Approximation of nonessential spectrum of transfer operators

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Article

Reference

BALADI, Viviane, HOLSCHNEIDER, Matthias

BALADI, Viviane, HOLSCHNEIDER, Matthias. Approximation of nonessential spectrum of transfer operators. Nonlinearity, 1999, vol. 12, no. 3, p. 525-538

DOI : 10.1088/0951-7715/12/3/006

Available at:

http://archive-ouverte.unige.ch/unige:12575

Disclaimer: layout of this document may differ from the published version.

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Approximation of nonessential spectrum of transfer operators

Viviane Baladi and Matthias Holschneider

December 1998

Abstract. We give sucient conditions to approximate the \nonessential" spectrum of a bounded operator^Lacting on a Banach space^Bby part of the spectra of a sequence of compact (or nite rank) operators^L_j = (Id^?_j)^L(Id^?_j), where Id^?_j is a suitable family of uniformly bounded operators which approach the identity. (By nonessential spectrum we mean here all the spectrum outside of the disc of radius equal to the essential spectral radius.) For this, we combine the formulas

ess(^L) = lim_m

!1

(limsup_j

!1

(inf)^k_j^L^m^k)¹^=m= lim_m

!1

(limsup_j

!1

(inf)^kL^m_j^k)¹^=m^; for the essential spectral radius with nonstandard perturbative results on the stability of the nonessential spectrum of quasicompact operators. We present concrete applications to transfer operators of smooth expanding maps using multiresolution analysis (large scale approximation projections).

1. Introduction

Matrices, and more generally linear operators on innite-dimensional vector spaces, are ubiquitous tools which permeate pure and applied mathematics. A natural problem, which has kept mathematicians busy for centuries, is to determine, or at least approximate, their spectrum (in innite-dimensional situations, sometimes only a discrete part of it). In this work, we are concerned with the innite-dimensional (Banach space) situation, and we deal with bounded linear operators which are not necessarily compact. Our main result (Proposition 3 in Section 2) is a list of conditions guarantee- ing that a subset of the eigenvalues of a sequence of compact or nite-rank operators (Id ^?j)^L(Id ^?j) (together with the corresponding eigenspaces) converges to those eigenvalues of the original operator ^L which are outside of a disc containing the essential spectrum. The simple proof combines a convenient exact formula for the essential spectral radius (Theorem 1 from [H1]) with a non-standard { and somewhat unex- pected { perturbative result (Theorem 2 from [BY]), which had originally been used to control the spectrum of randomly perturbed dynamical systems.

1991 Mathematics Subject Classication. 41A30 42C15 47A10 47B38 58F19.

V.B. is partially supported by the Fonds National Suisse de la Recherche Scientique.

Typeset by^A^M^S-TEX

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Sequences of compact or nite rank operators Id^?j for which our results hold can be explicited in some cases (sometimes via multiresolution analysis, using wavelets).

In Section 3, we explain a specic dynamical systems setting where our scheme works.

The linear operator there is the Ruelle transfer operator associated to a dierentiable uniformly expanding dynamical system on a torus. (Transfer operators, sometimes also called Perron-Frobenius operators, are very powerful tools to study the ergodic properties of dynamical systems. We refer e.g. to [R] and the references therein for the framework of Section 3.) The Banach spaces are Sobolev or Holder spaces, and the nite-rank operators Id ^?j are constructed using the Meyer [M] orthonormal wavelet basis (it would be interesting to actually run the algorithm on a computer).

An important and famous nite-dimensional matrix scheme used in ergodic theory of dynamical systems (to approximate the physical, or \SRB" measure, together with its rate of mixing) is the Ulam method. Recent numerical and theoretical work has shown that not only the maximal eigenvalue, but also further spectral values of Ulam matrices approximate well part of the spectrum associated to various types of chaotic dynamics (see in particular the sequence of papers and eective algorithms of Dellnitz and collaborators [DJ], and the rigorous results of Hunt [Hu], and Froyland [Fr]; we also mention the recent paper [KMY] | see also [K1] | for similar approximation results, together with quantied estimates on the speed of convergence, nally [BIS] contains results obtained using Theorem 2 below from [BY]). It seems to us, however, that since Ulam matrices are obtained by locally constant approximations, they cannot describe the action of the dynamics on observables smoother than Holder or Lipschitz. Our scheme, on the other hand, is applicable to a wider range of smoothness classes.

