A into A. We now derive µ : A → A explicitly. Let (z 0 , θ0 ) ∈ A. Writing κ(z 0 , θ0 ) = (z 1 , θ1 ), we have z 12 = z 02 + 2Lµρ2 −ρ1 z 0 sin(θ0 ) + L 2 µ2(ρ2 −ρ1 ) ,

z 0 cos(θ0 ) π −1 . θ1 = − tan 2 z 0 sin(θ0 ) + Lµρ2 −ρ1 Integrating (5.2) and writing G t ◦ κ = (z(t), θ (t)), we have z(t) = z 1 e−µt , θ (t) = θ1 + t (ω + γµ µ) +

βµ 2ρ1 −1 2 µ z 1 (1 − e−2µt ). 2

(5.4)


787

Evaluating θ (τ (µ)) using (5.3), we have

βµ 2ρ1 −1 2 z 12 µ θ (τ (µ)) = θ1 + (ω + γµ µ)τ (µ) + z1 − 2 . 2 z˜

(5.5)

Replacing the first occurrence of z 12 in (5.5) with the right side of (5.4), we obtain 2 z βµ θ (τ (µ)) = θ1 + ξ(µ) + µ2ρ1 −1 z 02 + 2Lz 0 sin(θ0 )µρ1 +ρ2 −1 − µ2ρ1 −1 12 , 2 z˜ where ξ(µ) = (ω + γµ µ)τ (µ) +

βµ 2 2ρ2 −1 L µ . 2

The second component of µ is given by z(τ (µ)) =

z1 . z˜

We wish to show that the family {µ } converges to a singular limit as µ → 0. This cannot be accomplished directly because ξ(µ) diverges as µ → 0, preventing the convergence of θ (τ (µ)). We overcome this difficulty by taking advantage of the fact that θ (t) is computed modulo 2π . Assume that ω > 0. For µ sufficiently small, ξ(µ) is monotone. In addition, ξ(µ) → ∞ as µ → 0. Let (µn ) be a sequence such that µn → 0 monotonically, ξ is monotone on (0, µ1 ], and ξ(µn ) ∈ 2π Z for all n ∈ N. We introduce the parameter a ∈ [0, 2π ) and write µ in terms of a. For n ∈ N and a ∈ [0, 2π ), let µ(a, n) = ξ −1 (ξ(µn ) + a). When referring to µ(a, n), we will henceforth suppress the dependence on n and simply write µ(a). The problematic term ξ(µ) becomes ξ(µ(a)) = a. Writing µ(a) = Ta,L ,µn , we have 1

Ta,L ,µn (z 0 , θ0 ) =

z1 , z˜

2

Ta,L ,µn (z 0 , θ0 ) = θ1 + a +

βµ(a) µ(a)2ρ1 −1 z 02 2

+ 2Lµ(a)ρ1 +ρ2 −1 z 0 sin(θ0 ) − µ(a)2ρ1 −1

z 12 , z˜ 2

where T 1 and T 2 are the components of T . Let ρ1 and ρ2 satisfy 21 < ρ1 < 1 and ρ1 + ρ2 = 1. Then as n → ∞, Ta,L ,µn converges in the C 0 topology to the map Ta,L ,0 defined by 1

Ta,L ,0 = 1, π 2 Ta,L ,0 = + β0 Lz 0 sin(θ0 ) + a. 2 The following lemma asserts that the convergence is strong enough for the application of rank one theory. Lemma 5.1. Fix L > 0. The maps (z 0 , θ0 , a) → Ta,L ,µn (z 0 , θ0 ) converge in the C 3 topology to the map (z 0 , θ0 , a) → Ta,L ,0 (z 0 , θ0 ) as n → ∞ on the domain A × [0, 2π ).

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Proof of Lemma 5.1. Holding a fixed, the derivatives of θ1 , zz˜1 , and z˜ 12 of orders 1, 2, and 3 with respect to z 0 and θ0 are O(µρ1 −ρ2 ). When differentiating with respect to a, use the fact that for i = 1, 2, 3, µi+1 n (i) ∂a µ(a) = O i . log(µ−1 n ) We finish this subsection with a distortion estimate. Lemma 5.2 (Distortion estimate). Let 0 < L 2 < L 3 . There exists K D > 0 such that for all n ∈ N, a ∈ [0, 2π ), L ∈ [L 2 , L 3 ], and (z 0 , θ0 ), (z 0 , θ0 ) ∈ A, we have | det DTa,L ,µn (z 0 , θ0 )| K D. | det DTa,L ,µn (z 0 , θ0 )| Proof of Lemma 5.2. Recall that Ta,L ,µn = G τ (µ(a)) ◦ κ. We bound the distortion by analyzing G and κ independently. Let (z 0 , θ0 ), (z 0 , θ0 ) ∈ A. Writing µ = µ(a), det Dκ(z 0 , θ0 ) is given by z 03 + µρ2 −ρ1 · Lz 0 (1 + z 0 ) sin(θ0 ) + µ2(ρ2 −ρ1 ) · L 2 (1 + (z 0 − 1) cos2 (θ0 )) . z 1 (z 0 sin(θ0 ) + Lµρ2 −ρ1 )2 + z 02 cos2 (θ0 ) This explicit formula implies the estimate det Dκ(z 0 , θ0 ) = O(1). det Dκ(z 0 , θ0 ) Now set G = G τ (µ(a)) . For any point in κ(A), the determinant of the derivative of G is precisely z˜ −1 . Therefore, det DG(z 1 , θ1 ) = 1. det DG(z 1 , θ1 ) 5.2. Inclusion of the higher-order terms in the normal form. We show that the inclusion of the higher-order terms in the differential equations defining the flow does not affect ˆ Written the form of the singular limit derived in Subsect. 5.1. Set r = µρ1 zˆ and θ = θ. ˆ in terms of zˆ and θ , the normal form (2.1) becomes

z˙ˆ = −µˆz + µ4ρ1 zˆ 5 gµ (µρ1 zˆ , θˆ ) (5.6) θ˙ˆ = ω + γ µ + β µ2ρ1 zˆ 2 + µ4ρ1 zˆ 4 h (µρ1 zˆ , θˆ ) µ

µ

µ

t denote the flow generated by (5.6). We define the family {T } on A by first applyLet G t -flow to return κ(A) to A. Set T a,L ,µn = ing the kick map κ and then allowing the G G τ (µ(a)) ◦ κ. a,L ,µn (z 0 , θ0 ) Lemma 5.3. Fix L > 0. If ρ2 > 13 , then the maps (z 0 , θ0 , a) → T 3 converge in the C topology to the map (z 0 , θ0 , a) → Ta,L ,0 (z 0 , θ0 ) as n → ∞ on the domain A × [0, 2π ).


789

t ◦ κ, we have Proof of Lemma 5.3. Computing the first component of G

t −1 µs −µt 4ρ1 5 ρ1 ˆ 1+µ z 1 e zˆ (s) gµ (µ zˆ (s), θ (s)) ds zˆ (t) = z 1 e 0

= z(t) + ζ (t), where the perturbative term ζ (t) is defined by t eµs zˆ (s)5 gµ (µρ1 zˆ (s), θˆ (s)) ds. ζ (t) = µ4ρ1 e−µt

(5.7)

0

t ◦ κ, we have θˆ (t) = θ (t) + θ(t), ˜ Computing the second component of G where t v 2µ4ρ1 z 1 e−2µv eµs zˆ (s)5 gµ (µρ1 zˆ (s), θˆ (s)) ds θ˜ (t) =βµ µ2ρ1 0

8ρ1 −2µv

+µ

e

0

v 0

t

+ µ4ρ1 0

e zˆ (s) gµ (µ zˆ (s), θˆ (s)) ds µs

5

ρ1

2 dv

(5.8)

zˆ (s)4 h µ (µρ1 zˆ (s), θˆ (s)) ds.

In order to establish C 0 convergence, it suffices to show that the perturbative terms ζ (τ (µ(a))) and θ˜ (τ (µ(a))) converge to 0 in the C 0 topology as n → ∞. Estimating the integrals in (5.7) and (5.8), we obtain ζ (τ (µ)) = O(µ5ρ2 −ρ1 −1 log(µ−1 )), θ˜ (τ (µ)) = O µ6ρ2 −2 (log(µ−1 ))2 + O µ10ρ2 −3 (log(µ−1 ))3 . Since ρ2 ∈ ( 13 , 21 ) and ρ1 ∈ ( 21 , 23 ), we have ζ (τ (µ)) C 0 → 0 and θ˜ (τ (µ)) C 0 → 0 as µ → 0. We complete the proof of Lemma 5.3 by showing that τ (µ(a)) ◦ κ − D i G τ (µ(a)) ◦ κ C 0 → 0 D i G for 1 i 3. In light of Lemma 5.1, this establishes the asserted C 3 convergence. τ (µ(a)) − Since D i κ C 0 is bounded for 1 i 3, it is sufficient to show that D i G D i G τ (µ(a)) C 0 → 0 for 1 i 3. We use the following elementary Gronwall-type lemma. be C 1 vector Lemma 5.4 ((10)). Let ⊂ R N be a convex open domain. Let W and W fields on . Suppose that for t ∈ [0, t0 ], ϕˆ and ϕ solve the equations dϕ d ϕˆ (ϕ) =W ˆ and = W (ϕ) dt dt with ϕ(0) ˆ = ϕ(0). Then for all t ∈ [0, t0 ], we have ϕ(t) ˆ − ϕ(t)

A1 A2 t (e − 1), A2

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where (x) − W (x) and A2 = A1 = sup W x∈

N

sup DW j (x) .

j=1 x∈

We rescale time in (5.2) and (5.6) by setting t = t τ (µ(a)). Let η and ηˆ denote the rescaled vector fields. We have

η1 = τ (µ(a))(−µz)

η2 = τ (µ(a))(ω + γµ µ + µ2ρ1 βµ z 2 ), ηˆ 1 = τ (µ(a))(−µˆz + µ4ρ1 zˆ 5 gµ (µρ1 zˆ , θˆ )) ηˆ 2 = τ (µ(a))(ω + γµ µ + βµ µ2ρ1 zˆ 2 + µ4ρ1 zˆ 4 h µ (µρ1 zˆ , θˆ ))

We explicitly treat the case i = 1. The cases i = 2 and i = 3 are handled using the same ϕ = DG, W = D η, technique. Apply Lemma 5.4 with ϕˆ = D G, ˆ W = Dη, and t = 1. The quantity A2 is bounded. Therefore, the estimate A1 = O(µ5ρ2 −ρ1 −1 log(µ−1 )) implies that 1 − DG 1 C 0 = O(µ5ρ2 −ρ1 −1 log(µ−1 )). (5.9) D G 5.3. Verification of (G1)-(G6). Theorem 2.1 follows from an application of Theorems 4.2 and 4.3. Statements (1) and (2) of Theorem 2.1 require the verification of (G1)-(G5) for a,L ,µn }. Statement (3) of Theorem 2.1 requires the additional verification the family {T of (G6). We proceed with the verification of statement (1) of Theorem 2.1. Properties (G1)(a) and (G1)(b) follow from the general theory of ordinary differential equations. For (G1)(c), τ (µ(a)) is bounded because the distortion of κ it suffices to show that the distortion of G is bounded. Using (5.9), we have ε2 z˜−1 + ε1 τ (µ(a)) (z 1 , θ1 ) = βµ 2ρ −1 DG , 1 2z 1 − 2zz˜ 21 + ε3 1 + ε4 2 µ where ε j = O(µ5ρ2 −ρ1 −1 log(µ−1 )) for 1 j 4. Since ρ2 > 38 , for 1 j 4. Therefore, we have

εj z˜ −1

→ 0 as µ → 0

τ (µ(a)) (z 1 , θ1 )) = z˜ −1 + O(µ5ρ2 −ρ1 −1 log(µ−1 )). det(D G τ (µ(a)) is bounded. This estimate implies that the distortion of G 2 Lemma 5.3 establishes (G2). Let f a,L denote the restriction of Ta,L ,0 to the circle S 1 = {(z 0 , θ0 ) : z 0 = 1}. We have f a,L (θ ) =

π + β0 L sin(θ ) + a. 2

Applying Theorem 3.4 with (θ ) = sin(θ ), c(1) = π2 , and c(2) = 3π 2 , if L is sufficiently large then there exist L ∗ ∈ [L , L + |βπ0 | ] and a ∗ ∈ [0, 2π ) such that f a ∗ ,L ∗ ∈ M. This is (G3). We establish parameter transversality (G4) by applying Lemma 4.1. Write f = f a ∗ ,L ∗ and f a = f a,L ∗ . We have ∂a f a (·) = 1 and |( f k ) ( f (x))| K k . Therefore,


the absolute value of the left side of (4.1) is bounded below by 1 − quantity is positive if K > 2. For (G5), observe that 2

791

∞

k=1

K −k . This

2

∂z 0 Ta,L ,0 (1, c(1) ) = β0 L = 0 and ∂z 0 Ta,L ,0 (1, c(2) ) = −β0 L = 0. This completes the verification of statement (1) of Theorem 2.1. Statement (2) of Theorem 2.1 follows from the fact that for all L sufficiently large, f a,L ∈ M for a O(L −1 )-dense set of values of a. Wang and Young (9) prove this result in a slightly different context. The proof for the family { f a,L } is essentially the same. Statement (3) of Theorem 2.1 requires the verification of the conditions for mixing (G6). Property (G6)(a) holds provided eλ0 = K > 8. Property (G6)(b) is satisfied with N = 1 provided L is sufficiently large. References 1. Guckenheimer, J., Wechselberger, M., Young, L.-S.: Chaotic attractors of relaxation oscillators. Nonlinearity 19(3), 701–720 (2006) 2. Lin, K.K.: Entrainment and chaos in a pulse-driven Hodgkin-Huxley oscillator, SIAM J. Appl. Dyn. Syst. 5(2), 179–204 (2006) (electronic) 3. Lin, K.K., Young, L.-S.: Shear-induced chaos. http://arxiv.org/pdf/0705.3294, 2007 4. Thieullen, Ph., Tresser, C., Young, L.-S.: Positive Lyapunov exponent for generic one-parameter families of unimodal maps. J. Anal. Math. 64, 121–172 (1994) 5. Thieullen, Ph., Tresser, C., Young, L.-S.: Exposant de Lyapunov positif dans des familles à un paramètre d’applications unimodales. C. R. Acad. Sci. Paris Sér. I Math. 315(1), 69–72 (1992) 6. Wang, Q., Oksasoglu, A.: Strange attractors in periodically kicked Chua’s circuit. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 15(1), 83–98 (2005) 7. Wang, Q., Young, L.-S.: Toward a theory of rank one attractors. Ann. Math. 16(2), 349–480 (2008) 8. Wang, Q., Young, L.-S.: Strange attractors with one direction of instability. Comm. Math. Phys. 218(1), 1–97 (2001) 9. Wang, Q., Young, L.-S.: From invariant curves to strange attractors. Comm. Math. Phys. 225(2), 275–304 (2002) 10. Wang, Q., Young, L.-S.: Strange attractors in periodically-kicked limit cycles and Hopf bifurcations. Comm. Math. Phys. 240(3), 509–529 (2003) Communicated by G. Gallavotti


Communications in


Deligne Products of Line Bundles over Moduli Spaces of Curves L. Weng1, , D. Zagier2,3 1 Graduate School of Mathematics, Kyushu University, Fukuoka, Japan.

