dimension theory of hyperbolic flows--luis barreira

Springer Monographs in Mathematics

Dimension Theory of Hyperbolic Flows

Luís Barreira

Springer Monographs in Mathematics

For further volumes:www.springer.com/series/3733

http://www.springer.com/series/3733

Luís Barreira

Dimension Theoryof Hyperbolic Flows

Luís BarreiraDepartamento de MatemáticaInstituto Superior TécnicoLisboa, Portugal

ISSN 1439-7382 Springer Monographs in MathematicsISBN 978-3-319-00547-8 ISBN 978-3-319-00548-5 (eBook)DOI 10.1007/978-3-319-00548-5Springer Cham Heidelberg New York Dordrecht London

Library of Congress Control Number: 2013942381

Mathematics Subject Classification: 37C45, 37Dxx, 37Axx

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To Claudia, for everything

Preface

The objective of this book is to provide a comprehensive exposition of the mainresults and main techniques of dimension theory and multifractal analysis of hyper-bolic flows. This includes the discussion of some recent results in the area as wellas some of its open problems.

The book is directed to researchers as well as graduate students specializing indynamical systems who wish to have a sufficiently comprehensive view of the the-ory together with a working knowledge of its main techniques. The discussion ofsome open problems, perhaps somewhat biased towards my own interests, is in-cluded also with the hope that it may lead to further developments.

Over the last two decades, the dimension theory of dynamical systems has pro-gressively developed into an independent and extremely active field of research.However, while the dimension theory and multifractal analysis for maps are verymuch developed, the corresponding theory for flows has experienced a steady al-though slower development. It should be emphasized that this is not because of lackof interest. For instance, geodesic flows and hyperbolic flows stand as cornerstonesof the theory of dynamical systems. Sometimes a result for flows can be reducedto the case of maps, for example with the help of symbolic dynamics, but often itrequires substantial changes or even new ideas. Because of this, many parts of thetheory are either only sketched or are too technical for a wider audience. In thisrespect, the present monograph is intended to have a unifying and guiding role.Moreover, the text is self-contained and with the exception of some basic results inChaps. 3 and 4, all the results in the book are included with detailed proofs.

On the other hand, there are topics that are not yet at a stage of developmentthat makes it reasonable to include them in detail in a monograph of this nature,either because there are only partial results or because they require very specifictechniques. This includes results for nonconformal flows, nonuniformly hyperbolicflows and flows modeled by countable symbolic dynamics. In such cases, I haveinstead provided a sufficient discussion with references to the relevant literature.

Luís BarreiraLisbon, PortugalJune 2013

vii

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Dimension Theory for Maps . . . . . . . . . . . . . . . . . . . . . 11.2 Dimension Theory for Flows . . . . . . . . . . . . . . . . . . . . 51.3 Pointwise Dimension . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Multifractal Analysis . . . . . . . . . . . . . . . . . . . . . . . . 81.5 Geodesic Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.6 Variational Principles . . . . . . . . . . . . . . . . . . . . . . . . 131.7 Multidimensional Theory . . . . . . . . . . . . . . . . . . . . . . 13

Part I Basic Notions

2 Suspension Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.1 Basic Notions and Cohomology . . . . . . . . . . . . . . . . . . . 192.2 The Bowen–Walters Distance . . . . . . . . . . . . . . . . . . . . 252.3 Further Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 Hyperbolic Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.1 Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2 Markov Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.3 Symbolic Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 36

4 Pressure and Dimension . . . . . . . . . . . . . . . . . . . . . . . . . 394.1 Topological Pressure and Entropy . . . . . . . . . . . . . . . . . . 39

4.1.1 Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . 394.1.2 Properties of the Pressure . . . . . . . . . . . . . . . . . . 414.1.3 The Case of Suspension Flows . . . . . . . . . . . . . . . 42

4.2 BS-Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.3 Hausdorff and Box Dimensions . . . . . . . . . . . . . . . . . . . 45

4.3.1 Dimension of Sets . . . . . . . . . . . . . . . . . . . . . . 454.3.2 Dimension of Measures . . . . . . . . . . . . . . . . . . . 46

ix

x Contents

Part II Dimension Theory

5 Dimension of Hyperbolic Sets . . . . . . . . . . . . . . . . . . . . . . 515.1 Dimensions Along Stable and Unstable Manifolds . . . . . . . . . 515.2 Formula for the Dimension . . . . . . . . . . . . . . . . . . . . . 58

6 Pointwise Dimension and Applications . . . . . . . . . . . . . . . . . 616.1 A Formula for the Pointwise Dimension . . . . . . . . . . . . . . 616.2 Hausdorff Dimension and Ergodic Decompositions . . . . . . . . 676.3 Measures of Maximal Dimension . . . . . . . . . . . . . . . . . . 70

Part III Multifractal Analysis

7 Suspensions over Symbolic Dynamics . . . . . . . . . . . . . . . . . . 817.1 Pointwise Dimension . . . . . . . . . . . . . . . . . . . . . . . . 817.2 Multifractal Analysis . . . . . . . . . . . . . . . . . . . . . . . . 837.3 Irregular Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877.4 Entropy Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

8 Multifractal Analysis of Hyperbolic Flows . . . . . . . . . . . . . . . 918.1 Suspensions over Expanding Maps . . . . . . . . . . . . . . . . . 918.2 Dimension Spectra of Hyperbolic Flows . . . . . . . . . . . . . . 948.3 Entropy Spectra and Cohomology . . . . . . . . . . . . . . . . . . 105

Part IV Variational Principles

9 Entropy Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1119.1 A Conditional Variational Principle . . . . . . . . . . . . . . . . . 1119.2 Analyticity of the Spectrum . . . . . . . . . . . . . . . . . . . . . 1159.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

9.3.1 Multifractal Spectra for the Local Entropies . . . . . . . . 1209.3.2 Multifractal Spectra for the Lyapunov Exponents . . . . . . 1219.3.3 Suspension Flows . . . . . . . . . . . . . . . . . . . . . . 122

9.4 Multidimensional Spectra . . . . . . . . . . . . . . . . . . . . . . 124

10 Multidimensional Spectra . . . . . . . . . . . . . . . . . . . . . . . . 12710.1 Multifractal Analysis . . . . . . . . . . . . . . . . . . . . . . . . 12710.2 Finer Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13410.3 Hyperbolic Flows: Analyticity of the Spectrum . . . . . . . . . . . 136

11 Dimension Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13911.1 Future and Past . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13911.2 Conditional Variational Principle . . . . . . . . . . . . . . . . . . 141

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

Chapter 1Introduction

This introductory chapter gives an overview of the dimension theory and the mul-tifractal analysis of dynamical systems, with emphasis on hyperbolic flows. Manyof the results presented here are proved later on in the book. We also discuss topicsthat are not yet sufficiently well developed to include in the remaining chapters ofa monograph of this nature. Finally, we include a discussion of open problems andsuggestions for further developments.

1.1 Dimension Theory for Maps

The dimension theory of dynamical systems is an extremely active field of research.Its main objective is to measure the complexity from the dimensional point of viewof the objects that remain invariant under the dynamics, such as the invariant setsand measures. We refer the reader to [3, 81] for detailed accounts of substantial partsof the dimension theory of dynamical systems (although these books make almostno reference to flows).

The objective of this book is to provide a comprehensive exposition of the mainresults and main techniques of dimension theory of hyperbolic flows. In this sectionwe present some of these results. As a motivation, we start with a brief discussionof the corresponding theory for maps.

We first consider expanding maps. Let g : M → M be a differentiable map of asmooth manifold M and let J ⊂ M be a compact g-invariant set. We say that J is arepeller of g and that g is an expanding map on J if there exist constants c > 0 andβ > 1 such that

‖dxgnv‖ ≥ cβn‖v‖

for every n ∈ N, x ∈ J and v ∈ TxM . The map g is said to be conformal on J if dxg

is a multiple of an isometry for every x ∈ J . We define a function ϕ : J → R by

ϕ(x) = − log‖dxg‖.L. Barreira, Dimension Theory of Hyperbolic Flows,Springer Monographs in Mathematics, DOI 10.1007/978-3-319-00548-5_1,© Springer International Publishing Switzerland 2013

1

http://dx.doi.org/10.1007/978-3-319-00548-5_1

2 1 Introduction

The following result expresses the Hausdorff dimension dimH J and the lower andupper box dimensions dimBJ and dimBJ (see Sect. 4.3 for the definitions) of arepeller J in terms of the topological pressure P (see Chap. 4).

Theorem 1.1 If J is a repeller of a C1+δ map g that is conformal on J , then

dimH J = dimBJ = dimBJ = s,

where s is the unique real number such that

P(sϕ) = 0. (1.1)

Ruelle showed in [93] that dimH J = s. The equality between the Hausdorff andbox dimensions is due to Falconer [39]. It is also shown in [93] that if μ is theequilibrium measure of −sϕ, then

dimH J = dimH μ, (1.2)

where

dimH μ = inf{dimH A : A ⊂ J and μ(J \ A) = 0

}.

Equation (1.1) was introduced by Bowen in [29] (in his study of quasi-circles) andis usually called Bowen’s equation. It is also appropriate to call it the Bowen–Ruelleequation, taking into account the fundamental role played by the thermodynamicformalism developed by Ruelle in [91] (see also [92]) as well as his article [93].Equation (1.1) establishes a connection between the thermodynamic formalism anddimension theory of dynamical systems. Since topological pressure and Hausdorffdimension are both Carathéodory characteristics (see [81]), the relation between thetwo is very natural. Moreover, equation (1.1) has a rather universal character: in-deed, virtually all known equations used to compute or to estimate the dimension ofan invariant set are particular cases of this equation or of an appropriate generaliza-tion (see [4] for a detailed discussion).

Now we consider hyperbolic sets. Let Λ be a hyperbolic set for a diffeomor-phism f . We define functions ϕs : Λ → R and ϕu : Λ →R by

ϕs(x) = log‖dxf |Es(x)‖ and ϕu(x) = − log‖dxf |Eu(x)‖,

where Es(x) and Eu(x) are respectively the stable and unstable subspaces at thepoint x. The set Λ is said to be locally maximal if there exists an open neighbor-hood U of Λ such that

Λ =⋂

n∈Zf n(U).

The following result is a version of Theorem 1.1 for hyperbolic sets.

1.1 Dimension Theory for Maps 3

Theorem 1.2 If Λ is a locally maximal hyperbolic set for a C1 surface diffeomor-phism and dimEs(x) = dimEu(x) = 1 for every x ∈ Λ, then

dimH Λ = dimBΛ = dimBΛ = ts + tu,

where ts and tu are the unique real numbers such that

P(tsϕs) = P(tuϕu) = 0.

It follows from work of McCluskey and Manning in [76] that dimH Λ = ts + tu.The equality between the Hausdorff and box dimensions is due to Takens [103] forC2 diffeomorphisms and to Palis and Viana [79] for arbitrary C1 diffeomorphisms.The result in Theorem 1.2 can be readily extended to the more general case of con-formal diffeomorphisms. We recall that f is said to be conformal on a hyperbolicset Λ if the maps dxf |Es(x) and dxf |Eu(x) are multiples of isometries for everyx ∈ Λ. It happens that for conformal diffeomorphisms the product structure is a Lip-schitz map with Lipschitz inverse (in general it is only a Hölder homeomorphismwith Hölder inverse). This allows us to compute the dimension of a hyperbolic setby adding the dimensions along the stable and unstable manifolds.

Palis and Viana [79] established the continuous dependence of the dimensionon the diffeomorphism. Mañé [72] obtained an even higher regularity. In higher-dimensional manifolds (and so in the nonconformal case) the Hausdorff dimensionof a hyperbolic set may vary discontinuously. Examples were given by Pollicott andWeiss in [86] followed by Bonatti, Díaz, and Viana in [25]. Díaz and Viana [34]considered one-parameter families of diffeomorphisms on the 2-torus bifurcatingfrom an Anosov map to a DA map and showed that for an open set of these familiesthe Hausdorff and box dimensions of the nonwandering set are discontinuous acrossthe bifurcation.

The study of the dimension of repellers and hyperbolic sets for nonconformalmaps is much less developed than the corresponding theory for conformal maps.Some major difficulties include a clear separation between different Lyapunov direc-tions, a small regularity of the associated distributions (that typically are only Höldercontinuous), and the existence of number-theoretical properties forcing a variationof the Hausdorff dimension with respect to a certain typical value. As a result ofthis, in many situations only partial results have been obtained. For example, someresults were obtained not for a particular transformation, but for Lebesgue almostall values of some parameter (although possibly without knowing what happens fora specific value of this parameter). Moreover, often only dimension estimates wereobtained instead of a formula for the dimension of an invariant set.

This brief discussion of the difficulties encountered in the study of nonconformalmaps motivates our first problem.

Problem 1.1 Develop a dimension theory for repellers and hyperbolic sets of non-conformal maps.

4 1 Introduction

This is a very ambitious problem and in fact it should correspond to a large re-search program. Thus, it is reasonable to start with less general problems. This mayinvolve, for example: to assuming that there is a clear separation between differentLyapunov exponents; to obtaining results for almost all values of some parameterand not for a specific transformation; or to obtaining sharp lower and upper boundsfor the dimension instead of exact values. Several new phenomena occur in thestudy of nonconformal transformations. For example, in general the Hausdorff andbox dimensions of a repeller do not coincide. An example was given by Pollicott andWeiss in [86], modifying a construction of Przytcki and Urbanski in [88] dependingon delicate number-theoretical properties.

Nevertheless, there exist many partial results towards a nonconformal theory, forseveral classes of repellers and hyperbolic sets, starting essentially with the semi-nal work of Douady and Oesterlé in [35]. In particular, Falconer [40] computed theHausdorff dimension of a class of repellers for nonconformal maps (building on hisformer work [38]). Related results were obtained by Zhang in [110] and in the caseof volume expanding maps by Gelfert in [49]. In another direction, Hu [59] com-puted the box dimension of a class of repellers for nonconformal maps leaving in-variant a strong unstable foliation. Related results were obtained earlier by Bedfordin [23] (see also [24]) for a class of self-similar sets that are graphs of continuousfunctions. In another direction, Falconer [37] studied a class of limit sets of geomet-ric constructions obtained from the composition of affine transformations that arenot necessarily conformal and he obtained a formula for the Hausdorff and box di-mensions for Lebesgue almost all values in some parameter space (see also [102]).Related ideas were applied by Simon and Solomyak in [101] to compute the Haus-dorff dimension of a class of solenoids in R

3. Bothe [26] and then Simon [100] (alsousing his work in [99] for noninvertible transformations) studied earlier the dimen-sion of solenoids. In particular, it is shown in [26] that under certain conditions onthe dynamics the dimension is independent of the radial section (even though theholonomies are typically not Lipschitz). More recently, Hasselblatt and Schmelingconjectured in [55] (see also [54]) that, in spite of the difficulties due to the possiblelow regularity of the holonomies, the Hausdorff dimension of a hyperbolic set canbe computed adding the dimensions along the stable and unstable manifolds. Theyprove this conjecture for a class of solenoids. The ideas developed in all these worksshould play an important role in the study of Problem 1.1.

There also exist some related results for nonuniformly expanding maps. In partic-ular, Horita and Viana [57] and Dysman [36] studied abstract models, called mapswith holes, which include examples of nonuniformly expanding repellers. In [58]Horita and Viana considered nonuniformly expanding repellers emerging from aperturbation of an Anosov diffeomorphism of the 3-torus through a Hopf bifurca-tion. Finally, we mention some related work in the case of nonuniformly hyperbolicinvariant sets. Hirayama [56] obtained an upper bound for the Hausdorff dimensionof the stable set of the set of typical points for a hyperbolic measure. Fan, Jiangand Wu [43] studied the dimension of the maximal invariant set of an asymptoti-cally nonhyperbolic family. Urbánski and Wolf [104] considered horseshoe mapsthat are uniformly hyperbolic except at a parabolic point, in particular establishinga dimension formula for the horseshoe.

1.2 Dimension Theory for Flows 5

In connection with identity (1.2) another interesting problem is the following.

Problem 1.2 Given a repeller for a nonconformal map, find whether there exists aninvariant measures of full dimension.

Identity (1.2) is due to Ruelle [93] and follows from the equivalence between μ

and the s-dimensional Hausdorff measure on J . The existence of an ergodic measureof full dimension on a repeller of a C1 conformal map was established by Gatzourasand Peres in [48]. The situation is much more complicated in the case of noncon-formal maps, and there exist only some partial results. In particular, it is shownin [48] that repellers of some maps of product type also have ergodic measures offull dimension. For piecewise linear maps, Gatzouras and Lalley [47] showed ear-lier that certain invariant sets, corresponding to full shifts in the symbolic dynam-ics, carry an ergodic measure of full dimension. Kenyon and Peres [64] obtainedthe same result for linear maps and arbitrary compact invariant sets. Bedford andUrbanski considered a particular class of self-affine sets in [24] and obtained condi-tions for the existence of a measure of full dimension. Related ideas appeared earlierin work of Bedford [22] and McMullen [77]. More recently, Yayama [107] consid-ered general Sierpinski carpets modeled by arbitrary topological Markov chains andLuzia [70, 71] considered expanding triangular maps of the 2-torus.

1.2 Dimension Theory for Flows

Now we turn to the case of flows. To a large extent the theory is analogous to thetheory for maps. Let Φ be a C1+δ flow with a locally maximal hyperbolic set Λ suchthat Φ|Λ is conformal and topologically mixing (see Sect. 5.1). Let also V s(x)

and V u(x) be the families of local stable and unstable manifolds (see Sect. 3.1).The following result of Pesin and Sadovskaya in [82] expresses the dimensionsof the sets V s(x) ∩ Λ and V u(x) ∩ Λ in terms of the topological pressure (seeTheorem 5.1). We define functions ζs, ζu : Λ → R by

ζs(x) = ∂

∂tlog‖dxϕt |Es(x)‖∣∣

t=0

and

ζu(x) = ∂

∂tlog‖dxϕt |Eu(x)‖∣∣

t=0.

Theorem 1.3 Let Φ be a C1+δ flow with a locally maximal hyperbolic set Λ suchthat Φ|Λ is conformal and topologically mixing. Then

dimH (V s(x) ∩ Λ) = dimB(V s(x) ∩ Λ) = dimB(V s(x) ∩ Λ) = ts

and

dimH (V u(x) ∩ Λ) = dimB(V u(x) ∩ Λ) = dimB(V u(x) ∩ Λ) = tu,

6 1 Introduction

where ts and tu are the unique real numbers such that

PΦ|Λ(tsζs) = PΦ|Λ(−tuζu) = 0.

It is also shown in [82] that

dimH Λ = dimBΛ = dimBΛ = ts + tu + 1

(see Theorem 5.2). This is a version of Theorem 1.2 for flows.We also formulate a version of Problem 1.1 for hyperbolic flows.

Problem 1.3 Develop a dimension theory for hyperbolic sets of nonconformalflows.

Similar comments to those in the case of maps apply to Problem 1.3. However,the theory is largely untouched ground. In particular, to the best of our knowledge,the first lower and upper bounds for the dimensions along the stable and unstablemanifolds of a hyperbolic set, for a nonconformal flow, appear for the first time inthis book (see (5.18)).

1.3 Pointwise Dimension

In the theory of dynamical systems, each global quantity can often be “constructed”with the help of a certain local quantity. Two examples are the Kolmogorov–Sinaientropy and the Hausdorff dimension, which are quantities of a global nature. Theycan be built (in a rigorous mathematical sense) using respectively the local entropyand the pointwise dimension. More precisely, in the case of the entropy this goesback to the classical Shannon–McMillan–Breiman theorem: the Kolmogorov–Sinaientropy is obtained by integrating the local entropy. On the other hand, the Hausdorffdimension of a measure is given by the essential supremum of the lower pointwisedimension. More precisely, given a measure μ in a set Λ ⊂ R

m, we have

dimH μ = ess sup

{lim infr→0

logμ(B(x, r))

log r: x ∈ Λ

},

where B(x, r) is the ball of radius r centered at x, with the essential supremumtaken with respect to μ. In particular, if there exists a real number d such that

limr→0

logμ(B(x, r))

log r= d (1.3)

for μ-almost every x ∈ Λ, then dimH μ = d . This criterion was established byYoung in [108]. The limit in (1.3), if it exists, is called the pointwise dimensionof μ at x. Let μ be a finite measure with compact support that is invariant under aC1+δ diffeomorphism f . It follows from work of Ledrappier and Young in [67, 68]

1.3 Pointwise Dimension 7

and work of Barreira, Pesin and Schmeling in [11] that if the measure μ is hyper-bolic (that is, if all Lyapunov exponents are nonzero μ-almost everywhere), thenthe pointwise dimension exists almost everywhere (see [10] for details). In the two-dimensional case this statement was established by Young in [108].

Let f : M → M be a C1+δ surface diffeomorphism. For each x ∈ M and v ∈TxM , we consider the Lyapunov exponent

λ(x, v) = lim supn→+∞

1

nlog‖dxf

nv‖.

Let μ be an f -invariant probability measure on M . We say that μ is hyperbolic ifλ(x, v) = 0 for μ-almost every x ∈ M and every v = 0. When μ is of saddle type,that is, when the function TxM \ {0} → λ(x, v) takes exactly one positive valueλu(x) and one negative value λs(x), for μ-almost every x ∈ M , we define

λu(μ) =∫

M

λu dμ and λs(μ) =∫

M

λs dμ.

Moreover, we denote by hμ(f ) the entropy of f with respect to μ. By work of Brinand Katok in [32], the limit

hμ(x) = limε→0

limn→∞−1

nlogμ

(n−1⋂

k=0

f −kB(f k(x), ε)

)

exists for μ-almost every x ∈ M and

hμ(f ) =∫

M

hμ(x)dμ.

The number hμ(x) is called the local entropy of μ at the point x.The following result was established by Young in [108].

Theorem 1.4 Let f be a C1+δ diffeomorphism. If μ is an ergodic f -invariant mea-sure, then

dimH μ = hμ(f )

(1

λu(μ)− 1

λs(μ)

). (1.4)

Barreira and Wolf [20] considered measures sitting on a hyperbolic set that arenot necessarily ergodic and established an explicit formula for the pointwise dimen-sion, which is a local version of identity (1.4).

Theorem 1.5 Let f be a C1+δ surface diffeomorphism with a locally maximal hy-perbolic set Λ and let μ be an f -invariant probability measure on Λ. For μ-almostevery x ∈ Λ, we have

limr→0

logμ(B(x, r))

log r= hμ(x)

(1

λu(x)− 1

λs(x)

).

8 1 Introduction

The novelty of the approach in [20] is not Theorem 1.5 itself, but instead theelementary method of proof. Indeed, the result also follows from work of Ledrappierand Young in [68], although with a rather involved proof in the general context ofnonuniform hyperbolicity (see [10] for details).

In [21], Barreira and Wolf established an analogous formula for conformal hy-perbolic flows (see Theorem 6.1).

Theorem 1.6 Let Φ be a C1+δ flow with a locally maximal hyperbolic set Λ suchthat Φ|Λ is conformal and let μ be a Φ-invariant probability measure on Λ. Forμ-almost every x ∈ Λ, we have

limr→0

logμ(B(x, r))

log r= hμ(x)

(1

λu(x)− 1

λs(x)

)+ 1. (1.5)

In [82], Pesin and Sadovskaya first established identity (1.5) in the special case ofequilibrium measures for a Hölder continuous function (we note that these measuresare ergodic and have a local product structure). Identity (1.5) can be used to describehow the Hausdorff dimension dimH μ behaves under an ergodic decomposition. Werecall that an ergodic decomposition of a measure μ can be identified with a proba-bility measure τ in the metrizable space of Φ-invariant probability measures on Λ

such that the subset of ergodic measures has full τ -measure (see Sect. 4.3). Namely,for any ergodic decomposition of μ we have

dimH μ = ess supν dimH ν,

with the essential supremum taken with respect to τ (see Theorem 6.3).The discussion in the case of maps motivates the following problem.

Problem 1.4 Establish identity (1.5) for an arbitrary hyperbolic measure.

We emphasize that this is a very ambitious problem. The main difficulty seems tobe that it should be necessary to develop appropriate tools in the context of a nonuni-form hyperbolicity theory for flows. Nevertheless, it is reasonable to conjecture thatidentity (1.5) indeed holds for any hyperbolic measure.

1.4 Multifractal Analysis

The multifractal analysis of dynamical systems can be considered a subfield of thedimension theory of dynamical systems. Roughly speaking, multifractal analysisstudies the complexity of the level sets of invariant local quantities obtained froma dynamical system. For example, one can consider Birkhoff averages, Lyapunovexponents, pointwise dimensions and local entropies. These functions are typicallyonly measurable and thus their level sets are rarely manifolds. Hence, in order to

1.4 Multifractal Analysis 9

measure their complexity it is appropriate to use quantities such as the topologicalentropy and the Hausdorff dimension.

The concept of multifractal analysis was suggested by Halsey, Jensen, Kadanoff,Procaccia and Shraiman in [51]. The first rigorous approach is due to Collet,Lebowitz and Porzio in [33] for a class of measures invariant under one-dimensionalMarkov maps. In [69], Lopes considered the measure of maximal entropy for hy-perbolic Julia sets, and in [89], Rand studied Gibbs measures for a class of repellers.We refer the reader to the books [3, 81] for detailed accounts of substantial parts ofthe theory.

We briefly describe the main elements of multifractal analysis. Let T : X → X

be a continuous map of a compact metric space and let g : X → R be a continuousfunction. For each α ∈ R, let

Kα ={

x ∈ X : limn→∞

1

n

n∑

i=0

g(T i(x)) = α

}

. (1.6)

We also consider the set

K ={

x ∈ X : lim infn→∞

1

n

n∑

i=0

g(T i(x)) < lim supn→∞

1

n

n∑

i=0

g(T i(x))

}

. (1.7)

Clearly,

X = K ∪⋃

α∈RKα. (1.8)

This union is formed by pairwise disjoint T -invariant sets. It is called a multifractaldecomposition of X. For each α ∈ R, let

D(α) = dimH Kα.

The function D is called the dimension spectrum for the Birkhoff averages of g.By Birkhoff’s ergodic theorem, if μ is an ergodic T -invariant finite measure

on X, and α = ∫X

g dμ/μ(X), then μ(Kα) = μ(X). That is, there exists a set Kα

in the multifractal decomposition of full μ-measure. However, the other sets in themultifractal decomposition need not be empty. In fact, for several classes of hyper-bolic dynamical systems (for example, a topological Markov chain, an expandingmap or a hyperbolic diffeomorphism) and certain functions g (for example, Höldercontinuous functions that are not cohomologous to a constant), it was proved that:

1. the set {α ∈ R : Kα = ∅} is an interval;2. the function D is analytic and strictly convex;3. the set K is everywhere dense and dimH K = dimH X.

In particular, the multifractal decomposition in (1.8) is often composed of an un-countable number of dense T -invariant sets of positive Hausdorff dimension. Forexample, for repellers and hyperbolic sets for C1+δ conformal maps, Pesin and

10 1 Introduction

Weiss [83, 84] obtained a multifractal analysis of the dimension spectrum. We referthe reader to [3, 4] for details and further references.

As in the case of the dimension theory of nonconformal maps, the study of thecorresponding multifractal analysis is still at an early stage of development and thefollowing is a challenging problem.

Problem 1.5 Obtain a multifractal analysis for repellers and hyperbolic sets fornonconformal maps.