We end this introduction by mentioning, in order of expected diculty, three direc- tions for future research. As soon as one proves that a mathematical object can be approximated by a sequence, one obvious question is the speed of convergence. For the case considered in Section 3, we believe that exponential speeds of convergence hold (by analogy to the results in [KMY], e.g.).

A second natural problem consists in extending our dynamical results from Section 3 to compact boundaryless manifolds more general than the n-torus ^Tⁿ. This should be possible by developing and/or applying the necessary multi-resolution analysis.

Last, but denitely not least, we have limited ourselves to uniformly expanding dynamical systems for which the transfer operator has nice spectral properties when acting on smooth functions. When the dynamics is uniformly hyperbolic, the inverse maps im- prove smoothness along unstable manifolds but make functions less smooth along stable manifolds. Although recent progress has been made in our understanding of analytic [Rg], but also dierentiable [Li, Ki], hyperbolic settings, one still does not have a good Banach space framework for the transfer operator. Perhaps our approximation scheme can be extended to the hyperbolic setting via the use of \directional" Banach spaces.

(Further extensions to nonuniformly hyperbolic dynamics would also be desirable.)

Acknowledgements:

V.B. thanks G. Courtois, G. Keller, and J. Buzzi for useful comments. She gratefully acknowledges the hospitality of IHES where part of this work was done. M.H. is thankful to the University of Geneva for its kind hospitality.

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2. Approximation of the discrete spectrum: two abstract results

We rst recall a few basic denitions and facts (see [K] and [DS] for more information).

Let (^B;^k^k) be a Banach space, that will always be assumed innite-dimensional. (Since we mostly have function spaces in mind, we denote vectors in^Bby', etc.). Denote by B(^B) the set of bounded linear operators acting in^B (noting^kLkfor the operator norm of ^L ² B(^B)), by K(^B) B(^B) the ideal of compact operators, and by F(^B) K(^B) the ideal of nite rank operators. For ^L ² B(^B) the resolvent set of ^L is the set of complex numbersz so that ^L^?zId : ^B^! ^B is an invertible operator with a bounded inverse (^L^?zId)^?1 ²B(^B). The spectrum (^L) of ^L is the set ofz ² ^C which are not in the resolvent set of ^L. The spectral radius (^L) of ^L is

(^L) = sup^fjz^j s.t. z ²(^L)^g:

As is well known, the spectral radius of ^L can be obtained as the following limit:

(^L) = lim_m

!1 kLmk

1=m:

An elementz ²(^L) is aneigenvalueof^Lif^L?zId has nontrivial kernel. Thegeometric multiplicity of an eigenvalue z is the dimension 1 m¹(z) ¹ of its eigenspace ^f' ²

B s.t. (^L^?z)'= 0^g, and its(algebraic) multiplicityis the dimensionm²(z)¹ of the generalised eigenspace^f'²^B s.t. ⁹m1;(^L^?z)^m'= 0^g. (We havem¹(z) m²(z).) The supremum 1i(z) ¹of thosemwhich occur in the denition of the generalised eigenspace of an eigenvalue z is called the index of z.

Theessential spectral radius ^ess(^L) of ^Lis the smallest number 0 such that any ²(^L) with modulus^j^j> is an isolated eigenvalue of nite (algebraic) multiplicity of ^L. We sometimes use the informal terminology \nonessential spectrum" or \discrete spectrum" to denote the spectrum of ^L outside of the disc of radius ^ess(^L).

There exist several denitions for theessential spectrumof a linear bounded operator.

Browder's [Br, Section 6] essential spectrum is the set of those z ²^C such that at least one of the three following possibilities holds: z is a limit point of (^L), or (^L^?zId)^B is not closed, or the generalised eigenspace ^f' ² ^B s.t. ⁹m 1;(^L^?z)^m' = 0^g has innite dimension. Wolf's [W] essential spectrum is the set of those z ² ^C such that

L?zId is not Fredholm (see [K]). In general the Wolf and Browder essential spectrum do not coincide (as noted in [N], the complement of the Browder essential spectrum is the union of those components of the complement of the Wolf essential spectrum which meet the resolvent set, in particular the Browder essential spectrum always contains the Wolf essential spectrum).