E-mail: [email protected]

2 Max-Planck-Institut für Mathematik, Bonn, Germany 3 Collège de France, Paris, France

Received: 22 June 2007 / Accepted: 9 October 2007 Published online: 20 May 2008 – © The Author(s) 2008

Abstract: We study Deligne products for forgetful maps between moduli spaces of marked curves by offering a closed formula for tautological line bundles associated to marked points. In particular, we show that the Deligne products for line bundles on the total spaces corresponding to “forgotten” marked points are positive integral multiples of the Weil-Petersson bundles on the base moduli spaces. 1. Introduction Let Mg,N denote the moduli space of curves C of genus g with N ordered marked points P1 , . . . , PN , and π = π N : Cg,N → Mg,N the universal curve over Mg,N . (We are using the language of stacks here [3].) The marked points give sections Pi : Mg,N → Cg,N , i = 1, . . . , N of π . The Picard group of Mg,N is known to be free of rank N + 1 [4] and has a Z-basis given by the Mumford class λ (the line bundle whose fiber at C is detH 0 (C, K C ) ⊗ detH 1 (C, K C )−1 ) and the “tautological line bundles” i := Pi∗ (K N ), where K N is the relative canonical line bundle (relative dualizing sheaf) of π [1,5]. The i carry metrics in such a way that their first Chern forms give the Kähler metrics on Mg,N defined by Takhtajan-Zograf [9,10] in terms of Eisenstein series associated to punctured Riemann surfaces [11,12]. There is a further interesting element ∈ Pic(Mg,N ), whose associated first Chern form (for a certain natural metric) gives the Kähler form for the Weil-Petersson metric on Mg,N [11,12]; it is given in terms of λ and the i N by the Riemann-Roch formula = 12λ + i=1 i . We by the for can also define mula := K N (P1 + · · · + P N ), K N (P1 + · · · + P N ) π , where ·, · π : Pic(Cg,N )2 → Pic(M g,N ) denotes the Deligne pairing. We recall that the Deligne pairing is a bilinear map ·, · p : Pic(Y )2 → Pic(X ) which is defined for any flat morphism p : Y → X Partially supported by the Japan Society for the Promotion of Science.

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of relative dimension 1 and that it can be generalized to a multilinear map ·, . . . , · p : Pic(Y )n+1 → Pic(X ), the Deligne product, defined for any flat morphism p : Y → X of relative dimension n. (The precise definitions will be given in §3.) We are interested in computing the Deligne product explicitly for the forgetful map π N ,m : Mg,N +m → Mg,N ,

(C; P1 , . . . , PN +m ) → (C; P1 , . . . , PN ).

˜ , ˜ ˜ j to denote the Mumford, Weil-Petersson and tautological In other words, if we use λ, line bundles on Mg,N +m , respectively, then we would like to compute L 1 , . . . , L m+1 π N ,m as a linear combination of 1 , . . . , N and λ (or ) on Mg,N , where each L ν is one of ˜ We have not solved this problem in general (though it is inter˜1 , . . . , Ñ +m and λ˜ (or ). ˜ esting and perhaps not intractable), but only in the case where each L ν is one of the i , ˜ does not appear. The formula we find expresses L 1 , . . . , L m+1 i.e., where λ˜ (or ) π N ,m in this case as a positive linear combination of and those i (i = 1, . . . , N ) for which if eachof L 1 , . . . , L m+1 is one of the last m line ˜i appear among the L ν . In particular, is simply a positive integer multiple bundles Ñ +1 , . . . , Ñ +m , then L 1 , . . . , L m+1 π N ,m of the Weil-Petersson class , giving an interesting relation between the Weil-Petersson and the tautological line bundles.

2. Statement of the Theorem As just explained, we want to compute the Deligne product L 1 , . . . , L m+1 π , where N ,m each L ν belongs to the set {˜1 , . . . , Ñ +m }. It turns out to be more convenient to use the multiplicities where the ˜i occur in {L 1 , . . . , L m+1 } as coordinates. We therefore introduce the notation TN ,m (a1 , . . . , a N +m ) := ˜1 , . . . , ˜1 , . . . , Ñ +m , . . . , Ñ +m π N ,m a1

a N +m

∈ Pic(Mg,N ),

(1)

where a1 , . . . , a N +m ∈ Z≥0 with a1 + · · · + a N +m = m + 1. We will sometimes denote this element by TN ,m (a1 , . . . , a N ; a N +1 , . . . , a N +m ) or even, setting a N +i =: di , by TN ,m (a1 , . . . , a N ; d1 , . . . , dm ), to emphasize the different roles played by the indices corresponding to the points which are also marked in Mg,N and to those which are “forgotten” by the projection map π N ,m . Our main result is then: Theorem. Let a1 , . . . , a N , d1 , . . . , dm ≥ 0 be non-negative integers with sum m + 1. Then the line bundle TN ,m (a1 , . . . , a N ; d1 , . . . , dm ) defined in (1) is given in terms of the elements i , ∈ Pic(Mg,N ) by the formula N

ai !

m

i=1

= C1 (d)

d j ! TN ,m (a1 , . . . , a N ; d1 , . . . , dm )

j=1 N i=1

+ C2 (d) , ai i − N

(2)

Deligne Products of Line Bundles over Moduli Spaces of Curves

795

= N + 2g − 3 and the coefficients Cν (d) = Cν ( N , d1 , . . . , dm ) (ν = 1, 2) where N and are given explicitly by depend only on the di and on N C1 (d) = C2 (d) =

m + n − 1)! (m − n)! ( N n=0 m n=0

− 1)! (N

σn ,

+ n − 2)! (m − n + 1)! ( N σn N ( N − 2)!

(3)

with σn = σn (d1 , . . . , dm ) the n th elementary symmetric polynomial in the di . = 0 or 1, then the factors 1/( N − 1)! and 1/ N ( N − 2)! occurring in Remark. 1. If N − 1)/ N !, the formulas for C1 (d) and C2 (d) are to be interpreted as N / N ! and ( N respectively. 2. The proof (or rather, the recursive description of the TN ,m on which it is based) will show that in the formula for TN ,m (a1 , . . . , a N ; d1 , . . . , dm ) in terms of i and , all the coefficients are non-negative and integral (even though 1 , . . . , N , is not a Z-basis of Pic(Mg,N )). Neither property is obvious from the formulas (2) and , (3), though one can see easily that both C1 (d) and C2 (d) − (m + 1 − σ1 )C1 (d)/ N the coefficient of on the right-hand side of (2), are polynomials in N . 3. In Sect. 6 we will give an alternative explicit formula for the coefficients C1 (d) and C2 (d). 4. Notice that, as already mentioned in the Introduction, formula (2) in the special case when all the ai are 0 says that TN ,m (0, . . . , 0; d1 , . . . , dm ) is a multiple of alone. In other words, all Deligne products of line bundles corresponding to points which are “forgotten” by π N ,m are positive integral multiples of the Weil-Petersson bundle . 3. The Deligne product We start with some basic facts about Deligne products [2]. Let π : X → S be a flat family of algebraic varieties of relative dimension n. Then for any n + 1 invertible sheaves L 0 , . . . , L n over X , following Deligne [2], we may introduce prod the Deligne uct, denoted by L 0 , . . . , L n (X/S) or L 0 , . . . , L n π or simply L 0 , . . . , L n , defined uniquely axioms: by the following (DP1) L 0 , . . . , L n (X/S) is an invertible sheaf on S, and is symmetric and multi-linear in the L i ’s. (DP2) L 0 , . . . , L n (X/S) is locally generated by the symbols t0 , . . . , tn , where the ti are sections of L i whose divisors have no common intersection, and these symbols satisfy the following

property: if one multiplies one of the sections ti by a rational function f on X , where j=i div(t j ) = n k Yk is finite over S and div( f ) has no intersection with any Yk , then NormYk /S ( f )n k · t0 , . . . , tn . t0 , . . . , f ti , . . . , tn = k

(Here NormYk /S ( f ) is defined as follows: Since Yk is finite over S, the function field of Yk is a finite extension of that of S, and hence can be viewed as a finite dimensional vector space. Since f is in the function field of X , via restriction we may view f as an

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element of the function field of Yk . Then multiplication by f defines a linear map of the vector space of the function field of Yk over the function field of S. By definition, the determinant of this linear map is called the norm of f with respect to Yk /S.) (DP3) If tn is a section of L n such that all components Dα of the divisor div(tn ) = α n α Dα are flat (of relative dimension n − 1) over S, then we have a canonical isomorphism

L 0 , . . . , L n (X/S) L 0 , . . . , L n−1 (div(tn )/S) := ⊗α L 0 , . . . , L n−1 (Dα /S)⊗n α .

Roughly speaking, the Deligne product may be built up as follows using the above axioms: one first uses (DP3) to make an induction on the relative dimension so as to reduce to special cases, say, n = 1, by using a certain choice of sections, and then shows with the help of axiom (DP2) that this construction does not depend on the choice of sections of line bundles and is symmetric by virtue of the Weil reciprocity law. As a consequence of these axioms and the uniqueness, we know that the Deligne product formalism is compatible with any base change, and that the products satisfy the following compatibility relations with respect to compositions of flat morphisms: Proposition 1. ([2]). Let f : X → Y and g : Y → Z be flat morphisms of relative dimension n and m, respectively. Then: (a) For invertible sheaves L 0 , . . . , L n on X and H1 , . . . , Hm on Y , we have

L 0 , . . . , L n f , H1 , . . . , Hm g L 0 , . . . , L n , f ∗ H1 , . . . , f ∗ Hm g◦ f . (4a)

(b) For invertible sheaves L 1 , . . . , L n on X and H0 , . . . , Hm on Y , we have

f (c (L )···c (L )) f ∗ H0 , L 1 . . . , L n f , H1 , . . . , Hm g H0 , H1 , . . . , Hm g∗ 1 1 1 n .

(4b)

A special case of Proposition 1 which will be needed later is the formula

L , f ∗ H0 , . . . , f ∗ Hm

g◦ f

deg f (L) H0 , . . . , Hm g

(5)

for f , g as in the proposition with n = 1 and for any bundles L on X and H0 , . . . , Hm on Y . ∗To get this, we use part (a) of the proposition to write the left-hand side as L , f H0 f , H1 , . . . , Hm g and then part (b) to write L , f ∗ H0 f as deg f (L)H0 . Remark. Recall that there is a map c1 from Pic(X ) to the codimension 1 part CH1 (X ) of the Chow group of X . If f : X → Y is flat of relative dimension n and L 0 , . . . , L n belong to Pic(X ), then the image of L 0 , . . . , L n under c1 is equal to the image of the product c1 (L 0 ) · · · c1 (L n ) under the push-forward map f ∗ : CHn+1 (X ) → CH1 (Y ). At this level, formulas (4a), (4b) and (5) are just specializations of the general projection formula g∗ ( f ∗ (A) · B) = (g f )∗ (A · f ∗ (B)), valid for any flat morphisms f : X → Y and g : Y → Z and elements A ∈ CH(X ), B ∈ CH(Y ).