We mention briefly some works containing partial results towards a solution ofthis problem. Feng and Lau [46] and Feng [44, 45] studied products of nonnegativematrices and their thermodynamic properties. Jordan and Simon [61] establishedformulas for the dimension spectra of almost all self-affine maps in the plane (wenote that their results generalize to any dimension). In [8], Barreira and Gelfertconsidered repellers of nonconformal maps satisfying a certain cone condition andobtained a multifractal analysis for the topological entropy of the level sets of theLyapunov exponents.

Another challenging problem concerns nonuniformly hyperbolic maps. We em-phasize that in this case even the conformal case is at an early stage of development.

Problem 1.6 Obtain a multifractal analysis for nonuniformly hyperbolic maps.

The intrinsic difficulty of this problem is not strictly related to the general classof dynamics under consideration. Indeed, there are essential differences between thethermodynamic formalisms for uniformly hyperbolic and nonuniformly hyperbolicdynamics. Due to the important role played by the thermodynamic formalism, thiscan be seen as the main reason behind important differences between the dimensiontheories for uniformly hyperbolic and nonuniformly hyperbolic dynamics.

We also mention some works related to Problem 1.6. We first observe that for uni-formly hyperbolic systems and their codings by finite topological Markov chains,the dimension and entropy spectra of an equilibrium measure has bounded domainand is analytic. In strong contrast, in the case of nonuniformly hyperbolic systemsand countable topological Markov chains the spectrum may have unbounded do-main and need not be analytic. In [87], Pollicott and Weiss considered the Gaussmap and the Manneville–Pomeau transformation. Related results were obtained byYuri in [109]. In [73–75], Mauldin and Urbanski developed the theory of infiniteconformal iterated function systems, studying in particular the Hausdorff dimensionof the limit set (see also [52]). Related results were obtained by Nakaishi in [78].In [66], Kesseböhmer and Stratmann established a detailed multifractal analysis forStern–Brocot intervals, continued fractions and certain Diophantine growth rates,building on their former work [65]. We refer to [85] for results concerning Fareytrees and multifractal analysis. In [60], Iommi obtained a multifractal analysis forcountable topological Markov chains. He uses the Gurevich pressure introduced bySarig in [94] (building on former work of Gurevich [50] on the notion of topologicalentropy for countable Markov chains).

1.5 Geodesic Flows 11

One can also obtain a multifractal analysis for a class of hyperbolic flows andfor suspension flows over topological Markov chains. In the multifractal analysisof a flow Φ = {ϕt }t∈R in X, the sets Kα and K in (1.6) and (1.7) are replacedrespectively by

Kα ={x ∈ X : lim

t→∞1

t

∫ t

0g(ϕτ (x)) dτ = α

}

and

K ={x ∈ X : lim inf

t→∞1

t

∫ t

0g(ϕτ (x)) dτ < lim sup

t→∞1

t

∫ t

0g(ϕτ (x)) dτ

}.

In particular:

1. Pesin and Sadovskaya [82] obtained a multifractal analysis of the dimensionspectrum for the pointwise dimensions of a Gibbs measure on a locally maximalhyperbolic set for a conformal flow (see Theorem 8.3);

2. Barreira and Saussol [12] obtained a multifractal analysis of the entropy spec-trum for the Birkhoff averages of a Hölder continuous function on a locally max-imal hyperbolic set (see Theorem 8.4).

The main idea of the proofs is to use Markov systems and the associated symbolicdynamics developed by Bowen [27] and Ratner [90] to reduce the setup to the caseof maps. This is done using suspension flows over topological Markov chains, ob-tained from a Markov system, and a careful analysis of the relation between thecohomology for the flow and the cohomology for the map in the base. Later onin the book, we describe more general results with proofs that do not use Markovsystems and the associated symbolic dynamics.

The following are versions of Problems 1.5 and 1.6 for flows.

Problem 1.7 Obtain a multifractal analysis for repellers and hyperbolic sets fornonconformal flows.

Problem 1.8 Obtain a multifractal analysis for nonuniformly hyperbolic flows.

In [9], Barreira and Iommi considered suspension flows over a countable topo-logical Markov chain, building also on work of Savchenko [95] on the notion oftopological entropy.

1.5 Geodesic Flows

In this section we discuss an application of the multifractal analysis for flows togeodesic flows on compact surfaces of negative curvature.

12 1 Introduction

Consider a compact orientable Riemannian surface M with (sectional) curva-ture K . The Gauss–Bonnet theorem says that

∫

M

K dλM = 2πχ(M), (1.9)

where λM is the volume in M and χ(M) is the Euler characteristic of M . LetΦ = {ϕt }t∈R be the geodesic flow in the unit tangent bundle SM . It preserves thenormalized Liouville measure λSM in SM , induced from the volume in M . ByBirkhoff’s ergodic theorem, the limit

κ(x) = limt→∞

1

t

∫ t

0K(ϕs(x)) ds (1.10)

exists for λSM -almost every x ∈ SM . It follows from (1.9) and (1.10) that∫

SM

κ dλSM =∫

M

K dλM = 2πχ(M). (1.11)

Now let us assume that M has strictly negative curvature. In this case M hasgenus at least 2. The geodesic flow is ergodic and hence, in addition to (1.11), wehave

κ(x) =∫

M

K dλM = 2πχ(M) (1.12)

for λSM -almost every x ∈ SM . More generally, identity (1.12) holds almost every-where with respect to any invariant probability measure. However, the level sets

SMα = {x ∈ SM : κ(x) = α}may still be nonempty for some values of α. The following result was establishedby Barreira and Saussol in [12].

Theorem 1.7 Given a compact orientable surface M with χ(M) < 0, for each met-ric g in an open set of C3 metrics in M of strictly negative curvature, there exists anopen interval Ig containing 2πχ(M) such that SMα ⊂ SM is a nonempty properdense subset with h(Φ|SMα) > 0 for every α ∈ Ig .

We have

SM = N ∪⋃

α

SMα,

where

N ={x ∈ SM : lim inf

t→∞1

t

∫ t

0K(ϕs(x)) ds < lim sup

t→∞1

t

∫ t

0K(ϕs(x)) ds

}

and the union is composed of pairwise disjoint sets. By Birkhoff’s ergodic theorem,the set N has zero measure with respect to any invariant measure. This stronglycontrasts with the following result also obtained in [12] (using ideas in [17]).

1.6 Variational Principles 13

Theorem 1.8 Given a compact orientable surface M with χ(M) < 0, for eachmetric g in an open set of C3 metrics in M of strictly negative curvature, the setN ⊂ SM is a nonempty proper dense subset with h(Φ|N) = h(Φ).

1.6 Variational Principles

In this section we describe another approach to the multifractal analysis of entropyspectra, based on what we call a conditional variations principle. For simplicityof the presentation, in order to avoid introducing extra material at this point, weconsider again the setup of Sect. 1.5 (the general case is considered in Sect. 1.7).

Let Φ = {ϕt }t∈R be a geodesic flow in the unit tangent bundle SM . For eachα ∈ R, let

E(α) = h(Φ|SMα)

be the topological entropy of Φ on the set SMα . The function E is called the entropyspectrum. In many works of multifractal analysis the function E is described interms of a Legendre transform involving the topological pressure. A conditionalvariational principle provides an alternative description.

The following result of Barreira and Saussol in [15] is a conditional variationalprinciple for the spectrum E. Let hμ(Φ) be the entropy of the geodesic flow withrespect to a measure μ ∈ M, where M is the set of all Φ-invariant probability mea-sures on SM .

Theorem 1.9 For a compact orientable surface M and a metric of strictly negativecurvature on M , for each

α ∈ int

{∫

SM

K dμ : μ ∈ M

},

we have

E(α) = max

{hμ(Φ) :

∫

SM

K dμ = α and μ ∈M

}.

Theorem 1.9 is a particular case of Theorem 9.1.

1.7 Multidimensional Theory

In this section we consider multidimensional versions of entropy and dimensionspectra for a flow Φ with upper semicontinuous entropy μ → hμ(Φ) and we de-scribe a conditional variational principle for these spectra. This allows us to studysimultaneously the level sets of several local quantities, instead of only one as inSect. 1.4.

14 1 Introduction

For example, the entropy of a C1 flow with a hyperbolic set is upper semicontin-uous. More generally, the entropy of any expansive flow is upper semicontinuous.On the other hand, there are many transformations without a hyperbolic set (andnot satisfying specification) for which the entropy is still upper semicontinuous. Forexample, all β-shifts are expansive and thus, the entropy is upper semicontinuous(see [63] for details), but for β in a residual set of full Lebesgue measure (althoughthe complement has full Hausdorff dimension) the corresponding β-shift does notsatisfy specification (see [96]). This motivates establishing results not only for flowswith a hyperbolic set but more generally for flows with upper semicontinuous en-tropy. Moreover, we consider functions with a unique equilibrium measure. It fol-lows from work of Walters [105] that for each β-shift any Lipschitz function hasa unique equilibrium measure. We recall that for topologically mixing hyperbolicflows each Hölder continuous function has a unique equilibrium measure.

Now we consider a continuous flow Φ in a compact metric space X. Given con-tinuous functions a1, a2 : X → R, we consider the level sets of Birkhoff averages

Kα1,α2 ={x ∈ X : lim

t→∞1

t

∫ t

0ai(ϕs(x)) ds = αi for i = 1,2

}.

The associated entropy spectrum is defined by

E(α1, α2) = h(Φ|Kα1,α2).

We also consider the set

P ={(∫

X

a1 dμ,

∫

X

a2 dμ

): μ ∈M

},

where M is the family of all Φ-invariant probability measures on X. The followingis a conditional variational principle for the spectrum E.

Theorem 1.10 Assume that the map μ → hμ(Φ) is upper semicontinuous and thatfor each c1, c2 ∈ R the function c1a1 +c2a2 has a unique equilibrium measure. Thenfor each (α1, α2) ∈ intP, we have

E(α1, α2) = max

{hμ(Φ) :

(∫

X

a1 dμ,

∫

X

a2 dμ

)= (α1, α2) and μ ∈M

}

and there exists an ergodic measure μ ∈M with μ(Kα1,α2) = 1 such that

hμ(Φ) = E(α1, α2) and

(∫

X

a1 dμ,

∫

X

a2 dμ

)= (α1, α2).

Theorem 1.10 is a particular case of Theorem 10.1 due to Barreira and Doutor [6].In the case when Φ is a hyperbolic flow, the statement in Theorem 1.10 was firstestablished by Barreira and Saussol in [15]. This study revealed new nontrivial phe-nomena absent in one-dimensional multifractal analysis. In particular, while the do-main of a one-dimensional spectrum is always an interval, for multidimensional

1.7 Multidimensional Theory 15

spectra it may not be convex and may have empty interior, although still containinguncountably many points. Moreover, the interior of the domain of a multidimen-sional spectrum may have more than one connected component. We refer to [16] fora detailed discussion.

The proof of Theorem 1.10 is based on techniques developed by Barreira, Saussoland Schmeling in [16] and Barreira and Saussol in [15]. We emphasize that thisapproach deals directly with the flows and in particular it does not require Markovsystems.

Part IBasic Notions

This part is of an introductory nature and serves as a reference for the remainingchapters. We recall in a pragmatic manner all the necessary notions and results fromhyperbolic dynamics, the thermodynamic formalism and dimension theory that areneeded in the book. In Chap. 2 we consider suspension flows, the notion of co-homology and the Bowen–Walters distance. Suspension flows serve as models forhyperbolic flows which are introduced in Chap. 3. Here we also recall the notion ofa Markov system and we describe how it can be used to associate a symbolic dy-namics to any locally maximal hyperbolic set. In Chap. 4 we recall all the necessarynotions and results from the thermodynamic formalism and dimension theory. Thisincludes the notions of topological pressure, BS-dimension, lower and upper boxdimensions and pointwise dimension.

Chapter 2Suspension Flows

In this chapter we present several basic notions and results regarding suspensionflows, as a preparation for many developments in later chapters. We note that anysmooth flow with a hyperbolic set gives rise to a suspension flow (see Chap. 3). Inparticular, we present the notions of cohomology and of Bowen–Walters distance.It happens that one can often describe the properties of a suspension flow in termsof corresponding properties in the base. This relation is considered in this chap-ter for the notion of cohomology. Several other relations of a similar type will beconsidered later in the book.

2.1 Basic Notions and Cohomology

We first introduce the notion of a suspension flow. Let T : X → X be a homeomor-phism of a compact metric space and let τ : X → (0,∞) be a Lipschitz function.Consider the space

Z = {(x, s) ∈ X ×R : 0 ≤ s ≤ τ(x)

},

and let Y be the set obtained from Z by identifying the points (x, τ (x)) and(T (x),0) for each x ∈ X. One can introduce in a natural way a topology on Z

and thus also on Y (obtained from the product topology on X × R), with respectto which Y is a compact topological space. This topology is induced by a certaindistance introduced by Bowen and Walters in [31] (see Sect. 2.2).

Definition 2.1 The suspension flow over T with height function τ is the flow Ψ ={ψt }t∈R in Y with the maps ψt : Y → Y defined by

ψt(x, s) = (x, s + t) (2.1)

(see Fig. 2.1).

L. Barreira, Dimension Theory of Hyperbolic Flows,Springer Monographs in Mathematics, DOI 10.1007/978-3-319-00548-5_2,© Springer International Publishing Switzerland 2013

19

http://dx.doi.org/10.1007/978-3-319-00548-5_2

20 2 Suspension Flows

Fig. 2.1 A suspension flowΨ = {ψt }t∈R over T

We note that any suspension flow is indeed a flow, that is, ψ0 = id and

ψt ◦ ψs = ψt+s for t, s ∈R.

The set X is called the base of the suspension flow. We extend τ to a functionτ : Y → R by

τ(y) = min{t > 0 : ψt(y) ∈ X × {0}},

and T to a map T : Y → X × {0} by T (y) = ψτ(y)(y) (since there is no danger ofconfusion, we continue to use the symbols τ and T for the extensions).

Now we introduce the notion of cohomology for flows and maps.

Definition 2.2 (Notion of Cohomology)

1. A function g : Y → R is said to be Ψ -cohomologous to a function h : Y → R ina set A ⊂ Y if there exists a bounded measurable function q : Y →R such that

g(x) − h(x) = limt→0

q(ψt (x)) − q(x)

t(2.2)

for every x ∈ A.2. A function G : Y →R is said to be T -cohomologous to a function H : Y →R in

a set A ⊂ Y if there exists a bounded measurable function q : Y →R such that

G(x) − H(x) = q(T (x)) − q(x)

for every x ∈ A.

One can easily verify that the two notions of cohomology in Definition 2.2 areequivalence relations. We refer to the corresponding equivalence classes as coho-mology classes.

Now we show that the notion of cohomology for a suspension flow can be de-scribed in terms of the notion of cohomology for the map in the base. This ob-servation will be very useful in some of the proofs. Given a continuous function

2.1 Basic Notions and Cohomology 21

g : Y → R, we define a new function Ig : Y → R by

Ig(y) =∫ τ(y)

0g(ψs(y)) ds. (2.3)

Theorem 2.1 If Ψ is a suspension flow over T : X → X and g,h : Y → R arecontinuous functions, then the following properties are equivalent:

1. g is Ψ -cohomologous to h in Y , with

g(y) − h(y) = limt→0

q(ψt (y)) − q(y)

tfor y ∈ Y ;

2. Ig is T -cohomologous to Ih in Y , with

Ig(y) − Ih(y) = q(T (y)) − q(y) for y ∈ Y ; (2.4)

3. Ig is T -cohomologous to Ih in X × {0}, with

Ig(y) − Ih(y) = q(T (y)) − q(y) for y ∈ X × {0}.

Proof We first assume that g is Ψ -cohomologous to h in Y . For each y ∈ Y , wehave

Ig(y) − Ih(y) =∫ τ(y)

0limt→0

q(ψt (ψs(y))) − q(ψs(y))

tds

= limt→0

1

t

(∫ τ(y)+t

t

q(ψs(y)) ds −∫ τ(y)

0q(ψs(y)) ds

)

= limt→0

1

t

(∫ τ(y)+t

0q(ψs(y)) ds −

∫ τ(y)

0q(ψs(y)) ds

)

− limt→0

1

t

∫ t

0q(ψs(y)) ds

= q(ψτ(y)(y)) − q(y)

= q(T (y)) − q(y).

Therefore, Ig is T -cohomologous to Ih in Y .Now we assume that Ig is T -cohomologous to Ih in Y . For each x ∈ Y , we have

τ(ψt (x)) = τ(x) − t

for any sufficiently small t > 0 (depending on x). Thus, T (ψt (x)) = T (x) and itfollows from (2.4) with y = ψt(x) that

Ig(ψt (x)) − Ih(ψt (x)) = q(T (x)) − q(ψt (x)).


Since

limt→0+

Ig(ψt (x)) − Ig(x)

t= lim

t→0−1

t

∫ t

0g(ψs(x)) ds = −g(x),

we obtain

g(x) − h(x) = limt→0+

(−Ig(ψt (x)) − Ih(ψt (x))

t+ Ig(x) − Ih(x)

t

)

= limt→0+

(−q(T (x)) − q(ψt (x))

t+ q(T (x)) − q(x)

t

)

= limt→0+

q(ψt (x)) − q(x)

t. (2.5)

Similarly, we have

τ(ψ−t (x)) ={

τ(x) + t if x /∈ X × {0},t if x ∈ X × {0}

for any sufficiently small t > 0 (depending on x). Now we consider two cases. Forx /∈ X × {0} we have T (ψ−t (x)) = T (x) and one can proceed in a similar mannerto show that

g(x) − h(x) = limt→0−

q(ψt (x)) − q(x)

t. (2.6)

On the other hand, for x ∈ X×{0} we have T (ψ−t (x)) = x and it follows from (2.4)with y = ψ−t (x) that

Ig(ψ−t (x)) − Ih(ψ−t (x)) = q(x) − q(ψ−t (x)).

Since

limt→0+

Ig(ψ−t (x))

t= lim

t→0+1

t

∫ 0

−t

g(ψs(x)) ds = g(x),

we obtain

g(x) − h(x) = limt→0−

Ig(ψt (x)) − Ih(ψt (x))

−t

= limt→0−

q(x) − q(ψt (x))

−t. (2.7)

By (2.5), (2.6) and (2.7), for each x ∈ Y we have


q(ψt (x)) − q(x)

t

Therefore, g is Ψ -cohomologous to h in Y .

2.1 Basic Notions and Cohomology 23

It remains to verify that Property 3 implies Property 2 (clearly, Property 2 impliesProperty 3). Let us assume that Property 3 holds for some function q : X×{0} →R.We extend q to a function q : Y → R by

q(ψt (y)) = q(y) −∫ t

0

[g(ψs(y)) − h(ψs(y))

]ds

for every y = (x,0) and t ∈ [0, τ (x)). For each t ∈ [0, τ (x)), we have T (ψt (y)) =T (y) and by (2.3) we obtain

q(T (ψt (y))) − q(ψt (y)) = q(T (y)) − q(ψt )

=∫ τ(y)

t

[g(ψs(y)) − h(ψs(y))

]ds

= Ig(ψt (y)) − Ih(ψt (y)),

which yields Property 2. This completes the proof of the theorem. �

By Theorem 2.1, each cohomology class for the dynamics in the base X induces acohomology class for the suspension flow in the whole space Y , and all cohomologyclasses in Y appear in this way.

One can show that cohomologous functions have the same Birkhoff averages.These averages are one of the main elements of ergodic theory and multifractalanalysis.

Theorem 2.2 Let Ψ be a flow in Y and let g,h : Y →R be continuous functions. Ifg and h are Ψ -cohomologous, then

lim inft→∞

1

t

∫ t

0g(ψs(x)) ds = lim inf

t→∞1

t

∫ t

0h(ψs(x)) ds (2.8)

and

lim supt→∞

1

t

∫ t

0g(ψs(x)) ds = lim sup

t→∞1

t

∫ t

0h(ψs(x)) ds (2.9)

for every x ∈ Y .

Proof By (2.2), we have

1

t

∫ t

0g(ψs(x)) ds − 1

t

∫ t

0h(ψs(x)) ds = 1

t

∫ t

0

d

dtq(ψt (ψs(x)))|t=0 ds

= 1

t

∫ t

0

d

dsq(ψs(x)) ds

= q(ψt (x)) − q(x)

t. (2.10)


The identities in (2.8) and (2.9) now follow readily from (2.10). �

It is also of interest to describe the convergence and divergence of the Birkhoffaverages of the flow Ψ in terms of the Birkhoff averages of the map T in the base.

Theorem 2.3 Let Ψ be a suspension flow over T : X → X with height function τ

and let g : Y → R be a continuous function. For each x ∈ X and s ∈ [0, τ (x)], wehave

lim inft→∞

1

t

∫ t

0g(ψr(x, s)) dr = lim inf

m→∞

∑mi=0 Ig(T

i(x))∑m

i=0 τ(T i(x))(2.11)

and

lim supt→∞

1

t

∫ t

0g(ψr(x, s)) dr = lim sup

m→∞

∑mi=0 Ig(T

i(x))∑m

i=0 τ(T i(x)). (2.12)

Proof Given m ∈ N, we define a function τm : Y →R by

τm(x) =m−1∑

i=0

τ(T i(x)). (2.13)

For each x ∈ Y and m ∈N, we have

∫ τm(x)

0g(ψs(x)) ds =

m−1∑

i=0

∫ τi+1(x)

τi (x)

g(ψs(x)) ds

=m−1∑

i=0

∫ τ(T i (x))

0g(ψs(T

i(x))) ds

=m−1∑

i=0

Ig(Ti(x)). (2.14)

Now we observe that given t > 0, there exists a unique integer m ∈ N such thatτm(x) ≤ t < τm+1(x). We have t = τm(x) + κ for some κ ∈ (inf τ, sup τ), and thus,

1

t

∫ t

0g(ψs(x)) ds =

∫ τm(x)

0 g(ψs(x)) ds + ∫ τm(x)+κ

τm(x)g(ψs(x)) ds

τm(x) + κ.

2.2 The Bowen–Walters Distance 25

Therefore,∣∣∣∣∣1

t

∫ t

0g(ψs(x)) ds − 1

τm(x)

∫ τm(x)

0g(ψs(x)) ds

∣∣∣∣∣

≤∣∣∣∣

1

τm(x) + κ− 1

τm(x)

∣∣∣∣

∫ τm(x)

0|g(ψs(x))|ds + κ sup|g|

τm(x) + κ

≤ κ

(τm(x) + κ)τm(x)· τm(x) sup|g| + κ sup|g|

τm(x) + κ

≤ 2 sup τ sup|g|τm(x)

.

Since τ is bounded (because it is a continuous function on the compact set X),letting t → ∞, we have m → ∞ and τm(x) → ∞. Hence, it follows from (2.14)that

∣∣∣∣1

t

∫ t

0g(ψs(x)) ds − 1

τm(x)

m−1∑

i=0

Ig(Ti(x))

∣∣∣∣→ 0

when t → ∞. This completes the proof of the theorem. �

We note that for each x ∈ X the limits in (2.11) and (2.12) are independent of s.More generally, one can consider a continuous map T : X → X that need not

be a homeomorphism. More precisely, let T be a local homeomorphism in an openneighborhood of each point of the compact metric space X. The suspension semi-flow over T with height function τ is the semiflow Ψ = {ψt }t∈R in Y with the mapsψt : Y → Y defined by (2.1). One can readily extend Theorems 2.1 and 2.3 to sus-pension semiflows.

2.2 The Bowen–Walters Distance

In this section we describe a distance introduced by Bowen and Walters in [31]for suspension flows. We also establish several properties of this distance that areneeded later on in the proofs.

Let T : X → X be a homeomorphism of a compact metric space (X,dX) andlet τ : X → (0,∞) be a Lipschitz function. We consider the suspension flow Ψ ={ψt }t∈R in Y with the maps ψt : Y → Y given by (2.1). Without loss of generality,one can always assume that the diameter diamX of the space X is at most 1. Whenthis is not the case, since X is compact, one can simply consider the new distancedX/diamX in X.

We proceed with the construction of the Bowen–Walters distance. We first as-sume that τ = 1. Given x, y ∈ X and t ∈ [0,1], we define the length of the horizontalsegment [(x, t), (y, t)] (see Fig. 2.2) by

ρh((x, t), (y, t)) = (1 − t)dX(x, y) + tdX(T (x), T (y)). (2.15)


Fig. 2.2 Horizontal segment[(x, t), (y, t)] and verticalsegment [(x, t), (y, s)]

Fig. 2.3 A finite chain ofhorizontal and verticalsegments between (x, t) and(y, s)

Clearly,

ρh((x,0), (y,0)) = dX(x, y) and ρh((x,1), (y,1)) = dX(T (x), T (y)).

Moreover, given points (x, t), (y, s) ∈ Y in the same orbit, we define the length ofthe vertical segment [(x, t), (y, s)] (see Fig. 2.2) by

ρv((x, t), (y, s)) = inf{|r| : ψr(x, t) = (y, s) and r ∈ R

}. (2.16)

For the height function τ = 1, the Bowen–Walters distance d((x, t), (y, s)) betweentwo points (x, t), (y, s) ∈ Y is defined as the infimum of the lengths of all pathsbetween (x, t) and (y, s) that are composed of finitely many horizontal and verticalsegments.

More precisely, for each n ∈N, we consider all finite chains

z0 = (x, t), z2, . . . , zn−1, zn = (y, s) (2.17)

of points in Y such that for each i = 0, . . . , n − 1 the segment [zi, zi+1] is eitherhorizontal or vertical (see Fig. 2.3). The lengths of horizontal and vertical segmentsare defined respectively by (2.15) and (2.16). We remark that when the segment[zi, zi+1] is simultaneously horizontal and vertical, since by hypothesis the space X

has diameter at most 1, when computing the length of [zi, zi+1] it is considered tobe a horizontal segment. Finally, the length of the chain from z0 to zn in (2.17) isdefined as the sum of the lengths of the segments [zi, zi+1] for i = 0, . . . , n − 1.

2.2 The Bowen–Walters Distance 27

Now we consider an arbitrary height function τ : X → (0,∞) and we introducethe Bowen–Walters distance dY in Y .

Definition 2.3 Given (x, t), (y, s) ∈ Y , we define

dY ((x, t), (y, s)) = d((x, t/τ (x)), (y, s/τ(s))

),

where d is the Bowen–Walters distance for the height function τ = 1.

For an arbitrary function τ , a horizontal segment takes the form

w = [(x, tτ (x)), (y, tτ (y))],and its length is given by

�h(w) = (1 − t)dX(x, y) + tdX(T (x), T (y)).

Moreover, the length of a vertical segment w = [(x, t), (x, s)] is now

�v(w) = |t − s|/τ(x),

for any sufficiently close t and s.It is sometimes convenient to measure distances in another manner. Namely,

given (x, t), (y, s) ∈ Y , let

dπ((x, t), (y, s)) = min

⎧⎨

⎩

dX(x, y) + |t − s|,dX(T (x), y) + τ(x) − t + s,

dX(x,T (y)) + τ(y) − s + t

⎫⎬

⎭. (2.18)

We note that dπ may not be a distance. Nevertheless, the following result relates dπ

to the Bowen–Walters distance dY .

Proposition 2.1 If T is an invertible Lipschitz map with Lipschitz inverse, then thereexists a constant c ≥ 1 such that

c−1dπ(p,q) ≤ dY (p, q) ≤ cdπ (p, q) (2.19)

for every p,q ∈ Y .