Our denition for the essential spectral radius is consistent both with Browder's and Wolf's denition of essential spectrum as we explain now. Firstly,^ess(^L) is the radius of the smallest disc containing the Browder essential spectrum (because of [Br, Lemma 17, p. 110]). Secondly, ^ess(^L) is also the radius of the smallest disc containing the Wolf essential spectrum. This second property can be deduced from two facts: on the one hand z is in the Wolf essential spectrum if and only if ^L^?zId is invertible modulo

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K(^B) if and only if ^L^?zId is invertible modulo F(^B), see [L, Chapter IX, Theorem 6]. On the other hand Nussbaum's formula [N] states that the radius ^ess(^L) of the smallest disc containing the Browder essential spectrum coincides with

^ess(^L) = lim_m

!1

(inf^fkL^m^?^{K k} s.t. ^K²K(^B)^g)¹^=m: (2.1) By the above equivalent formulations of the Wolf spectrum, Nussbaum's formula can be modied to

^ess(^L) = lim_m

!1

(inf^fkL^m^?^F^k s.t. ^F ²F(^B)^g)¹^=m: (2.2) This is a non-trivial result since the set F(^B) of nite rank operators is not necessarily dense in the ideal K(^B) of compact operators. (The paper [F] of Fried was extremely useful in clarifying the above points.)

We denote by^B the dual space of the Banach space^B, i.e., the space of bounded linear functionals :^B^!^C. (Having in mind mainly complex measures and distributions we write , for elements of ^B.) For ² ^B, we use the notation (') = (^j'). For

L2B(^B), we write ^L ²B(^B), for the dual operator dened by (^L^j') = (^jL(')).

Recall that operators of nite rank on ^B can be written as

7!

X

²A( )'; where A is a nite index set, ²^B, and ' ²^B.

Our rst abstract result is a list of exact formulas (probably well-known \in spirit") for the essential spectral radius of a bounded operator ^L (see [H1] for proofs). They hold for Banach spaces possessing suitable families of bounded operators j converging to zero and for which (Id ^?j)^L is compact. Candidates for such families of operators can be constructed via multiresolution analysis for many classical function spaces (most notably Sobolev, but more generally Triebel, and partly Besov-Holder classes), of periodic functions on the torus ^Tⁿ, say (see e.g. [H1, H2], see also Section 3 below for applications to transfer operators, where the Id ^?j are in fact projections).

Denition (Compact approximation of the identity for

(^L;^B)

).

A sequence of operators Id ^?j ²B(^B), for j ²^Z⁺, is a compact uniformly bounded approximation of the identity for (^L;^B) if:

(i) ⁹ const >0 s.t. ^kj^kconst ;⁸j²^Z⁺; (ii) (Id ^?j)^L is compact,⁸j²^Z⁺;

and (iii) ^B⁰= ^B⁰(^fj^g) =^f'²^B s.t. lim_j

!1

j(') = 0^g is dense in ^B: We shall also need a dual notion:

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Denition (

-compact approximation of the identity for

(^L;^B)

).

A sequence of operators Id^?j ²B(^B), forj ²^Z⁺, is a-compact uniformly bounded approximation of the identity for (^L;^B) if it satises (i) together with:

(ii) ^L(Id ^?j) is compact, ⁸j²^Z⁺; and (iii) ^B⁰ =^B⁰(^fj^g) =^f ²^B s.t. lim_j

!1

_j() = 0^g is dense in ^B:

Stronger results will hold for sequences of operators satisfying certain hierarchical constraints (which hold in particular in the setting of Section 3):

Denition (Hierarchical compact approximation of the identity for

(^L;^B)

).

A sequence of operators Id ^? j ² B(^B) is called a hierarchical compact (respectively

-compact) approximation of the identity for (^L;^B) if it satises (i), (ii) and (iii) (respectively (i), (ii);(iii)) together with

(iv) jj⁺¹ = j⁺¹j = j⁺¹; ⁸j²^Z⁺: (2.3) The operator ^L is said to act in scales, respectively -scales, for k with respect to the hierarchical compact approximation of the identity, if there is k ²^Z⁺ such that

(v) lim_j

!1

kj⁺k^L(Id ^?j)^k= 0;

respectively

(v) lim_j

!1

k(Id ^?j)^Lj⁺k^k= 0: ^(2.4) If (v) (respectively (v)) in (2.4) holds fork = 0 then^Lis said to actexactly(respectively

*-exactly) in scales on ^fj^g.

Theorem 1 (Holschneider [H1], 1996).