797

4. Geometric Preparations We now apply the Deligne product to universal curves over moduli spaces. As in §1, we denote by K N the relative canonical line bundle of π = π N : Cg,N → Mg,N , by Pi (1 ≤ i ≤ N ) the N sections of π and by i = Pi∗ (K N ) the i th tautological line bundle on Mg,N . We write L i (1 ≤ i ≤ N + 1) for the line bundles on Cg,N defined in the same way, where Cg,N is identified with Mg,N +1 . Also for convenience, we denote the bundle OCg,N (Pi ) (1 ≤ i ≤ N ) simply by Pi . The following properties can be found in [6,7]. (Deligne products are not used in Knudsen’s original papers, but the verbatim change is rather trivial. See e.g., [11,12].) Proposition 2. ([6]) With the above notations, we have (a) Pi , P j π O (i, j = 1, . . . , N , i = j); (b) K N (Pi ), Pi π O (i = 1, . . . , N ); (c) L i π ∗ i + Pi (i = 1, . . . , N ); (d) L N +1 K N (P1 + · · · + P N ). The next proposition, which is a slight extension of Proposition 2, contains all the geometric information which we will need to compute the Deligne products in (1). We use the same notations as above, but also denote by π = π N −m,m the forgetful map from Mg,N to Mg,N −m for some m ≥ 0 and use ξ1 , . . . , ξm to denote m general elements of Pic(Cg,N ). Proposition 3. With the above notations, we have (a) Pi , P j , ξ1 , . . . , ξm π ◦π O (i, j = 1, . . . , N , i = j); (b) Pi , L i , ξ1 , . . . , ξm π ◦π O (i = 1, . . . , N ); (c) Pi , L N +1 π O (i = 1, . . . , N ); (d) degπ L N +1 = 2g − 2 + N . Proof. Since the sections Pi and P j are disjoint for i = j, the pull-back of O(Pi ) to P j is trivial (Prop. 2(a)). Therefore axiom (DP3) from §3 implies (a). Next, we use Prop. 2(c) to write L i , Pi , ξ1 , . . . , ξm π ◦π π ∗ i , Pi , ξ1 , . . . , ξm π ◦π + Pi , Pi , ξ1 , . . . , ξm π ◦π . By (DP3), the first term equals i , ξ1 P , . . . , ξm P π (actually multiplied by degπ (Pi ), i i but the relative degree of a section is 1), while the second term is − i , ξ1 , . . . , ξm Pi

Pi π

by Prop. formula). This proves (b). Part (c) follows from 2(b) (adjunction Prop. 2(d), since Pi , K N (Pi ) π vanishes by the adjunction formula and all Pi , P j π with j = i vanish by (a). Part (d) also follows from Prop. 2(d) by taking the relative degree of both sides. We also mention the stronger statement that L N +1 , L i π (2g − 2 + N ) i for i = 1, . . . , N . The proof of this is similar to the other parts of the proposition, but we omit it since this result will not be used in the sequel.

5. The Recursion Formula for TN,m In this section, we use the results of §4 to give a recursion formula and initial data for the line bundles (1) which determine them completely in Pic(Mg,N ). These recursions will be solved in §6. The recursion formula which we will prove for the TN ,m is as follows.

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L. Weng, D. Zagier

Proposition 4. (String Equation) For m ≥ 0 and any integers a1 , . . . , a N +m ≥ 0, we have TN ,m+1 (a1 , . . . , a N +m , 0) =

N +m

TN ,m (a1 , . . . , ai − 1, . . . , a N +m ),

i=1

with the convention that TN ,m+1 (a1 , . . . , a N +m ) = 0 if any ai < 0. Recall that the indices ai with i > N in TN ,m (a1 , . . . , a N +m ) play a different role than the ai with i ≤ N and that we also use the notations d j for a N + j (1 ≤ j ≤ m) and TN ,m (a1 , . . . , a N ; d1 , . . . , dm ) for TN ,m (a1 , . . . , a N +m ). Proposition 4 lets us reduce the calculation of these bundles by induction to the case when every di is strictly positive. (If any d j is zero, we can put it in the last position, because TN ,m is symmetric in the d’s.) N But since i=1 ai + mj=1 d j = m +1, this can only happen if (d1 , . . . , dm ) = (1, . . . , 1) or (2, 1, . . . , 1). There are therefore only two initial cases which have to be considered. The values of TN ,m in these two cases are given by the following: Proposition 5. The line bundles TN ,m (a1 , . . . , a N ; d1 , . . . , dm ) in the two cases when all the di are strictly positive are given by the formulas TN ,m (1, 0, . . . , 0; 1, . . . , 1) = N −1

m

+ m)! (N 1 ! N

(m ≥ 0)

and + m)! (N TN ,m (0, . . . , 0; 2, 1, . . . , 1) = + 1)! (N

(m ≥ 1),

m−1

N

= N + 2g − 3. where N Proposition 5 in turn can be deduced by induction over m from the special cases TN ,0 (1, 0, . . . , 0; ) = 1 , N −1

TN ,1 (0, . . . , 0; 2) = N

(the first of which is trivial because the Deligne product is simply the identity map, and the second by the very definition of ) and from the following companion result to Proposition 4. Proposition 6. (Dilaton Equation) For m ≥ 0 and any integers a1 , . . . , a N +m ≥ 0, we have TN ,m+1 (a1 , . . . , a N +m , 1) = (N + m + 2g − 2) TN ,m (a1 , . . . , a N +m ). The proofs of Propositions 4 and 6 are similar to one another and will be given together. For convenience, we use the abbreviated notation S1◦k1 , S2◦k2 , . . . , Sn◦kn f to denote the Deligne product S , . . . , S , S , . . . , S , . . . , Sn , . . . , Sn f 1 1 2 2 k1

k2

kn


799

for any line bundles Si and any integers ki ≥ 0. We also replace the “N ” of §4 by “N + m” and use the same conventions as there, i.e., i (1 ≤ i ≤ N + m) denotes the i th tautological line bundle on Mg,N +m and L i (1 ≤ i ≤ N + m + 1) the i th tautological line bundle on Mg,N +m+1 , while π and π denote the projections from Mg,N +m+1 to Mg,N +m and from Mg,N +m to Mg,N , respectively. Finally, we set M = N + m. With these notations, the two formulas to be proved become

◦a M 1 L ◦a 1 ,..., LM

π ◦π

=

M

◦(ai −1)

1 ◦a 1 , . . . , i

M , . . . , ◦a M

π

(6)

i=1

◦(a −1) (with the usual convention that · · · , i i , · · · = 0 if ai = 0) and ◦a1 ◦a1 ◦a M M L 1 , . . . , L ◦a M , L M+1 π ◦π = (2g − 2 + M) 1 , . . . , M π .

(7)

To prove these equations, we proceed as follows. Using Prop. 2(c), Prop. 3(b) and then Prop. 2(c) again (all of them with N replaced by M), we obtain ◦a1 ◦(a −1) L 1 , ξ1 , . . . , ξr π ◦π = L 1 , π ∗ 1 + P1 1 , ξ1 , . . . , ξr π ◦π ◦(a −1) = L 1 , π ∗ 1 1 , ξ1 , . . . , ξr π ◦π ◦a = π ∗ 1 1 , ξ1 , . . . , ξr π ◦π ◦(a −1) + P1 , π ∗ 1 1 , ξ1 , . . . , ξr π ◦π for any line bundles ξ1 , . . . , ξr in Pic(Cg,M ), where a1 + r = m + 2. Now if there are a2 indices i with ξi = L 2 , then we can do the same with L 2 as we did with L 1 . This gives ◦(a −1) ◦(a2 −1) four terms a priori, but one of them is P1 , π ∗ 1 1 , P2 , π ∗ 2 , . . . π ◦π and this vanishes by Prop. 3(a). Continuing, we find ◦a1 ◦a M L 1 , . . . , L ◦a M , L M+1 π ◦π ◦a ◦a = π ∗ 1 1 , . . . , π ∗ M M , L ◦a M+1 π ◦π +

M

◦a ◦(ai −1) ◦a , . . . , π ∗ M M , L ◦a Pi , π ∗ 1 1 , . . . , π ∗ i M+1 π ◦π

(8)

i=1

for any ai , a ≥ 0. We have to evaluate this in two cases, when a = 0 and when a = 1. If a = 0, then the first term in (8) vanishes, because each of the arguments of the Deligne product is a pull-back under π . For the second term, we note that Pi , π ∗ ξ1 , . . . , π ∗ ξm+1 π ◦π = ξ1 , . . . , ξm+1 π for any ξ1 , . . . , ξm+1 ∈ Pic(Mg,M ). (This follows from (DP3), because Pi is a section of π .) This gives Eq. (6) and hence Proposition 4 (string equation). If a = 1, then the second term of (8) vanishes, because Pi , π ∗ ξ1 , . . . , π ∗ ξm , L M+1 π ◦π = Pi , L M+1 π , ξ1 , . . . , ξm π = 0 for any ξ1 , . . . , ξm ∈ Pic(Mg,M ), by Prop. 1 (a) and Prop. 3 (c). The first term in (8) is equal to the right-hand side of (7) by Eq. (5) and Prop. 3 (d). This proves Eq. (7) and hence Proposition 6 (dilaton equation).

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L. Weng, D. Zagier

6. Proof of the Main Theorem Since the recursion formula and initial values given in Propositions 4 and 5 determine the elements TN (a1 , . . . , a N +m ) ∈ Pic(Mg,N ) uniquely, we can prove Theorem 1 by showing that the elements TN (a1 , . . . , a N +m ) defined by (2) and (3) satisfy these three equations. A direct proof of this is possible, but rather complicated, involving a series of lemmas about elementary symmetric polynomials and multinomial coefficients. This proof can be simplified considerably by a judicious use of generating functions, but remains quite complicated. A much simpler proof is obtained using the following trick. By the substitution t = x/(1 + x) and Euler’s formula for the beta integral (or simply by integration by parts and induction on α and β) we see that ∞ 1 xα dx α! β! = t α (1 − t)β dt = α+β+2 (x + 1) (α + β + 1)! 0 0 for any integers α, β ≥ 0. Hence, if we define a polynomial F(x) = Fd1 ,...,dm (x) by F(x) = then we have ∞ (x

0

m

m x + dj = σn x m−n ,

j=1

n=0

F(x) d x + 1) N +m+1

=

m

σn

n=0

σn = σn (d1 , . . . , dm ),

+ n − 1)! − 1)! (m − n)! ( N (N = C (d) + m)! + m)! 1 (N (N

(9a)

and

∞ 0

x F(x) d x (x

+ 1) N +m+1

=

m

σn

n=0

+ n − 2)! (N − 2)! (m − n + 1)! ( N N = C (d) + m)! + m)! 2 (N (N (9b)

; d1 , . . . , dm ) as in Eq. (3) (or the first remark after Theorem 1 if N with Cν (d) = Cν ( N is 0 or 1). In particular, we have + m)! ∞ + m)! dx (N (N = C1 ( N ; 1, . . . , 1) = , − 1)! 0 (x + 1) N +1 ! (N N m ∞ + m)! x dx (N ; 1, . . . , 1) = ( N + m)! = C2 ( N , ( N − 2)! 0 (x + 1) N +1 · N ! N N m ∞ + m)! x (x + 2) d x 2 (N ; 2, 1, . . . , 1) = ( N + m)! = C2 ( N , + 1)! ( N − 2)! 0 (N N (x + 1) N +2 m−1

so Eq. (2) in the two cases when all di are strictly positive reduces to TN ,m (1, 0, . . . , 0; 1, . . . , 1) N −1

m

; 1, . . . , 1) 1 − + C2 ( N ; 1, . . . , 1) = ( N + m)! 1 = C1 ( N ! N N


801

and TN ,m (0, . . . , 0; 2, 1, . . . , 1) m−1

N

1 ; 2, 1, . . . , 1) = ( N + m)! , C2 ( N + 1)! 2! (N in accordance with the initial values in Proposition 5. To prove the recursion formula of Proposition 4 (string equation), we will show that it is equivalent to a pair of recurrences for the coefficients Cν (d) (Eq. (11) below) and then prove these recurrences using the integral representation (9). Denote the right-hand side of (2) by t N ,m (a1 , . . . , a N +m ) or t N ,m (a1 , . . . , a N ; d1 , . . . , dm ), with d j = a N + j for N +m 1 ≤ j ≤ m. Since we want TN ,m (a1 , . . . , a N +m ) = t N ,m (a1 , . . . , a N +m )/ i=1 ai !, we have to prove the recursion =

t N ,m+1 (a1 , . . . , a N +m , 0) =

N +m

ai t N ,m (a1 , . . . , ai − 1, . . . , a N +m ).

i=1

(The extra factor ai in front of t N ,m (a1 , . . . , ai − 1, . . . , a N +m ) comes from the change N +m in i=1 ai ! when ai is decreased by 1.) Now again separating the ai (1 ≤ i ≤ N ) and the d j = a N + j (1 ≤ j ≤ m), we can write this out more explicitly as t N ,m+1 (a1 , . . . , a N ; d1 , . . . , dm , 0) =

N

ai t N ,m (a1 , . . . , ai − 1, . . . , a N ; d1 , . . . , dm )

i=1 m

+

d j t N ,m (a1 , . . . , a N ; d1 , . . . , d j − 1, . . . , dm ).

(10)

j=1

We can write the definition of t N ,m (a1 , . . . , a N ; d1 , . . . , dm ) in an abbreviated notation as t N ,m (a1 , . . . , a N ; d1 , . . . , dm ) = C1 (d)

N

ak k + C2 (d) ,

k=1

; d1 , . . . , dm ) as before ( N does not change when we change m where Cν (d) = Cν ( N of by 1, so can be omitted from the notation) and where k is the element k − / N Pic(Mg,N ) ⊗ Q. Then the left-hand side of (10) equals C1 (d, 0)

N

ak k + C2 (d, 0) ,

i=1

while the right-hand side equals N N ak − δki k + C2 (d) ai C1 (d) i=1

k=1

+

m j=1

N d j C1 (d1 , . . . , d j − 1, . . . , dm ) ak k + C2 (d1 , . . . , d j −1, . . . , dm ) k=1

802

=

L. Weng, D. Zagier

N N m ai − 1 + d j C1 (d1 , . . . , d j − 1, . . . , dm ) ak k C1 (d) k=1

i=1

j=1

N m + C2 (d) ai + d j C2 (d1 , . . . , d j − 1, . . . , dm ) . i=1

j=1

N ai = m + 2 − σ1 (d) (because Comparing these two expressions, and recalling that i=1 the sum of all the indices a1 , . . . , a N , d1 , . . . , dm , 0 in Eq. (10) must equal m + 2), we find that the theorem will follow from the two identities: m d j C1 (d1 , . . . , d j − 1, . . . , dm ), C1 (d, 0) = m + 1 − σ1 C1 (d) +

(11a)

j=1 m C2 (d, 0) = m + 2 − σ1 C2 (d) + d j C2 (d1 , . . . , d j − 1, . . . , dm ).