Proof Let (x, t), (y, s) ∈ Y . One can easily verify that

L−1|t − s| − L2dX(x, y) ≤∣∣∣∣

t

τ (x)− s

τ (y)

∣∣∣∣≤ L|t − s| + L2dX(x, y), (2.20)

where L ≥ max{1/min τ, sup τ,1} is a Lipschitz constant simultaneously for T ,T −1 and τ . Now we consider the chain formed by the points (x, t), (y, tτ (y)/τ(x))


and (y, s). It is composed of a horizontal segment and a vertical segment, and thus,using (2.20),

dY ((x, t), (y, s)) ≤ �h

((x, t), (y, tτ (y)/τ(x))

)+ �v

((y, tτ (y)/τ(x)), (y, s)

)

≤(

1 − t

τ (x)

)dX(x, y) + t

τ (x)dX(T (x), T (y))

+∣∣∣∣

t

τ (x)− s

τ (y)

∣∣∣∣

≤ LdX(x, y) + L|t − s| + L2dX(x, y). (2.21)

Therefore,

dY ((x, t), (y, s)) ≤ c[dX(x, y) + |t − s|] (2.22)

taking c ≥ L + L2. Similarly, considering the chain formed by the points (x, t),(x, τ (x)) = (T (x),0), (y,0) and (y, s), we obtain

dY ((x, t), (y, s)) ≤ τ(x) − t

τ (x)+ dX(T (x), y) + s

τ (y)

≤ L[dX(T (x), y) + τ(x) − t + s

]. (2.23)

By (2.22), (2.23) and the symmetry of dY , we conclude that

dY ((x, t), (y, s)) ≤ cdπ ((x, t), (y, s))

taking c ≥ L + L2.For the other inequality in (2.19), consider a chain z0, . . . , zn between (x, t) and

(y, s) not intersecting the roof {(x, τ (x)) : x ∈ X} of Y . Let

�H =∑

i∈H

�h(zi, zi+1) and �V =∑

i∈V

�v(zi, zi+1),

where H is the set of all is such that [zi, zi+1] is a horizontal segment and V ={0, . . . , n−1}\H . We also write zi = (xi, ri) ∈ Y . Since the chain does not intersectthe roof, we obtain

�H =∑

i∈H

(1 − ri)dX(xi, xi+1) + ridX(T (xi), T (xi+1))

≥ L−1∑

i∈H

(1 − ri)dX(xi, xi+1) + ridX(xi, xi+1)

≥ L−1dX(x, y). (2.24)

On the other hand, by (2.20) we have

�V ≥ |t/τ (x) − s/τ(y)| ≥ L−1|t − s| − L2dX(x, y). (2.25)

2.3 Further Properties 29

It follows from (2.24) and (2.25) that

2L4�(z0, . . . , zn) ≥ (L4 + L)�H + L�V ≥ dX(x, y) + |t − s|, (2.26)

where �(z0, . . . , zn) is the length of the chain z0, . . . , zn.It is easy to verify that for any chain of length � there exists a chain with the

same endpoints, and of length at most L�, such that at most one segment of thischain intersects the roof. We notice that if a chain intersects the roof at least twicein the same direction, then its length is at least 2, which is larger than the length ofthe chain used to establish (2.21). This implies that LdY ((x, t), (y, s)) is boundedfrom below by the infimum of the lengths of all chains between (x, t) and (y, s)

intersecting the roof at most once. Now let z0, . . . , zn be a chain intersecting theroof exactly once. Without loss of generality, one can assume that there exists aj ∈ {1, . . . , n} such that rj = τ(xj ), with zj = (xj , rj ), and [zj−1, zj ] is a verticalsegment. If the point zj is after zj−1 on the same orbit, then by (2.26) we obtain

2L4[�(z0, . . . , zj ) + �(zj , . . . , zn)]≥ dX(x, xj ) + τ(x) − t + dX(T (xj ), y) + s.

Since

Ld(x, xj ) + d(T (xj ), y) ≥ d(T (x), T (xj )) + d(T (xj ), y) ≥ d(T (x), y),

we conclude that

2L5�(z1, . . . , zn) ≥ dX(T (x), y) + τ(x) − t + s. (2.27)

Similarly, if the point zj is before zj−1 on the same orbit, then

2L5�(z1, . . . , zn) ≥ dX(x,T (y)) + τ(y) − s + t. (2.28)

By (2.26), (2.27) and (2.28), we obtain

dπ((x, t), (y, s)) ≤ cdY ((x, t), (y, s))

provided that c ≥ 2L6. Since 2L6 ≥ L + L2, taking c = 2L6 we obtain the inequal-ities in (2.19). �

2.3 Further Properties

In this section we establish some additional properties of suspension flows, relatedto the existence and uniqueness of equilibrium and Gibbs measures. We continueto assume that T is an invertible Lipschitz map with Lipschitz inverse. Given acontinuous function g : Y → R, we consider the function Ig : X → R given by (2.3).

Proposition 2.2 If g is Hölder continuous, then Ig is Hölder continuous in X.


Proof Take x, y ∈ X with τ(x) ≥ τ(y). We have

|Ig(x) − Ig(y)| =∣∣∣∣∣

∫ τ(x)

τ(y)

g(ψs(x)) ds +∫ τ(y)

0[g(ψs(x)) − g(ψs(y))]ds

∣∣∣∣∣

≤ sup|g| · |τ(x) − τ(y)| + sup τ · sups∈(0,τ (y))

|g(ψs(x)) − g(ψs(y))|

≤ sup|g| · LdX(x, y) + b sups∈(0,τ (y))

dY ((x, s), (y, s))α (2.29)

for some positive constants α and b. It follows from Proposition 2.1 and (2.29)(together with (2.18)) that

|Ig(x) − Ig(y)| ≤ sup|g| · LdX(x, y) + b(cdπ ((x, s), (y, s))

)α

≤ [sup|g| · L + bcα

]dX(x, y)α.

This yields the desired result. �

Now we consider Bowen balls in X and Y , defined respectively by

BX(x,m, ε) =⋂

0≤n≤m

T −nBX(T n(x), ε) (2.30)

for each x ∈ X, m ∈N and ε > 0, and

BY (y,ρ, ε) =⋂

0≤t≤ρ

ψ−tBY (ψt (y), ε) (2.31)

for each y ∈ Y and ρ, ε > 0.

Definition 2.4 The map T is said to have bounded variation if for each Höldercontinuous function g : X →R there exists a constant D > 0 such that

∣∣∣∣∣

m−1∑

k=0

g(T k(x)) −m−1∑

k=0

g(T k(y))

∣∣∣∣∣≤ Dε

for every x ∈ X, m ∈N, ε > 0 and y ∈ BX(x,m, ε).

The following result establishes a relation between the Bowen balls in (2.30)and (2.31). We recall the function τm given by (2.13).

Proposition 2.3 If T has bounded variation, then there exists a κ ≥ 1 such that

BY

((x, s), τm(x), ε/κ

)⊂ BX(x,m, ε) × (s − ε, s + ε) ⊂ BY

((x, s), τm(x), κε

)

(2.32)for every x ∈ X, s ∈ [0, τ (x)] and m ∈N, and any sufficiently small ε > 0.

2.3 Further Properties 31

Proof Take ε ∈ (0,1/(2c)) with c as in Proposition 2.1. Moreover, take (x, t) ∈ Y

with t ∈ (cε, τ (x) − cε) and (y, t) ∈ BY ((x, s), τm(x), ε).If m = 0, then

dπ((x, t), (y, s)) ≤ cε,

by Proposition 2.1. Since

τ(x) − t + s ≥ τ(x) − t ≥ cε and τ(y) − s + t ≥ t ≥ cε,

we obtain

dX(x, y) + |t − s| = dπ((x, t), (y, s)) ≤ cε,

which implies that dX(x, y) ≤ cε and |t −s| ≤ cε. This establishes the first inclusionin (2.32) for m = 0.

Now take n ∈ {1, . . . ,m}, and let tn = τn(x) − t and sn = τn(y) − s. One caneasily verify that

ψtn(x, t) = (T n(x),0) and ψsn(y, s) = (T n(y),0).

By Proposition 2.1, we obtain

dX(T n(x), T n(y)) ≤ cdY (ψtn(x, t),ψsn(y, s))

≤ cdY (ψtn(x, t),ψtn(y, s)) + cdY (ψtn(y, s),ψsn(y, s))

≤ cε + c|tn − sn|. (2.33)

Moreover, by (2.18), we have

dπ(ψtn(x, t),ψtn(y, s)) ≤ cε.

Thus, there exist yn ∈ X and rn ∈ (tn −cε, tn +cε) such that ψrn(y, s) = (yn,0). Thesequence rn is strictly increasing, because tn+1 − tn > 2cε. Hence sn ≤ rn ≤ tn + cε.By symmetry, we obtain tn ≤ sn + cε, and hence |tn − sn| ≤ cε. Finally, by (2.33),we conclude that

dX(T n(x), T n(y)) ≤ c(1 + c)ε.

Taking κ ≥ c(1 + c), this establishes the first inclusion in (2.32).Now let y ∈ BX(x,m, ε) and s ∈ (t − ε, t + ε). Take r ∈ (0, τm(x)) and n ∈ N

such that

τn(x) ≤ r + t < τn+1(x).


We also write r ′ = r + t −τn(x) ≥ 0. Since T has bounded variation, it follows fromProposition 2.1 that

dY (ψr(x, t),ψr(y, s)) ≤ dY

((T n(x), r ′), (T n(y), r ′)

)+ dY

((T n(y), r ′),ψr(y, s)

)

≤ cdπ

((T n(x), r ′), (T n(y), r ′)

)

+ cdπ

((T n(y), r ′),ψr(y, s)

)

≤ cdX(T n(x), T n(y)) + c|r ′ + τn(y) − r − s|≤ cdX(T n(x), T n(y)) + c|t − s| + c|τn(x) − τn(y)|≤ c(2 + D)ε.

Taking κ ≥ c(2 + D), this establishes the second inclusion in (2.32). �

Chapter 3Hyperbolic Flows

In this chapter we recall in a pragmatic manner all the necessary notions and resultsfrom hyperbolic dynamics, starting with the notion of a hyperbolic set for a flow. Inparticular, we consider the Markov systems constructed by Bowen and Ratner for alocally maximal hyperbolic set, and we describe how they can be used to associate asymbolic dynamics to the hyperbolic set. This allows one to see the restriction of anysmooth flow to a hyperbolic set as a factor of a suspension flow over a topologicalMarkov chain.

3.1 Basic Notions

Let Φ = {ϕt }t∈R be a C1 flow in a smooth manifold M . This means that ϕ0 = id,

ϕt ◦ ϕs = ϕt+s for t, s ∈ R,

and that the map (t, x) → ϕt (x) is of class C1. We first introduce the notion of ahyperbolic set.

Definition 3.1 A compact Φ-invariant set Λ ⊂ M is said to be a hyperbolic setfor Φ if there exists a splitting

TΛM = Es ⊕ Eu ⊕ E0,

(see Fig. 3.1) and constants c > 0 and λ ∈ (0,1) such that for each x ∈ Λ:

1. the vector (d/dt)ϕt (x)|t=0 generates E0(x);2. for each t ∈ R we have

dxϕtEs(x) = Es(ϕt (x)) and dxϕtE

u(x) = Eu(ϕt (x));3. ‖dxϕtv‖ ≤ cλt‖v‖ for v ∈ Es(x) and t > 0;4. ‖dxϕ−t v‖ ≤ cλt‖v‖ for v ∈ Eu(x) and t > 0.


33

http://dx.doi.org/10.1007/978-3-319-00548-5_3

34 3 Hyperbolic Flows

Fig. 3.1 The splitting of ahyperbolic set

For example, for any geodesic flow in a compact Riemannian manifold withstrictly negative sectional curvature the whole unit tangent bundle is a hyperbolicset. Furthermore, time changes and small C1 perturbations of a flow with a hyper-bolic set also have a hyperbolic set.

Now let Λ be a hyperbolic set for Φ . For each x ∈ Λ and any sufficiently smallε > 0, we consider the sets

As(x) = {y ∈ B(x, ε) : d(ϕt (y),ϕt (x)) ↘ 0 when t → +∞}

and

Au(x) = {y ∈ B(x, ε) : d(ϕt (y),ϕt (x)) ↘ 0 when t → −∞}

.

Let V s(x) ⊂ As(x) and V u(x) ⊂ Au(x) be the largest connected components con-taining x. These are smooth manifolds, called respectively (local) stable and unsta-ble manifolds (of size ε) at the point x. Moreover:

1.

TxVs(x) = Es(x) and TxV

u(x) = Eu(x);2. for each t > 0 we have

ϕt (Vs(x)) ⊂ V s(ϕt (x)) and ϕ−t (V

u(x)) ⊂ V u(ϕ−t (x));3. there exist κ > 0 and μ ∈ (0,1) such that for each t > 0 we have

d(ϕt (y),ϕt (x)) ≤ κμtd(y, x) for y ∈ V s(x), (3.1)

and

d(ϕ−t (y), ϕ−t (x)) ≤ κμtd(y, x) for y ∈ V u(x).

We also introduce the notion of a locally maximal hyperbolic set.

Definition 3.2 A set Λ is said to be locally maximal (with respect to a flow Φ) ifthere exists an open neighborhood U of Λ such that

Λ =⋂

t∈Rϕt (U). (3.2)

3.2 Markov Systems 35

Now let Λ be a locally maximal hyperbolic set. For any sufficiently small ε > 0,there exists a δ > 0 such that if x, y ∈ Λ are at a distance d(x, y) ≤ δ, then thereexists a unique t = t (x, y) ∈ [−ε, ε] for which the set

[x, y] = V s(ϕt (x)) ∩ V u(y) (3.3)

is a single point in Λ.

3.2 Markov Systems

In order to establish some of the results we need the notion of a Markov systemand its associated symbolic dynamics. These were developed by Bowen [27] andRatner [90].

Let Φ be a C1 flow with a locally maximal hyperbolic set Λ. Consider an opensmooth disk D ⊂ M of dimension dimM − 1 that is transverse to the flow Φ , andtake x ∈ D. Let also U(x) be an open neighborhood of x diffeomorphic to theproduct D × (−ε, ε). The projection πD : U(x) → D defined by πD(ϕt (y)) = y isdifferentiable.

Definition 3.3 A closed set R ⊂ Λ ∩ D is said to be a rectangle if R = intR(with the interior computed with respect to the induced topology on Λ ∩ D) andπD([x, y]) ∈ R for x, y ∈ R.

Now we consider a collection of rectangles R1, . . . ,Rk ⊂ Λ (each contained insome open disk transverse to the flow) such that

Ri ∩ Rj = ∂Ri ∩ ∂Rj for i = j.

Let Z =⋃ki=1 Ri . We assume that there exists an ε > 0 such that:

1. Λ =⋃t∈[0,ε] ϕt (Z);

2. for each i = j either

ϕt (Ri) ∩ Rj = ∅ for every t ∈ [0, ε],or

ϕt (Rj ) ∩ Ri = ∅ for every t ∈ [0, ε].We define the transfer function τ : Λ → [0,∞) by

τ(x) = min{t > 0 : ϕt (x) ∈ Z

}, (3.4)

and the transfer map T : Λ → Z by

T (x) = ϕτ(x)(x). (3.5)


The set Z is a Poincaré section for the flow Φ . One can easily verify that the restric-tion of the map T to Z is invertible. We also have T n(x) = ϕτn(x)(x), where

τn(x) =n−1∑

i=0

τ(T i(x)). (3.6)

Now we introduce the notion of a Markov system.

Definition 3.4 The collection of rectangles R1, . . . ,Rk is said to be a Markov sys-tem for Φ on Λ if

T(int(V s(x) ∩ Ri)

)⊂ int(V s(T (x)) ∩ Rj

)

and

T −1(int(V u(T (x)) ∩ Rj ))⊂ int

(V u(x) ∩ Ri

)

for every x ∈ intT (Ri) ∩ intRj .

It follows from work of Bowen and Ratner that any locally maximal hyperbolicset Λ has Markov systems of arbitrary small diameter (see [27, 90]). Furthermore,the map τ is Hölder continuous on each domain of continuity, and

0 < inf{τ(x) : x ∈ Z} ≤ sup{τ(x) : x ∈ Λ} < ∞. (3.7)

3.3 Symbolic Dynamics

In this section we describe how a Markov system for a hyperbolic set gives rise to asymbolic dynamics.

Given a Markov system R1, . . . ,Rk for a flow Φ on a locally maximal hyperbolicset Λ, we consider the k × k matrix A with entries

aij ={

1 if intT (Ri) ∩ intRj = ∅,

0 otherwise,(3.8)

where T is the transfer map in (3.5). We also consider the set ΣA ⊂ {1, . . . , k}Zgiven by

ΣA = {(· · · i−1i0i1 · · · ) : ainin+1 = 1 for n ∈ Z

},

and the shift map σ : ΣA → ΣA defined by σ(· · · i0 · · · ) = (· · · j0 · · · ), where jn =in+1 for each n ∈ Z.

Definition 3.5 The map σ |ΣA is said to be a (two-sided) topological Markov chainwith transition matrix A.

3.3 Symbolic Dynamics 37

We define a coding map π : ΣA →⋃ki=1 Ri for the hyperbolic set by

π(· · · i0 · · · ) =⋂

j∈Z(T |Z)−j (intRij ).

One can easily verify that

π ◦ σ = T ◦ π. (3.9)

Given β > 1, we equip ΣA with the distance d given by

d((· · · i−1i0i1 · · · ), (· · · j−1j0j1 · · · ))=

∞∑

n=−∞β−|n||in − jn|. (3.10)

As observed in [27], it is always possible to choose the constant β so that the func-tion τ ◦ π : ΣA → [0,∞) is Lipschitz.

By (3.9), the restriction of a smooth flow to a locally maximal hyperbolic set isa factor of a suspension flow over a topological Markov chain. Namely, to eachMarkov system one can associate the suspension flow Ψ = {ψt }t∈R over σ |ΣA

with (Lipschitz) height function τ ◦ π . We extend π to a finite-to-one onto mapπ : Y → Λ by

π(x, s) = (ϕs ◦ π)(x) (3.11)

for (x, s) ∈ Y . Then

π ◦ ψt = ϕt ◦ π (3.12)

for every t ∈R.We denote by Σ+

A the set of (one-sided) sequences (i0i1 · · · ) such that

(i0i1 · · · ) = (j0j1 · · · ) for some (· · · j−1j0j1 · · · ) ∈ ΣA,

and by Σ−A the set of (one-sided) sequences (· · · i−1i0) such that

(· · · i−1i0) = (· · · j−1j0) for some (· · · j−1j0j1 · · · ) ∈ ΣA.

The set Σ−A can be identified with Σ+

A∗ , where A∗ is the transpose of A, by the map

Σ−A � (· · · i−1i0) → (i0i−1 · · · ) ∈ Σ+

A∗ .

We also consider the shift maps σ+ : Σ+A → Σ+

A and σ− : Σ−A → Σ−

A defined by

σ+(i0i1 · · · ) = (i1i2 · · · ) and σ−(· · · i−1i0) = (· · · i−2i−1).

Now we describe how distinct points in a stable or unstable manifold can becharacterized in terms of the symbolic dynamics. Given x ∈ Λ, take ω ∈ ΣA suchthat π(ω) = x. Let R(x) be a rectangle of the Markov system that contains x. Foreach ω′ ∈ ΣA, we have

π(ω′) ∈ V u(x) ∩ R(x) whenever π−(ω′) = π−(ω),


and

π(ω′) ∈ V s(x) ∩ R(x) whenever π+(ω′) = π+(ω),

where π+ : ΣA → Σ+A and π− : ΣA → Σ−

A are the projections defined by

π+(· · · i−1i0i1 · · · ) = (i0i1 · · · ) (3.13)

and

π−(· · · i−1i0i1 · · · ) = (· · · i−1i0). (3.14)

Therefore, writing ω = (· · · i−1i0i1 · · · ), the set V u(x)∩R(x) can be identified withthe cylinder set

C+i0

= {(j0j1 · · · ) ∈ Σ+

A : j0 = i0}⊂ Σ+

A , (3.15)

and the set V s(x) ∩ R(x) can be identified with the cylinder set

C−i0

= {(· · · j−1j0) ∈ Σ−

A : j0 = i0}⊂ Σ−

A . (3.16)

Chapter 4Pressure and Dimension

In this chapter we recall in a pragmatic manner all the necessary notions and re-sults from the thermodynamic formalism and dimension theory. In particular, weintroduce the notions of topological pressure, BS-dimension, Hausdorff dimension,lower and upper box dimensions and pointwise dimension. We emphasize that weconsider the general case of the topological pressure for noncompact sets, which iscrucial in multifractal analysis.

4.1 Topological Pressure and Entropy

This section is dedicated to the notion of topological pressure and some of its basicproperties, including the variational principle and its regularity properties. We referthe reader to [30, 63, 80, 92, 106] for details and proofs.

4.1.1 Basic Notions

We first introduce the notions of topological pressure and topological entropy.Let Φ = {ϕt }t∈R be a continuous flow in a compact metric space (X,d). Given

x ∈ X, t > 0 and ε > 0, we consider the Bowen ball

B(x, t, ε) = {y ∈ X : d(ϕs(y),ϕs(x)) < ε for s ∈ [0, t]}. (4.1)

Now let a : X → R be a continuous function and write

a(x, t, ε) = sup

{∫ t

0a(ϕs(y)) ds : y ∈ B(x, t, ε)

}. (4.2)

For each set Z ⊂ X and α ∈ R, we define

M(Z,a,α, ε) = limT →∞ inf

Γ

∑

(x,t)∈Γ

exp(a(x, t, ε) − αt

),


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40 4 Pressure and Dimension

where the infimum is taken over all finite or countable sets Γ = {(xi, ti )}i∈I suchthat xi ∈ X and ti ≥ T for i ∈ I , and

⋃i∈I B(xi, ti , ε) ⊃ Z. One can easily verify

that the limit

PΦ|Z(a) = limε→0

inf{α ∈ R : M(Z,a,α, ε) = 0

}

exists.

Definition 4.1 The number PΦ|Z(a) is called the topological pressure of a on theset Z (with respect to the flow Φ).

We note that Z need not be compact nor Φ-invariant. This observation is crucialin multifractal analysis since typically the sets under consideration are not com-pact. For simplicity of notation, we also write PΦ(a) = PΦ|X(a). When the set Z iscompact and Φ-invariant, the topological pressure is given by

PΦ|Z(u) = limε→0

lim inft→∞

1

tlog inf

Γ

∑

x∈Γ

exp(u(x, t, ε))

= limε→0

lim supt→∞

1

tlog inf

Γ

∑

x∈Γ

exp(u(x, t, ε)), (4.3)

where the infimum is taken over all finite or countable sets Γ = {xi}i∈I ⊂ X suchthat

⋃i∈I B(xi, t, ε) ⊃ Z.

Definition 4.2 The number h(Φ|Z) = PΦ|Z(0) is called the topological entropyof Φ on the set Z.

When Z is compact and Φ-invariant, we recover the usual notion of topologicalentropy, that is,

h(Φ|Z) = limε→0

lim inft→∞

logNZ(t, ε)

t= lim

ε→0lim supt→∞

logNZ(t, ε)

t,

where NZ(t, ε) is the least number of sets D(x) = B(x, t, ε) that are needed tocover Z.

Now let M be the set of all Φ-invariant probability measures on X. We recallthat a measure μ in X is said to be Φ-invariant if μ(ϕt (A)) = μ(A) for every setA ⊂ X and t ∈R. When equipped with the weak∗ topology, the space M is compactand metrizable. We also recall that a measure μ on X is said to be ergodic if for anyΦ-invariant set A ⊂ X (that is, any set such that ϕt (A) = A for every t ∈ R) we haveμ(A) = 0 or μ(X \ A) = 0.

Moreover, for each measure μ ∈ M the limit

hμ(Φ) = limε→0

inf{h(Z, ε) : μ(Z) = 1} (4.4)

4.1 Topological Pressure and Entropy 41

exists, where

h(Z, ε) = inf{α ∈ R : M(Z,0, α, ε) = 0

}.

Proposition 4.1 If Φ is a continuous flow in a compact metric space and μ ∈ M isan ergodic measure, then the number hμ(Φ) in (4.4) coincides with the entropy ofΦ with respect to μ, that is, the entropy of the time-1 map ϕ1 with respect to μ.

The proof of Proposition 4.1 can be obtained from a simple modification of theproof of an analogous result established by Pesin in [81] in the case of discrete time.

4.1.2 Properties of the Pressure

In this section we recall some basic properties of the topological pressure, startingwith the variational principle.

Proposition 4.2 (Variational Principle) If Φ is a continuous flow in a compact met-ric space X and a : X → R is a continuous function, then

PΦ(a) = sup

{hμ(Φ) +

∫

X

a dμ : μ ∈ M

}. (4.5)

A measure μ ∈ M is said to be an equilibrium measure for the function a (withrespect to the flow Φ) if the supremum in (4.5) is attained at this measure, that is,

PΦ(a) = hμ(Φ) +∫

X

a dμ.

We denote by C(X) the space of all continuous functions a : X →R equipped withthe supremum norm, and by D(X) ⊂ C(X) the family of continuous functions witha unique equilibrium measure. Given a finite set K ⊂ C(X), we denote by spanK ⊂C(X) the linear space generated by the functions in K .

Proposition 4.3 If Φ is a continuous flow in a compact metric space X such thatthe map μ → hμ(Φ) is upper semicontinuous, then:

1. any function a ∈ C(X) has equilibrium measures, and D(X) is dense in C(X);2. given a, b ∈ C(X), the map R � t → PΦ(a + tb) is differentiable at t = 0 if

and only if a ∈ D(X), in which case the unique equilibrium measure μa for thefunction a is ergodic and satisfies

d

dtPΦ(a + tb)|t=0 =

∫

X

b dμa; (4.6)

3. if span{a, b} ⊂ D(X), then the function t → PΦ(a + tb) is of class C1.


In order to give some examples of flows with an upper semicontinuous entropy,we introduce the notion of an expansive flow.

Definition 4.3 A flow Φ in a metric space X is said to be expansive if there exists anε > 0 such that given x, y ∈ X and a continuous function s : R → R with s(0) = 0satisfying

d(ϕt (x),ϕs(t)(x)) < ε and d(ϕt (x),ϕs(t)(y)) < ε

for every t ∈R, we have x = y.

If Φ is an expansive flow, then the map μ → hμ(Φ) is upper semicontinuous(see [106]). For example, if Λ is a hyperbolic set for a flow Φ , then Φ|Λ is expan-sive.

Also for flows with a hyperbolic set, Proposition 4.3 can be strengthened as fol-lows. We first recall that Φ is said to be topologically mixing on Λ (or simply Φ|Λis said to be topologically mixing) if for any nonempty open sets U and V intersect-ing Λ there exists an s ∈ R such that ϕt (U) ∩ V ∩ Λ = ∅ for every t > s.

Proposition 4.4 If Φ is a C1 flow with a locally maximal hyperbolic set Λ suchthat Φ|Λ is topologically mixing, then:

1. the map μ → hμ(Φ) is upper semicontinuous;2. each Hölder continuous function a : Λ →R has a unique equilibrium measure;3. given Hölder continuous functions a, b : Λ → R, the function R � t → PΦ(a +

tb) is analytic and

d2

dt2PΦ(a + tb) ≥ 0 for t ∈ R, (4.7)

with equality if and only if b is Φ-cohomologous to a constant.