^Let ^L²B(^B). Suppose there is a compact uniformly bounded approximation of the identity ^fId ^?j^g for (^L;^B) (i.e., satisfying (i;ii;iii)). Then ^B⁰(^fj^g) =^B, and

^ess(^L) = lim_m

!1

(limsup_j

!1

kj^Lm

k)¹^=m = lim_m

!1

(liminf_j

!1

kj^Lm

k)¹^=m: (2.5) If there is^fId ^?j^ga -compact uniformly bounded approximation of the identity in (^L;^B) (i.e., satisfying (i;ii;iii)), then ^B⁰(^fj^g) =^B and

^ess(^L) = lim_m

!1

(limsup_j

!1

kLmj^k)¹^=m = lim_m

!1

(liminf_j

!1

kLmj^k)¹^=m: (2.6) If there is a hierarchical compact (or -compact) approximation of the identity^fId ^? j^g, and if there is k ² ^Z⁺ such that ^L acts in scales for k on ^fj^g (i.e., (i;iv) and

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either(ii;iii;v) or (ii^?iii;v) hold), then any of the four limits in (2.5{2.6)coincide with ^ess(^L).

Remark 1. In the last assertion of the theorem, if j satises^kj^k1, then the interior limit actually exists. Indeed, for any bounded^A we have by (iv)

kAj^k=^kAj^?1j^k^kAj^?1^k:

Therefore the sequence^kAj^kis non-increasing. The same argument applies to^kj^Ak. Remark 2. If the sequence j satises only (i) and (ii) (respectively (ii)) then the Nussbaum formula (2.1) clearly gives the upper bounds in (2.5), respectively (2.6) (see also the proof in the Appendix). In some applications neither (iii) nor (iii) can be assumed to hold, but an assumption similar to the one appearing in [K2], and that we state now, may be used. (Note that a key idea to obtain lower bounds for the essential spectral radius by constructing almost eigenvalues was contained in the beautiful short paper [Ma] of Mather.) Suppose that j and^L are such that (i;ii) hold and that there is a double sequence of innite dimensional closed spacesV_j^m j^B, forj;m²^Z⁺ and a constant C > 0 so that for each xedm and all sequences '_mj ² V_j^m with ^k'_mj^k= 1 we have

limsup_j

!1

kLm'_mj^kClimsup_j

!1

kLmj^k: (2.7)

Then ^ess(^L) = lim_m

!1

(limsup_j

!1

kLmj^k)¹^=m:

(See [H2, Theorem 3.3, and also Section 9] for an analysis of transfer operators acting on homogeneous (s+)-Holder spaces, with 0 < < 1 and s ² ^Z⁺, using condition (2.7).)

The idea to use families of projection operators to exploit the essential spectrum is not new. For example, Persson [P] gives a formula for the lower bound of the essential spectrum of a self-adjoint operator in L²(^Rⁿ) by using restrictions of the operator to complements of compact sets of ^Rⁿ. More recently, Keller [K3] developed a theory of quasi-nuclear operators and applied it to construct dynamical Fredholm determinants associated to transfer operators in an analytic setting (his Proposition 2.2 is related to our Theorem 1, but it requires additional assumptionson the approximatingprojections, in particular the nontrivial hypothesis [K3, (2.11)]).

Theorem 1 will be used in combination with the following result on spectral approximations (which was originally proved in view of understanding small stochastic perturbations):

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Theorem 2 (Baladi{Young [BY], 1993).

For ^L ² B(^B), let ^Lj ² B(^B) be a sequence of operators with

kLj^k const;⁸ j²^Z⁺ and _jlim

!1

Lj(') =^L(');⁸'²^B:

Assume that for any > ^ess(^L), there is m⁰ ²^Z⁺ such that for each mm⁰, there is j⁰(m) such that for all j j⁰(m)

kL

m

?Lmj^k^m: (2.8)

Then for any > ^ess there is J() such that the essential spectral radius ^ess(^Lj)<

for each jJ.