(11b)

j=1

To prove the first of these, we use Eq. (9a). Replacing d = (d1 , . . . , dm ) by (d, 0) = (d1 , . . . , dm , 0) increases m by 1 and replaces the polynomial F(x) by x F(x), whereas replacing d by (d1 , . . . , d j − 1, . . . , dm ) leaves m unchanged and replaces F(x) by F(x)(x + d j − 1)/(x + d j ). Therefore substituting (9a) into (11a) and dividing both sides + m)!/( N − 1)! gives by ( N ∞ x F(x) d x + m + 1) (N (x + 1) N +m+2 0

∞ m x + dj − 1 F(x) d x m+1+ = d j −1 + x + dj (x + 1) N +m+1 0 j=1 as the identity to be proved. But this is immediate by integration by parts, since the expres (x) x sion in square brackets equals 1 + mj=1 x+d = 1 + x FF(x) . The proof of Eq. (11b) is j exactly the same, using (9b) instead of (9a), with F(x) replaced by x F(x). This completes the proof of the theorem. 7. Final Remark A formula very similar to Eq. (2) (in the case when all ai = 0) appears in §4.6 of [8], but in a somewhat different situation: the formula there is for the moduli space of curves of genus 1 with N marked points and deals with the Gromov-Witten invariants, which are integers, whereas our formula is for arbitary genus (although in the final result the ) and gives the Deligne products, genus does not appear except in the shift from N to N which take values in Pic(Mg,N ). Both proofs are based on the string and dilaton equations, which are valid in both contexts. This suggests a possible common generalization. Our situation concerns codimension one cycles, while Gromov-Witten invariants have to do with zero-dimensional cycles. It therefore seems reasonable to ask whether (2) and the equation in [8] are special cases of a more general result valid for intermediate dimensions, for which the string and dilaton equations still hold.


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References 1. Arbarello, E., Cornalba, M.: The Picard groups of the moduli spaces of curves. Topology 26(2), 153–171 (1987) 2. Deligne, P.: Le déterminant de la cohomologie. In: Current trends in arithmetic algebraic geometry, Contemporary Math. Vol. 67, Providence, RI: Amer. Math. Soc., pp. 93–178, 1987 3. Deligne, P., Mumford, D.: The irreducibility of the space of curves of given genus. IHES Publ. Math. 36, 75–109 (1969) 4. Harer, J.: The second homology group of the mapping class group of an orientable surface. Invent. Math. 72(2), 221–239 (1983) 5. Kaufmann, R., Manin, Yu., Zagier, D.: Higher Weil-Petersson volumes of moduli spaces of stable n-pointed curves. Commun. Math. Phys. 181(3), 763–787 (1996) 6. Knudsen, F.: The projectivity of the moduli space of stable curves, II. Math. Scand. 52, 161–199 (1983) 7. Knudsen, F., Mumford, D.: The projectivity of the moduli space of stable curves. I. Math. Scand. 39, 19–55 (1976) 8. Lando, S.K., Zvonkin, A.K.: Graphs on Surfaces and Their Applications, In: Encyclopedia of Mathematical Sciences, Berlin–Heidelberg: Springer-Verlag, 2004 9. Takhtajan, L., Zograf, P.: The Selberg zeta function and a new Kähler metric on the moduli space of punctured Riemann surfaces. J. Geo. Phys. 5, 551–570 (1988) ¯ 10. Takhtajan, L., Zograf, P.: A local index theorem for families of ∂-operators on punctured Riemann surfaces and a new Kähler metric on their moduli spaces. Commun. Math. Phys. 137, 399–426 (1991) 11. Weng, L.: Hyperbolic Metrics, Selberg Zeta Functions and Arakelov Theory for Punctured Riemann Surfaces. Lecture Note Series in Mathematics 6, Osaka: Osaka University, 1998 12. Weng, L.: -admissible theory, II.. Math. Ann. 320, 239–283 (2001) Communicated by L. Takhtajan


Communications in


The Paraboson Fock Space and Unitary Irreducible Representations of the Lie Superalgebra osp(1|2n) S. Lievens, N. I. Stoilova, , J. Van der Jeugt Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281-S9, B-9000 Gent, Belgium. E-mail: [email protected]; [email protected]; [email protected] Received: 2 July 2007 / Accepted: 30 November 2007 Published online: 15 May 2008 – © Springer-Verlag 2008

Abstract: It is known that the defining relations of the orthosymplectic Lie superalgebra osp(1|2n) are equivalent to the defining (triple) relations of n pairs of paraboson operators bi± . In particular, with the usual star conditions, this implies that the “parabosons of order p” correspond to a unitary irreducible (infinite-dimensional) lowest weight representation V ( p) of osp(1|2n). Apart from the simple cases p = 1 or n = 1, these representations had never been constructed due to computational difficulties, despite their importance. In the present paper we give an explicit and elegant construction of these representations V ( p), and we present explicit actions or matrix elements of the osp(1|2n) generators. The orthogonal basis vectors of V ( p) are written in terms of Gelfand-Zetlin patterns, where the subalgebra u(n) of osp(1|2n) plays a crucial role. Our results also lead to character formulas for these infinite-dimensional osp(1|2n) representations. Furthermore, by considering the branching osp(1|2n) ⊃ sp(2n) ⊃ u(n), we find explicit infinite-dimensional unitary irreducible lowest weight representations of sp(2n) and their characters. 1. Introduction The classical notion of Bose operators or bosons has been generalized a long time ago to parabose operators or parabosons [8]. These parabosons are of interest in many applications, in particular in quantum field theory [6,24,25], generalizations of quantum statistics (para-statistics) [8,10–12], and in Wigner quantum systems [17,19,21,31]. The generalization of the usual boson Fock space is characterized by a parameter p, referred to as the order. For a single paraboson, n = 1, the structure of the paraboson Fock space is well known [26]. Surprisingly, for a system of n parabosons with n > 1, the structure Permanent address: Institute for Nuclear Research and Nuclear Energy, Boul. Tsarigradsko Chaussee 72, 1784 Sofia, Bulgaria. NIS was supported by a project from the Fund for Scientific Research – Flanders (Belgium) and by project P6/02 of the Interuniversity Attraction Poles Programme (Belgian State – Belgian Science Policy).

806

S. Lievens, N. I. Stoilova, J. Van der Jeugt

of the paraboson Fock space is not known, even though it can in principle be constructed by means of the so-called Green ansatz [8]. The computational difficulties of the Green ansatz are related to finding a proper basis of an irreducible constituent of a p-fold tensor product [10]. An important result was given by Ganchev and Palev [7], who observed that the triple relations for n pairs of parabosons are in fact defining relations for the orthosymplectic Lie superalgebra osp(1|2n) [15]. This implies also that the paraboson Fock space of order p is a certain infinite-dimensional unitary irreducible representation (unirrep) of osp(1|2n). However, the construction of this representation, for n > 1 and arbitrary p, also turned out to be difficult. In the present paper we present a solution to this problem. Our solution is based upon group theoretical techniques, in particular related to the branching osp(1|2n) ⊃ sp(2n) ⊃ u(n). This allows us to construct a proper GelfandZetlin (GZ) basis for some induced representation [16], from which the basis for the irreducible representation follows. Our result is thus a complete solution to the problem: for the paraboson Fock space we give not only an orthogonal GZ-basis but also the action (matrix elements) of the paraboson operators in this basis. The structure of the paper is as follows. In Sect. 2, we recall the definition of parabosons and of the paraboson Fock space V ( p). In Sect. 3, we discuss the important relation between parabosons and the Lie superalgebra osp(1|2n), and give a description of V ( p) in terms of representations of osp(1|2n). The following section is devoted to the first non-trivial example of osp(1|4). The analysis of the representations V ( p) for osp(1|4) is performed in detail, and the techniques used here can be lifted to the general case osp(1|2n). This general case is investigated in Sect. 5, where the main computational result (matrix elements) is given in Proposition 6 and the main structural result (characters) in Theorem 7. Section 6 is devoted to the branching to the even subalgebra sp(2n) = sp(2n, R), and gives some sp(2n) characters. We end the paper with some final remarks. 2. The Paraboson Fock Space V ( p) Before introducing the paraboson Fock space, let us recall some aspects of the usual boson Fock space. For a single pair (n = 1) of boson operators B + , B − , the defining relation is given by [B − , B + ] = 1. (2.1) The boson Fock space is defined as a Hilbert space with vacuum vector |0, in which the action of the operators B + , B − is defined and satisfies 0|0 = 1,

B − |0 = 0,

(B ± )† = B ∓ .

Moreover, under the action of the algebra spanned by {B + ,

(2.2)

B − , 1} (subject to (2.1)), the

Hilbert space is irreducible. A set of basis vectors of this space, denoted by V (1), is given by (B + )k k ∈ Z+ = {0, 1, 2 . . .}. (2.3) |k = √ |0, k! These vectors are orthogonal and normalized. The space V (1) is a unitary irreducible representation (unirrep) of the Lie superalgebra osp(1|2) [26] (see the next paragraph). It is also the direct sum of two unirreps of the Lie algebra sp(2) = sp(2, R) (one with even k’s and one with odd k’s), known as the metaplectic representations (certain positive discrete series representations of sp(2)) [3,14,29].

Paraboson Fock Space and Unitary Irreducible Representations

807

For a single pair (n = 1) of paraboson operators b+ , b− [8], the defining relation is a triple relation (with anticommutator {., .} and commutator [., .]) given by [{b− , b+ }, b± ] = ±2b± .

(2.4)

The paraboson Fock space [26] is again a Hilbert space with vacuum vector |0, defined by means of 0|0 = 1, b− |0 = 0, {b− , b+ }|0 = p |0,

(b± )† = b∓ , (2.5)

and by irreducibility under the action of b+ , b− . Herein, p is a parameter, known as the order of the paraboson. In order to have a genuine inner product, p should be positive and real: p > 0. A set of basis vectors for this space, denoted by V ( p), is given by |2k =

(b+ )2k |0, √ 2k k!( p/2)k

|2k + 1 =

(b+ )2k+1 |0. √ 2k k!2( p/2)k+1

(2.6)

This basis is orthogonal and normalized; the symbol (a)k = a(a + 1) · · · (a + k − 1) is the common Pochhammer symbol. If one considers b+ and b− as odd generators of a Lie superalgebra, then the elements + {b , b+ }, {b+ , b− } and {b− , b− } form a basis for the even part of this superalgebra. Using the relations (2.4) it is easy to see that this superalgebra is the orthosymplectic Lie superalgebra osp(1|2), with even part sp(2). The paraboson Fock space V ( p) is then a unirrep of osp(1|2). It splits as the direct sum of two positive discrete series representations of sp(2): one with lowest weight vector |0 (lowest weight p/2) and basis vectors |2k, and one with lowest weight vector |1 (lowest weight 1 + p/2) and basis vectors |2k + 1. For p = 1 the paraboson Fock space coincides with the ordinary boson Fock space. This also follows from the general action (b− b+ − b+ b− )|2k = p |2k,

(b− b+ − b+ b− )|2k + 1 = (2 − p) |2k + 1.

(2.7)

Let us now consider the case of n pairs of boson operators Bi± (i = 1, 2, . . . , n), satisfying the standard commutation relations [Bi− , B +j ] = δi j .

(2.8)

The n-boson Fock space is again defined as a Hilbert space with vacuum vector |0, with 0|0 = 1,

Bi− |0 = 0,

(Bi± )† = Bi∓

(i = 1, . . . , n).

(2.9)

The Hilbert space is irreducible under the action of the algebra spanned by the elements 1, Bi+ , Bi− (i = 1, . . . , n), subject to (2.8). A set of (orthogonal and normalized) basis vectors of this space is given by |k1 , . . . , kn =

(B1+ )k1 · · · (Bn+ )kn |0, √ k1 ! · · · kn !

k1 , . . . , kn ∈ Z+ .

(2.10)

We shall see that this Fock space is a certain unirrep of the Lie superalgebra osp(1|2n), with lowest weight ( 21 , . . . , 21 ).