The first property in Proposition 4.4 follows from the fact that Φ|Λ is expansive.We recall that a function a : Λ → R is said to be Φ-cohomologous to a functionb : Λ →R if there exists a bounded measurable function q : Λ →R such that

a(x) − b(x) = limt→0

q(ϕt (x)) − q(x)

t

(see Definition 2.2), in which case PΦ|Λ(a) = PΦ|Λ(b). In particular, if b is con-stant, then PΦ|Λ(a) = h(Φ|Λ) + b.

4.1.3 The Case of Suspension Flows

In this section we consider the particular case of suspension flows and we explainhow the topological pressure and the invariant measures for the flow are related tothe corresponding notions in the base.

4.2 BS-Dimension 43

Let Ψ = {ψt }t∈R be a suspension flow in Y , over a homeomorphism T : X → X

of the compact metric space X, and let μ be a T -invariant probability measure on X.One can show that μ induces a Ψ -invariant probability measure ν on Y such that

∫

Y

g dν =∫

X

∫ τ(x)

0g(x, s) dsdμ(x)

/∫

X

τ dμ (4.8)

for every continuous function g : Y → R. We notice that locally ν is the product ofμ and Lebesgue measure. Moreover, any Ψ -invariant probability measure ν on Y

is of this form for some T -invariant probability measure μ on X. We remark thatidentity (4.8) is equivalent to

∫

Y

g dν =∫

X

Ig dμ/∫

X

τ dμ, (4.9)

where Ig is the function given by (2.3). Moreover, Abramov’s entropy formula(see [1]) says that

hν(Ψ ) = hμ(T )∫X

τ dμ. (4.10)

Now let g : Y → R be a continuous function. By (4.9) and (4.10), we have

hν(Ψ ) +∫

Y

g dν = hμ(T ) + ∫X

Ig dμ∫X

τ dμ(4.11)

for any T -invariant probability measure μ on X, where ν is the Ψ -invariant proba-bility measure induced by μ on Y . Since τ > 0, it follows from (4.11) that

PΨ (g) = 0 if and only if PT (Ig) = 0,

where PT (Ig) is the topological pressure of Ig with respect to T . Therefore, whenPΨ (g) = 0 the measure ν is an equilibrium measure for g (with respect to Ψ ) if andonly if μ is an equilibrium measure for Ig|X (with respect to T ). Moreover, since

supμ

hμ(T ) + ∫X(Ig − PΨ (g)τ) dμ∫X

τ dμ= sup

μ

hμ(T ) + ∫X

Ig dμ∫X

τ dμ− PΨ (g)

= supν

(hν(Φ) +

∫

Y

g dν

)− PΨ (g) = 0,

we have

PT (Ig − PΨ (g)τ) = 0.

4.2 BS-Dimension

In this section we recall a Carathéodory characteristic introduced by Barreira andSaussol in [12]. It is a generalization of the notion of topological entropy and is


a version of a Carathéodory characteristic introduced by Barreira and Schmelingin [17] in the case of discrete time.

Let Φ be a continuous flow in a compact metric space X and let u : X → R+ be

a continuous function. For each set Z ⊂ X and α ∈ R, we define

N(Z,u,α, ε) = limT →∞ inf

Γ

∑

(x,t)∈Γ

exp(−αu(x, t, ε)

),

where the infimum is taken over all finite or countable sets Γ = {(xi, ti )}i∈I suchthat xi ∈ X and ti ≥ T for i ∈ I , and

⋃i∈I B(xi, ti , ε) ⊃ Z. Writing

dimu,ε Z = inf{α ∈ R : N(Z,u,α, ε) = 0

},

one can show that the limit

dimu Z = limε→0

dimu,ε Z

exists.

Definition 4.4 dimu Z is called the BS-dimension of Z (with respect to u).

When u = 1 the BS-dimension coincides with the topological entropy, that is,dimu Z = h(Φ|Z).

It follows easily from the definitions that the topological pressure and the BS-dimension are related as follows.

Proposition 4.5 The unique root of the equation PΦ|Z(−αu) = 0 is α = dimu Z.

Given a probability measure μ in X and ε > 0, let

dimu,ε μ = inf{dimu,ε Z : μ(Z) = 1

}.

One can easily verify that the limit

dimu μ = limε→0

dimu,ε μ

exists.

Definition 4.5 dimu μ is called the BS-dimension of μ (with respect to u).

For each ergodic measure μ ∈M, we have

dimu μ = hμ(Φ)∫X

udμ.

This identity can be obtained in a similar manner to that in the case of discrete time(see [3, Proposition 7.2.7]).

4.3 Hausdorff and Box Dimensions 45

4.3 Hausdorff and Box Dimensions

In this section we review some notions and results from dimension theory, both forsets and measures. In particular, we introduce the notions of Hausdorff dimension,lower and upper box dimensions, and pointwise dimension. We refer the readerto [3, 41] for details and proofs.

4.3.1 Dimension of Sets

We define the diameter of a set U ⊂ Rm by

diamU = sup{‖x − y‖ : x, y ∈ U

},

and the diameter of a collection U of subsets of Rm by

diamU= sup{diamU : U ∈U

}.

Given a set Z ⊂ Rm and α ∈ R, the α-dimensional Hausdorff measure of Z is de-

fined by

m(Z,α) = limε→0

infU

∑

U∈U(diamU)α,

where the infimum is taken over all finite or countable collections U with diamU ≤ ε

such that⋃

U∈U U ⊃ Z.

Definition 4.6 The Hausdorff dimension of Z ⊂ Rm is defined by

dimH Z = inf{α ∈R : m(Z,α) = 0}.

The lower and upper box dimensions of Z ⊂ Rm are defined respectively by

dimBZ = lim infε→0

logN(Z,ε)

− log εand dimBZ = lim sup

ε→0

logN(Z,ε)

− log ε,

where N(Z,ε) is the least number of balls of radius ε that are needed to cover theset Z.


dimH Z ≤ dimBZ ≤ dimBZ. (4.12)


4.3.2 Dimension of Measures

Now we introduce corresponding notions for measures and we relate them to thepointwise dimension. Let μ be a finite measure on a set X ⊂ R

m.

Definition 4.7 The Hausdorff dimension and the lower and upper box dimensionsof μ are defined respectively by

dimH μ = inf{dimH Z : μ(X \ Z) = 0

},

dimBμ = limδ→0

inf{dimBZ : μ(Z) ≥ μ(X) − δ

},

dimBμ = limδ→0

inf{dimBZ : μ(Z) ≥ μ(X) − δ

}.


dimH μ = limδ→0

inf{dimH Z : μ(Z) ≥ μ(X) − δ

}(4.13)

(see [3]). Moreover, it follows from (4.12) and (4.13) that

dimH μ ≤ dimBμ ≤ dimBμ. (4.14)

The following quantities allow us to formulate a criterion for the coincidence ofthe three numbers in (4.14).

Definition 4.8 The lower and upper pointwise dimensions of the measure μ at thepoint x ∈ X are defined respectively by

dμ(x) = lim infr→0

logμ(B(x, r))

log rand dμ(x) = lim sup

r→0

logμ(B(x, r))

log r.

The following criterion is due to Young [108].

Proposition 4.6 If μ is a finite measure on X and there exists a constant d ≥ 0 suchthat

dμ(x) = dμ(x) = d

for μ-almost every x ∈ X, then

dimH μ = dimBμ = dimBμ = d.

The following result expresses the Hausdorff dimension of a measure in terms ofthe lower pointwise dimension.

Proposition 4.7 If μ is a finite measure on X, then the following properties hold:

4.3 Hausdorff and Box Dimensions 47

1. if dμ(x) ≥ α for μ-almost every x ∈ X, then dimH μ ≥ α;2. if dμ(x) ≤ α for every x ∈ Z ⊂ X, then dimH Z ≤ α;3. we have

dimH μ = ess sup{dμ(x) : x ∈ X}.

By Whitney’s embedding theorem, Proposition 4.7 can be readily extended tomeasures on subsets of smooth manifolds.

We also want to describe how the Hausdorff dimension of an invariant measureis related to its ergodic decompositions. Let Φ be a continuous flow in a metricspace M and let X ⊂ M be a compact Φ-invariant set. We continue to denote byM the set of all Φ-invariant probability measures on X and we endow it with theweak∗ topology. Let also ME ⊂ M be the subset of all ergodic measures.

Definition 4.9 Given μ ∈ M, a probability measure τ in M (or, more precisely,in the Borel σ -algebra generated by the weak∗ topology) is said to be an ergodicdecomposition of μ if τ(ME) = 1 and

∫

X

ϕ dμ =∫

M

(∫

X

ϕ dν

)dτ(ν)

for any continuous function ϕ : X → R.

It is well known that any measure μ ∈ M has ergodic decompositions. The fol-lowing statement is a simple consequence of the definitions.

Proposition 4.8 If τ is an ergodic decomposition of a measure μ ∈ M, then

dimH μ ≥ ess sup{dimH ν : ν ∈ ME}, (4.15)

where the essential supremum is taken with respect to τ .

We emphasize that in general inequality (4.15) may be strict. For example, if Φ isa rational linear flow in the 2-torus T2, then the Lebesgue measure μ has Hausdorffdimension dimH μ = 2 but clearly

ess sup{dimH ν : ν ∈ ME} = 1.

Part IIDimension Theory

This part is dedicated to the dimension theory of hyperbolic flows, both for invariantmeasures and invariant sets. In Chap. 5 we study the dimension of a locally max-imal hyperbolic set for a conformal flow in terms of the topological pressure. Thearguments use Markov systems. Chapter 6 is dedicated to the study of the pointwisedimension of an arbitrary invariant measure sitting on a locally maximal hyperbolicset for a conformal flow. The pointwise dimension is expressed in terms of the localentropy and the Lyapunov exponents. We also describe the Hausdorff dimension ofa nonergodic measure in terms of an ergodic decomposition and we establish theexistence of invariant measures of maximal dimension.

Chapter 5Dimension of Hyperbolic Sets

This chapter is dedicated to the study of the dimension of a locally maximal hyper-bolic set for a conformal flow. We first consider the dimensions along the stable andunstable manifolds and we compute them in terms of the topological pressure. Wealso show that the Hausdorff dimension and the lower and upper box dimensionsof the hyperbolic set coincide and that they are obtained by adding the dimensionsalong the stable and unstable manifolds, plus the dimension along the flow. This isa consequence of the conformality of the flow. The proofs are based on the use ofMarkov systems.

5.1 Dimensions Along Stable and Unstable Manifolds

In this section we obtain formulas for the dimensions of a locally maximal hyper-bolic set for a conformal flow along the stable and unstable manifolds. These areexpressed in terms of the topological pressure.

Let Φ = {ϕt }t∈R be a C1 flow and let Λ be a locally maximal hyperbolic setfor Φ . We first introduce the notion of a conformal flow.

Definition 5.1 The flow Φ is said to be conformal on Λ (or simply Φ|Λ is said tobe conformal) if the maps

dxϕt |Es(x) : Es(x) → Es(ϕt (x)) and dxϕt |Eu(x) : Eu(x) → Eu(ϕt (x))

are multiples of isometries for every x ∈ Λ and t ∈R.

This means that the flow contracts and expands equally in all directions. Forexample, if

dimEs(x) = dimEu(x) = 1 (5.1)

for every x ∈ Λ, then the flow is conformal on Λ. Specific examples satisfying (5.1)are given by any geodesic flow on the unit tangent bundle of a compact surface M


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52 5 Dimension of Hyperbolic Sets

with negative (sectional) curvature. Certainly, the dimension theory of this particularclass of examples is trivial (because the whole unit tangent bundle is a hyperbolic setfor the geodesic flow and thus, its dimension is simply 2 dimM − 1). On the otherhand, the multifractal analysis of dimension spectra, also developed for conformalflows (see Chap. 8), is nontrivial even in this particular class of examples.

Now we assume that Φ|Λ is conformal and we consider the families of localstable and unstable manifolds V s(x) and V u(x) for x ∈ Λ. The following resultof Pesin and Sadovskaya [82] expresses the dimensions of the sets V s(x) ∩ Λ andV u(x)∩ Λ in terms of the topological pressure. We define functions ζs, ζu : Λ →R

by

ζs(x) = ∂

∂tlog‖dxϕt |Es(x)‖

∣∣∣t=0

= limt→0

1

tlog‖dxϕt |Es(x)‖ (5.2)

and

ζu(x) = ∂

∂tlog‖dxϕt |Eu(x)‖

∣∣∣t=0

= limt→0

1

tlog‖dxϕt |Eu(x)‖. (5.3)

Since the flow Φ is of class C1, these functions are well defined.

Theorem 5.1 Let Φ be a C1+δ flow with a locally maximal hyperbolic set Λ suchthat Φ|Λ is conformal and topologically mixing. Then

dimH (V s(x) ∩ Λ) = dimB(V s(x) ∩ Λ) = dimB(V s(x) ∩ Λ) = ts (5.4)

and

dimH (V u(x) ∩ Λ) = dimB(V u(x) ∩ Λ) = dimB(V u(x) ∩ Λ) = tu (5.5)

for every x ∈ Λ, where ts and tu are the unique real numbers such that

PΦ|Λ(tsζs) = PΦ|Λ(−tuζu) = 0. (5.6)

Proof The idea of the proof is to first compute the dimensions along the stableand unstable directions inside the elements of some Markov system. We start byverifying that the numbers ts and tu are well defined. It follows from (4.6) that

d

dtPΦ|Λ(tζs) =

∫

Λ

ζs dμts

andd

dtPΦ|Λ(−tζu) = −

∫

Λ

ζu dμtu,

where μts and μt

u are respectively the equilibrium measures for tζs and −tζu. ByBirkhoff’s ergodic theorem, we have

∫

Λ

ζs dμts =

∫

Λ

limτ→∞

1

τ

∫ τ

0ζs(ϕv(x)) dv dμt

s(x)

5.1 Dimensions Along Stable and Unstable Manifolds 53

=∫

Λ

limτ→∞

1

τlog‖dxϕτ |Es(x)‖dμt

s(x)

≤ logλ < 0, (5.7)

and hence, the function t → PΦ|Λ(tζs) is strictly decreasing. Moreover, PΦ|Λ(0) =h(Φ|Λ) ≥ 0. Thus, there exists a unique real number ts such that PΦ|Λ(tsζs) = 0 andts ≥ 0. A similar argument shows that tu is also uniquely defined and that tu ≥ 0.

Let R1, . . . ,Rk be a Markov system for Φ on Λ. We assume that the diameterof the rectangles Ri is small when compared to the sizes of the stable and unstablemanifolds. We also consider the function τ in (3.4) and the map T in (3.5), whereZ =⋃k

i=1 Ri .We only establish the identities in (5.5). The argument for (5.4) is entirely anal-

ogous. For i = 1, . . . , k, let

Vi = T (V u(x) ∩ Λ) ∩ Ri (5.8)

and V =⋃ki=1 Vi . Let also S be the invertible map T |Z : Z → Z. We define

Ri0···in =n⋂

j=0

S−jRij and Vi0···in = V ∩ Ri0···in (5.9)

for each (· · · i0 · · · ) ∈ ΣA and n ∈N, where A is the transition matrix obtained fromthe Markov system as in (3.8).

We first obtain an upper bound for the upper box dimension. Since T nVi0···in ⊂Vin , if U is a cover of Vin , then S−nU is a cover of Vi0···in . Therefore,

N(Vi0···in , r) ≤ N(Vin, r/λi0···in )

for r > 0, where

λi0···in = max{‖dxS

−n|Eu(x)‖ : x ∈ Ri0···in},

and hence,

N(V, r) ≤∑

i0···inN(Vi0···in , r) ≤

∑

i0···inN(V, r/λi0···in ).

Now let us take s > dimBV . Then there exists an r0 > 0 such that N(V, r) < r−s

for r ∈ (0, r0). Letting

cn(s) =∑

i0···inλ

s

i0···in ,

we obtain N(V, r) ≤ r−scn(s) for r < λnr0, where

λn = mini0···in

λi0···in .


It follows by induction that

N(V, r) ≤ r−scn(s)m

for m ∈N and r < λmn r0. Therefore,

logN(V, r)

− log r≤ s + m log cn(s)

− log r≤ s + m log cn(s)

− log(λmn r0)

,

and letting r → 0 yields the inequality

dimBV ≤ s + lim supm→∞

m log cn(s)

− log(λmn r0)

= s − log cn(s)

logλn

.

Letting s ↘ dimBV , it follows from this inequality that

cn(dimBV ) ≥ 1 (5.10)

for any sufficiently large n (because then λn < 1). Now we observe that

cn(s) =∑

i0···inexp max

x∈Ri0 ···in

(

−s

∫ τn(x)

0ζu(ϕv(x)) dv

)

,

with τn(x) as in (3.6). It follows from (4.3) and (5.10) that

PΦ|Λ(−sζu) = limn→∞

1

nlog cn(s) ≥ 0

(we note that in the present context the limits when ε → 0 in (4.3) are not necessary).Since the function s → PΦ|Λ(−sζu) is strictly decreasing and PΦ|Λ(−tuζu) = 0, weconclude that

s ≤ tu for s > dimBV.

Finally, letting s → dimBV yields the inequality dimBV ≤ tu.Now we consider the Hausdorff dimension and we proceed by contradiction. Let

us assume that dimH V < tu and take s > 0 such that

dimH V < s < tu. (5.11)

Then m(V, s) = 0, and since V is compact, given δ > 0, there exists a finite opencover U of V such that

∑

U∈U(diamU)s < δs. (5.12)

For each n ∈ N, take δn > 0 such that

pn(U) = card{(i0 · · · in) : U ∩ Ri0···in = ∅

}< k


whenever diamU < δn (we recall that k is the number of elements of the Markovsystem). We note that δn → 0 when n → ∞. It follows from (5.12) with δ = δn thatdiamU < δn and hence, pn(U) < k for every U ∈ U. Now let N = n + m − 1 forsome m ∈ N such that all entries of the matrix Am are positive (we recall that Φ|Λis topologically mixing). For each (i0 · · · ) ∈ Σ+

A and n ∈ N, let Ui0···iN be the coverof V composed of the sets T N(U) with U ∈U such that U ∩ Ri0···in = ∅. We have

∑

U∈Ui0 ···iN

(diamU)s ≤ λ−si0···iN

∑

U∈U,U∩Ri0 ···in =∅

(diamU)s,

where

λi0···in = min{‖dxS

−n|Eu(x)‖ : x ∈ Ri0···in}.

Now let us assume that∑

U∈Ui0···iN

(diamU)s ≥ δsn

for every (i0 · · · ) ∈ Σ+A and n ∈N. We obtain

kδsn > k

∑

U∈U(diamU)s ≥

∑

U∈Upn(U)(diamU)s

=∑

i0···in

∑

U∈U,U∩Ri0 ···in =∅

(diamU)s

≥ k−m+1∑

i0···iN

∑

U∈U,U∩Ri0 ···in =∅

(diamU)s

≥ k−m+1∑

i0···iN

(λi0···iN

∑

U∈Ui0···iN

(diamU)s)

≥ k−m+1δsn

∑

i0···iNλs

i0···iN

and hence,∑

i0···iNλs

i0···iN ≤ km. (5.13)

Since the map

ψ(x) = ‖dxS−1|Eu(x)‖

is Hölder continuous, for each x, y ∈ Ri0···in we have

‖dxS−n|Eu(x)‖

‖dyS−n|Eu(y)‖ =n−1∏

j=0

ψ(T j (x))

ψ(T j (y))


≤n−1∏

j=0

(1 + |ψ(T j (x)) − ψ(T j (y))|

infψ

)

≤n−1∏

j=0

(1 + Kd

(T j (x), T j (y)

)δ)

≤n−1∏

j=0

(1 + Kd

(T n(x), T n(y)

)δλδ(n−j)

), (5.14)

for some constants K > 0 and λ, δ ∈ (0,1). Since T n(x), T n(y) ∈ Ri0 , it followsfrom (5.14) that

‖dxS−n|Eu(x)‖

‖dyS−n|Eu(y)‖ ≤n−1∏

j=0

(1 + K ′λδ(n−j)

)

≤∞∏

j=1

(1 + K ′λδj

)< ∞ (5.15)

for some constant K ′ > 0. Hence, there exists an L > 0 such that

λi0···in ≤ Lλi0···in

for every (i0 · · · ) ∈ Σ+A and n ∈N. By (5.13), we obtain

PΦ|Λ(−sζu) = limN→∞

1

N

∑

i0···iNλ

s

i0···in

≤ limN→∞

1

N

∑

i0···iNλs

i0···in ≤ 0.

Since the function s → PΦ|Λ(−sζu) is strictly decreasing and PΦ|Λ(−tuζu) = 0,this contradicts (5.11). Therefore,

∑

U∈Ui0 ···iN

(diamU)s < δsn (5.16)

for some sequence i0 · · · iN and any sufficiently large n (recall that N = n+m− 1).Now we restart the process using the cover V1 = Ui0···iN to find inductively finitecovers Vl of V for each l ∈ N. By (5.16), we have diamVl < δn and hence, pn(U) <

k for every U ∈ Vl . This implies that cardVl+1 < cardVl and thus, cardVl = 1 forsome l = l(n). Writing Vl(n) = {Un}, we obtain

diamV ≤ diamUn < δn → 0


when n → ∞, which is impossible. This contradiction shows that dimH V ≥ tu.We have shown that

dimH V = dimBV = dimBV = tu. (5.17)

Now we observe that the map F = T |(V u(x) ∩ Λ) and its inverse are Lipschitz ineach domain of continuity (we recall that each rectangle is contained in a smoothdisk, which ensures that the restriction τ |(V u(x) ∩ Λ) is Lipschitz in each domainof continuity). In particular, F preserves the Hausdorff and box dimensions, andhence, the identities in (5.5) follow readily from (5.17). �

We emphasize that the dimensions of the sets V s(x) ∩ Λ and V u(x) ∩ Λ areindependent of the point x (see (5.4) and (5.5)). Our proof of Theorem 5.1 is basedon corresponding arguments of Barreira [2] in the case of discrete time (see [3]for details and further references). The original argument of Pesin and Sadovskayain [82] uses Moran covers instead.

More generally, if Φ|Λ is not conformal but the maps

dxϕt |Eu(x) : Eu(x) → Eu(ϕt (x))

are multiples of isometries for every x ∈ Λ and t ∈ R, then the identities in (5.5)still hold (without modifications in the proof of Theorem 5.1). A similar observationholds for the dimensions along the stable manifolds.

Now let Φ be a C1+δ flow with a locally maximal hyperbolic set Λ such that Φ|Λis topologically mixing but not necessarily conformal. It follows from the proof ofTheorem 5.1 that if R1, . . . ,Rk is a Markov system for Φ on Λ, then

dimH (V u(x) ∩ Λ) ≥ r and dimB(V u(x) ∩ Λ) ≤ r (5.18)

for every x ∈ Λ, where r and r are the unique real numbers such that

limn→∞

1

nlog

∑

i0···inexp min

x∈Ri0 ···in

(

−r

∫ τn(x)

0ζu(ϕv(x)) dv

)

= 0

and

limn→∞

1

nlog

∑

i0···inexp max

x∈Ri0 ···in

(

−r

∫ τn(x)

0ζu(ϕv(x)) dv

)

= 0, (5.19)

with the sets Ri0···in as in (5.9). We note that the limit in (5.19) coincides withPΦ|Λ(−rζu), and hence, r = tu. Similar observations hold for the dimensions alongthe stable manifolds.


5.2 Formula for the Dimension

In this section we establish a formula for the dimension of a locally maximal hyper-bolic set for a conformal flow. Due to the conformality, the dimension is obtained byadding the dimensions along the stable and unstable manifolds, plus the dimensionalong the flow.

Theorem 5.2 ([82]) If Φ is a C1+δ flow with a locally maximal hyperbolic set Λ

such that Φ|Λ is conformal and topologically mixing, then

dimH Λ = dimBΛ = dimBΛ = ts + tu + 1. (5.20)

Proof Again, the idea of the proof is to reduce the problem to a Markov system. Wethen use the conformality of Φ|Λ to show that the dimensions in Theorem 5.1 canbe added.

Let R1, . . . ,Rk be a Markov system for Φ on Λ. Each rectangle Ri is containedin a smooth disk Di and has the product structure

{x, y} = πi([x, y]),

where πi is the projection onto Di , with [x, y] as in (3.2). In general, the map(x, y) → {x, y} is only Hölder continuous and has a Hölder continuous inverse.However, since Φ is conformal on Λ, it follows from results of Hasselblatt in [53]that the distributions x → Es(x) ⊕ E0(x) and x → Eu(x) ⊕ E0(x) are Lipschitz.This implies that

Ri × Ri � (x, y) → {x, y} ∈ Ri

is a Lipschitz map with Lipschitz inverse. Therefore, letting

Wsi = πi(V

s(x)) ∩ Ri and Wui = πi(V

u(x)) ∩ Ri,

we obtain

dimH Ri = dimH {Wsi ,Wu

i } = dimH (Wsi × Wu

i ) (5.21)

for every x ∈ Ri , with analogous identities for the lower and upper box dimensions.By Theorem 5.1, we have

dimH Wsi = dimBWs

i = dimBWsi = ts (5.22)

and

dimH Wui = dimBWu

i = dimBWui = tu. (5.23)

Indeed, V s(x) ∩ Λ is taken onto⋃k

i=1 Wsi by a map that together with its inverse

are Lipschitz in each domain of continuity (compare with (5.8)). Hence, it follows

5.2 Formula for the Dimension 59

from (5.21) (and the analogous identities for the lower and upper box dimensions)together with (5.22) and (5.23) that

dimH Ri = dimBRi = dimBRi = ts + tu. (5.24)

Since Λ is locally diffeomorphic to the product (−ε, ε) × ⋃ki=1 Ri , for any suffi-

ciently small ε > 0, identity (5.20) follow readily from (5.24). �

Chapter 6Pointwise Dimension and Applications

In this chapter, again for conformal hyperbolic flows, we establish an explicit for-mula for the pointwise dimension of an arbitrary invariant measure in terms of thelocal entropy and the Lyapunov exponents. In particular, this formula allows us toshow that the Hausdorff dimension of a (nonergodic) invariant measure is equal tothe essential supremum of the Hausdorff dimensions of the measures in each ergodicdecomposition. We also discuss the problem of the existence of invariant measuresof maximal dimension. These are measures at which the supremum of the Hausdorffdimensions over all invariant measures is attained.

6.1 A Formula for the Pointwise Dimension

In this section we consider hyperbolic flows and we establish a formula for thepointwise dimension of an arbitrary invariant measure. As a consequence, we alsoobtain a formula for the Hausdorff dimension of the measure.

Let Φ = {ϕt }t∈R be a C1+δ flow in a smooth manifold M and let Λ ⊂ M be alocally maximal hyperbolic set for Φ . We always assume in this chapter that theflow Φ is conformal on Λ (see Definition 5.1). Let also μ be a Φ-invariant proba-bility measure on Λ. By Birkhoff’s ergodic theorem, the limits

λs(x) = limt→+∞

1

tlog‖dxϕt |Es(x)‖ and λu(x) = lim

t→+∞1

tlog‖dxϕt |Eu(x)‖

(6.1)exist for μ-almost every x ∈ Λ. These are respectively the negative and positivevalues of the Lyapunov exponent

λ(x, v) = lim supt→+∞

1

tlog‖dxϕtv‖


61

http://dx.doi.org/10.1007/978-3-319-00548-5_6

62 6 Pointwise Dimension and Applications

for x ∈ Λ and v ∈ TxM (at μ-almost every point). On the other hand, by the Brin–Katok formula for flows, we have

hμ(x) = limε→0

limt→∞−1

tlogμ(B(x, t, ε)) (6.2)

for μ-almost every x ∈ Λ, where

B(x, t, ε) = {y ∈ M : d(ϕτ (y),ϕτ (x)) < ε for τ ∈ [0, t]}.