Furthermore, writingX, respectively Xj (withjJ()), for the (nite-dimensional) direct sum of the generalised eigenspaces associated to the eigenvalues of ^L, respectively

Lj, of modulus larger than , and denoting the corresponding spectral projections by ^P,

Pj, we have

(1) The norms ^{kP ?}^Pj^k tend to zero as j^! ¹. More precisely, there is >0 such that

kP?Pj^kexp^?^m⁽^j⁾ ; ⁸jJ(); (2.9) where

m(j) = max^fm²^Z⁺ s.t. j j⁰(m)^g:

(2) The Hausdor distance between the spectrum of ^LjX and that of ^Lj^jXj tends to zero as j ^!¹. More precisely, for all j J()

Hdist((^LjX);(^Lj^jXj)) const(C_X⁽^m⁽^j⁾⁾(j) +C_X⁽¹⁾(j))¹^=d; (2.10) where d 1is the maximum of the indices of eigenvalues of ^L of modulus larger than and

C_X⁽^m⁾(j) = max_'

2X

k'^k=1

k(^L_mj^?^L^m)'^k: (2.11) (Note that C_X⁽^m⁽^j⁾⁾(j)^m⁽^j⁾ for jJ().)

(3) If ' ² X is an eigenvector for ^L and an eigenvalue of algebraic multiplicity da and index di 1 , then for each j J() the operator ^Lj has 1 ` da

eigenvectors 'i;j for eigenvalues i;j (i = 1;::: ;`), with sum over i = 1;::: ;`

of the algebraic multiplicities of i;j equal to da, and max

1i`max(^j^?i;j^j;^k'^?'i;j^k)const(C_X⁽^m⁽^j⁾⁾(j) +C_X⁽¹⁾(j))¹^=dⁱ: (2.12)

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Theorem 2 is obtained by combining Lemmas 1, 2 and 3 from Section 2 in [BY], noting that assumptions (A.1) and (A.3) there follow from the denition of the essential spectral radius, while assumption (A.2) in [BY] is just our hypothesis (2.8).

We would like to attract the reader's attention to very recent results of Keller and Liverani [KL] which reinforce Theorem 2 in certain cases.

The main new result of this paper is obtained by putting together Theorem 1 and Theorem 2:

Proposition 3.

For ^L²B(^B), let ^fId^?j^g be either both a compact and a-compact uniformly bounded approximation of the identity, or a compact or -compact uniformly bounded hierarchical approximation of the identity on which ^Lacts in scales for some k, i.e., either (a) [(i;ii;ii) and (i;ii;iii)], or (b) (i;ii;iii;iv;v), or (c) (i;ii;iii;iv;v) hold.

Assume further that for some large enough M ² ^Z⁺, ^L^M acts exactly in scales on the sequence j (More precisely: in case (a) we assume [(v) and (v)] for k = 0, in case (b) we assume (v) for k= 0, and in case (c) the convergence (v) for k= 0.)

Then for any xed > ^ess(^L), the spectrum and generalised eigenspaces of the compact operators

Lj = (Id ^?j)^L(Id ^?j)

outside of the disc of radiusconverge to those of ^Lin the sense of (1)-(3) of Theorem 2 (including bounds (2.9-2.12)).

Obviously, Theorems 1 and 2 as well as Proposition 3 are of interest only when ^ess(^L) < (^L) (especially when there are \many" eigenvalues between ^ess(^L) and (^L)). Proposition 3 is especially interesting if^L(Id ^?j) or (Id ^?j)^L are nite rank operators. Section 3 contains specic examples where both conditions in this paragraph are satised.

Proof of Proposition 3. We start by replacing ^L by ^M= ^L^M and ^Lj by ^Mj = (Id ^? j)^L^M(Id ^?j), at the end of the proof we shall see how to recover results about ^L itself. (Note that if (ii) respectively (ii) is satised for ^L then it also holds for ^L^M. Iterating assumption (iv) does not lead into diculties.)

The only thing which requires checking is assumption (2.8) of Theorem 2. For this, we observe that the conditions for the strongest statement of Theorem 1 are satised.

Thus

^ess(^L^M) = lim_m

!1

(limsup_j

!1

kj^Mmk)¹^=m = lim_m

!1

(limsup_j

!1

kMmj^k)¹^=m:

In other words, for any ~ with ~ > ^ess(^L^M), there is m⁰ 1 such that for allmm⁰ and each > 0 there isj¹(m) such that for all jj¹(m)

kj^Mmk~^m+ and ^kM^mj^k~^m+: (2.13)

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(We may, and will, take = ~^m.) For a constantC >^b 0 (independent ofj andm) to be dened below, and for each xed mm⁰(~), we set ,