808


We are primarily interested in a system of n pairs of paraboson operators b±j ( j = 1, . . . , n). The defining triple relations for such a system are given by [8] ξ

η

η

ξ

[{b j , bk }, bl ] = ( − ξ )δ jl bk + ( − η)δkl b j ,

(2.11)

where j, k, l ∈ {1, 2, . . . , n} and η, , ξ ∈ {+, −} (to be interpreted as +1 and −1 in the algebraic expressions − ξ and − η). The paraboson Fock space V ( p) is the Hilbert space with vacuum vector |0, defined by means of ( j, k = 1, 2, . . . , n) 0|0 = 1,

b−j |0 = 0,

(b±j )† = b∓j ,

{b−j , bk+ }|0 = pδ jk |0,

(2.12)

and by irreducibility under the action of the algebra spanned by the elements b+j , b−j ( j = 1, . . . , n), subject to (2.11). The parameter p is referred to as the order of the paraboson system. In general p is thought of as a positive integer, and for p = 1 the paraboson Fock space V ( p) coincides with the boson Fock space V (1). We shall see that also certain non-integer p-values are allowed. Constructing a basis for the Fock space V ( p) turns out to be a difficult problem, unsolved so far. Even the simpler question of finding the structure of V ( p) (weight structure) is not solved. In the present paper we shall unravel the structure of V ( p), determine for which p-values V ( p) is actually a Hilbert space, construct an orthogonal (normalized) basis for V ( p), and give the actions of the generators b±j on the basis vectors. 3. The Lie Superalgebra osp(1|2n) The Lie superalgebra osp(1|2n) [15] consists of matrices of the form ⎛ ⎞ 0 a a1 ⎝ at b c ⎠ , 1 −a t d −bt

(3.1)

where a and a1 are (1 × n)-matrices, b is any (n × n)-matrix, and c and d are symmetric (n × n)-matrices. The even elements have a = a1 = 0 and the odd elements are those with b = c = d = 0. It will be convenient to have the row and column indices running from 0 to 2n (instead of 1 to 2n+1), and to denote by ei j the matrix with zeros everywhere except a 1 on position (i, j). Then the Cartan subalgebra h of osp(1|2n) is spanned by the diagonal elements h j = e j j − en+ j,n+ j

( j = 1, . . . , n).

(3.2)

In terms of the dual basis δ j of h∗ , the odd root vectors and corresponding roots of osp(1|2n) are given by: e0,k − en+k,0 ↔ −δk , k = 1, . . . , n, e0,n+k + ek,0 ↔ δk , k = 1, . . . , n. The even roots and root vectors are e j,k − en+k,n+ j ↔ δ j − δk , j = k = 1, . . . , n, e j,n+k + ek,n+ j ↔ δ j + δk , j ≤ k = 1, . . . , n, en+ j,k + en+k, j ↔ −δ j − δk , j ≤ k = 1, . . . , n.


809

If we introduce the following multiples of the odd root vectors bk+ =

√ 2(e0,n+k + ek,0 ),

bk− =

√

2(e0,k − en+k,0 )

(k = 1, . . . , n)

(3.3)

then it is easy to verify that these operators satisfy the triple relations (2.11). Since all η ξ even root vectors can be obtained by anticommutators {b j , bk }, the following holds [7]: Theorem 1 (Ganchev and Palev). As a Lie superalgebra defined by generators and relations, osp(1|2n) is generated by 2n odd elements bk± subject to the following (paraboson) relations: ξ

η

η

ξ

[{b j , bk }, bl ] = ( − ξ )δ jl bk + ( − η)δkl b j .

(3.4)

The paraboson operators b+j are the positive odd root vectors, and the b−j are the negative odd root vectors. Recall that the paraboson Fock space V ( p) is characterized by ( j, k = 1, . . . , n) (b±j )† = b∓j ,

b−j |0 = 0,

{b−j , bk+ }|0 = p δ jk |0.

(3.5)

Furthermore, it is easy to verify that {b−j , b+j } = 2h j

( j = 1, . . . , n).

(3.6)

Hence we have the following: Corollary 2. The paraboson Fock space V ( p) is the unitary irreducible representation of osp(1|2n) with lowest weight ( 2p , 2p , . . . , 2p ). In order to construct the representation V ( p) [27] one can use an induced module construction. The relevant subalgebras of osp(1|2n) are easy to describe by means of the odd generators b±j . Proposition 3. A basis for the even subalgebra sp(2n) of osp(1|2n) is given by the elements {b±j , bk± } (1 ≤ j ≤ k ≤ n), {b+j , bk− } (1 ≤ j, k ≤ n).

(3.7)

The n 2 elements {b+j , bk− }

( j, k = 1, . . . , n)

(3.8)

are a basis for the sp(2n) subalgebra u(n). Note that with {b+j , bk− } = 2E jk , the triple relations (3.4) imply the relations [E i j , E kl ] = δ jk E il − δli E k j . In other words, the elements {b+j , bk− } form, up to a factor 2, the standard u(n) or gl(n) basis elements. So the odd generators b±j clearly reveal the subalgebra chain osp(1|2n) ⊃ sp(2n) ⊃ u(n). Note that u(n) is, algebraically, the same as the general linear Lie algebra gl(n). But the condition (b±j )† = b∓j implies that we are dealing here with the “compact form” u(n).

810


The subalgebra u(n) can be extended to a parabolic subalgebra P of osp(1|2n) [27]: P = span{{b+j , bk− }, b−j , {b−j , bk− } | j, k = 1, . . . , n}.

(3.9)

Recall that {b−j , bk+ }|0 = p δ jk |0, with {b−j , b+j } = 2h j . This means that the space spanned by |0 is a trivial one-dimensional u(n) module C|0 of weight ( 2p , . . . , 2p ). Since b−j |0 = 0, the module C|0 can be extended to a one-dimensional P module. Now we are in a position to define the induced osp(1|2n) module V ( p): osp(1|2n)

V ( p) = IndP

C|0.

(3.10)

This is an osp(1|2n) representation with lowest weight ( 2p , . . . , 2p ). By the PoincaréBirkhoff-Witt theorem [16,27], it is easy to give a basis for V ( p): + , b+ })kn−1,n |0, (b1+ )k1 · · · (bn+ )kn ({b1+ , b2+ })k12 ({b1+ , b3+ })k13 · · · ({bn−1 n k1 , . . . , kn , k12 , k13 . . . , kn−1,n ∈ Z+ .

(3.11)

This is all rather formal, and it sounds easy. The difficulty however comes from the fact that in general V ( p) is not a simple module (i.e. not an irreducible representation) of osp(1|2n). Let M( p) be the maximal nontrivial submodule of V ( p). Then the simple module (irreducible module), corresponding to the paraboson Fock space, is V ( p) = V ( p)/M( p).

(3.12)

The purpose is now to determine the vectors belonging to M( p), and hence to find the structure of V ( p). Furthermore, we want to find explicit matrix elements of the osp(1|2n) generators in an appropriate basis of V ( p). As an illustrative example, we shall first treat the case of osp(1|4). 4. Paraboson Fock Representations of osp(1|4) We shall examine the induced module V ( p) in the case n = 2, with basis vectors |k, l, m ≡ (b1+ )k (b2+ )l ({b1+ , b2+ })m |0

(k, l, m ∈ Z+ ).

Note that the weight of this vector is p p , + kδ1 + lδ2 + m(δ1 + δ2 ). 2 2

(4.1)

(4.2)

The level of this vector is defined as k + l + 2m. The basis vectors |k, l, m are easy to define, and the actions of the generators b1± , b2± on these vectors can be computed, using the triple relations. For the positive root vectors this is easy: b1+ |k, l, m = |k + 1, l, m, b2+ |2k, l, m = |2k, l + 1, m, b2+ |2k

+ 1, l, m = |2k, l, m + 1 − |2k + 1, l + 1, m.

(4.3)


811

For the negative root vectors this requires some tough computations, yielding: b1− |2k, l, m = 2k|2k − 1, l, m + 2m|2k, l + 1, m − 1,

b1− |2k + 1, l, m = ( p + 2m + 2k)|2k, l, m − 2m|2k + 1, l + 1, m − 1,

b2− |2k, 2l, m = 2l|2k, 2l − 1, m + 2m|2k + 1, 2l, m − 1,

b2− |2k, 2l + 1, m = ( p + 2l)|2k, 2l, m + 2m|2k + 1, 2l + 1, m − 1,

(4.4)

b2− |2k + 1, 2l, m = 2l|2k, 2l − 2, m + 1 − 2l|2k + 1, 2l − 1, m + 2m|2k + 2, 2l, m − 1,

b2− |2k + 1, 2l + 1, m = 2l|2k, 2l − 1, m + 1 − ( p + 2l − 2)|2k + 1, 2l, m + 2m|2k + 2, 2l + 1, m − 1. It is now possible to compute “inner products” of vectors |k, l, m, using 0|0 = 1 and (bi± )† = bi∓ . Clearly, vectors of different weight have inner product zero. Let us compute a number of the nonzero inner products. At weight ( 2p , 2p ) there is one vector only, |0, 0, 0 = |0, with 0, 0, 0|0, 0, 0 = 1. (4.5) At level 1 there is one vector of weight ( 2p + 1, 2p ) and one of weight ( 2p , inner products respectively: 1, 0, 0|1, 0, 0 = p,

p 2

+ 1), with

0, 1, 0|0, 1, 0 = p.

At level 2 there is one vector of weight ( 2p + 2, 2p ), one of weight ( 2p , vectors of weight ( 2p + 1, 2p + 1). The inner products are given by:

(4.6) p 2

+ 2), and two

2, 0, 0|2, 0, 0 = 0, 2, 0|0, 2, 0 = 2 p, 1, 1, 0|1, 1, 0 = p 2 , 1, 1, 0|0, 0, 1 = 2 p, 0, 0, 1|0, 0, 1 = 4 p.

(4.7)

From (4.6) it follows already that p should be a positive number, otherwise the inner product (bilinear form) is not positive definite. The matrix of inner products of the vectors of weight ( 2p + 1, 2p + 1) has determinant det

p2 2 p 2p 4p

= 4 p 2 ( p − 1).

(4.8)

So this matrix is positive definite only if p > 1. Thus, for p > 1 both vectors of weight ( 2p + 1, 2p + 1) belong to V ( p); but for p = 1 one vector (2|1, 1, 0 − |0, 0, 1) belongs to M( p) and the subspace of V ( p) of weight ( 2p + 1, 2p + 1) is one-dimensional. One could continue this analysis level by level, but the computations become rather complicated and in order to find a technique that works for arbitrary n one should find a better way of analysing V ( p). For this purpose, we shall construct a different basis for V ( p). This new basis is indicated by the character of V ( p): this is a formal infij j nite series of terms µx11 x22 , with ( j1 , j2 ) a weight of V ( p) and µ the dimension of this weight space. So the vacuum vector |0 of V ( p), of weight ( 2p , 2p ), yields a term p

p

x12 x22 = (x1 x2 ) p/2 in the character char V ( p). Since the basis vectors are given by (b1+ )k (b2+ )l ({b1+ , b2+ })m |0, where k, l, m ∈ Z+ , it follows that

812


char V ( p) =

(x1 x2 ) p/2 . (1 − x1 )(1 − x2 )(1 − x1 x2 )

(4.9)

Such expressions have an interesting expansion in terms of Schur functions, valid for general n. Proposition 4 (Cauchy, Littlewood). Let x1 , . . . , xn be a set of n variables. Then [22] n

i=1 (1 − x i )

1 1≤ j 1, all u(2) representations (m) survive, and V ( p) = V ( p). Theorem 5. The osp(1|4) representation V ( p) with lowest weight ( 2p , 2p ) is a unirrep if and only if p ≥ 1. For p > 1, V ( p) = V ( p) and char V ( p) = (x1 x2 ) p/2 /((1 − x1 )(1 − x2 )(1 − x1 x2 )). For p = 1, V ( p) = V ( p)/M( p) with M( p) = 0. V (1) is the boson Fock space, and char V (1) = (x1 x2 )1/2 /((1 − x1 )(1 − x2 )).

816


The explicit action of the osp(1|4) generators in V ( p) is given by (4.27)-(4.30). The basis is orthogonal and normalized. Note that also for p = 1 this action remains valid, provided one keeps in mind that all vectors with m 22 = 0 must vanish. We end this section mentioning that more general osp(1|4) irreducible representations with no unique “vacuum” were investigated in [4,13]. However their matrix elements were not determined. 5. Paraboson Fock Representations of osp(1|2n) Similarly as in the previous section, we start our analysis by considering the induced module V ( p). First, a new basis for V ( p) will be introduced. In this basis, matrix elements are computed, and from these expressions it will be clear which vectors belong to M( p). A basis for V ( p) was already given in (3.11). From this expression, one finds (x1 · · · xn ) p/2 . i=1 (1 − x i ) 1≤ j n − 1.