The number hμ(x) is called the local entropy of μ at x. Moreover, the functionx → hμ(x) is μ-integrable and Φ-invariant μ-almost everywhere, and

hμ(Φ) =∫

Λ

hμ(x)dμ(x). (6.3)

Now we present an explicit formula for the pointwise dimension of μ in terms ofthe local entropy and the Lyapunov exponents.

Theorem 6.1 [21] Let Φ be a C1+δ flow with a locally maximal hyperbolic set Λ

such that Φ|Λ is conformal and let μ be a Φ-invariant probability measure on Λ.For μ-almost every x ∈ Λ, we have

dμ(x) = dμ(x) = hμ(x)

(1

λu(x)− 1

λs(x)

)+ 1. (6.4)

Proof Once more, the idea of the proof is to reduce the problem to a Markov system,and then use the conformality of Φ|Λ to show that there exists a Moran cover offinite multiplicity. Other than this more technical aspect, all the remaining argumentsare at the foundational level of ergodic theory.

Let R1, . . . ,Rk be a Markov system for Φ on Λ. For each x ∈ Λ and n ∈ N, wedefine

τn(x) =n−1∑

k=0

τ(T k(x)).

By Birkhoff’s ergodic theorem, the limit

χ(x) = limn→∞

τn(x)

n(6.5)

exists for μ-almost every x ∈ Λ. It follows from (3.7) that χ(x) > 0 for μ-almostevery x ∈ Λ. Now let η be the measure induced by μ on the set Z = ⋃k

i=1 Ri . Itfollows from Proposition 2.3 and (4.8) that

hμ(y) = limε→0

limt→∞−1

tlogμ(B(y, t, ε))

6.1 A Formula for the Pointwise Dimension 63

= limε→0

limn→∞− 1

τn(x)logμ(B(y, τn(x), ε))

= 1

χ(x)limε→0

limn→∞−1

nlog[η(BZ(x,n, ε))2ε]

= 1

χ(x)limε→0

limn→∞−1

nlogη(BZ(x,n, ε)) (6.6)

for μ-almost every y = ϕs(x) ∈ Λ, with x ∈ Z and s ∈ [0, τ (x)].Given i−m, . . . , in ∈ {1, . . . , k}, we define the rectangle

Ri−m···in = {x ∈ Z : T j (x) ∈ Rij for j = −m, . . . , n

}, (6.7)

where T is the transfer map in (3.5). By (6.6) and the Shannon–McMillan–Breimantheorem, for μ-almost every x ∈ Λ we have

hμ(x) = 1

χ(x)lim

n,m→∞− 1

n + mlogη(Rn,m(x)), (6.8)

where Rn,m(x) = Ri−m···in is any rectangle such that x ∈ Rn,m(x). We assume thatfor each x ∈ Λ a particular choice of rectangles Rn,m(x) is made from the begin-ning, for all n,m ∈ N. Let X ⊂ Λ be a full μ-measure Φ-invariant set such that foreach x ∈ Λ:

1. the numbers λs(x) and λu(x) in (6.1) and χ(x) in (6.5) are well defined;2. the number hμ(x) in (6.2) is well defined and identity (6.8) holds.

We proceed with the proof of the theorem. Take ε > 0. For each x ∈ X \ Z, thereexists a p(x) ∈N such that if t ≥ p(x), then

λs(x) − ε <1

tlog‖dxϕt |Es(x)‖ < λs(x) + ε, (6.9)

λu(x) − ε <1

tlog‖dxϕt |Eu(x)‖ < λu(x) + ε, (6.10)

and if n,m ≥ p(x), then

χ(x) − ε <τn(x)

n< χ(x) + ε, (6.11)

−hμ(x)χ(x) − ε <1

n + mlogη(Rn,m(x)) < −hμ(x)χ(x) + ε. (6.12)

Given � ∈N, we consider the set

Q� = {x ∈ X : p(x) ≤ �

}.


Clearly,⋃

�∈N Q� = X. For each x ∈ X, there exists an r(x) > 0 such that forany r ∈ (0, r(x)) one can choose integers m = m(x, r) and n = n(x, r) withτm(x), τn(x) ≥ p(x) for which

‖dxϕτm(x)|Es(x)‖ ≥ r, ‖dxϕτm+1(x)|Es(x)‖ < r (6.13)

and

‖dxϕτn(x)|Eu(x)‖−1 ≥ r, ‖dxϕτn+1(x)|Eu(x)‖−1 < r. (6.14)

Combining (6.9) with (6.13), and (6.10) with (6.14), we obtain

τm(x)(λs(x) − ε) < log r + a, log r < τm(x)(λs(x) + ε) (6.15)

and

− log r − a < τn(x)(λu(x) + ε), τn(x)(λu(x) − ε) < − log r, (6.16)

where

a = max

{− inf

x∈Zlog‖dxϕτ(x)|Es(x)‖, sup

x∈Z

log‖dxϕτ(x)|Eu(x)‖}

.

We write R(x, r) = Rn(x,r),m(x,r)(x).We first establish an upper bound for the pointwise dimension. By Proposition 2.3

and the conformality of Φ on Λ, there exists a c > 0 (independent of x and r) suchthat

B(x, cr) ⊃ R(x, r) × Ir(x),

where Ir (x) is some interval of length 2r . By (4.8), (6.11) and (6.12), for eachx ∈ X \ Z and any sufficiently small r , we obtain

μ(B(x, cr)) ≥ η(R(x, r))2r

≥ exp[(−hμ(x)χ(x) − ε)(n + m)]2r

≥ exp[−hμ(x)(τn(x) + τm(x)) − (hμ(x) + 1)ε(n + m)]2r

≥ exp[(−hμ(x) − (hμ(x) + 1)ε/σ )(τn(x) + τm(x))]2r,

where σ = infZ τ > 0 (see (3.7)). Using (6.15) and (6.16), we conclude that

μ(B(x, cr)) ≥ exp

[(hμ(x) + (hμ(x) + 1)

ε

σ

)( log r

λu(x) − ε− log r

λs(x) + ε

)]2r.

Taking logarithms and letting r → 0, we finally obtain

dμ(x) ≤(hμ(x) + (hμ(x) + 1)

ε

σ

)( 1

λu(x) − ε− 1

λs(x) + ε

)+ 1.

6.1 A Formula for the Pointwise Dimension 65

The arbitrariness of ε implies that

dμ(x) ≤ hμ(x)

(1

λu(x)− 1

λs(x)

)+ 1

for every x ∈ X and hence for μ-almost every x ∈ Λ.Now we establish a lower bound for the pointwise dimension. Take ε > 0. Given

x ∈ X, we define

Γ (x) = {y ∈ X : |λs(y) − λs(x)| < ε, |λu(y) − λu(x)| < ε,

|χ(y) − χ(x)| < ε, and |hμ(y) − hμ(x)| < ε}. (6.17)

We note that the sets Γ (x) are Φ-invariant. Moreover, they cover X and one canchoose points yi ∈ X for i = 1,2, . . . such that Γi = Γ (yi) has measure μ(Γi) > 0for each i, and

⋃i∈N Γi has full μ-measure.

Take i, � ∈N. We proceed in a similar manner to that in [81, Sect. 22] to constructa cover of Γi ∩ Q� ∩ Z by sets R(x, r) (we recall that Z = ⋃k

i=1 Ri ). For eachx ∈ Γi ∩ Q� and r > 0, we denote by R′(x, r) the largest rectangle containing x

(among those in (6.7)) with the property that

R′(x, r) = R(y, r) for some y ∈ R′(x, r) ∩ Γi ∩ Q� ∩ Z,

and

R(z, r) ⊂ R′(x, r) for every z ∈ R′(x, r) ∩ Γi ∩ Q� ∩ Z.

Two sets R′(x, r) and R′(y, r) either coincide or intersect at most along their bound-aries. It follows from the Borel density lemma (see [42, Theorem 2.9.11]) that forμ-almost every x ∈ Γi ∩ Q� there exists an r(x) > 0 such that

μ(B(x, r)) ≤ 2μ(B(x, r) ∩ Γi ∩ Q�)

for every r ∈ (0, r(x)).Again by the conformality of Φ on Λ and the uniform transversality of the stable

and unstable manifolds, there exist a K > 0 (independent of x and r) and pointsx1, . . . , xk ∈ Γi ∩ Q� with k ≤ K such that

B(x, r) ∩ Γi ∩ Q� ⊂k⋃

j=1

(R′(xj , r) × Ir (xj )

).


Therefore,

μ(B(x, r)) ≤ 2μ(B(x, r) ∩ Γi ∩ Q�)

≤ 4r

k∑

j=1

μ(R′(xj , r))

≤ 4r

k∑

j=1

exp[(−hμ(xj )χ(xj ) + ε)(n(xj , r) + m(xj , r))]

≤ 4r

k∑

j=1

exp[(

−hμ(xj ) + (hμ(xj ) + 1)ε

σ

)(τn(xj ) + τm(xj ))

],

using (6.12). By (6.15), (6.16) and the definition of Γi , we conclude that

μ(B(x, r)) ≤ 4r

k∑

j=1

exp

[a(yi)

(log r + a

λu(xj ) + ε− log r + a

λs(xj ) − ε

)]

≤ 4rK exp

[b(x)

(log r + a

λu(x) + 2ε− log r + a

λs(x) − 2ε

)],

where

a(yi) = hμ(yi) − ε − (hμ(yi) + ε + 1)ε/σ

and

b(x) = hμ(x) − 2ε − (hμ(x) + 2ε + 1)ε/σ.

Taking logarithms and letting r → 0 we find that

dμ(x) ≥(hμ(x) − 2ε − (hμ(x) + 2ε + 1)

ε

σ

)( 1

λu(x) + 2ε− 1

λs(x) − 2ε

)+ 1

for μ-almost every x ∈ Γi ∩ Q�. Letting � → ∞, we conclude that this inequalityholds for μ-almost every x ∈ Γi , and the arbitrariness of ε implies that

dμ(x) ≥ hμ(x)

(1

λu(x)− 1

λs(x)

)+ 1, (6.18)

also for μ-almost everyx ∈ Γ . Since⋃

�∈N Γi has full μ-measure, inequality (6.18)holds for μ-almost every x ∈ Λ. This completes the proof of the theorem. �

In [82], Pesin and Sadovskaya established the identities in (6.4) for equilibriummeasures for a Hölder continuous function. We note that these measures are ergodicand have a local product structure, while Theorem 6.1 considers arbitrary invariantmeasures.

The following result is a simple consequence of Theorem 6.1.

6.2 Hausdorff Dimension and Ergodic Decompositions 67

Theorem 6.2 If Φ is a C1+δ flow with a locally maximal hyperbolic set Λ such thatΦ|Λ is conformal and μ is a Φ-invariant probability measure on Λ, then

dimH μ = ess sup

{hμ(x)

(1

λu(x)− 1

λs(x)

)+ 1 : x ∈ Λ

}. (6.19)

If, in addition, μ is ergodic, then

dimH μ = hμ(Φ)

(1

∫Λ

ζu dμ− 1

∫Λ

ζs dμ

)+ 1. (6.20)

Proof Combining Theorem 6.1 with Proposition 4.7 yields identity (6.19). Whenμ is ergodic, since the functions hμ, λs and λu are Φ-invariant they are constantμ-almost everywhere. It follows from (6.3) that hμ(x) = hμ(Φ) for μ-almost everyx ∈ Λ. Moreover, proceeding as in (5.7), we have

∫

Λ

ζu dμ =∫

Λ

λu(x) dμ(x) and∫

Λ

ζs dμ =∫

Λ

λs(x) dμ(x).

Hence,

λu(x) =∫

Λ

ζu dμ and λs(x) =∫

Λ

ζs dμ

for μ-almost every x ∈ Λ. Therefore,

hμ(x)

(1

λu(x)− 1

λs(x)

)+ 1 = hμ(Φ)

(1

∫Λ

ζu dμ− 1

∫Λ

ζs dμ

)+ 1,

also for μ-almost every x ∈ Λ. Identity (6.20) now follows readily from Theo-rem 6.1 together with Proposition 4.6. �

6.2 Hausdorff Dimension and Ergodic Decompositions

In this section we describe the behavior of the Hausdorff dimension of an invariantmeasure with respect to an ergodic decomposition.

Theorem 6.3 ([21]) Let Φ be a C1+δ flow with a locally maximal hyperbolic set Λ

such that Φ|Λ is conformal and let μ be a Φ-invariant probability measure on Λ.For each ergodic decomposition τ of μ, we have

dimH μ = ess sup{dimH ν : ν ∈ ME},

with the essential supremum taken with respect to τ .


Proof By Proposition 4.8, we have

dimH μ ≥ ess sup{dimH ν : ν ∈ME}.Now we establish the reverse inequality. Take ε > 0. For each x in the full μ-measure set X considered in the proof of Theorem 6.1, let Γ (x) be the set in (6.17).We also take points yi ∈ X for i = 1,2, . . . such that the Φ-invariant sets Γi = Γ (yi)

have measure μ(Γi) > 0 for each i, and⋃

i∈N Γi has full μ-measure. For each i weconsider the normalized restriction μi of μ to Γi . It follows from (6.3) and (6.17)that

hμi(Φ|Γi) = 1

μ(Γi)

∫

Γi

hμ(x) dμ(x) ≥ hμ(yi) − ε. (6.21)

We note that a measure ν ∈ M is ergodic (with respect to Φ) if and only if theinduced measure ην on Z is ergodic (with respect to T ). Moreover, the ergodicdecomposition τ induces a measure τZ on the set MZ of all T -invariant probabil-ity measures on Z. We consider a new measure τZ on MZ with Radon–Nikodymderivative

dτZ

dτZ

(η) =∫Z

τ dημ∫Z

τ dη. (6.22)

Now let G : Z → R be a continuous function. We define another functiong : Λ → R by g(ϕs(x)) = G(x)/τ(x) for x ∈ Z and s ∈ [0, τ (x)). Clearly,

G(x) =∫ τ(x)

0g(ϕs(x)) ds,

and we have∫

MZ

∫

Z

GdηdτZ(η) =∫

Z

τ dημ

∫

MZ

∫Z

Gdη∫Z

τ dηdτZ(η)

=∫

Z

τ dημ

∫

MZ

∫

Λ

g dν dτ(ν)

=∫

Z

τ dημ

∫

Λ

g dμ =∫

Z

Gdημ.

This shows that τZ is an ergodic decomposition of ημ. Since the Radon–Nikodymderivative in (6.22) is bounded and bounded away from zero, a subset of MZ haspositive τZ-measure if and only if it has positive τZ-measure.

Now let

Mi = {ν ∈M : ν(Λ \ Γi) = 0}.Since Γi is Φ-invariant, there exists a one-to-one correspondence between the er-godic Φ-invariant probability measures on Γi and the measures in Mi ∩ME . There-fore, τ(Mi ∩ ME) > 0 and the normalized restriction τi of τ to Mi is an ergodicdecomposition of μ|Γi with respect to Φ . We have

6.2 Hausdorff Dimension and Ergodic Decompositions 69

hηi(T |Γi ∩ Z) =

∫

Mi

hη(T ) dτi(ν),

where η and ηi are the measures induced respectively by ν and μi on Z. Hence,there exists a set Ai ⊂ Mi ∩ ME of positive τi -measure, and thus also of positiveτ -measure, such that

hη(T ) > hηi(T |Γi ∩ Z) − ε

for each ν ∈ Ai . Since∫

Mi

∫

Γi∩Z

τ dηdτi(η) =∫

Γi∩Z

τ dμi,

one can also assume that∫

Γi∩Z

τ dη ≤∫

Γi∩Z

τ dηi + ε (6.23)

for every ην ∈ MZ with ν ∈ Ai . Therefore, for each ν ∈ Ai and x ∈ Γi , we have

hη(T ) + ε∫Γi∩Z

τ dη − ε≥ hμi

(T |Γi ∩ Z)∫Γi∩Z

τ dηi

= hμi(Φ|Γi)

≥ hμ(yi) − 2ε > hμ(x) − 3ε, (6.24)

using (6.21). On the other hand, for ν ∈ Ai and x ∈ Γi , we have∫

Λ

ζs dν ≥ λs(yi) − ε ≥ λs(x) − 2ε

and∫

Λ

ζu dν ≤ λu(yi) + ε ≤ λu(x) + 2ε.

For each x ∈ X, combining (6.23) and (6.24) with (6.20), we conclude that

hμ(x)

(1

λu(x)− 1

λs(x)

)+ 1 ≤

(hη(T ) + ε

∫Γi∩Z

τ dη − ε+ 3ε

)

×(

1∫Λ

ζu dν − 2ε− 1

∫Λ

ζs dν + 2ε

)+ 1

≤dimH ν + C(ε)

for every ν ∈ Ai , where C is a function (independent of i and ν) such that C(ε) → 0when ε → 0. Since Ai has positive τ -measure, it follows from (6.19) that

dimH μ ≤ ess sup{dimH ν : ν ∈ ME} + C(ε),

and the arbitrariness of ε yields the desired result. �


6.3 Measures of Maximal Dimension

In this section we establish the existence of measures of maximal dimension on alocally maximal hyperbolic set for a conformal flow.

We first introduce the notion of a measure of maximal dimension. Let Φ be a C1

flow with a locally maximal hyperbolic set Λ.

Definition 6.1 A measure μ ∈ M such that

dimH μ = sup{dimH ν : ν ∈ M}is called a measure of maximal dimension.

Now we assume that Φ|Λ is conformal. For each ν ∈M, let

λs(ν) =∫

Λ

λs(x) dν(x) and λu(ν) =∫

Λ

λu(x) dν(x),

with λs(x) and λu(x) as in (6.1). Since


1

t

∫ t

0ζs(ϕτ (x)) dτ and λu(x) = lim

t→+∞1

t

∫ t

0ζu(ϕτ (x)) dτ,

it follows from Birkhoff’s ergodic theorem that

λs(ν) =∫

Λ

ζs dν and λu(ν) =∫

Λ

ζu dν. (6.25)

The following result establishes the existence of measures of maximal dimensionon a locally maximal hyperbolic set for a conformal flow.

Theorem 6.4 If Φ is a C1+δ flow with a locally maximal hyperbolic set Λ such thatΦ|Λ is conformal and topologically mixing, then there exists an ergodic measure ofmaximal dimension.

Proof The proof closely follows arguments of Barreira and Wolf in [19] for discretetime. We define a function Q : R2 → R by

Q(p,q) = PΦ|Λ(−pζu + qζs).

By Proposition 4.4, since ζs and ζu are Hölder continuous, the function Q is ana-lytic. Moreover, for each (p, q) ∈ R

2 the Hölder continuous function −pζu + qζs

has a unique equilibrium measure, which we denote by νp,q . We also write

λu(p,q) = λu(νp,q), λs(p, q) = λs(νp,q), h(p, q) = hνp,q (Φ),

and we think of λu, λs and h as functions on R2. By Proposition 4.2, we have

Q(p,q) = h(p,q) − pλu(p,q) + qλs(p, q). (6.26)

6.3 Measures of Maximal Dimension 71

Since the maps ν → λs(ν) and ν → λu(ν) given by (6.25) are continuous on thecompact set M (when equipped with the weak∗ topology), one can define

λmins = minλs(M), λmax

s = maxλs(M)

and

λminu = minλu(M), λmax

u = maxλu(M).

We also consider the intervals

Is = (λmins , λmax

s ) and Iu = (λminu , λmax

u ).

We note that Is = ∅ (respectively Iu = ∅) if and only if ζs (respectively ζu) is notcohomologous to a constant.

Now we consider the functions

du(p, q) = h(p,q)/λu(p, q) and ds(p, q) = −h(p,q)/λs(p, q). (6.27)

It follows from (4.6) that

∂Q

∂p= −λu and

∂Q

∂q= λs. (6.28)

Since Q is analytic, the functions λu and λs are also analytic. We concludefrom (6.26) that h is analytic, and it follows from (6.27) that the functions du and ds

are also analytic.

Lemma 6.1 The following properties hold:

1. if ζu is not cohomologous to a constant, then for each q ∈R:

a. λu(·, q) is strictly decreasing and {λu(p,q) : p ∈R} = Iu;b. h(·,0) is strictly decreasing in [0,∞);c. du(·,0) is strictly increasing in (−∞, tu] and strictly decreasing in [tu,∞).

2. if ζs is not cohomologous to a constant, then for each p ∈R:

a. λs(p, ·) is strictly decreasing and {λs(p, q) : q ∈ R} = Is ;b. h(0, ·) is strictly decreasing in [0,∞);c. ds(0, ·) is strictly increasing in (−∞, ts] and strictly decreasing in [ts ,∞).

Proof of the lemma Let us assume that ζu is not cohomologous to a constant andtake q ∈R. By (4.7) and (6.28), we have

∂λu

∂p= −∂2Q

∂p2< 0, (6.29)

and thus λu(·, q) is strictly decreasing. Since the function λu(·, q) is continuous, theset {λu(p,q) : p ∈ R} is an open interval. We claim that

limp→∞λu(p,q) = λmin

u and limp→−∞λu(p,q) = λmax

u . (6.30)


If the first identity did not hold, then there would exist ν ∈ M and δ > 0 such thatλu(ν) + δ < λu(p,q) for p ∈R. Now take p > 0 satisfying

pδ > h(Φ|Λ) − qλs(ν) + qλs(p, q)

(such a p always exists, because the function λs(·, q) is bounded). We obtain

Q(p,q) = h(p,q) − pλu(p,q) + qλs(p, q)

< h(Φ|Λ) − p(λu(ν) + δ) + qλs(p, q)

< hν(Φ) − pλu(ν) + qλs(ν),

which contradicts Proposition 4.2. This establishes the first identity in (6.30). A sim-ilar argument establishes the second identity.

It follows from (6.26) that

h(p,0) = Q(p,0) + pλu(p,0).

Using (6.28) and (6.29), one can easily verify that

∂h

∂p(p,0) = p

∂λu

∂p(p,0). (6.31)

This establishes property 1b.Finally, using (6.26), (6.29) and (6.31), we obtain

∂du

∂p(p,0) = p∂λu/∂p(p,0)λu(p,0) − h(p,0)∂λu/∂p(p,0)

λu(p,0)2

= −∂λu

∂p(p,0)

Q(p,0)

λu(p,0)2

= ∂2Q

∂p2(p,0)

Q(p,0)

λu(p,0)2.

(6.32)

On the other hand, it follows from Proposition 4.2 that the function Q(·, q) is strictlydecreasing. This implies that

Q(p,0) > Q(tu,0) = 0 for p < tu,

and

Q(p,0) < Q(tu,0) = 0 for p > tu.

Property 1c follows now immediately from (6.29) and (6.32).The proofs of the remaining statements are analogous. �

Using Lemma 6.1 one can introduce two curves that are crucial to our approach.

Lemma 6.2 The following properties hold:


1. for each a ∈ Iu there exists a unique function γu : R →R satisfying

λu(γu(q), q) = a for q ∈ R,

and γu is analytic;2. for each b ∈ Is there exists a unique function γs : R →R satisfying

λs(p, γs(p)) = b for p ∈R,

and γs is analytic.

Proof of the lemma We only prove the second statement. The proof of the firststatement is analogous. Let b ∈ Is . In particular, Is = ∅ and ζs is not cohomologousto a constant. By statement 2a in Lemma 6.1 and (6.28), for each p ∈R there existsa unique number γs(p) ∈R such that

∂Q

∂q(p,γs(p)) = λs(p, γs(p)) = b.

Since ζs is not cohomologous to a constant, we have ∂2Q/∂q2(p, q) > 0 for every(p, q) ∈ R

2 and it follows from the Implicit function theorem that the map p →γs(p) is analytic. �

We proceed with the proof of the theorem. Let (νn)n∈N be a sequence of measuresin ME (that is, a sequence of ergodic measures in M) such that

limn→∞ dimH νn = sup{dimH ν : ν ∈ ME}. (6.33)

Since M is compact, one can also assume that (νn)n∈N converges to some mea-sure m ∈ M. Since the map M � ν → hν(Φ) is upper semicontinuous, it followsfrom (6.19) and the continuity of the maps ν → λu(ν) and ν → λs(ν) that

limn→∞ dimH νn ≤ d(m), (6.34)

where

d(m) = hm(Φ)

(1

∫Λ

ζu dm− 1

∫Λ

ζs dm

)+ 1.

By (6.33) and (6.34), we obtain

sup{dimH ν : ν ∈ ME} ≤ d(m). (6.35)

Hence, in order to establish the existence of a measure μ ∈ME satisfying

dimH μ = sup{dimH ν : ν ∈ ME} (6.36)

it is sufficient to show that there exists a μ ∈ME such that

dimH μ = d(m). (6.37)


Clearly, any measure μ ∈ ME satisfying (6.37) also satisfies (6.36). When m isergodic, it follows from (6.19) that dimH m = d(m), and hence identity (6.36) holdsfor μ = m. However, the measure m may not be ergodic.

Let a = λu(m) and b = λs(m). By Lemma 6.2, when a ∈ Iu (respectively b ∈ Is )one can consider the curve γu (respectively γs ) associated to the number a (respec-tively b). Now we prove some auxiliary statements.

Lemma 6.3 If λs(m) ∈ Is , then there exists a p ∈ [0, hm(Φ)/λu(m)] such that

λu(p,γs(p)) = λu(m).

Proof of the lemma The assumption λs(m) ∈ Is guarantees that the function γs iswell defined. Since νp,γs(p) is the equilibrium measure of −pζu + γs(p)ζs , we have

h(p,γs(p))−pλu(p,γs(p))+γs(p)λs(p, γs(p)) ≥ hm(Φ)−pλu(m)+γs(p)λs(m)

(6.38)for p ∈R. One can easily verify that

h(p,γs(p))

λu(p, γs(p))− hm(Φ)

λu(m)≥(

1 − λu(m)

λu(p,γs(p))

)(p − hm(Φ)

λu(m)

). (6.39)

Let κ = hm(Φ)/λu(m). It follows from (6.39) with p = κ that

h(κ, γs(κ))/λu(κ, γs(κ)) ≥ hm(Φ)/λu(m). (6.40)

Now let us assume that

λu(κ, γs(κ)) > λu(m).

By (6.40), we obtain h(κ, γs(κ)) > hm(Φ). It follows from (6.19) and (6.40) thatdimH νκ,γs(κ) > d(m). This contradicts (6.35), and thus, we must have

λu(κ, γs(κ)) ≤ λu(m). (6.41)

On the other hand, it follows from (6.19) and (6.35) that

h(0, γs(0))

λu(0, γs(0))− h(0, γs(0))

λs(m)≤ hm(Φ)

λu(m)− hm(Φ)

λs(m). (6.42)

Taking p = 0 in (6.38), we obtain h(0, γs(0)) ≥ hm(Φ), and it follows from (6.42)that

λu(0, γs(0)) ≥ λu(m). (6.43)

By the continuity of the function p → λu(p,γu(p)) together with (6.41) and (6.43),there exists a p ∈ [0, κ] such that λu(p,γs(p)) = λu(m). This completes the proofof the lemma. �

Lemma 6.4 Assume that neither ζu nor ζs are cohomologous to a constant. Thenλu(m) ∈ Iu if and only if λs(m) ∈ Is .