=(m) = 1

(m^?1)(C^b^kMk)^m^?1 ; and take j² max(j⁰(m);j¹(m)) large enough so that

kj^M(Id ^?j)^k~^m (2.14)

for all j j² (recall that ^Macts exactly in scales). We then have the bounds

kMm?Mmj^k=^kM^m^?[(Id ^?j)^M(Id ^?j)]^m^k

=^kM^m^?(Id ^?j)^M^m(Id ^?j)

?

m^X^?1

k⁼¹(Id ^?j)^M^k(2Id ^?j)j^M[(Id ^?j)²^M]^m^?^k^?1(Id ^?j)^k

kj^Mmj+ (Id ^?j)^M^mj + j^Mm(Id ^?j)^k + (m^?1)~^m(C^b^kMk)^m^?1

(sup_j ^kj^k)²2~^m+ 2(sup_j ^kj^k)(1 + sup_j ^kj^k)2~^m + ~^m;

(2.15) (the second equality can be proved by induction onm since Id ^?(Id ^?j)² = (2Id ^? j)j; we also use ^C = (1 + sup_j^kj^k)(2 + sup_j^kj^k)²). Since all constants in (2.15) are uniform inmwe have obtained an estimate which is equivalent to (2.8) for^M=^L^M. We now explain how to prove our statements for the original operator^L. First note that, since êss(^L^M) = (êss(^L))^M, if ~ > êss(^L) then = ~^M > êss(^L^M). Also, since the spectral projection associated to ^L and an eigenvalue coincides with that of ^L^M and ^M, the assertion regarding ^kP^?^Pj^k in (1) of Theorem 2 is clearly valid.

Since X is nite-dimensional, we may conclude by applying perturbation theory of nite dimensional matrices as in the proof of Lemma 3 in [BY, pp. 361-362]. (Note that algebraic multiplicity is preserved when taking powers of an operator, whereas the index, which is the size of the largest Jordan block, may only decrease.)

Nonconvergence of the determinants

In Section 3 we shall discuss situations where Proposition 3 furnishes us with a sequence of nite rank operators^Lj whose eigenvalues of large enough modulus (together with the corresponding generalised eigenspaces) converge to the corresponding data for a given noncompact bounded operator ^L. It is tempting to consider the associated sequence of \Fredholm determinants"dj(z) = det(Id ^?z^Lj). The functions dj(z) are polynomials, of degree increasing with j, and whose zeroes of small enough modulus

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converge as j ^! ¹ to the inverse eigenvalues of ^L. In many situations, however, the functions dj(z) do not converge as holomorphic functions in any disc. (The \small (essential) spectrum" of ^L seems to be the reason for this lack of convergence.) We believe that such counter-examples can be obtained in the framework of smooth expanding maps of the circle, by considering a sequence j associated to locally constant approximations (Haar basis) or approximations of higher but nite smoothness.

3. Dynamical transfer operators and wavelet approximations: an application

A typical situation where Theorem 1 applies is when ^L is the transfer operator associated to a smooth expanding map of then-dimensional torus^Tⁿ, the Banach space is a space of smooth distributions, a Sobolev space, or more generally a Triebel space (see [H2]), and the j are obtained by orthogonal projections on a multi-resolution analysis (see [Me]). We now give precise denitions and statements, without striving for the fullest generality.

Denition of the transfer operator

Let f : ^Tⁿ ^! ^Tⁿ be a C¹ uniformly expanding map (i.e., there exists > 1 with

kDfv^k ^kv^k for each x ² ^Tⁿ and each v ² Tx^Tn). Let g : ^Tⁿ ^! ^C be a C¹ function. The Ruelle transfer operator^L associated to the pair (f;g) is dened (e.g. on L²(^Tⁿ) =L²(^Tⁿ;d), with Lebesgue measure on the torus) by

L'(x) = ^X

f⁽y⁾⁼x

g(y)'(y)

jdetDf(y)^j : (3.1)

(The choiceg1 leads to the usual Perron-Frobenius-type transfer operator, for which there exists a maximal eigenfunction which is the density of the unique absolutely continuous invariant probability measure for f.)