819

Fig. 2. Structure of the osp(1|2n) representations V ( p) and V ( p), here illustrated for n = 3 (so only partitions of length at most 3 appear). The boldface line indicates that the corresponding reduced matrix element has a factor ( p − 1); the dotted line indicates that it has a factor ( p − 2)

For p > n − 1, V ( p) = V ( p) and (x1 · · · xn ) p/2 i (1 − x i ) j n − 1, the character is written in the form (5.14), i.e. with a genuine osp(1|2n) denominator (each factor in this denominator corresponds to a positive root α of osp(1|2n) for which 2α is not a root). Can the characters of V ( p) for p ∈ {1, 2, . . . , n − 1} be written in a similar form? In other words, what is E p in the expression char V ( p) = x p/2

sλ (x) = x p/2

i (1 − x i )

λ, (λ)≤ p

Ep

jl−1 0. Furthermore, its contribution can also be computed as Res P

2 e(µiC )eλρi 2 e(µC ) X = (−1)n res Xj21 . . . res jnn−1 resiX n i j ρj j ρi 2 e(µC ) = (−1)n Res P , j ρj

is the path {(X n , i), (X n−1 , jn ), . . . , (X 1 , j2 )}. where P Proof. For the first statement it suffices to remark the following. We each time want to test the function ρi in the exponent after consecutive residues: ρi >n 0, ρi >n−1 0, . . . . However the poles at which we take the consecutive residues are exactly

840

J. Martens

ρ ijn = ρ jn −ρi , ρ ijn−1 = ρ jn−1 − ρi , . . . . As taking a residue at such a pole essentially sets that term to zero, we might as well test for ρi >n 0, ρ jn >n−1 0, ρ jn−1 >n−2 0, . . . . For the second statement, notice that calculating one-dimensional residues res Xil can ρj

essentially be understood in terms of elementary linear algebra as Gaussian elimination for a matrix. Indeed, if you organize all the occurring linear functions (expressed in coordinates) as rows in a matrix as follows: variables used for residues

⎡X n . . . X 1

factors used for poles

all other factors

⎧ ρ ijn ⎪ ⎪ ⎪ ⎪. ⎢ ⎨ .. ⎢ ⎢ ⎢ ⎪ ⎪ρ ij2 ⎢ ⎪ ⎪ ⎩ i ⎢ ⎢ ρi ⎢ ⎢ ⎣

remaining variables

Y1 . . . Ym⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

then taking a residue in a variable can be interpreted as doing the column-elimination for the corresponding column (given by the variable used for the residue) and row (given by the factor), and replacing all but the used row in the original function with the corresponding new rows. The factor of the pivotal entry is all that remains of the row used. From this one immediately sees that taking the consecutive one-dimensional residues comes down to doing row-reduction for the first n rows, replacing the remaining factors in the original function by the corresponding new rows, and dividing the function by the determinant of the upper n × n submatrix in the above matrix. With this in mind it is now clear that one can in fact change the ordering of the poles used for the consecutive residues, as well as the ordering of the variables used, with the sign change that occurs through the determinant as the only penalty. The single condition that needs to be satisfied is that for the new ordering, all the pivotal elements need to be non-zero. Now, given our ordering of the poles (i.e. ρ ijn , . . . , ρ ij2 , ρii ), change the ordering by making the last pole used first (hence obtaining ρii , ρ ijn , . . . , ρ ij2 ). As we have used ρii = −ρi for the first pole, this immediately implies that we might as well change all the other factors (i.e. ρ ijn , . . . , ρ ij2 as well as e(µiC )) to the corresponding factors (i.e. C) ρ jn , . . . , ρ j2 and e(µC )) for the ‘central function’ e(µ . This ordering also has the j ρj advantage that it exactly corresponds to the ordering of the testing explained above, and furthermore the consecutive conditions that the pivotal elements are non-zero is indeed implied through the consecutive tests ρ jl >l+1 0. 2

With this it makes sense to define the following: a complex vector Definition 1. Let X 1 , . . . , X n be a choice of linear coordinates on space V . Then for any rational function f = qp on V , where q = j ρ j , and where each ρ j comes with a preferred polarization, we define Res+X 1 ,...X n inductively as Res+X 1 ,...X n = res+X 1 . . . res+X n

p , q

Equivariant Volumes of Quotients & Instanton Counting

841

where in each step the other variables are treated as generic constants, and where the single-variable residues are defined as res+X

p p = res Xj , q q ρ j >X 0

ρ j in the sum on the right hand side being the factors of q. With this definition we now can summarize the discussion of this section: Theorem 4. Let a compact connected Lie group G act tri-Hamiltonianly on a hyperKähler vector space V , such that the group action commutes with the action of a torus T . Assume that T acts Hamiltonianly with respect to one of the Kähler structures on V . Let ω and µT denote the corresponding Kähler form and moment map on the hyper-Kähler quotient V ///G. Then with the notation as above we have V ///(0,0,0) G

where K˜ is the constant

eω+µT =

2 e(µC ) K˜ Res+X 1 ,...,X n , |WG | j ρj

(−1)n + +n vol(TG ) .

Here X 1 , . . . , X n is a linear coordinate system on the Lie algebra of G, suitable in the sense of (9). For the volume of a symplectic reduction of a vector space with a linear action, the formula is identical except that e(µC ) is missing. We remark here the difference in the definitions of the various residue operations: JKResΛ and the one-dimensional jkres+X on the one side, and Res+X 1 ,...X n on the other. The former, originally coming from a Fourier-transform, involve a positivity test for the coefficient in the exponent of the numerator (hence the choice of the position of Λ and + in the notation). The latter involves a test of positivity on the factors of the denominator. The advantage of the reordering of the poles is that in practice it offers an easier way to go through the calculations of residues: at each step (i.e. when taking the residue for each successive variable) just take the sum of the residues at the poles corresponding to factors where the variable has a positive coefficient. We remark that the way we have written the residue above does not simply apply to rational functions—in particular each factor in the denominator has to come with a preferred sign, which is similar to the choice of polarization used in the Guillemin-Lerman-Sternberg approach to the DuistermaatHeckman formula [GLS96]. In the next section we will use this result further to compare our calculational method with the one given by Nekrasov and Shadchin [NS04] in the physics literature. 5. Instanton Counting We are now ready to apply the calculational techniques described above to our guiding examples: the calculation of the equivariant volumes of the ADHM spaces as they occur in instanton counting. We discuss the corresponding residue formulas, both for their own sake and as examples of how to implement the calculational heuristic given above in various settings, and rederive the results of Nekrasov and Shadchin [NS04]. The most important aspect still requiring settlement is the matter of the singularities of the quotients.

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5.1. Physical background. The recent work regarding instanton counting [Nek03] is a direct answer to an open problem in physics. In particular, it is concerned with N = 2 supersymmetric quantum Yang-Mills theory. One now wants to describe the low-energy behavior of this theory, and using the ideas of the Wilson renormalization group, one finds that this low-energy limit is described by means of another action, the effective action. In general it is extremely difficult to explicitly write down the effective Lagrangian. However, in the case of N = 2 quantum Yang-Mills theory it was shown by Seiberg and Witten in the seminal [SW94] that the entire Lagrangian for the low energy effective theory is determined by a single (multivalued) holomorphic function F SW , the SeibergWitten prepotential. Furthermore, using an assumption that a form of electric-magnetic duality (called Montonen-Olive duality) holds for the quantum theory, Seiberg and Witten managed to describe this prepotential analytically, in terms of period integrals of a family of curves. Their work was a major breakthrough in physics, but led to spectacular advances in mathematics as well, in particular in the framework of the Seiberg-Witten invariants in differential geometry. The original work of Seiberg and Witten was based on an assumption of duality, and though this duality is widely believed to hold, it is conjectural even for physicists as no derivation from first principles is known. In the last 10 years attempts were made to derive and verify their results directly. It was shown that this reduces to calculating certain integrals over moduli spaces of framed instantons on R4 (see e.g. [DHKM02] for background), but the actual calculations of these integrals were very difficult in general. The solution to this outstanding problem was finally accomplished by Nekrasov in 2002 [Nek03]. The strategy Nekrasov employs is to use maximal symmetry on the moduli spaces, induced by change of framing of instantons and rotations in R4 , and then compute equivariant volumes with respect to this group action by means of localization techniques in equivariant cohomology. In particular Nekrasov considers the following generating function3 ∞ Z inst (q, τ, 1 , 2 ) = qk eω+µT ,(1 ,2 ,τ ) , (15) k=0

Mon,c

where Mon,c is the instanton-moduli space of rank n, charge c framed instantons on R4 . The torus T = T 2 × TG is the product of T 2 , the maximal torus of SU (2) that acts diagonally on R4 after the identification R4 = C2 and hence has an induced action on Mon,c , and TG , the maximal torus of the gauge group that acts on Mon,c by changing the framing of the instantons at infinity. Here i and τ are coordinates on the Lie algebras of T 2 and TG respectively. With the interpretation given in Sect. 2 above this is a mathematically well-defined object, a function on an open subset of the Lie algebra of T2 × TG , which by analytic continuation gives a meromorphic (even rational) function on the whole of the complexified Lie algebra of T2 × TG . Nekrasov argues that one can write Z inst (q, τ, 1 , 2 ) as

inst F (q, τ, 1 , 2 ) Z inst (q, τ, 1 , 2 ) = exp , 1 2 where the function F inst (q, τ, 1 , 2 ) is analytic and regular near 1 , 2 = 0. Furthermore, he claims that F inst |1 ,2 =0 corresponds to the instanton part of the prepotential 3 The full generating function Nekrasov studies Z = Z pert Z inst also has an extra factor, Z pert , the perturbative one-loop contribution, which is of no concern to us.


843

of Seiberg-Witten. As the latter can be defined rigorously in terms of periods of certain families of curves, this correspondence gives rise to a remarkable conjecture in geometry. In 2003, this conjecture was proven independently by Nakajima and Yoshioka [NY05a,NY04] and Nekrasov and Okounkov [NO06] for the gauge group SU (n), using very different techniques. In [NS04] Nekrasov and Shadchin indicated how the proof of [NO06] could be adapted to the other classical gauge groups. More recently, a noncomputational proof for all simple gauge groups was given by Braverman and Etingof in [Bra04,BE06]. 5.2. Equivariant volumes of ADHM spaces for SU (n) instantons. 5.2.1. ADHM spaces The space Mon,c we are considering here is the moduli-space of framed SU (n) instantons with instanton number c (cf. [AHDM78,NY04]), constructed as a hyper-Kähler quotient of the vector space Mn,c of ADHM data Mn,c = {(A, B, i, j)}, where A, B, i, j are linear maps between complex hermitian vector spaces V and W of dimension c and n respectively as follows: 8 VT f

A

j

B

i

W and the quotient is taken by the group action of U (V ) which acts on A and B by conjugation and on i and j by left and right multiplication. Strictly speaking, the moduli space of framed instantons (or, equivalently, framed rank n holomorphic bundles on CP2 with second Chern number c) is the non-singular locus of this quotient. This non-singular locus doesn’t satisfy the conditions to apply either the Prato-Wu theorem (2) or the method described above—in particular the moment map for the torus actions we are considering is not proper, due to the fact that the space is not metrically complete in the Kähler metric. One can however extend (‘partially compactify’) these spaces in various ways: first of all, one can allow for so-called ideal instantons, which are interpreted in differential geometry as being (framed) connections whose curvature is concentrated at certain points. This leads to the Uhlenbeck space, which is the full hyper-Kähler quotient: −1 −1 Mun,c = Mn,c ///(0,0) U (V ) = µC (0)//0 U (V ) = µ−1 R (0) ∩ µC (0)/U (V ).

The Uhlenbeck space is highly singular, however. A better option is to extend the nonsingular locus in another way, which provides a desingularization of the Uhlenbeck space. As the group U (V ) occurring in the hyper-Kähler quotient has characters, we can vary the symplectic reduction or GIT quotient (cf. [Tha96,DH98]) by changing the value of the real moment map. This gives the Gieseker space −1 −1 Mn,c = Mn,c ///(+,0) U (V ) = µ−1 C (0)//+ U (V ) = µR (+) ∩ µC (0)//U (V ). g

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J. Martens

The Gieseker space has a modular interpretation in algebraic geometry as the modulispace of framed torsion-free sheaves on CP2 , i.e. torsion-free sheaves that come with a fixed trivialization on a line ∞ ⊂ CP2 (in physics this space is also thought of as the moduli space of instantons on a non-commutative R4 [NS98]). Of particular relevance g for us is that Mn,c is smooth. 5.2.2. Torus actions on ADHM spaces In instanton counting one is interested in the group action given by T 2 × TU (n) , where TU (n) changes the framing at infinity by acting by the maximal torus of the gauge group (for notational convenience it is easier to allow this to be U (n) rather than SU (n)), and T 2 acts by rescaling C2 ⊂ CP2 . In order to lift this action to the space of ADHM data, we must examine the monad constructed out of ADHM data (see e.g. [Nak99, Chap. 2 Don84]). The monad is a sequence of vector bundle maps over CP2 :

x A−x V ⊗O a = x00 B−x12 ⊕ x0 j b =(−x0 B+x2 x0 A−x1 x0 i ) V ⊗ O(−1) −−−−−−−−→ V ⊗ O −−−−−−−−−−−−−−→ V ⊗ O(1). ⊕ W ⊗O

If a set of ADHM data satisfies the vanishing of the complex moment map, this sequence of bundles on CP2 is actually a complex, and the cohomology E = ker b/im a is a bundle (or sheaf) on CP2 , which indeed satisfies the required triviality on the line at infinity. By examining the effect of T 2 on the monad constructed out of the ADHM data, and by interpreting W ∼ = H 0 ( ∞ , E| ∞ ) as the trivialization on the line at infinity ∞ by means of the Beilinson spectral sequence (see e.g.[Nak99,OSS80]), one can lift the action of TU (n) × T 2 to Mn,c as was done in [NY05a]: for (e1 , e2 ) ∈ T 2 , t ∈ TUn this gives (e1 , e2 , t).(A, B, i, j) = (e1 A, e2 B, it −1 , e1 e2 t j).