Proof of the lemma Let us assume that λs(m) ∈ Is . By Lemma 6.3, there exists ap such that λu(p,γs(p)) = λu(m). By Lemma 6.1, we have λu(p,γs(p)) ∈ Iu andhence λu(m) ∈ Iu. A similar argument together with the corresponding version ofLemma 6.3 show that λs(m) ∈ Is whenever λu(m) ∈ Iu. �

By Lemma 6.4, it is sufficient to consider four cases:

1. λs(m) ∈ Is and λu(m) ∈ Iu;2. λs(m) ∈ Is and ζu is cohomologous to a constant;3. λu(m) ∈ Iu and ζs is cohomologous to a constant;4. λs(m) ∈ Is and λu(m) ∈ Iu.

We still need another auxiliary statement.

Lemma 6.5 If p,q ∈R are such that

λu(p,q) = λu(m) and λs(p, q) = λs(m),

then m = νp,q .

Proof of the lemma We have

h(p,q) +∫

Λ

(−pζu + qζs) dνp,q = h(p,q) − pλu(m) + qλs(m)

≥ hm(Φ) +∫

Λ

(−pζu + qζs) dm.

Hence, h(p,q) ≥ hm(Φ), with equality if and only if νp,q = m. On the other hand,combining (6.19) with (6.35) we obtain h(p,q) ≤ hm(Φ). Therefore h(p,q) =hm(Φ) and m = νp,q . �

Now we consider each of the above four cases.

Lemma 6.6 If λu(m) ∈ Iu and λs(m) ∈ Is , then there exist p,q ∈ R such that(p, γs(p)) = (γu(q), q) and m = νp,q .

Proof of the lemma The hypotheses of the lemma guarantee that the curves γu andγs are well defined. Since λs(p, γs(p)) = λs(m), it follows from Lemma 6.3 andthe uniqueness of γu that (p, γs(p)) = (γu(q), q) for some p,q ∈R. In particular,

λu(p,q) = λu(m) and λs(p, q) = λs(m).

Hence, it follows from Lemma 6.5 that m = νp,q . �

Lemma 6.7 If λs(m) ∈ Is and ζu is cohomologous to a constant, then there existp,q ∈ R such that m = νp,q .


Proof of the lemma Since λs(m) ∈ Is , the curve γs is well defined, and λs(p, γs(p)) =λs(m) for every p. On the other hand, the cohomological assumption ensures thatλu(p,γs(p)) = λu(m). Taking q = γs(p) we obtain

λu(p,q) = λu(m) and λs(p, q) = λs(m).

Hence, it follows from Lemma 6.5 that m = νp,q . �

An analogous argument establishes the following result.

Lemma 6.8 If λu(m) ∈ Iu and ζs is cohomologous to a constant, then there existp,q ∈ R such that m = νp,q .

Finally we consider the fourth case.

Lemma 6.9 If λu(m) /∈ Iu and λs(m) /∈ Is , then:

1. λu(m) = λminu and λs(m) = λmax

s ;2. there exists a measure ν ∈ME such that

λu(ν) = λu(m), λs(ν) = λs(m) and hν(Φ) = hm(Φ).

Proof of the lemma We first establish property 1. When Iu = Is = ∅ (that is, whenζu and ζs are both cohomologous to constants) there is nothing to prove. Now let usassume that

Iu = ∅, Is = ∅ and λs(m) = λmins . (6.44)

Since ν0,0 is the measure of maximal entropy, we have h(0,0) ≥ hm(Φ). Hence,it follows from λu(0,0) = λmin

u , statement 2a in Lemma 6.1 and (6.19) thatdimH ν0,0 > d(m). But this contradicts (6.35), and hence (6.44) cannot occur. Analo-gously, one can show that it is impossible to have Is = ∅, Iu = ∅ and λu(m) = λmax

u .To complete the proof of property 1, it remains to consider the case when Iu = ∅

and Is = ∅. Then

λu(m) ∈ ∂Iu = {λminu , λmax

u } and λs(m) ∈ ∂Is = {λmins , λmax

s }.We first assume that

λu(m) = λmaxu and λs(m) = λmin

s . (6.45)

Since ν0,0 is the measure of maximal entropy, we have h(0,0) ≥ hm(Φ). On theother hand, it follows from Lemma 6.1 that

λu(0,0) < λu(m) and λs(0,0) > λs(m).

By (6.19), we obtain dimH ν0,0 > d(m). But this contradicts (6.35), and hence (6.45)cannot occur. Now let us assume that

λu(m) = λminu and λs(m) = λmin

s . (6.46)


We claim that

h(p,0) > hm(Φ) (6.47)

for p > 0. Otherwise, if h(p,0) ≤ hm(Φ) for some p > 0, then it would follow fromLemma 6.1 that

h(p,0) − pλu(p,0) < hm(Φ) − pλu(m).

But this is impossible, because νp,0 is the equilibrium measure of −pζu. We alsoclaim that

du(p,0) ≥ hm(Φ)/λu(m) (6.48)

for any sufficiently large p (see (6.27) for the definition of the function du). Other-wise, by Lemma 6.1, there would exist p0 ∈ R and δ > 0 such that

du(p,0) + δ < hm(Φ)/λu(m)

for p ≥ p0. Then it would follow from (6.30) that hm(Φ) > h(p,0) for any suf-ficiently large p. But this contradicts (6.47), and hence (6.48) holds for any suffi-ciently large p. By (6.46), (6.47) and (6.48), we obtain

dimH νp,0 = du(p,0) + ds(p,0) ≥ hm(Φ)

λu(m)− h(p,0)

λs(p,0)> d(m),

also for any sufficiently large p. This contradicts (6.35), and hence (6.46) cannotoccur. Analogously, one can show that it is impossible to have

λu(m) = λmaxu and λs(m) = λmax

s .

This establishes property 1.To establish property 2, consider an ergodic decomposition τ of the measure m

(see Definition 4.9). Then

λminu = λu(m) =

∫

M

λu(ν) dτ(ν).

Since λu(ν) ≥ λminu for ν ∈ M, there exists a set A1 ⊂ ME with τ(A1) = 1 such that

λu(ν) = λminu for ν ∈ A1. Analogously, there exists a set A2 ⊂ ME with τ(A2) = 1

such that λs(ν) = λmaxs for ν ∈ A2. Hence, it follows from (6.19) and (6.35) that

hν(Φ) ≤ hm(Φ) for ν ∈ A1 ∩ A2. On the other hand, since

τ(A1 ∩ A2) = 1 and hm(Φ) =∫

M

hν(Φ)dτ(ν),

there exists a set A ⊂ A1 ∩ A2 with τ(A) = 1 such that hν(Φ) = hm(Φ) for ν ∈ A.This completes the proof of the lemma. �

By Lemmas 6.6, 6.7, 6.8 and 6.9, in each of the above four cases there exists ameasure μ ∈ ME satisfying (6.37) (namely, the measure νp,q in the first three lem-mas, and the measure ν in Lemma 6.9). This completes the proof of the theorem. �

Part IIIMultifractal Analysis

This part is dedicated to the multifractal analysis of hyperbolic flows. In Chap. 7we consider the simpler case of suspension flows over topological Markov chains.This allows us to present the main ideas without the additional technical complica-tions that appear when one considers hyperbolic flows. We also show that for everyHölder continuous function noncohomologous to a constant the set of points withoutBirkhoff average has full topological entropy. In Chap. 8 we describe the multifrac-tal analysis of hyperbolic flows. In the particular case of the entropy spectra, weshow that the cohomology assumptions in the study of irregular sets are genericallysatisfied.

Chapter 7Suspensions over Symbolic Dynamics

In this chapter we initiate the study of multifractal analysis for flows. This corre-sponds to giving a detailed description of the entropy of the level sets of the point-wise dimension for an invariant measure. We consider suspension flows over a topo-logical Markov chain, which can be seen as a model for hyperbolic flows, althoughwithout certain additional technical complications. We refer to Chap. 8 for a multi-fractal analysis of hyperbolic flows. A nontrivial consequence of the results in thischapter is that for every Hölder continuous function that is not cohomologous toa constant, the set of points without Birkhoff average has full topological entropy.These results can essentially be proven in two different ways, either reducing theproblem to the discrete-time dynamics on the base, or in an intrinsic manner, with-out leaving the context of flows. In order to include both approaches, in this chapterwe consider the first approach, which consists of first reducing the problem to thedynamics on the base and then applying the existing results for discrete time. InChap. 10 we obtain generalizations of the results in this chapter, using intrinsic ar-guments.

7.1 Pointwise Dimension

In this section we introduce the notion of pointwise dimension, in the general con-text of the BS-dimension. This is the local quantity that we will be considering inthis chapter (we recall that any multifractal analysis consists of studying the com-plexity of the level sets of some local quantity).

Let Ψ = {ψt }t∈R be a suspension flow in Y , over a homeomorphism T : X → X

of the compact metric space X, and let μ be a T -invariant probability measureon X. We equip the space Y with the Bowen–Walters distance (see Sect. 2.2), andwe consider the Φ-invariant probability measure ν induced by μ on Y .

Now we introduce the notion of pointwise dimension.


81

http://dx.doi.org/10.1007/978-3-319-00548-5_7

82 7 Suspensions over Symbolic Dynamics

Definition 7.1 Given a continuous function u : Y → R+, we define the lower and

upper u-pointwise dimensions of ν at x ∈ Y respectively by

dν,u(x) = limε→0

lim inft→∞ − logν(B(x, t, ε))

u(x, t, ε)(7.1)

and

dν,u(x) = limε→0

lim supt→∞

− logν(B(x, t, ε))

u(x, t, ε), (7.2)

with u(x, t, ε) as in (4.2).

For example, when Ψ is a suspension flow over a conformal expanding map T

(see Sect. 8.1), for u = log‖dT ‖ the numbers dν,u(x) and dν,u(x) coincide respec-tively with the lower and upper pointwise dimensions of the measure ν, and foru = 1 these numbers coincide respectively with the lower and upper local entropies.

When ν is an ergodic Φ-invariant probability measure on Y , we have

dν,u(x) = dν,u(x) = dimu ν = hν(Φ)∫Y

udν

for ν-almost every x ∈ Y . These identities can be obtained in a similar manner tothat in the case of discrete time (see [3, Proposition 7.2.7]).

The following result shows that when T is a topological Markov chain the limitswhen ε → 0 in (7.1) and (7.2) are not necessary.

Proposition 7.1 ([12]) Let Ψ be a suspension flow over a topologically mixing two-sided topological Markov chain, let ν be an equilibrium measure for a Hölder con-tinuous function g (with respect to Ψ ), and let u : Y → R

+ be a Hölder continuousfunction. Then

dν,u(y) = lim inft→∞ − logν(B(y, t, ε))

∫ t

0 u(ψτ (y)) dτ(7.3)

and

dν,u(y) = lim supt→∞

− logν(B(y, t, ε))∫ t

0 u(ψτ (y)) dτ

for every y ∈ Y and any sufficiently small ε > 0.

Proof Given ε > 0, let

δ(ε) = sup{|u(y1) − u(y2)| : dY (y1, y2) < ε

}.


Clearly, δ(ε) → 0 when ε → 0. We have

1 ≤ u(y, t, ε)∫ t

0 u(ψτ (y)) dτ

≤∫ t

0 [u(ψτ (y)) + δ(ε)]dτ∫ t

0 u(ψτ (y)) dτ

≤ 1 + δ(ε)

infu,

and thus,

dν,u(y) = limε→0

lim inft→∞ − logν(B(y, t, ε))

∫ t

0 u(ψτ (y)) dτ. (7.4)

For each m ∈ N, let τm : X → R be the function in (2.13). Given x ∈ X, letm = m(x, t) ∈ N be the unique integer such that τm−1(x) ≤ t < τm(x). By Proposi-tion 2.3, there exists a constant c ≥ 1 such that

BX(x,m, ε) ×(s − ε

c, s + ε

c

)⊂ B(y, t, ε) ⊂ BX(x,m − 1, ε) × (s − cε, s + cε)

(7.5)for every y = (x, s) ∈ Y and t > 0, and any sufficiently small ε > 0, where

BX(x,m, ε) = {z ∈ X : dX(T k(z), T k(x)) < ε for k = 0, . . . ,m

}. (7.6)

By (4.8), this implies that

lim inft→∞ − logν(B(y, t, ε))

∫ t

0 u(ψτ (y)) dτ= lim inf

t→∞ − logμ(BX(x,m, ε))∫ t

0 u(ψτ (y)) dτ. (7.7)

On the other hand, by Proposition 2.2, the function Ig is Hölder continuous in X.Hence, since μ is an equilibrium measure for Ig it has the Gibbs property, that is,given ε > 0, there exists a d ≥ 1 such that

d−1 ≤ μ(BX(x,m, ε))

exp(−mPT (Ig) +∑m−1

k=0 Ig(T k(x))) ≤ d (7.8)

for every x ∈ X and m ∈N. Thus, the limit in (7.7) is independent of ε, and it followsfrom (7.4) that identity (7.3) holds. A similar argument applies to dν,u(y). �


In this section we present a multifractal analysis of the dimension spectrum for theu-pointwise dimension for suspension flows over a topological Markov chain.


Let Ψ be a suspension flow over T and let μ be a Ψ -invariant probability measureon Y . For each α ∈ R, let

Kα = {y ∈ Y : dν,u(y) = dν,u(y) = α

}. (7.9)

For y ∈ Kα , the common value α of dν,u(y) and dν,u(y) is denoted by dν,u(y) andis called the u-pointwise dimension of ν at the point y.

Definition 7.2 The function Du defined by

Du(α) = dimu Kα

is called the u-dimension spectrum for the u-pointwise dimensions (with respect tothe measure ν).

For example, if u = 1, then Du(α) = h(Φ|Kα).Now let g : Y → R be a Hölder continuous function. For each q ∈ R, we define

a function gq : Y → R by

gq = −Tu(q)u + qg,

where Tu(q) is the unique real number such that PΨ (gq) = 0. We denote respec-tively by νq and mu the equilibrium measures for gq and −dimu Y · u (with re-spect to Ψ ). The conditions in Theorem 7.1 below ensure that Tu(q), νq and mu areuniquely defined.

The following is a multifractal analysis of the spectrum Du for suspension flowsover topological Markov chains.

Theorem 7.1 ([12]) Let Ψ be a suspension flow over a topologically mixing two-sided topological Markov chain, let u : Y → R

+ be a Hölder continuous function,and let ν be an equilibrium measure for a Hölder continuous function g such thatPΨ (g) = 0. Then the following properties hold:

1. the function Tu is analytic, T ′u(q) ≤ 0 and T ′′

u (q) ≥ 0 for every q ∈ R, Tu(0) =dimu Y and Tu(1) = 0;

2. the domain of Du is a closed interval in [0,∞) and coincides with the range ofthe function αu = −T ′

u;3. for each q ∈ R we have νq(Kαu(q)) = 1,

Du(αu(q)) = Tu(q) + qαu(q) = dimu νq,

dνq ,u(x) = Tu(q) + qαu(q)

for νq -almost every x ∈ Kαu(q), and

dνq,u(x) ≤ Tu(q) + qαu(q)

for every x ∈ Kαu(q);


4. if ν = mu, then Du and Tu are analytic strictly convex functions.

Proof Again, the idea of the proof is to reduce the problem to the case of maps. Wefirst express the pointwise dimension in terms of the dynamics in the base.

Lemma 7.1 If y = (x, s) ∈ Y , then

dν,u(y) = lim infm→∞ −

∑mi=0 Ig(T

i(x))∑m

i=0 Iu(T i(x))

and

dν,u(y) = lim supm→∞

−∑m

i=0 Ig(Ti(x))

∑mi=0 Iu(T i(x))

.

Proof of the lemma Let τm : Y → R be the function in (2.13). Given t > 0, let m ∈N

be the unique integer such that τm(x) ≤ t < τm+1(x), and write t = τm(x) + κ withκ ∈ (inf τ, sup τ). Proceeding as in the proof of Theorem 2.3, we obtain

∣∣∣∣∣1

t

∫ t

0u(ψτ (y)) dτ − 1

τm(y)

m−1∑

i=0

Iu(Ti(y))

∣∣∣∣∣→ 0 (7.10)

when t → ∞. Now let BX(x,m, ε) be the Bowen ball in (7.6). By (7.5), we have∣∣∣∣− logν(B(y, t, ε))

t+ logμ(BX(x,m, ε))

τm(x)

∣∣∣∣→ 0 (7.11)

when t → ∞. Moreover, T i(x, s) = T i(x,0) for every i ∈ N, and hence,

m−1∑

i=0

Iu(Ti(y)) =

m−1∑

i=0

Iu(Ti(x)).

Now let

A = − logν(B(y, t, ε))∫ t

0 u(ψτ (y)) dτ+ logμ(BX(x,m, ε))

∑m−1i=0 Iu(T i(x))

.

Since 0 < infu ≤ supu < ∞, it follows from (7.10) and (7.11) that

A =(− logμ(BX(x,m, ε))

τm(x)+ o(t)

)t

∫ t

0 u(ψτ (y)) dτ

+ logμ(BX(x,m, ε))

τm(x)

(t

∫ t

0 u(ψτ (y)) dτ+ o(t)

)

,

and hence,

|A| ≤(

1

infu+ hμ(T )

inf τ

)o(t).


This completes the proof of the lemma. �

We also express the BS-dimension in terms of a Carathéodory characteristic inthe base. Given a set Z ⊂ X and β ∈R, let

Nβ(Z) = lim�→∞ inf

Γ

∑

C∈Γ

exp

⎛

⎝−β sup

⎧⎨

⎩

m(C)−1∑

i=0

Iu(Ti(x)) : x ∈ C

⎫⎬

⎭

⎞

⎠ , (7.12)

where the infimum is taken over all finite or countable covers Γ of Z by cylinder sets

Ci−n···im = {(· · · j0 · · · ) : jk = ik for −n ≤ k ≤ m

}, with m,n > �. (7.13)

Lemma 7.2 If the set Z ⊂ X is T -invariant, then

dimu

{(x, s) ∈ Y : x ∈ Z and s ∈ [0, τ (x)]}= inf{β ∈R : Nβ(Z) = 0}.

Proof of the lemma Using the same notation as in the proof of Lemma 7.1, weobtain the inequality

∣∣∣∣∣

∫ t

0u(ψτ (x)) dτ −

m−1∑

i=0

Iu(Ti(x))

∣∣∣∣∣≤ κ supu,

which yields the desired result. �

The above lemmas allow us to reduce the study of the spectrum Du to the studyof corresponding properties of the dynamics in the base. Namely, by Lemma 7.1,we have

Kα = {(x, s) ∈ Y : x ∈ Zα and s ∈ [0, τ (x)]},

where

Zα ={

x ∈ X : limm→∞−

∑m−1i=0 Ig(T

i(x))∑m−1

i=0 Iu(T i(x))= α

}

,

and it follows from Lemma 7.2 that

Du(α) = inf{β ∈ R : Nβ(Zα) = 0}.In other words, the u-dimension spectrum Du for the u-pointwise dimensions (withrespect to the measure ν) coincides with the Ig-dimension spectrum studied by Bar-reira and Schmeling in [17] (in the case of discrete time). Hence, the desired resultfollows readily from Theorem 6.6 in that paper (see [3] for a detailed discussion). �

Theorem 7.1 is a continuous-time version of Theorem 6.6 in [17], which followsfrom work of Pesin and Weiss [83] and Schmeling [97].

7.3 Irregular Sets 87

Taking u = 1 in Theorem 7.1, we obtain a multifractal analysis of the spectrum

E(α) = h(Ψ |{y ∈ Y : hν(y) = α}),

where

hν(y) = limt→∞− logν(B(y, t, ε))

t= lim

t→∞1

t

∫ t

0g(ψτ (y)) dτ (7.14)

(see Theorem 7.3). The function E is called the entropy spectrum for the local en-tropies (with respect to the measure ν), and coincides with the entropy spectrum forthe Birkhoff averages of g (see Sect. 7.4).

7.3 Irregular Sets

In this section we consider the complement of the sets Kα in (7.9), that is, theirregular set Z = Y \ ⋃α∈R Kα , Even though Z has zero measure with respect toany Φ-invariant probability measure it also has full BS-dimension.

Let Ψ = {ψt }t∈R be a continuous flow in Y .

Definition 7.3 Given continuous functions g1, . . . , gk : Y → R and u : Y → R+,

we consider the irregular set

Fu(g1, . . . , gk)

=k⋂

j=1

{

y ∈ Y : lim inft→∞

∫ t

0 gj (ψs(y)) ds∫ t

0 u(ψs(y)) ds< lim sup

t→∞

∫ t

0 gj (ψs(y)) ds∫ t

0 u(ψs(y)) ds

}

.

We have

Fu(g1, . . . , gk) = {(x, s) : x ∈ Cu(g1, . . . , gk) and s ∈ [0, τ (x)]},

where

Cu(g1, . . . , gk)

=k⋂

j=1

{

x ∈ X : lim infm→∞

∑mi=0 Igj

(T i(x))∑m

i=0 Iu(T i(x))< lim sup

m→∞

∑mi=0 Igj

(T i(x))∑m

i=0 Iu(T i(x))

}

.

This is a consequence of the following result.

Proposition 7.2 Let Ψ be a suspension flow over a map T : X → X and leta, b : Y →R be continuous functions with b > 0. If x ∈ X and s ∈ [0, τ (x)], then

lim inft→∞

∫ t

0 a(ψτ (x, s)) dτ∫ t

0 b(ψτ (x, s)) dτ= lim inf

m→∞

∑mi=0 Ia(T

i(x))∑m

i=0 Ib(T i(x))


and

lim supt→∞

∫ t

0 a(ψτ (x, s)) dτ∫ t

0 b(ψτ (x, s)) dτ= lim sup

m→∞

∑mi=0 Ia(T

i(x))∑m

i=0 Ib(T i(x)).

Proof The argument is a modification of the proof of Theorem 2.3. Given m ∈ N,we consider the function τm : Y →R in (2.13). For each t > 0, there exists a uniqueinteger m ∈ N such that τm(x) ≤ t < τm+1(x). Writing t = τm(x) + κ with κ ∈(inf τ, sup τ), we obtain

∣∣∣∣∣

∫ t

0 a(ψs(x)) ds∫ t

0 b(ψs(x)) ds−

∫ τm(x)

0 a(ψs(x)) ds∫ τm(x)

0 b(ψs(x)) ds

∣∣∣∣∣

=∣∣∣∣∣∣

∫ t

τm(x)a(ψs(x)) ds

∫ τm(x)

0 b(ψs(x)) ds − ∫ τm(x)

0 a(ψs(x)) ds∫ t

τm(x)b(ψs(x)) ds

∫ t

0 b(ψs(x)) ds∫ τm(x)

0 b(ψs(x)) ds

∣∣∣∣∣∣

≤ κ sup|a| · τm(x) supb + τm(x) sup|a| · κ supb

τm(x) supb · τm(x) supb

= 2κ sup|a|τm(x) supb

.

Letting t → ∞, we have m → ∞ and τm(x) → ∞. Hence, it follows from (2.14)that

∣∣∣∣∣

∫ t

0 a(ψs(x)) ds∫ t

0 b(ψs(x)) ds−

∑m−1i=0 Ia(T

i(x))∑m−1

i=0 Ib(T i(x))

∣∣∣∣∣→ 0

when t → ∞. This yields the desired result. �

In particular, Proposition 7.2 allows us to reduce the study of irregular sets to thestudy of corresponding sets in the base.

The following result gives a necessary and sufficient condition so that the irreg-ular set Fu(g1, . . . , gk) has full BS-dimension.

Theorem 7.2 ([12]) Let Ψ be a suspension flow over a topologically mixing two-sided topological Markov chain and let g1, . . . , gk, u : Y → R be Hölder continuousfunctions with u > 0. Then the following properties are equivalent:

1. gj is not Ψ -cohomologous to a multiple of u in Y , for j = 1, . . . , k;2. dimu Fu(g1, . . . , gk) = dimu Y .

Proof In a similar manner to that in the proof of Theorem 7.1, we first reduce theproblem to the case of maps. By Lemma 7.2, we have

dimu Y = inf{β ∈R : Nβ(X) = 0} (7.15)

7.4 Entropy Spectra 89

and

dimu Fu(g1, . . . , gk) = inf{β ∈R : Nβ(Cu(g1, . . . , gk)) = 0

}, (7.16)

with Nβ(Z) as in (7.12). We note that the set Cu(g1, . . . , gk) is defined entirely interms of the map T and the functions Iu and Igj

for j = 1, . . . , k. On the other hand,by Theorem 2.1, the function gj is Ψ -cohomologous to a multiple of u in Y if andonly if Igj

is T -cohomologous to a multiple of Iu in X, and hence, if and only ifIgj

is T -cohomologous to Iαj u = αj Iu in X, where αj is the unique real numbersuch that PT (Igj

) = PT (αj Iu). These are precisely the cohomology assumptions inTheorem 7.1 in [17] in the case of discrete time (see [3] for a detailed discussion),which tell us that gj is not Ψ -cohomologous to a multiple of u in Y , for j = 1, . . . , k,if and only if the right-hand sides of (7.15) and (7.16) are equal. This yields thedesired result. �

7.4 Entropy Spectra

This section considers the particular case of entropy spectra. As a consequence ofthe results in the former sections, we obtain a multifractal analysis of these spectraand we study the corresponding irregular sets.

Let Ψ be a suspension flow over a map T : X → X and let g : Y → R be acontinuous function. For each α ∈R, let

E(α) = h(Ψ |Kα),

where

Kα ={x ∈ Y : lim

t→∞1

t

∫ t

0g(ψτ (x)) dτ = α

}.

The topological entropy is computed with respect to the Bowen–Walters distancein Y . The function E is called the entropy spectrum for the Birkhoff averages of g.For each q ∈ R, let νq be the equilibrium measure for qg and write

T (q) = PΨ (qg).

The following result is a multifractal analysis of the spectrum E.

Theorem 7.3 ([12]) Let Ψ be a suspension flow over a topologically mixing two-sided topological Markov chain and let g : Y →R be a Hölder continuous functionwith PΨ (g) = 0. Then the following properties hold:

1. the domain of E is a closed interval in [0,∞) coinciding with the range of thefunction α = −T ′, and for each q ∈R we have

E(α(q)) = T (q) + qα(q) = hνq (Ψ );


2. if g is not Ψ -cohomologous to a constant in Y , then E and T are analytic strictlyconvex functions.

Proof In view of (7.14), the result follows from Theorem 7.1 taking u = 1. �

Now we consider the irregular sets. More precisely, given a continuous functiong : Y → R, let

B(g) ={y ∈ Y : lim inf

t→∞1

t

∫ t

0g(ψτ (y)) dτ < lim sup

t→∞1

t

∫ t

0g(ψτ (y)) dτ

}.

It follows from Theorem 2.3 that

B(g) = {(x, s) ∈ Y : x ∈ C and s ∈ [0, τ (x)]},

where

C ={

x ∈ X : lim infm→∞

∑mi=0 Ig(T

i(x))∑m

i=0 τ(T i(x))< lim sup

m→∞

∑mi=0 Ig(T

i(x))∑m

i=0 τ(T i(x))

}

.

The following result gives a necessary and sufficient condition so that the irregularsets have full topological entropy.

Theorem 7.4 ([12]) Let Ψ be a suspension flow over a topologically mixing two-sided topological Markov chain and let gj : Y → R be Hölder continuous functionsfor j = 1, . . . , k. Then the following properties are equivalent:

1. gj is not Ψ -cohomologous to a constant in Y , for j = 1, . . . , k;2. h(Ψ |⋂k

j=1 B(gj )) = h(Ψ ).