Denition of the hierarchical approximation of the identity

We now give an explicit example of operators j that will satisfy the assumptions of Proposition 3. We start with an orthonormal wavelet basis forL²(^Rⁿ), e.g., the Meyer basis obtained from multiresolution analysis of L²(^Rⁿ) (see [M]). This is a set of 2ⁿ^?1 functions k (k = 1;::: ;2ⁿ ^?1), in the Schwartz space S(^Rⁿ) of rapidly decreasing functions

S(^Rⁿ) =^f'²L²(^Rⁿ) s.t. sup

Rn ^jx^m@^p'(x)^j<¹;⁸p;m²(^Z⁺)ⁿ^g

such that all moments of each k vanish (i.e., ^R^Rnx^m k(x)d = 0), and such that

f

kj;` = 2^jn=² k(2^jx^?`) for k = 1;::: ;2ⁿ^?1;j ²^Z;`²^Zⁿ^g

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is an orthonormal basis ofL²(^Rⁿ). Moreover, the k have compactly supported Fourier transforms. Let ^P :S(^Rⁿ)^!L²(^Tⁿ) be the periodisation operator

P'(x) = ^X

`^2Zⁿ'(x+`): Then we get an orthonormal basis of L²(^Tⁿ) by considering

fP ^j;`_k ;k = 1;::: ;2ⁿ^?1;j ²^Z^?;`²^f0;::: ;2^?^j ^?1^gⁿ^g^[^f ⁰ 1^g:

Finally, we dene the sequence j : L²(^Tⁿ)^! L²(^Tⁿ), for j ²^Z⁺, by setting Id ^?j

to be the (nite-rank) orthogonal projection on the nite dimensional space generated by the ^P _k^j⁰^;` fork = 1;::: ;2ⁿ^?1, ^?j j⁰ 0, and `²^f0;::: ;2^?^j⁰^?1^gⁿ:

(Id ^?j)(') =^Z

Tn ' ⁰d +²^Xⁿ^?1

k⁼¹

0

X

j⁰^=?j

X

`^2f0;:::;²^?^j⁰^?1gⁿ

P

j⁰;`

k

Z

Tn '^P _k^j⁰^;`d ^(3.2)

Two (scales of) Banach spaces

We shall consider the transfer operator^Lacting on two scales of Banach spaces (the sequence of projections just introduced will work for both). The rst is the well-known scale of Sobolev spaces H^s(^Tⁿ), for xed s²^R⁺, i.e.,

H^s(^Tⁿ) =^f'²L²(^Tⁿ) s.t. ^k'^kH^s :=

s

j'^(0)^j²+ ^X

m^2Zⁿ

jm^j²^s^j'^(m)^j² <^1g: _(3.3) where we used the notation ^'(m), for the Fourier coecients of ', and we write ^jm^j=

pP

im²_i.

The second is derived from the Holder { Zygmund scale of spaces (^Tⁿ). This is the space of periodic functions f ² C^[^s^](^Tⁿ), for s = [s] +^fs^g, [s]²^N, ^fs^g ²[0;1), for which for all multi-indices, ^j^j= [s] we have

k'^k¹+ sup_;x;y ^j@'(x)^?@'(y)^j

jx^?y^j^f^s^g <¹ (3.4) The left-hand side denes the norm of ^s(^Tⁿ) Here we suppose s =² ^N, otherwise we have to use the Zygmund spaces (see, e.g., [Tr] for a denition). In Lemma 4 below we will use the closure of C¹(^Tⁿ) in ^s(^Tⁿ), denoted by ^s(^Tⁿ). The point is that both Sobolev and Holder scales of spaces (together with many others) are well characterised through wavelet coecients. More precisely, a distribution ' is in H^s(^Tⁿ) if and only if itswavelet coecients

^j;`_k := (^P _k^j;`^j')

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satisfy ^s

X

j;k;`2^?2^js^j^j;`_k ^j² <¹:

The left-hand side denes a norm equivalent to the standard Sobolev norm. The scale ^s(^Tⁿ) is characterised in wavelet space via

j;k;`sup2^?^sj^j^j;`_k ^j <¹

Again we have equivalence of norms. The closed subspace ^s(^Tⁿ) consists of precisely those functions for which, in addition,

j^!?1lim max_`;k 2^?^sj^j^j;`_k ^j= 0:

The projections Id ^?j from (3.2) are nite-rank on both the Sobolev and Holder spaces. Since our transfer operator ^L is associated with a smooth map f and weight g, it is also bounded on these spaces (see [H1]). As the following key technical lemma shows, we may apply Proposition 3:

Lemma 4. (Exact scaling for suitable Banach spaces)

^Let^Lbe a transfer operator associated to a smooth expanding map f and a smooth weight g as in (3.1), acting on a Banach space ^B = H^s(^Tⁿ) or ^B = ^s(^Tⁿ). Let Id ^?j be the sequence of nite- rank projections dened by (3.2). Then j and ^L satisfy properties (i;ii;iii;iv;v). Furthermore, there is M ² ^Z⁺ such that ^L^M acts exactly in scales on the hierarchical sequence j.