(16)

5.2.3. Volumes For this torus action one can now calculate the regularized or equivariant volume of the moduli space. We remark that the original question asks for the equivariant volume of the moduli space with respect to the Kähler form it inherits from being included in the Uhlenbeck space (which is an affine variety). Nevertheless, we can work with the Gieseker space by pulling back the symplectic form from the Uhlenbeck space to the Gieseker space—so we desingularize in algebraic geometry but not in symplectic g geometry. This gives a closed 2-form on Mn,c which is degenerate on the exceptional g g set of Mn,c → Mun,c ; on Mon,c ⊂ Mn,c it is exactly the form we are concerned with. Alternatively, one could think of the degenerate symplectic form as a limit of proper symplectic (even Kähler) forms that degenerate in the limit, and the regularized volume that we are interested in as the limit of the corresponding equivariant volumes for the Gieseker space. Despite the degeneracy, one can speak of moment maps with respect to this form. As usual, the sum of the 2-form and the moment map determines a cohomology class in the Cartan model of equivariant cohomology, and one can look at the localization formula for the formal exponential of this class, as was already remarked, even with the degeneracy, in [AB84]. From our viewpoint, we look at the integral of a function with g respect to a volume form, and as Mn,c \Moc,n has measure zero we have ω+µ e = eω+µ . g Mon,c

Mn,c


845

For the torus action described above, one can now give explicit descriptions of the fixed points (which are all isolated), and the isotropy representations on their tangent g spaces, as was done for Mn,c in e.g. [NY05a]. In particular, they can be described very nicely by n-tuples of partitions Y = (Y1 , . . . , Yn ) such that i |Yi | = k. As all these fixed points lie above the same unique fixed point in Mun,c under the desingularization g Mn,c → Mun,c , they all take the same value (i.e. 0) under the moment map for the g degenerate symplectic form on Mn,c , and hence we have by the Prato-Wu theorem g

Mn,c

eω+µ =

1 eµ(Y ) = = 1, g eT (νY ) eT (νY ) Mn,c Y Y

(17)

where the right-hand side has to be interpreted as a formal integral (of the class 1) defined by a localization formula. This is the viewpoint taken by several authors: the integrals that form the coefficients of the Nekrasov partition function are the formal equivariant g integrals of 1 over the Gieseker space Mn,c . On the other hand, we could apply the technique described in the sections above. In order to do this, however, it is very crucial that one thinks of the integral as an equivariant volume rather than the formal integral of 1, as one now has to use the new fixed points ‘at infinity’ introduced by the cut. The value of these new fixed points under the moment map will not be zero, and through the residue the geometry of the moment map gives the recipe for obtaining their contribution. From (16) we can see that the necessary condition for our cutting construction to work is clearly satisfied: there is a subgroup in T 2 × TU (n) acting with global weight 1 on Mn,c . With this lift we can now implement the calculational method described above. The resulting formula is Mon,c

eω+µ =

2 eT (µC ) 1 Resσ+i , c! eT (Mn,c )

where ! " σe − σ f ,

2 = e = f

! " 1 + 2 + σ g − σ h ,

eT (µC ) = 1≤g,h≤c

and eT (Mn,c ) =

! " 1 +σi −σ j

1≤i, j≤c

(2 +σk −σl )

(σm −τo )

! " 1 +2 −σ p +τq ,

1≤k,l≤c 1≤m≤c 1≤o≤n

1≤ p≤c 1≤q≤n

where we use a maximal torus of the form diag(s1 , . . . , sc ) = (eσ1 , . . . , eσc ) for U (c).

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5.3. Equivariant volumes for the other classical gauge groups. We can now try to implement the same method to compute the regularized volumes of the moduli spaces of S O(n) and Sp(n) instantons. We remark that for these other classical groups no equivalent of the Gieseker space is known; hence a direct localization calculation as (17) is not available. The Uhlenbeck spaces do exist here, but as before they are highly singular. We repeat that the integrals we are interested in are the indefinite integrals of n a function, eµT , depending on a parameter in tC with respect to a volume form ωn! on the non-singular locus of these Uhlenbeck spaces. When implementing our method, two minor issues need to be addressed: we need to treat the singularities differently because of this lack of a Gieseker space, and the lifting of the torus action to the space of ADHM data is not automatic. The ADHM construction for gauge groups Sp(n) and S O(n) was discussed in [Don84] and described in greater detail in e.g. [BS00]. In both of these cases the groups occurring by which one has to quotient are simple, and hence one cannot hope to obtain a desingularization of the hyper-Kähler quotient by means of a variation of GIT quotient as was the case for the gauge group SU (n). However, in [Kir85], Kirwan describes a method for constructing desingularizations of singular quotients M//G by blowing up certain and then constructing the desingularizasubvarieties in M to obtain a new space M, tion of M//G as M//G. One could therefore use this approach to obtain an equivariant desingularization of the Uhlenbeck spaces for gauge groups Sp(n) and S O(n), and use these to calculate the equivariant volumes. Nevertheless, since an interpretation of these desingularizations as moduli spaces is at least a priori lacking (see however [Fre05] for related discussions), determining the fixed point data in the hope of applying a direct localization formula is a non-trivial matter. In [JKKW03] the Kirwan desingularization construction was used to develop a residue formula as for intersection pairings in the intersection cohomology of a singular GIT quotient. While we are not directly interested in the intersection cohomology of the Uhlenbeck spaces, we can take a similar approach for calculating the equivariant volumes of the ADHM spaces for symplectic and special orthogonal gauge groups, to obtain the equivalent formula for these volumes to the one derived in the previous section for SU (n), by considering a degenerate form on the Kirwan desingularization.

5.3.1. Sp(n) Following [BS00], we can describe the ADHM construction of the Uhlenbeck space for gauge group Sp(n) as a hyper-Kähler quotient as follows: look at the diagram of linear maps

A

j

&

x V D B DD DD D Φ DD !

WC CC CC C J CC! W∗

j∗

(18)

VJ ∗


847 Φ

where Φ is a (fixed) real structure on V , i.e. an isomorphism V → V ∗ such that J ∼ =

Φ ∗ = Φ, and J is a (fixed) symplectic structure on W (W → W ∗ , J ∗ = −J ). The Sp

space of ADHM data Mn,c here consists of {(A, B, j)}, with the extra conditions that Φ A, Φ B ∈ S 2 V ∗ , and the group divided by O(V ), is determined by Φ. We can write the vanishing of the complex moment map as Φ[A, B] − j ∗ J j = 0, and we can again put together a monad:

x A−x V ⊗O a = x00 B−x12 ⊕ x0 j b =( x2 Φ−x0 B ∗ Φ −x1 Φ+x0 A∗ Φ −x0 j ∗ J ) V ⊗ O(−1) −−−−−−−−→ V ⊗ O −−−−−−−−−−−−−−−−−−−−−−→ V ⊗ O(1). ⊕ W ⊗O (19)

Composing this with Φ on the right gives a self-dual monad, whose cohomology is an Sp(n) instanton. As in the case of SU (n), we are again interested in a torus action given by changing the framing at infinity and rescaling C2 ⊂ CP2 . The former action is readily lifted to Sp Mn,c : t ∈ TSp(n) ⇒ t.(A, B, j) = (A, B, t j). As for the scaling, things become a bit more cumbersome—a lift of the torus action to the space of ADHM data does not seem to be available. However, we can still proceed as before if we temper our ambition and only try to lift the action with weight two—that is to say the action induced by the scaling of C2 ⊂ CP2 given by (e1 , e2 ).(x0 , x1 , x2 ) = (x0 , e12 x1 , e22 x2 ). Indeed, if we now introduce the bundle isomorphism ⎛e ⎞ V ⊗O V ⊗O 1 e v1 v1 ⊕ ⊕ ⎜ e22 ⎟ φ : V ⊗ O −→ V ⊗ O : v2 → ⎝ e v2 ⎠, 1 ⊕ ⊕ w w W ⊗O W ⊗O then by using this isomorphism we obtain

x0 A−e1−2 x1 φ 1 x0 (e12 A)−x1 im x0 B−e2−2 x2 ∼ = im x0 (e22 B)−x2 e1 e2 x0 (e1 e2 j) x0 j ker ker

!

e2−2 x2 Φ−x0 B ∗ Φ −e1−2 x1 Φ+x0 A∗ Φ x0 j

∼ =

and

φ

" ,

1 x Φ−x0 (e22 B)Φ −x1 Φ+xo (e22 B)∗ Φ x0 (e1 e2 j)). e1 e2 ( 2

Hence we can lift the scaling to (e1 , e2 ).(A, B, j) = (e12 A, e22 B, e1 e2 j). As we are just interested in calculating the equivariant volumes there is no problem in lifting a ‘higher weight’—as the equivariant volumes for the different weights are related

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J. Martens

by a scaling of the Lie algebra. Again we can remark that there is a U (1) in T whose action is given by a constant global weight (two, in this case): U (1) → T = T 2 × TSp(n) : s → (s, s, 1). If we now apply the symplectic cut with respect to this circle action — using a weight 2 action however on the copy of C used in the symplectic cut — to compactify we again get a projective space, Sp Sp Sp Mn,c = Mn,c PMn,c , λ

··

·

and we can implement the method as before. Two remarks here: there is an extra factor 1 2 needed to account for the fact that the moduli space is the hyper-Kähler quotient by O(V ) rather than just S O(V ) (the residue formula assumes that the group is connected), and secondly, because we only lifted the ‘weight 2’ action the space of ADHM data the corresponding variables 1 , 2 need to be rescaled. For the calculations below we will fix isomorphisms V ∼ = Cc , W ∼ = C2n , and

0 1 0 I represent Φ by the off-diagonal matrix , and J by −I 0n . Using maximal n 1 0 −1 tori of the form diag(s1 , . . . , sm , sm , . . . , s1 ) for even c = 2m, diag(s1 , . . . , sm , 1, sm−1 , . . . , s1−1 ) for odd c = 2m + 1 for S O(c) and diag(t1 , . . . , tn , t1−1 , . . . , tn−1 ) for Sp(n), we can easily write down all the weights involved. Implementing the calculational method we get 2 1 1 σi eT (µC ) ω+µ Res e = , + Sp Sp 2 |W | Mn,c eT (Mn,c ) where for c = 2m even |W | = 2m−1 m!, 2 =

(σi2 − σ j2 )2 , i< j

"2 "2 ! ! , (1 + 2 )2 − σi + σ j (1 + 2 )2 − σi − σ j

eT (µC ) = (1 + 2 )m i< j

and

⎛ ⎝

Sp

eT (Mn,c ) = k=1,2

(k + σi − σ j ) i, j

i

⎞

(k )2 − (σi + σ j )2 ⎠

i≤ j

l

2 2 1 + 2 1 + 2 2 2 + τl − σi − τl − σi , 2 2

and for c = 2m + 1 odd |W | = 2m m!,


2 =

849

(σi2 − σ j2 )2

σi2 ,

i< j

i

(1 + 2 )2 − σi2

eT (µC ) = (1 + 2 )m i

"2 "2 ! ! , (1 + 2 )2 − σi + σ j (1 + 2 )2 − σi − σ j

i< j

and

⎝k k=1,2

i

l

⎞ (k )2 − (σ1 + σ2 )2 ⎠

⎛

Sp eT Mn,c =

(k2 − σi2 ) i

(k + σi − σ j ) i, j

i≤ j

2 2 1 + 2 1 + 2 1 + 2 2 2 2 2 + τl − σi − τl − σi − τl . 2 2 2 l

5.3.2. SO(n) For the gauge group S O(n) the ADHM construction goes similarly: again using the diagram (18), where in this case Φ : V → V ∗ is a symplectic structure, and S O consists of the triples J : W → W ∗ is a real structure, the space of ADHM data Mc,n % of linear maps (A, B, j), this time with the extra condition that Φ A, Φ B ∈ 2 V ∗ . With this understood, (and the vanishing of the complex moment map again being Φ[A, B] − j ∗ J j = 0) nominally the same monad (19) again gives the desired bundle, from which we can directly see that the same lift of the torus action works (again with ‘double weight’ for the 2-torus that scales): (e1 , e2 , t).(A, B, j) = (e12 A, e22 B, e1 e2 t j) for (e1 , e2 ) ∈ T 2 and j ∈ TS O(n) . With maximal tori written as before it is again just a matter of identifying the weights S O and the complex moment map. We obtain again for all tori involved on the space Mn,c 2 1 σi eT (µC ) , eω+µ = Res + SO ) SO c!2c eT (Mn,c Mn,c where 2 =

(σi2 − σ j2 )2 i< j

i

(1 + 2 )2 − (σi − σ j )2

n

eT (µC ) = (1 + 2 ) 2 i< j

for even n = 2m

(2σi )2 ,

i≤ j

⎛

k=1,2

i, j

i

⎞ k2 − (σi + σ j )2 ⎠

! " k + σi − σ j

⎝

SO eT Mn,c =

(1 + 2 )2 − (σi + σ j )2 ,

i< j

l

2 2 1 + 2 1 + 2 + τl − σi2 − τl − σi2 , 2 2

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J. Martens

and for odd n = 2m + 1 ⎛ SO ⎝ = eT Mn,c k=1,2

i

⎞

+ 2 1 2 2 2⎠ 2 k − (σi + σ j ) − σi 2 i, j i< j i

2 2 1 + 2 1 + 2 + τl − σi2 − τl − σi2 . 2 2 ! " k +σi −σ j

l

The above formulas exactly correspond to those found in [NS04, §5.3]. 5.4. Comparison. We will now compare the calculational strategy outlined above with the work of Nekrasov-Shadchin [NS04]. In [NY05a, §4] a K -theoretic interpretation of the coefficients of Nekrasov’s partition g function (15) is given as follows. Let E be a T -equivariant coherent sheaf on Mn,c and look at 2nr g (−1)i chH i (Mn,c , E),

(20)

i=0

where the character ch denotes the trace of the representation, & defined as a Hilbert series for a representation M with weight decomposition M = µ Mµ as chM =

dim Mµ eµ ,

µ

where the sum is over all characters. For the case of the structure sheaf O, (20) reduces to chH 0 (Muc,n , O), and the following K -theoretic localization formula is explained [NY05a, Prop. 4.1]:

g

(−1)i chH i (Mn,c , E) =

i

! g "T F⊂ Mn,c

i∗ E %F ∗ , −1 ν F

% where as usual the sum in the right-hand side is over all fixed points, and −1 stands for the alternating sum of exterior powers as an element in the equivariant K -group K T (for all details see [NY05a]). For the case of the structure sheaf O this reduces to chH 0 (Mun,c , O) =

! g "T F⊂ Mn,c

%

1 ∗ −1 ν F

.