Proof The result follows from Theorem 7.2 taking k = 1, g1 = g and u = 1. �

Chapter 8Multifractal Analysis of Hyperbolic Flows

In this chapter we continue the study of multifractal analysis for flows. The empha-sis is now on dimension spectra of hyperbolic flows. We first consider the somewhatsimpler case of suspension semiflows over expanding maps. It is presented mainlyas a motivation for the case of hyperbolic sets for conformal flows, without the addi-tional complication of simultaneously having contraction and expansion. In the caseof entropy spectra for hyperbolic flows, we show that the cohomology assumptionsrequired in the study of irregular sets are generically satisfied.

8.1 Suspensions over Expanding Maps

In this section we consider suspension semiflows over conformal expanding mapsand we obtain a multifractal analysis of the dimension spectra of Gibbs measures.This can be seen as a simplified version of the multifractal analysis of the dimensionspectra of Gibbs measures on locally maximal hyperbolic sets, without simultane-ously having expansion and contraction.

Let f : M → M be a C1 map of a smooth manifold M and let Λ ⊂ M be acompact f -invariant set such that f is expanding on Λ. This means that there existconstants c > 0 and β > 1 such that

‖dxfnv‖ ≥ cβn‖v‖

for all x ∈ Λ, v ∈ TxM and n ∈ N. The set Λ is said to be a repeller of f . We notethat any repeller of f is also a repeller of f n for each n ∈ N. Thus, passing eventu-ally to a power of f , without loss of generality one can take c = 1. For simplicity ofthe exposition we always make this assumption.

We also introduce the notion of conformality.

Definition 8.1 The map f is said to be conformal on Λ if dxf is a multiple of anisometry for every x ∈ Λ.


91

http://dx.doi.org/10.1007/978-3-319-00548-5_8

92 8 Multifractal Analysis of Hyperbolic Flows

Any repeller has Markov partitions of arbitrarily small diameter. Each Markovpartition has associated a one-sided topological Markov chain σ : X → X and a cod-ing map π : X → Λ for the repeller. The map π is onto, finite-to-one, and satisfiesf ◦ π = π ◦ σ . We refer to [5] for details.

Consider a Markov partition of Λ and the associated coding map π : X → Λ.Let Ψ be the suspension semiflow over the one-sided topological Markov chainσ : X → X. We introduce a distance dX in X (inducing the usual topology) with theproperty that for a repeller Λ of a conformal map the coding map π : (X,dX) → Λ

is locally Lipschitz. Let u : X →R+ be the continuous function

u(x) = log‖dπ(x)f ‖. (8.1)

The distance dX is defined by

dX

((i0 · · · ), (j0 · · · ))= |i0 − j0| + exp

(−u(Ci0···in )),

where

n = max{m ∈N : ik = jk for k ≤ m

}

and

u(Ci0···in) = sup

{ n∑

k=0

u(σ k(x)) : x ∈ Ci0···in}.

The set Y is now equipped with the corresponding Bowen–Walters distance. It fol-lows from work of Schmeling [98] that if f is a C1+δ expanding map that is con-formal on Λ, then

dimH Z = 1 + dimu π−1Z

for any Ψ -invariant set Z ⊂ Λ, with u as in (8.1). We note that here dimu is theBS-dimension introduced by Barreira and Schmeling in [17] (for discrete time) andnot the corresponding notion for continuous time described in Sect. 4.2.

Now we introduce the dimension spectrum. Let ν be a Ψ -invariant probabilitymeasure on Y . For each α ∈R, let

Kα ={y ∈ Y : lim

r→0

logν(B(y, r))

log r= α

},

where B(y, r) ⊂ Y is the Bowen–Walters ball of radius r centered at y ∈ Y .

Definition 8.2 The function

D(α) = dimH Kα

is called the dimension spectrum for the pointwise dimensions (with respect to themeasure ν).

8.1 Suspensions over Expanding Maps 93

Let also μ be the measure on X associated to ν as in (4.8). By Proposition 2.3,there exists a c ≥ 1 such that

BX(x, r/c) × (s − r/c, s + r/c) ⊂ B(y, r) ⊂ BX(x, cr) × (s − cr, s + cr)

for every y = (x, s) ∈ Y and any sufficiently small r (taking the distance dX in X).Therefore,

Kα ={(x, s) ∈ Y : lim

r→0

logμ(BX(x, r))

log r= α − 1

}.

Since each set Kα is Ψ -invariant, we obtain

D(α) = 1 + dimu

{x ∈ X : lim

r→0

logμ(BX(x, r))

log r= α − 1

}, (8.2)

with u as in (8.1).Now let g : Y → R be a Hölder continuous function. For each q ∈ R, we define

a function gq : Y → R by

gq = −Tu(q)u + qg,

where Tu(q) is the unique real number such that PΨ (gq) = 0. We denote respec-tively by νq and mu the equilibrium measures for gq and −dimu Y · u (with respectto Ψ ).

Proceeding in a similar manner to that in Sect. 7.1 one can obtain a multifractalanalysis of the spectrum D. Using the same notation as in Sect. 7.1, the followingresult is a simple consequence of Theorem 7.1 and the above discussion, togetherwith appropriate versions of Propositions 2.1 and 2.3 for locally invertible maps.

Theorem 8.1 ([12]) Let Λ be a repeller of a C1+δ map that is conformal and topo-logically mixing on Λ and let Ψ be the suspension semiflow over the one-sidedtopological Markov chain associated to some Markov partition of Λ. If ν is anequilibrium measure for a Hölder continuous function g (with respect to Ψ ), thenthe following properties hold:

1. for ν-almost every y ∈ Y we have

limr→0

logν(B(y, r))

log r= 1 + hμ(f )

∫X

udμ;

2. the domain of D is a closed interval in [0,∞) and coincides with the range ofthe function α = −T ′, where T (q) = Tu(q) − q + 1;

3. for each q ∈ R we have νq(Kα(q)) = 1,

D(α(q)) = T (q) + qα(q) = dimH νq, (8.3)

limr→0

logνq(B(y, r))

log r= T (q) + qα(q)


for νq -almost every x ∈ Kα(q), and

lim supr→0

logνq(B(y, r))

log r≤ T (q) + qα(q)

for every x ∈ Kα(q);4. if ν = mu, then D and T are analytic strictly convex functions.

Let αu(q) = −T ′u(q). By (8.2), we have

T (q) + qα(q) = Tu(q) − q + 1 + q(−T ′u(q) + 1)

= 1 + Tu(q) + qαu(q)

= 1 + dimu

{x ∈ X : lim

r→0

logμ(BX(x, r))

log r= αu(q)

}

= D(αu(q) + 1) = D(α(q)),

which establishes the first equality in (8.3).In a similar manner, the following result follows easily from an appropriate ver-

sion of Theorem 7.2 for suspension semiflows over one-sided topological Markovchains.

Theorem 8.2 Under the hypotheses of Theorem 8.1, we have ν = mu if and only if

dimH

{y ∈ Y : lim inf

r→0

logν(B(y, r))

log r< lim sup

r→0

logν(B(y, r))

log r

}= dimH Y.

8.2 Dimension Spectra of Hyperbolic Flows

In this section we obtain a multifractal analysis of the dimension spectrum for thepointwise dimensions of a Gibbs measure (on a locally maximal hyperbolic set for aconformal flow). This can be seen as an elaborate version of Theorem 8.1, in whichcase only expansion is present.

Let Φ be a C1 flow with a locally maximal hyperbolic set Λ. Given a Markovsystem, we consider the associated two-sided topological Markov chain σ : X → X

and the coding map π : X → Λ (see Sect. 3.2).Now let βs,βu : X → R

+ be Hölder continuous functions. For each cylinder setCi−n···im in (7.13), we define

βs(Ci−n···im) = sup

{ m∑

k=0

βs(σk(x)) : x ∈ Ci−n···im

}

8.2 Dimension Spectra of Hyperbolic Flows 95

and

βu(Ci−n···im) = sup

{ n∑

k=0

βu(σ−k(x)) : x ∈ Ci−n···im

}.

For each set Z ⊂ X and α ∈ R, let

M(Z,α) = lim�→0

infΓ

∑

C∈Γ

exp(−αβs(C) − αβu(C)

),

where the infimum is taken over all finite or countable covers Γ of Z by cylindersets Ci−n···im with m,n > �.

Definition 8.3 The (βs, βu)-dimension of the set Z is defined by

dimβs,βu Z = inf{α ∈R : M(Z,α) = 0

}.

Now let Φ be a C1+δ flow with a locally maximal hyperbolic set Λ such thatΦ|Λ is conformal. We continue to consider a Markov system for Φ on Λ and theassociated symbolic dynamics. We define functions βs,βu : X → R by

βs = −Iζs ◦ π and βu = Iζu ◦ π, (8.4)

with ζs and ζu as in (5.2) and (5.3). One can easily verify that

βs(x) = − log‖dπ(x)ϕτ(π(x))|Es(π(x))‖and

βu(x) = log‖dπ(x)ϕτ(π(x))|Eu(π(x))‖.Without loss of generality, one can always assume that βs and βu are positive func-tions (simply consider an adapted metric). Since Φ is conformal on Λ, we have

n−1∑

k=0

βs(σk(x)) = − log‖dπ(x)ϕτn(π(x))|Es(π(x))‖

andn−1∑

k=0

βu(σ−k(x)) = log‖dπ(x)ϕ−τn(π(x))|Eu(π(x))‖,

where

τn(π(x)) =n−1∑

k=0

τ(π(σ k(x))).

It follows from work of Schmeling [98] that

dimH Z = 1 + dimβs,βu π−1Z


for any Ψ -invariant set Z ⊂ Λ.When X is equipped with the distance d in (3.10), in general the map π is only

Hölder continuous. We introduce a new distance dX in X (inducing the same topol-ogy as d) such that for flows that are conformal on Λ the map π : (X,dX) → Λ islocally Lipschitz. The new distance dX is defined by

dX

((· · · i0 · · · ), (· · · j0 · · · )) = |i0 − j0| + exp

(−βs(Ci−nu ···ins))

+ exp(−βu(Ci−nu ···ins

)),

where

ns = max{n ∈ N : ik = jk for k ≤ n

}

and

nu = max{n ∈ N : ik = jk for k ≥ −n

}.

Since

diamdXC = βs(C) + βu(C)

for any cylinder set C, the (βs, βu)-dimension of a set Z ⊂ X coincides with itsHausdorff dimension with respect to the distance dX . This distance induces a newBowen–Walters distance in Y .

Now we introduce the dimension spectrum. We continue to consider a locallymaximal hyperbolic set for a C1 flow Φ . Let also ν be a Φ-invariant probabilitymeasure on Λ. For each α ∈R, let

Kα ={y ∈ Λ : lim

r→0

logν(B(y, r))

log r= α

}.

Definition 8.4 The dimension spectrum for the pointwise dimensions (with respectto the measure ν) is defined by

D(α) = dimH Kα.

In a similar manner to that in (8.2), if Φ is conformal on Λ, then

D(α) = 1 + dimβs,βu

{x ∈ X : lim

r→0

logμ(BX(x, r))

log r= α − 1

},

with βs and βu as in (8.4).Given a continuous function g : Λ → R, for each q ∈ R let Ts(q) and Tu(q) be

the unique real numbers such that

PΦ|Λ(Ts(q)ζs + qg

)= PΦ|Λ(−Tu(q)ζu + qg

)= 0,

or equivalently,

Pσ

(−Ts(q)βs + qIg ◦ π)= Pσ

(−Tu(q)βu + qIg ◦ π)= 0. (8.5)


We write

T (q) = Ts(q) + Tu(q) − q + 1. (8.6)

The following result is due to Pesin and Sadovskaya [82]. It is a multifractal analysisof the dimension spectrum for the pointwise dimensions of a Gibbs measure on alocally maximal hyperbolic set.

Theorem 8.3 Let Φ be a C1+δ flow with a locally maximal hyperbolic set Λ suchthat Φ|Λ is conformal and topologically mixing and let ν be an equilibrium measurefor a Hölder continuous function g (with respect to Φ) such that PΦ|Λ(g) = 0. Thenthe following properties hold:

1. for ν-almost every y ∈ Λ we have

limr→0

logν(B(y, r))

log r= hμ(Φ)

(1

∫Λ

ζu dμ− 1

∫Λ

ζs dμ

)+ 1;

2. if α = −T ′, then

D(α(q)) = T (q) + qα(q) for q ∈ R.

Proof Since the measure ν is ergodic, the first property is an immediate conse-quence of Theorem 6.2.

To establish the second property, we proceed in a similar manner to that in theproof of Theorem 5.1. Namely, let R1, . . . ,Rk be a Markov system for Φ on Λ. Wealso consider the function τ in (3.4) and the map T in (3.5), where Z = ⋃k

i=1 Ri .Let S be the invertible map T |Z : Z → Z and let A be the transition matrix obtainedfrom the Markov system as in (3.8).

We recall the projections π+ and π− in (3.13) and (3.14). It follows from a con-struction described by Bowen in [28] (see Proposition 4.2.11 in [3]) that there existfunctions ψs, ds : Σ−

A → R and ψu,du : Σ+A →R such that:

1. Ig ◦ π , ψs ◦ π− and ψu ◦ π+ are cohomologous;2. log‖dS−1|Es‖ ◦ π and ds ◦ π− are cohomologous;3. log‖dT |Eu‖ ◦ π and du ◦ π+ are cohomologous.

Given ω ∈ ΣA and r ∈ (0,1), let n = n(ω, r) and m = m(ω, r) be the unique posi-tive integers such that

‖dxS−n|Es(x)‖−1 < r ≤ ‖dxS

−(n−1)|Es(x)‖−1 (8.7)

and

‖dxTm|Eu(x)‖−1 < r ≤ ‖dxT

m−1|Eu(x)‖−1, (8.8)

where x = π(ω). For each q ∈ R, let Jq be the set of sequences ω ∈ ΣA such that

− limr→0

(∑n(ω,r)−1k=0 ψs(σ k−(ω−))

∑n(ω,r)−1k=0 ds(σ k−(ω−))

+∑m(ω,r)−1

k=0 ψu(σ k+(ω+))∑m(ω,r)−1

k=0 du(σ k+(ω+))

)

= α(q),


where ω− = π−(ω) and ω+ = π+(ω).Moreover, for each q ∈ R, let μs

q and μuq be respectively the equilibrium mea-

sures of the functions

−Ts(q)ds + qψs and −Tu(q)du + qψu.

By (4.6), we obtain

0 = −T ′s (q)

∫

Σ−A

ds dμsq +

∫

Σ−A

ψs dμsq

and

0 = −T ′u(q)

∫

Σ+A

du dμuq +

∫

Σ+A

ψu dμuq.

This implies that

αs(q) = −T ′s (q) = −

∫Σ−

Aψs dμs

q∫Σ−

Ads dμs

q

and

αu(q) = −T ′u(q) = −

∫Σ+

Aψu dμu

q∫Σ+

Adu dμu

q

.

Since the measures μsq and μu

q are ergodic, by Birkhoff’s ergodic theorem we have

limn→∞−

∑n−1k=0 ψs(σ k−(ω−))

∑n−1k=0 ds(σ k−(ω−))

= αs(q)

for μsq -almost every ω− ∈ Σ−

A , and

limm→∞−

∑m−1k=0 ψu(σ k+(ω+))

∑m−1k=0 du(σ k+(ω+))

= αu(q)

for μuq -almost every ω+ ∈ Σ+

A . Therefore, given ω ∈ ΣA and δ > 0, there exists anr(ω) > 0 such that

αs(q) − δ < −∑n(ω,r)−1

k=0 ψs(σ k−(ω−))∑n(ω,r)−1

k=0 ds(σ k−(ω−))< αs(q) + δ (8.9)

and

αu(q) − δ < −∑m(ω,r)−1

k=0 ψu(σ k+(ω+))∑m(ω,r)−1

k=0 du(σ k+(ω+))< αu(q) + δ (8.10)

for r ∈ (0, r(ω)).


On the other hand, since μsq and μu

q are equilibrium measures of Hölder contin-uous functions they are Gibbs measures. Moreover, it follows from (8.5) that

Pσ−|Σ−A

(−Ts(q)ds + qψs)= Pσ+|Σ+

A

(−Tu(q)du + qψu)= 0.

Hence, there exist constants D1,D2 > 0 such that

D1 ≤ μsq(C−

i−n···i0)

exp(−Ts(q)

∑n−1k=0 ds(σ k−(ω−)) + q

∑n−1k=0 ψs(σ k−(ω−))

) ≤ D2 (8.11)

and

D1 ≤ μuq(C+

i0···im)

exp(−Tu(q)

∑m−1k=0 du(σ k+(ω+)) + q

∑m−1k=0 ψu(σ k+(ω+))

) ≤ D2, (8.12)

for every n,m ∈N and ω = (· · · i−1i0i1 · · · ) ∈ ΣA, where

C−i−n···i0 ⊂ Σ−

A and C+i0···im ⊂ Σ+

A

are cylinder sets.Given x ∈ Z, let R(x) be a rectangle of the Markov system that contains x. We

have R(x) = π(Ci0), where x = π(· · · i0 · · · ). We also consider the measures

νsq = π∗(μs

q |C−i0

) in As(x) = π(π−1− C−i0

) ∩ R(x),

and

νuq = π∗(μu

q |C+i0

) in Au(x) = π(π−1+ C+i0

) ∩ R(x).

Finally, we define a product measure νq in R(x) = [As(x),Au(x)] by

νq = νsq × νu

q ,

using (3.3) to define the product structure. Given l > 0, consider the sets

Ql = {ω ∈ Jq : r(ω) ≥ 1/l

}.

Clearly,

Ql ⊂ Ql+1 and Jq =⋃

l>0

Ql. (8.13)

Lemma 8.1 For each x ∈ Z, we have

dνsq(y) ≥ Ts(q) + q(αs(q) − δ)

for νsq -almost every y ∈ As(x) ∩ π(Jq), and

dνuq(z) ≥ Tu(q) + q(αu(q) − δ)


for νuq -almost every z ∈ Au(x) ∩ π(Jq).

Proof of the lemma Given ω = (i0 · · · ) ∈ Σ+A and r ∈ (0,1), let

Δ(ω, r) = π(π−1+ C+

i0···im),

where n = m(ω, r). We note that these sets intersect at most along their boundariesfor each given r . Proceeding in a similar manner to that in (5.14) and (5.15), nowalong the unstable direction, and using (8.8), one can show that diamΔ(ω, r) < r ,provided that the diameter of the Markov system is sufficiently small. Furthermore,since each set Ri is the closure of its interior, there exists a ρ > 0 such that Ri

contains a ball Bi of radius ρ for i = 1, . . . , k. This implies that there exists a con-stant κ ∈ (0,1) (independent of ω and r) such that each set Δ(ω, r) contains a ballof radius κr (see [3] for details). Since the sets Δ(ω, r) intersect at most alongtheir boundaries, it follows from elementary geometry that there exists a constantC > 0 (independent of r) such that each ball B(x, r) intersects at most C of thesets Δ(ω, r).

Now we construct a special cover of π(Ql). For each ω ∈ Ql , let Δ(ω, r) be thelargest set containing π(ω) such that:

1. Δ(ω, r) = Δ(ω′, r) for some ω′ ∈ Ql with π(ω′) ∈ Δ(ω, r);2. Δ(ω′, r) ⊂ Δ(ω, r) whenever π(ω′) ∈ Δ(ω, r).

By construction, for a given r the sets Δ(ω, r) form a cover of π(Ql). Let Δj =Δ(ωj , r) with ωj ∈ Ql , for j = 1, . . . ,N(r), be the elements of this cover. Givenr < 1/l, it follows from (8.8), (8.10) and (8.12) that

νuq (B(x, r) ∩ π(Ql))

≤∑

Δj ∩B(x,r)=∅

νuq (Δ(ωj , r))

≤ D2

∑

Δj ∩B(x,r)=∅

‖dπ(ωj )Tm(ωj ,r)|Eu(π(ωj ))‖−Tu(q)

× exp

(

q

m(ωj ,r)−1∑

k=0

ψu(σ k+(ω+))

)

≤ D2

∑

Δj ∩B(x,r)=∅

‖dπ(ωj )Tm(ωj ,r)|Eu(π(ωj ))‖−Tu(q)−q(αu(q)−δ).

Using (8.8) again, we conclude that there exists a C′ > 0 such that

νuq (B(x, r) ∩ π(Ql)) ≤ C′rTu(q)+q(αu(q)−δ) (8.14)

for every x ∈ J and r ∈ (0,1/l). By the Borel density lemma, for νuq -almost every

x ∈ π(Ql) we have

limr→0

νuq (B(x, r) ∩ π(Ql))

νuq (B(x, r))

= 1,


and thus, there exists a ρ(x) > 0 such that

νuq (B(x, r)) ≤ 2νu

q (B(x, r) ∩ π(Ql))

for every r ∈ (0, ρ(x)). Together with (8.14) this implies that

dνuq(x) = lim inf

r→0

logνuq (B(x, r))

log r

≥ lim infr→0

logνuq (B(x, r) ∩ π(Ql))

log r

≥ Tu(q) + q(αu(q) − δ)

for νuq -almost every x ∈ π(Ql). By (8.13), we conclude that

dνuq(x) ≥ Tu(q) + q(αu(q) − δ)

for νuq -almost every x ∈ π(Jq). Since δ is arbitrary, this yields the desired result

along the unstable direction. The corresponding result along the stable direction canbe obtained in a similar manner. �

It follows from (8.13) and the arbitrariness of δ that

dνsq(y) ≥ Ts(q) + qαs(q)

for νsq -almost every y ∈ As(x) ∩ π(Jq), and

dνuq(z) ≥ Tu(q) + qαu(q)

for νuq -almost every z ∈ Au(x) ∩ π(Jq). Since νq = νs

q × νuq , we obtain

dνq(x) = lim inf

r→0

logνq(B(x, r))

log r

≥ dνsq(x) + dνu

q(x)

≥ Ts(q) + Tu(q) + q(αs(q) + αu(q))

for νq -almost every x ∈ π(Jq). It follows from Proposition 4.7 that

dimH π(Jq) ≥ T (q) + qα(q). (8.15)

Lemma 8.2 For each x ∈ π(Jq), we have

dνq (x) ≤ T (q) + qα(q).


Proof of the lemma By the choice of n and m in (8.7) and (8.8), there exists aconstant c > 0 such that

D(ω, r) =m(ω,r)⋃

j=−n(ω,r)

S−kRij ⊂ B(x, cr) (8.16)

for every x = π(ω) ∈ Λ and r ∈ (0,1). It follows from (8.11) and (8.12) that forevery x = π(ω) with ω ∈ Ql and r < 1/l, taking n = n(ω, r) and m = m(ω, r) weobtain

νq(B(x, cr)) ≥ νq(D(ω, r)) = μsq(C−

i0···in )μuq(C+

i0···im)

≥ D21 exp

(

−Ts(q)

n−1∑

k=0

ds(σ k−(ω−)) + q

n−1∑

k=0

ψs(σ k−(ω−))

)

× exp

(

−Tu(q)

m−1∑

k=0

du(σ k+(ω+)) + q

m−1∑

k=0

ψu(σ k+(ω+))

)

= D21 exp

(

−Ts(q) log‖dxS−n|Es(x)‖ + q

n−1∑

k=0

Ig(S−k(x))

)

× exp

(

−Tu(q) log‖dxTm|Eu(x)‖ + q

m−1∑

k=0

Ig(Tk(x))

)

.

Therefore, by (8.7), (8.8), (8.9) and (8.10), we have

dνq (x) = lim supr→0

logνq(B(x, r))

log r

≤ Ts(q) lim supr→0

− log‖dxS−n(ω,r)|Es(x)‖log r

+ Tu(q) lim supr→0

− log‖dxTm(ω,r)|Eu(x)‖

log r

+ lim supr→0

q∑m(ω,r)−1

k=−(n(ω,r)−1)Ig(S

k(x))

log r

≤ Ts(q) + Tu(q) + q(αs(q) + αu(q) + 2δ)

for every x ∈ π(Ql). It follows from (8.13) and the arbitrariness of δ that

dνq (x) ≤ Ts(q) + Tu(q) + q(αs(q) + αu(q))

for every x ∈ π(Jq). �


By Lemma 8.2, it follows from Proposition 4.7 that

dimH π(Jq) ≤ Ts(q) + Tu(q) + q(αs(q) + αu(q)).

Together with (8.15) this implies that

dimH π(Jq) = Ts(q) + Tu(q) + q(αs(q) + αu(q)). (8.17)

Lemma 8.3 Given γ > 0, there exists a K > 0 such that

ν(B(y, γ r)) ≤ Kν(B(y, r))

for every y ∈ R(x) and any sufficiently small r > 0.

Proof of the lemma For ν-almost every y ∈ Λ, let νsy and νu

y be respectively theconditional measures of ν in As(x) and Au(x). Repeating verbatim the argumentsin the case of repellers of conformal maps (see Lemma 6.1.5 in [3]), we find thatthere exists a C > 0 such that

νsy(B

s(y,2r)) ≤ Cνsy(B

s(y, r)) and νuy (Bu(y,2r)) ≤ Cνu

y (Bu(y, r)) (8.18)

for every y ∈ Λ and any sufficiently small r > 0, where Bs(y, r) and Bu(y, r) arethe open balls centered at y of radius r with respect to the distances induced respec-tively on the local stable and unstable manifolds V s(y) and V u(y).

Now we observe that there exists a κ > 1 such that

Λ ∩ B(y, γ r) ⊂ [Λ ∩ Bs(y, κr),Λ ∩ Bu(y, κr)

](8.19)

and[Λ ∩ Bs(y, r/κ),Λ ∩ Bu(y, r/κ)

]⊂ Λ ∩ B(y, r) (8.20)

for every y ∈ Λ and any sufficiently small r > 0. It follows from (8.19) that

ν(B(y, γ r)) ≤ cνuy (Bu(y, κr))νs

y(Bs(y, κr))

for some constant c > 0 (independent of y and r). Applying (8.18) a number n oftimes such that κ2−n < 1/κ , we obtain

ν(B(y, γ r)) ≤ C2nνuy (Bu(y, r/κ))νs

y(Bs(y, r/κ)).

Hence, it follows from (8.20) that

ν(B(y, γ r)) ≤ cC2nν(B(y, r))

for some constant c > 0 (independent of y and r) and the desired inequality holdswith K = cC2n. �

The following result relates the level sets Kα to the sets Jq .


Lemma 8.4 We have π(Jq) = Kα(q) ∩ Z.

Proof of the lemma Let r ∈ (0,1) and take n = n(ω, r) and m = m(ω, r) as in (8.7)and (8.8). Proceeding as in the proof of Lemma 8.1, we find that there exists a κ > 0(independent of r) such that for each x = π(ω) ∈ Z there exists a y ∈ D(ω, r)

(see (8.16)) for which

B(y, κr) ⊂ D(ω, r) ⊂ B(x, cr). (8.21)

Moreover, B(x, r) ⊂ B(y, dr) for some constant d > 0 (independent of x and r).By Lemma 8.3, we obtain

ν(D(ω, r)) ≤ ν(B(x, cr)) ≤ K1ν(B(x, r))

≤ K1ν(B(y, dr)) ≤ K2ν(B(y, κr))

≤ K2ν(D(ω, r))

for some constants K1,K2 > 0. This implies that if either of the two limits

limr→0

logν(D(ω, r))

log rand lim

r→0

logν(B(x, r))

log r(8.22)

exists, then the other also exists and has the same value. On the other hand, since ν

is the equilibrium measure of g and PΦ|Λ(g) = 0, if the first limit exists, then

a := limr→0

logν(D(ω, r))

log r= lim

r→0

∑m(ω,r)−1k=−(n(ω,r)−1) Ig(S

k(x))

log r.