Proof of Lemma 4.

Assertions (ii) and (iv) are obvious consequences of the denitions. Properties (i) and (iii) follow from the above wavelet characterisation of the scalesH^s(^Tⁿ) and^s(^Tⁿ) in wavelet space. (Note that property (iii) does not hold for ^s(^Tⁿ).)

To prove the exact scaling property we recall a result proved in [H1, Sections 7-8].

There it was shown that the transfer operator may be written as

L=^L^@ +^R;

where^R is smoothing: it is continuous from H^r(^Tⁿ) to H^r⁺¹(^Tⁿ), and from^r(^Tⁿ) to ^r⁺¹(^Tⁿ) for all r >^?1.

L@ is a linearised version of^L, its precise denition can be found in [H1, Lemma 7.3].

(Note that the operators analogous to ^L^@ and ^R in Sections 7{8 of [H1] were dened in the wavelet coordinates, while here we work with their versions in the original space

Tn; also, [H1] considered the non periodic case of^Rⁿ, this does not lead to diculties.) It suces to recall that^L^@ is completely determined by the derivatives of the dynamics f at all points of ^Tⁿ. In particular, since, by hypothesis ^L is obtained by composing

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with uniform contractions, ^L^@ maps functions supported by a disk in Fourier space to functions supported by a -times smaller disk, where > 1 is our bound for the expansion rate of the dynamics. On the other hand, the image of (Id ^?j) consists of functions supported by a disk of radius c2^j in Fourier space, whereas the image of j is supported outside a disk of radius C2^j with some C > c > 0. Upon replacing ^L by ^L^m (m depends on C=c) we see that

j(^L^m)^@(Id ^?j) = 0:

It remains to analyse the eect of^R. Since^Ris acting in scales, it is continuous from H^s^?1(^Tⁿ) toH^s(^Tⁿ) and from^s^?1(^Tⁿ) to^s(^Tⁿ). It can thus be written as^R= ?^R⁰, where ? is dened via ? : ^P ^j;`_k ^! 2^j^P ^j;`_k . Thus ^R⁰ is continuous from H^s(^Tⁿ) to H^s(^Tⁿ). Now, by denition

kj?^kH^s^(Tⁿ⁾=^kj?^k^s^(Tⁿ⁾ const 2^?^j ^!0 (j^!¹); and the exact scaling property follows.

Remark 3. For y ²^Tⁿ we introduce the bounded linear operator^Df;y acting either on the Sobolev spaceH^s(^Rⁿ) or the derived from Holder space^s(^Tⁿ) (dened analogously to (3.3{3.4)) by

Df;y'= 'D(f_f^?1⁽_y⁾)

jdetDf(y)^j ; (3.5)

(where the inverse branchf_f^?1⁽_y⁾ is unambiguously dened by f^?1(f(y)) =y). For each m1 an operator^Df^m;y can be dened similarly as in (3.5).

In fact, Holschneider [H1, Theorems 21.{2.2] applies Theorem 1 in wavelet coordinates to show a more explicit formula for the essential spectral radius. The following bounds are easy consequences of his formula:

^ess(^L)

8

>

<

>

:

limm^!1

sup_x^2Tⁿ^P_fm⁽y⁾⁼x^jg⁽^m⁾(y)^j^kDf^m;y^k^s^(Rⁿ⁾

1=m

for^B=^s(^Tⁿ); limm^!1

sup_x^2Tⁿ^P_fm⁽y⁾⁼x^jg⁽^m⁾(y)^j²^kDf^m;y^k²H^s^(Rⁿ⁾

1=⁽²m⁾

for^B=H^s(^Tⁿ);

(3.6)

(for m 1, we write g⁽^m⁾(x) =^Q^m_i⁼⁰^?1g(fⁱ(x))). Using that each inverse branch of f^m is a contraction by ^?^m, it is not dicult to show that for any y

kDf^m;y^k^s^(Rⁿ⁾^?^ms and, fors > n₂ ;^kDf^m;y^kH^s^(Rⁿ⁾^?^m⁽^s^?ⁿ⁼²⁾: (3.7)