In the (co)-homological setting, in the mathematics literature on instanton counting [NY05a,NY05b,Bra04] most authors use the equivariant integral 1 as the coefficients for the generating function, where the integral is defined in practice through a localization theorem 1 = F e(ν1F ) . In [NY05a] the link between this and the K -theoretic approach is made by β 2nc 1 % = lim lim β 2nc chH 0 . 1 = ∗ = β→0 g e(ν F) β→0 Mn,c −1 (ν F )


851

One can think of the limit β → 0 as formally inducing a multiplication by the inverse Todd class prescribed by a Riemann-Roch theorem; this has the effect of changing the denominators of the localization theorem for K -theory to those of (co)homology. In [NS04], Nekrasov–Shadchin use the same philosophy to calculate the partition functions for all classical groups. They physically interpret the limit β → 0 as arising from considering the 4-dimensional theory as the limit of a 5-dimensional theory compactified on a circle of radius β. Then they calculate chH 0 (Mun,c ) by means of the ADHM construction as follows. Let ρ1 , . . . , ρ. be the characters of the T × TG action on Mc,n . Then the T × TG -equivariant character is 1 . chH 0 (Mn,c , O) = (1 − ρi ) Furthermore, the T × TG action on µC is homogeneous, say with weights ν1 , . . . , ν L for L = dim G. Hence we have (1 − νi ) 0 −1 . chH (µC (0), O) = (1 − ρi ) −1 In order to get the character over µC (0)//G, we need to get the G-invariant part of this. The projection onto the G-invariant part is given by averaging over the whole of G, 1 0 −1 chH (µC (0)//G, O) = chH 0 (µ−1 C (0), O), volG G

but by the Weyl integral formula this can be reduced to an integral over the maximal torus, 1 chH 0 (µ−1 (0)/ /G, O) = chH 0 (µ−1 C C (0), O). |WG |volTG TG Now factor the torus as T = U (1) and break up the integral into circle integrals. The integrand in each lives on C∗ , and one can think of these as contour integrals in the plane. These contour integrals can be calculated by Cauchy’s theorem, for each integral keeping the other variables as generic constants. Finally, take the limit β → 0+ to reduce everything to cohomology. Nekrasov-Shadchin interpret this limit as a contour integral of a meromorphic top degree form over the complexified Lie algebra. In either case the actual evaluation happens by means of iterated residues, for each variable choosing a half-space in which to consider the poles. It is thus that the formula gives exactly the same result as our method outlined above. It is interesting to remark that the non-compactness manifests itself in different ways in the two methods. Let us illustrate this by calculating the regularized volume of the simplest hyperKähler quotient, C4 ///C∗ , where s ∈ C∗ acts as s.(x1 , x2 , x3 , x4 ) = (sx1 , sx2 , s −1 x3 , s −1 x4 ). Furthermore we consider a T 2 action on C4 by (t1 , t2 )(x1 , x2 , x3 , x4 ) = (t1 x1 , t2 x2 , t2 x3 , t1 x4 ). As the complex moment map µC for the C∗ action is x1 x3 + x2 x4 , the T 2 action indeed 2 4 preserves µ−1 C (0), and furthermore the moment map for the T action on C clearly has a component that is proper and bounded below, hence all the conditions are satisfied.

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For the calculation according to Nekrasov-Shadchin we need to find the Hilbert-series of O on µ−1 C (0), which is 1 − t1 t2 . (1 − st1 )(1 − st2 )(1 − t2 s −1 )(1 − t1 s −1 ) ∗ In order to calculate the Hilbert series over the quotient µ−1 C (0)//C we then have to take the contour integral ' 1 − t1 t 2 1 ds ch = 2πi s (1 − st1 )(1 − st2 )(s − ts2 )(1 − ts1 ) 1 − t1 t2 1 − t1 t 2 . = t1 + 2 2 (1 − t1 t2 )(1 − t2 )(1 − t2 ) (1 − t1 )(1 − t1 t2 )(1 − tt21 )

Finally set t1 = e−βτ1 , t1 = e−βτ2 and s = e−βσ and take limβ→0 β 2 ch, which gives τ1 + τ2 τ1 + τ2 1 + = . (τ2 + τ1 )(2τ2 )(τ1 − τ2 ) 2τ1 (τ1 + τ2 )(τ2 − τ1 ) 2τ1 τ2

(21)

On the other hand, in the method we have outlined above, we can use the symplectic cut on C4 with respect to the diagonal T 1 ⊂ T 2 , which gives (C4 )λ = CP3 C4 . For clarity we shall not take the calculational shortcut discussed in Sect. 4.2, but actually go through the procedure of making the symplectic cut and taking the limit as λ → ∞. There are five fixed points for the C∗ × T 2 action on this space, of which we only need to consider [1 : 0 : 0 : 0] and [0 : 1 : 0 : 0] for the Jeffrey-Kirwan residue formula. Applying this gives eλ(σ +τ1 ) (−2σ + τ2 − τ1 ) (−σ − τ1 )(τ2 − τ1 )(τ2 − τ1 − 2σ )(−2σ ) eλ(σ +τ2 ) (−2σ + τ1 − τ2 ) . + jkres+σ (τ1 − τ2 )(−σ − τ2 )(−2σ )(τ1 − τ2 − 2σ )

jkres+σ

As usual, we are only interested in the terms that survive the λ → ∞ limit on the open cone of Lie(T 2 ); hence for each of the two terms above we only need to consider the residue at the pole that cancels the exponent, which gives the exact same expression as (21). A. Proof of Theorem 3 One could essentially try to adapt any proof of Theorem 3 to the equivariant setting. The approach we take here is almost completely based on [JK05], and we restrict ourselves to commenting on how to adjust it. Proof. We can break up the proof in two steps: first reduce the case where G is a general compact connected Lie group to the corresponding statement for a maximal torus TG of G, and then prove the theorem just for the case when G is a torus. The first step is achieved by the abelianization theorem:


853

Theorem 5. Let M as above be equipped with commuting Hamiltonian group actions of compact connected Lie groups G, H , let TG be a maximal torus of G, and let a be an ∗ element of HG×H M. Then in H H∗ (∗), 1 κG (a) = κTG ( 2 a). |W | G M//G M//TG This theorem is a corollary of the Jeffrey-Kirwan residue theorem [JK95b], but was also proven directly by Martin [Mar]. The original theorem of Martin was not formulated in an equivariant setting (i.e. no H was present), but it suffices to remark here that the proof given in [Mar] is valid without modifications in the equivariant case as well. Now, to prove Theorem 3 for a torus G = T , we will first outline the strategy of the proof given in [JK05], and then comment on how it can be used for our equivariant purposes. Begin by choosing a cone Λ, as follows. Look at the weights of the torus T at all the components of M T . Each of these determines a hyperplane in t. Choose a connected component of the complement of their union; call this Λ. Then look at a polyhedral cone containing the dual cone, say Λ∗ ⊂ Σ. The next step is to take the symplectic cut MΣ of M with respect to Σ—having chosen Σ in such a way that this is smooth. For technical purposes we here first alter our moment map for the T -action a little by choosing a point p ∈ Σ, and replacing µT by µ = µT − p. As we inherit T. a commuting action of T and H on this, we can investigate the fixed-point set MΣ The fixed-point components come in three flavors: ‘old ones,’ i.e. components F with ˜ ∈ ∂(Σ), and the component µ(F) ∈ Σ 0 , ‘new ones,’ i.e. components F˜ with 0 = µ( F) at the vertex of Σ, which we can identify with (M//p T ). Applying the localization theorem to MΣ gives

eω˜ η = MΣ

eω˜ i ∗ η eω˜ i ∗˜ ηΣ eω˜ i ∗ η F F + + . F˜ eT1 (νG ) F eT (ν F ) M//p T eT1 (ν M//T ) F

(22)

F˜

As a function on t⊕h this is ‘holomorphic’ (i.e. the analytic continuation is holomorphic on the complexification tC ⊕ hC ), and for such functions it is proven by Jeffrey and Kogan [JK05, Lemma 3.3] that the residue with respect to Λ and −Λ give the same results. This leads to the equality ⎛ ⎞ ω˜ i ∗ η eω˜ i ∗ η eω˜ i ∗˜ ηΣ e Λ⎝ F F ⎠ [d x] JKRes + + eT (ν F ) eT1 (ν M//T ) ˜ eT1 (νG ) F M/ / T F p F F˜ ⎛ ⎞ eω˜ i ∗ η ∗ ω ˜ ω ˜ ∗ e iF η e i η ⎠ F˜ Σ + + = JKRes−Λ ⎝ [d x]. e (ν ) e (ν ) e ˜ F T F M//p T T1 (ν M//T ) F T1 G F

F˜

(23) The modification of the moment map from µT to µ is done exactly to ensure that the residue operation is valid for all the terms appearing above. Next it is shown in [JK05, §5.2] that in fact, for small enough > 0 and with the choices of Λ, Σ and µ as above, 4 terms in this equality are zero, leading to eω˜ i ∗ η eω˜ i ∗ η F JKRes−Λ [d x] = JKResΛ [d x]. M//p T eT1 (ν M//T ) F eT (ν F ) F

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Finally, take the limit as → 0. By identifying the weights of T on ν M//T (which is a bundle of rank dim t), one obtains lim+ JKRes−Λ

→0

M//T

eω˜ i ∗ η [d x] = eT1 (ν M//T )

M//T

eω κη,

(24)

and finally this gives M//T

κ(η)eω =

JKResΛ

F⊂M T

F

eω˜ i F∗ η [d x]. eT (ν F )

This is how Theorem 2 is proved in the non-equivariant case in [JK05]. Now in order to make this valid in the presence of the extra group action of H , we remark that because the actions of T and H on M commute, the action of H descends to MΣ . Now we can use our approximation of the Borel model by smooth finite-dimensional spaces M × E i H/H , determined by finite-dimensional approximations E i H → Bi H to the classifying space E H → B H . These spaces are not symplectic, but they still are Poisson manifolds, with the fibers of M × H E i H → Bi H as the symplectic leaves. Furthermore, as the moment map µG : M → g∗ -action is invariant under H , we get moment maps M ×G E i H → g∗ in the sense of Poisson geometry; see e.g. [OR04]. Also, as the proof of the localization theorem used in (22) essentially only uses the functoriality of the integral as a push-forward (see [AB84]), we can apply it to other (proper) push-forwards as well. In particular, consider (MΣ × E i H )/H → Bi H. Denote the corresponding push-forward by i , so that i

: HT∗G (((MΣ × E i H )/H ) → HT∗G (Bi H ) = HT∗G (∗) ⊗ H ∗ (Bi H )

and i

: HT∗G ((F × E i H )/H ) → HT∗G (∗) ⊗ H ∗ (Bi H ).

The equivalent of the localization formula is then i F∗ α . α= i i eT (ν F×Ei H ) T F×E i H H

⊂

M ×E i H H

H

We can now apply the residue operation JKResΛ to both sides (or rather the obvious extension of the residue from k(t) to the various k(t) ⊗ H ∗ (Bi H )), and we observe that for the same reasons as in [JK05], the equivalences of (23) and the cancellations of the four terms are still valid—this time interpreted as happening in H ∗ (Bi H ). The same is the case for the equivalent statement of (24): essentially the only thing we need to remark for this is that the weights for the action of T occurring on the normal bundles of M//T iH . in MΣ are the same as the weights for the normal bundle of M//TH×Ei H in MΣ ×E H


855

Furthermore, as all the spaces are finite-dimensional, the inverses of the Euler classes exist in the usual way. So we have η [d x]. κ(η) = −JKResΛ e (ν F×E i,M iH ) T i T F⊂M

H

Now we want to take the limit i → ∞ to obtain the desired H -equivariant result. We need to observe in which ring we allow all of the manipulations to take place. Begin by remarking that in the proof of the abelian localization theorem [AB84], which was the basis of all of our localization results, one had to kill some of the torsion in the equivariant cohomology ring, but tensoring with the full fraction field k(t) isa bit excessive. In particular, we can do with just inverting all products of linear factors i βi , with βi ∈ t∗ . Doing this has the advantage that we still have a grading on the ring we ∗ ∗ ∗ ∗ obtain (say H T (∗)). This gives a bigrading on HT ×H (F) = HT (∗) ⊗ H H F for all T F ⊂ M . Now complete these rings by means of the filtration · · · ⊂ Rk+1 ⊂ Rk ⊂ Rk−1 ⊂ . . . , where Rk consists of finite sums of elements of bidegree (i, j) such that i + 2 j ≥ k. This completion has all the desired properties (though it is of course not unique as such): it contains the formal cohomology classes eω˜ , and it allows us to invert the equivariant Euler classes — which because of the splitting principle we can write without loss of generality as eT ×H (ν F ) = i (βi + c1 νi,F ) with βi ∈ t, c1 νi,F ∈ H H2 F—as 1 = eT ×H (ν F )

i

∞ c1 (νi,F ) j 1 − . βi βi j=0

The residue JKResΛ of an integral of such a class will lie in the usual completion of H H∗ (∗) by degree. This completes the proof of Theorem 3. Acknowledgements. I would like to thank Lisa Jeffrey, Mikhail Kogan, Nikita Nekrasov, Michèle Vergne and Peter Woit for useful conversations. The bulk of the work presented here was done while I was a graduate student at Columbia University, and I would like to thank my advisor Michael Thaddeus for continuous guidance and encouragement, as well as a careful reading of the final version of this article. Finally I would like to add that this work was largely inspired by reading the early paper [MNS00]. This article was partially written during a stay at the Max-Planck-Institut für Mathematik, Bonn, whose hospitality and support is gratefully acknowledged.

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