It follows from (8.7) and (8.8) that x = π(ω) ∈ π(Jq) if and only if

a = limr→0

(∑n(ω,r)−1k=0 Ig(f

−k(x))

log r+

∑m(ω,r)−1k=0 Ig(f

k(x))

log r

)

= limr→0

( ∑n(ω,r)−1k=0 Ig(S

−k(x))

− log‖dxS−n(ω,r)|Es(x)‖ +∑m(ω,r)−1

k=0 Ig(Tk(x))

− log‖dxT m(ω,r)|Eu(x)‖

)

= α(q).

Hence, x ∈ π(Jq) if and only if the second limit in (8.22) is equal to α(q). �

By (8.17) and Lemma 8.4, in view of (8.6) we obtain

dimH Kα(q) = 1 + dimH (Kα(q) ∩ Z)

= 1 + Ts(q) + Tu(q) + q(αs(q) + αu(q))

= T (q) + qα(q).

This completes the proof of the theorem. �

It is also shown in [82] that if ν is not a measure of full dimension, that is,dimH ν = dimH Λ, then the functions D and T are analytic and strictly convex.

8.3 Entropy Spectra and Cohomology 105

8.3 Entropy Spectra and Cohomology

In this section we consider the particular case of entropy spectra for hyperbolicflows. We emphasize that unlike in Chap. 7, the results for these spectra cannot beobtained from the results for dimension spectra in Sect. 8.2 (in Chap. 7 we wereinstead considering BS-dimension spectra, not dimension spectra). In particular, wedescribe appropriate versions of the results in Sect. 7.4. We also show that the co-homology assumptions required in the study of the irregular sets are genericallysatisfied.

Let Φ be a C1 flow with a locally maximal hyperbolic set Λ and let g : Λ → R

be a continuous function. For each α ∈R, we consider the set

Kα ={x ∈ Λ : lim

t→∞1

t

∫ t

0g(ϕτ (x)) dτ = α

}.

One can easily verify that Kα is Φ-invariant.We recall that a function g : Λ → R is said to be Φ-cohomologous to a function

h : Λ → R if there exists a bounded measurable function q : Λ → R such that


q(ϕt (x)) − q(x)

t

for every x ∈ Λ. In particular, if g : Λ →R is Φ-cohomologous to a constant c ∈R

in Λ, then

∣∣∣∣1

t

∫ t

0g(ϕτ (x)) dτ − c

∣∣∣∣ = 1

tlims→0

1

s

∣∣∣∣

∫ s+t

s

q(ϕτ (x)) dτ −∫ t

0q(ϕτ (x)) dτ

∣∣∣∣

= 1

tlims→0

1

s

∣∣∣∣

∫ s+t

t

q(ϕτ (x)) dτ −∫ s

0q(ϕτ (x)) dτ

∣∣∣∣

≤ 2 sup|q|t

(8.23)

for every x ∈ Λ and t > 0, and hence, Kc = Λ. This shows that it is only interestingto consider the case when g is not cohomologous to a constant.

Now we introduce the entropy spectrum. Given α ∈ R, let

E(α) = h(Φ|Kα).

The function E is called the entropy spectrum for the Birkhoff averages of g. Foreach q ∈ R, let νq be the equilibrium measure for qg and write

T (q) = PΦ(qg),

where PΦ(qg) is the topological pressure of qg with respect to Φ .The following is a multifractal analysis of the spectrum E.



such that Φ|Λ is topologically mixing and let g : Λ → R be a Hölder continuousfunction with PΦ(g) = 0. Then the following properties hold:

1. the domain of E is a closed interval in [0,∞) coinciding with the range of thefunction α = −T ′, and for each q ∈R we have

E(α(q)) = T (q) + qα(q) = hνq (Φ|Λ);2. if g is not Φ-cohomologous to a constant in Λ, then the functions E and T are

analytic and strictly convex.

Proof Consider a Markov system for Φ on Λ, the associated suspension flow Ψ

and the coding map π : Y → Λ defined by (3.11). It follows from (3.12) that

limt→∞

1

t

∫ t

0(g ◦ π)(ψτ (x)) dτ = α

if and only if

limt→∞

1

t

∫ t

0g(ϕτ (π(x))) dτ = α.

This shows that E = D1, with Du as in Sect. 7.1. Hence, the desired result followsfrom Theorem 7.1 taking u = 1. �

Given a continuous function g : Λ → R, the irregular set for the Birkhoff aver-ages of g (with respect to Φ) is defined by

B(g) ={x ∈ Λ : lim inf

t→∞1

t

∫ t

0g(ϕτ (x)) dτ < lim sup

t→∞1

t

∫ t

0g(ϕτ (x)) dτ

}.

One can easily verify that B(g) is Φ-invariant. By Birkhoff’s ergodic theorem, theset B(g) has zero measure with respect to any Φ-invariant finite measure. Moreover,by (8.23), if g is Φ-cohomologous to a constant in Λ, then B(g) = ∅.

The following result gives a necessary and sufficient condition so that the irreg-ular set has full topological entropy.


such that Φ|Λ is topologically mixing and let g : Λ → R be a Hölder continuousfunction. Then the following properties are equivalent:

1. g is not Φ-cohomologous to a constant in Λ;2. h(Φ|B(g)) = h(Φ|Λ).

Proof If g is Φ-cohomologous to a constant, then B(g) = ∅.Now we assume that g is not Φ-cohomologous to a constant. Consider a Markov

system, the associated suspension flow Ψ and the coding map π : Y → Λ. It follows

8.3 Entropy Spectra and Cohomology 107

from (3.12) that

lim inft→∞

1

t

∫ t

0(g ◦ π)(ψτ (x)) dτ < lim sup

t→∞1

t

∫ t

0(g ◦ π)(ψτ (x)) dτ

if and only if

lim inft→∞

1

t

∫ t

0g(ϕτ (π(x))) dτ < lim sup

t→∞1

t

∫ t

0g(ϕτ (π(x))) dτ.

Therefore,

B(g) = π(B(g ◦ π)). (8.24)

To complete the proof we proceed as in [17]. Let R ⊂ Λ be the set of points y ∈ Λ

such that ϕt (x) is on the boundary of some element of the Markov system for somet ∈R. We note that R is Φ-invariant and that

π : π−1(Λ \ R) → Λ \ R

is a homeomorphism. Moreover, since there exist cylinder sets C ⊂ X such thatπ(C) is disjoint from R, we have

h(Ψ |π−1R) < h(Ψ ) and h(Φ|R) < h(Φ|Λ).

By (8.24), we obtain

h(Φ|B(g)) = h(Ψ |B(g ◦ π)),

and it follows from Theorem 7.4 that

h(Φ|Λ) = h(Ψ ) = h(Ψ |B(g ◦ π)) = h(Φ|B(g)).


Theorem 8.5 is a counterpart of results of Barreira and Schmeling in [17] fordiscrete time.

Now we show that most Hölder continuous functions are not Φ-cohomologousto a constant. Let Cγ (Λ) be the space of Hölder continuous functions in Λ withHölder exponent γ ∈ (0,1). We define the norm of a function ϕ ∈ Cγ (Λ) by

‖ϕ‖γ = sup{|ϕ(x)| : x ∈ Λ} + sup

{ |ϕ(x) − ϕ(y)|d(x, y)γ

: x, y ∈ Λ and x = y

}, (8.25)

where d is the distance on M . We recall that Φ is said to be topologically transitiveon Λ (or simply Φ|Λ is said to be topologically transitive) if for any nonempty opensets U and V intersecting Λ there exist a t ∈ R such that ϕt (U) ∩ V ∩ Λ = ∅.

Theorem 8.6 ([12]) Let Φ be a C1 flow with a locally maximal hyperbolic set Λ

such that Φ|Λ is topologically transitive. For each γ ∈ (0,1), the set of all functionsin Cγ (Λ) that are not Φ-cohomologous to a constant is open and dense in Cγ (Λ).


Proof Let

G = {g ∈ Cγ (Λ) : g is not Φ-cohomologous to a constant

}

and take g ∈ G. By Livschitz’s theorem (see Theorem 19.2.4 in [62]), there existpoints xi = ϕTi

(xi) for i = 0,1 for which

c =∣∣∣∣

1

T0

∫ T0

0g(ϕτ (x0)) dτ − 1

T1

∫ T1

0g(ϕτ (x1)) dτ

∣∣∣∣ = 0.

For each f ∈ Cγ (Λ) such that ‖f − g‖γ < c/2, we have

∣∣∣∣

1

Ti

∫ Ti

0(f − g)(ϕτ (xi)) dτ

∣∣∣∣≤ sup

{|f (x) − g(x)| : x ∈ Λ}≤ ‖f − g‖γ <

c

2

for i = 0,1, and hence,

1

T0

∫ T0

0f (ϕτ (x0)) dτ = 1

T1

∫ T1

0f (ϕτ (x1)) dτ.

This implies that f is not Φ-cohomologous to a constant, and thus, the set G isopen.

Now let Γ0 and Γ1 be distinct periodic orbits, and let h ∈ Cγ (Λ) be a Höldercontinuous function such that h|Γi = i for i = 0,1. Take g ∈ G. For each ε > 0, thefunction gε = g + εh ∈ Cγ (Λ) is not Φ-cohomologous to a constant, because theaverages on Γ0 and Γ1 differ by ε. Moreover,

‖gε − g‖γ ≤ ε‖h‖γ ,

and hence the function g can be arbitrarily approximated by functions in G. There-fore, G is dense in Cγ (Λ). �

Theorems 8.5 and 8.6 readily imply the following result.

Theorem 8.7 Let Φ be a C1+δ flow with a locally maximal hyperbolic set Λ suchthat Φ|Λ is topologically mixing. Given γ ∈ (0,1), for an open and dense family offunctions g ∈ Cγ (Λ) we have h(Φ|B(g)) = h(Φ|Λ).

Part IVVariational Principles

This final part is dedicated to the study of conditional variational principles. Thiscorresponds to describing the topological entropy or the Hausdorff dimension of thelevel sets of a given function. In Chap. 9 we obtain a conditional variational princi-ple for flows with a locally maximal hyperbolic set and we study the analyticity ofseveral classes of multifractal spectra. In particular, we consider spectra defined bylocal entropies and Lyapunov exponents. In Chap. 10 we obtain a multidimensionalversion of multifractal analysis for hyperbolic flows. This corresponds to comput-ing the topological entropy of the multidimensional level sets associated to severalBirkhoff averages. In Chap. 11 we establish a conditional variational principle forthe dimension spectra of Birkhoff averages, considering simultaneously averagesinto the future and into the past.

Chapter 9Entropy Spectra

In this chapter we establish a conditional variational principle for flows with a lo-cally maximal hyperbolic set. In other words, we express the topological entropy ofthe level sets of the Birkhoff averages of a given function in terms of a conditionalvariational principle. As an application of this principle, we establish the analyticityof several classes of multifractal spectra for hyperbolic flows. In particular, we con-sider the multifractal spectra for the local entropies and for the Lyapunov exponents.

9.1 A Conditional Variational Principle

This section is dedicated to establishing a conditional variational principle for hy-perbolic flows. We consider multifractal spectra obtained from ratios of Birkhoffaverages.

Let Φ = {ϕt }t∈R be a continuous flow and let Λ be a Φ-invariant set. We denoteby C(Λ) the space of all continuous functions a : Λ → R. Given a, b ∈ C(Λ) withb > 0 and α ∈ R, let

Kα = Kα(a, b) ={

x ∈ Λ : limt→∞

∫ t

0 a(ϕs(x)) ds∫ t

0 b(ϕs(x)) ds= α

}

.

One can easily verify that the set Kα is Φ-invariant.Now we introduce the entropy spectrum.

Definition 9.1 The function F = F(a,b) defined by

F(α) = h(Φ|Kα) (9.1)

is called the entropy spectrum for the pair of functions (a, b).


111

http://dx.doi.org/10.1007/978-3-319-00548-5_9

112 9 Entropy Spectra

The following result is a conditional variational principle for the spectrum F. Let

α = inf

{∫Λ

a dμ∫Λ

b dμ: μ ∈M

}

and

α = sup

{∫Λ

a dμ∫Λ

b dμ: μ ∈M

},

where M is the set of all Φ-invariant probability measures on Λ.


such that Φ|Λ is topologically mixing and let a, b : Λ → R be Hölder continuousfunctions with b > 0. Then the following properties hold:

1. if α ∈ [α,α], then Kα = ∅;2. if α ∈ (α,α), then Kα = ∅,

F(α) = max

{hμ(Φ) :

∫Λ

a dμ∫Λ

b dμ= α and μ ∈ M

}(9.2)

and

F(α) = min{PΦ(qa − qαb) : q ∈R}. (9.3)

Proof Let us assume that Kα = ∅ and take x ∈ Kα . The sequence of probabilitymeasures (μn)n∈N in Λ such that

∫

Λ

udμn = 1

n

∫ n

0u(ϕs(x)) ds

for every u ∈ C(Λ) has an accumulation point μ ∈ M. Therefore,

α = limt→∞

∫ t

0 a(ϕs(x)) ds∫ t

0 b(ϕs(x)) ds

= limn→∞

∫Λ

a dμn∫Λ

b dμn

=∫Λ

a dμ∫Λ

b dμ∈ [α,α].

This establishes the first property.Now we establish the second property. By Proposition 4.2, for each α ∈ R we

have

infq∈RPΦ(qa − qαb) = inf

q∈R sup

{hμ(Φ) +

∫

Λ

(qa − qαb)dμ : μ ∈M

}

9.1 A Conditional Variational Principle 113

≥ sup

{hμ(Φ) :

∫Λ

a dμ∫Λ

b dμ= α and μ ∈M

}. (9.4)

On the other hand, given α ∈ (α,α), there exist measures ν−, ν+ ∈M such that∫Λ

a dν−∫Λ

b dν−< α <

∫Λ

a dν+∫Λ

b dν+.

Moreover, for each q ∈R and μ ∈M, we have

PΦ(qa − qαb) ≥ hμ(Φ) + q

(∫

Λ

a dμ − α

∫

Λ

b dμ

),

and hence,

lim infq→±∞PΦ(qa − qαb) ≥ hν±(Φ) + lim inf

q→±∞q

(∫

Λ

a dν± − α

∫

Λ

b dν±)

= +∞.

In particular, the map q → PΦ(qa − qαb) attains its infimum at some pointq = q(α) ∈ R. We note that this map is analytic (see [92]). Denoting by μα theequilibrium measure for the function q(α)(a − αb), we obtain

0 = d

dqPΦ(qa − qαb)

∣∣∣q=q(α)

=∫

Λ

a dμα − α

∫

Λ

b dμα.

Therefore, inequality (9.4) is in fact an equality and

infq∈RPΦ(qa − qαb) = hμα (Φ)

= max

{hμ(Φ) :

∫Λ

a dμ∫Λ

b dμ= α and μ ∈ M

}. (9.5)

Since Λ is hyperbolic, the flow Φ|Λ is expansive, in which case identity (4.4) takesthe simpler form

hμ(Φ) = inf{h(Φ|Z) : μ(Z) = 1

}

(that is, the limits in ε in (4.4) are not necessary provided that ε is sufficiently small).Since μα is ergodic, we have μα(Kα) = 1 and thus,

hμα (Φ) ≤ F(α). (9.6)

In view of (9.5) and (9.6), it remains to show that

F(α) ≤ infq∈RPΦ(qa − qαb).

Otherwise, there would exist q ∈ R, δ > 0 and c > 0 such that

F(α) − δ > c > PΦ(qa − qαb). (9.7)


Let u = qa − αqb and

Kα,δ,τ ={x ∈ Λ :

∣∣∣∣

∫ t

0u(ϕs(x)) ds

∣∣∣∣< δt for t ≥ τ

}.

We have

Kα ⊂⋃

τ∈NKα,δ,τ = Kα,δ

and it follows from the basic properties of the topological entropy (see [81]) that

limτ→+∞h(Φ|Kα,δ,τ ) = h(Φ|Kα,δ)

≥ h(Φ|Kα) = F(α).

In particular, there exists a τ ∈ N such that

c + δ < h(Φ|Kα,δ,τ ). (9.8)

For each y ∈ B(x, t, ε) and s ∈ [0, t], we have d(ϕs(x),ϕs(y)) < ε and thus,

|u(x, t, ε)| ≤∣∣∣∣

∫ t

0u(ϕs(y)) ds

∣∣∣∣+ η(ε)t,

where

η(ε) = sup{|u(x) − u(y)| : d(x, y) < ε

}.

Moreover, if B(x, t, ε) ∩ Kα,δ,τ = ∅, then there exists a y ∈ B(x, t, ε) such that∣∣∣∣

∫ t

0u(ϕs(y)) ds

∣∣∣∣< δt

whenever t ≥ τ . This implies that

|u(x, t, ε)| ≤ [δ + η(ε)]twhenever B(x, t, ε) ∩ Kα,δ,τ = ∅ and t ≥ τ . Hence,

M(Kα,δ,τ , u, c, ε) = limT →∞ inf

Γ

∑

(x,t)∈Γ

exp(u(x, t, ε) − ct)

≥ limT →∞ inf

Γ

∑

(x,t)∈Γ

exp(−[δ + η(ε)]t − ct)

= M(Kα,δ,τ ,0, c + δ + η(ε), ε

),

where the infimum is taken over all finite or countable sets Γ = {(xi, ti )}i∈I suchthat (xi, ti ) ∈ X × [T ,∞) for i ∈ I , and

⋃i∈I B(xi, ti , ε) ⊃ Kα,δ,τ . Since u is con-

tinuous, we have η(ε) → 0 when ε → 0, and in view of (9.8) it follows from the

9.2 Analyticity of the Spectrum 115

definition of the topological entropy that

M(Kα,δ,τ , u, c, ε) > 0

for any sufficiently small ε > 0. Therefore,

c ≤ PΦ|Kα,δ,τ (qa − qαb) ≤ PΦ(qa − qαb),

which contradicts (9.7). This completes the proof of the theorem. �

Identity (9.2) is called a conditional variational principle for the entropy spec-trum. Theorem 9.2 below gives a necessary and sufficient condition in terms of thefunctions a and b so that α < α.

We also explain how to obtain a measure at which the maximum in (9.2) is at-tained. Let q(α) ∈ R be a point where the function q → PΦ(qa − qαb) attains itsinfimum (it is shown in the proof of Theorem 9.1 that the infimum is indeed at-tained). Then the unique equilibrium measure μα for the function q(α)(a − αb)

satisfies

F(α) = hμα (Φ) and

∫Λ

a dμα∫Λ

b dμα

= α.

Using similar arguments to those in [14] in the case of discrete time, one can ex-tend Theorem 9.1 to the case when the entropy is upper semicontinuous, for continu-ous functions with unique equilibrium measures. For example, for a locally maximalhyperbolic set for a topologically mixing C1 flow the entropy is upper semicontin-uous and any continuous function with bounded variation has a unique equilibriummeasure. We recall that a continuous function a : X → R is said to have boundedvariation if there exist ε > 0 and κ > 0 such that

∣∣∣∣

∫ t

0a(ϕs(x)) ds −

∫ t

0a(ϕs(y)) ds

∣∣∣∣< κ

whenever d(ϕs(x),ϕs(y)) < ε for every s ∈ [0, t].

9.2 Analyticity of the Spectrum

Let Φ be a C1 flow with a locally maximal hyperbolic set Λ. Given functionsa, b : Λ → R as in Theorem 9.1, the following result shows that when a is notΦ-cohomologous to a multiple of b the spectrum F in (9.1) is analytic.


such that Φ|Λ is topologically mixing and let a, b : Λ → R be Hölder continuousfunctions with b > 0. Then the following properties hold:

1. if a is Φ-cohomologous to cb in Λ for some c ∈R, then α = α = c and Kc = Λ;


2. if a is not Φ-cohomologous to a multiple of b in Λ, then α < α and the function F

is analytic in the interval (α,α).

Proof Let us assume that there exists a constant c such that a is Φ-cohomologousto cb in Λ. We have

∣∣∣∣

∫ t

0a(ϕτ (x)) dτ − c

∫ t

0b(ϕτ (x)) dτ

∣∣∣∣

= lims→0

1

s

∣∣∣∣

∫ s+t

s

q(ϕτ (x)) dτ −∫ t

0q(ϕτ (x)) dτ

∣∣∣∣

= lims→0

1

s

∣∣∣∣

∫ s+t

t

q(ϕτ (x)) dτ −∫ s

0q(ϕτ (x)) dτ

∣∣∣∣

≤ 2 sup|q|, (9.9)

and hence,∣∣∣∣∣

∫ t

0 a(ϕτ (x)) dτ∫ t

0 b(ϕτ (x)) dτ− c

∣∣∣∣∣≤ 2 sup|q|

t inf|b|for x ∈ Λ and t > 0. Therefore, Kc = Λ. Moreover, by (9.9), for each μ ∈ M wehave

0 =∫

Λ

limt→∞

(1

t

∫ t


t

∫ t

0b(ϕτ (x)) dτ

)dμ(x)

=∫

Λ

a dμ − c

∫

Λ

b dμ

and α = α = c. This establishes the first property.For the second property, let us assume that a is not Φ-cohomologous to a multiple

of b. If α = α = c, then the function

μ →∫

Λ

a dμ − c

∫

Λ

b dμ

is identically zero. In particular, if μ is the invariant measure supported on the peri-odic orbit of a point x = ϕT (x), then

1

T

∫ T

0a(ϕs(x)) ds = c

1

T

∫ T

0b(ϕs(x)) ds.

By Livschitz’s theorem (see Theorem 19.2.4 in [62]), we conclude that the functionsa and cb are Φ-cohomologous. This contradiction implies that α < α.

Now we establish the analyticity of the spectrum.

Lemma 9.1 If for each α ∈ R the function a − αb is not Φ-cohomologous to aconstant, then the spectrum F is analytic in the interval (α,α).

9.2 Analyticity of the Spectrum 117

Proof of the lemma Take α ∈ (α,α) and let

F(q,α) = PΦ(qa − qαb).

By Theorem 9.1, the number F(α) coincides with minq∈R F(q,α). Moreover, thefunction F is analytic in both variables. We want to apply the Implicit function theo-rem to show that the minimum is attained at a point q = q(α) depending analyticallyon α.

We have

∂qF (q,α) =∫

Λ

(a − αb)dνq,α,

where νq,α is the equilibrium measure for qa − qαb. By Theorem 9.1, there exists aq = q(α) ∈ R at which the function q → PΦ(qa − qαb) attains a minimum. Thus,we have

∂qF (q(α),α) = 0.

Since a − αb is not Φ-cohomologous to a constant, the function q → F(q,α) isstrictly convex. Hence, q = q(α) is the unique real number satisfying ∂qF (q,α) = 0.Again since a − αb is not Φ-cohomologous to a constant, the derivative ∂2

qF doesnot vanish (see [92]). Thus, it follows from the Implicit function theorem that thefunction α → q(α) is analytic. This completes the proof of the lemma. �

In view of Lemma 9.1, it remains to consider the case when there exist c, d ∈ R

with d = 0 and a bounded measurable function q : Λ → R such that

a(x) − cb(x) = d + limt→0

q(ϕt (x)) − q(x)

t(9.10)

for x ∈ Λ. One can easily verify that

x ∈ Kα(a, b) if and only if x ∈ Kd/(α−c)(b,1).

Moreover, it follows from (9.9) and (9.10) that∣∣∣∣

∫ t


∫ t

0b(ϕτ (x)) dτ − dt

∣∣∣∣≤ 2 sup|q|.

Since b > 0 and d = 0, we conclude that c = α for every α ∈ R with Kα(a, b) = ∅.Hence, the function α → d/(α − c) is analytic in (α,α).

Now we observe that b is not Φ-cohomologous to a constant ρ ∈ R. Otherwisethe function a would be Φ-cohomologous to cb + d = (c + d/ρ)b, which yieldsa contradiction. Hence, one can apply Lemma 9.1 to the pair of functions (b,1) toconclude that the spectrum F(b,1) is analytic in the nonempty interval (κ, κ), where

κ = inf

{∫

Λ

b dμ : μ ∈ M

}and κ = sup

{∫

Λ

b dμ : μ ∈ M

}.


Since b > 0, we have κ > 0. The spectrum F(a,b) is the composition of the analyticfunctions α → d/(α − c) and F(b,1), and thus it is also analytic. Moreover,

(α,α) ={

(c + d/κ, c + d/κ) when d > 0,

(c + d/κ, c + d/κ) when d < 0.


In the special case when b = 1 the statement in Theorem 9.2 was first establishedin [12] (using a different method).

Now we show that most Hölder continuous functions satisfy the second alterna-tive in Theorem 9.2. Let Cγ (Λ) be the space of Hölder continuous functions in Λ

with Hölder exponent γ ∈ (0,1) equipped with the norm in (8.25). We denote byC

γ+(Λ) the set of all positive functions in Cγ (Λ).


such that Φ|Λ is topologically transitive. For each γ ∈ (0,1), the set of all functions(a, b) ∈ Cγ (Λ) × C

γ+(Λ) such that a is not Φ-cohomologous to a multiple of b is

open and dense in Cγ (Λ) × Cγ+(Λ).

Proof Let H = Cγ (Λ) × Cγ+(Λ). Let also G ⊂ H be the set of all pairs (a, b) ∈ H

such that a is not Φ-cohomologous to a multiple of b. Take (a, b) ∈ H \ G and letΓi be distinct periodic orbits of points xi = ϕTi

(xi) for i = 0,1. We write

〈g〉i = 1

Ti

∫ Ti

0g(ϕt (xi)) dt

for each continuous function g : Λ → R and i = 0,1. Consider a function h ∈Cγ (Λ) such that h|Γ0 = 〈b〉0 and h|Γ1 = 〈b〉1 + 1. This is always possible becauseΓ0 and Γ1 are closed and disjoint. Now we consider the pair of functions

(a, b) = (a, b) + (εh,0) ∈ H

for some constant ε > 0. For each c ∈R, we have

a − cb = a − cb + (c − c)b + εh.

Thus, if a − cb is Φ-cohomologous to zero, then

0 = 〈a − cb〉0 = (c − c + ε)〈b〉0

and

0 = 〈a − cb〉1 = (c − c + ε)〈b〉1 + ε.

Since 〈b〉0 ≥ minb > 0, we obtain c − c + ε = 0. But this is impossible, in view ofthe second identity. This contradiction implies that (a, b) ∈ G. Since ε is arbitrary,

9.3 Examples 119

the pair of functions (a, b) can be arbitrarily approximated in H by pairs in G, andthus G is dense in H .

Now we show that G is open. Let (a, b) ∈ G. Since b > 0, there exists a uniquec = c(a, b) ∈ R such that PΦ(a − cb) = PΦ(0). By Livschitz’s theorem, there alsoexists a periodic orbit Γ0 such that 〈a −cb〉0 = 0. Take ε ∈ (0,minb/2) and (a, b) ∈H such that

‖a − a‖γ + ‖b − b‖γ < ε.

We have

|PΦ(a − cb) − PΦ(0)| ≤ ‖a − a − c(b − b)‖γ < (1 + |c|)ε. (9.11)

Now let c ∈ R be the unique real number such that PΦ(a− cb) = PΦ(0). We observethat if a is Φ-cohomologous to a multiple of b, then a is Φ-cohomologous to cb andto no other multiple of b. Since b > minb/2 > 0, it follows from (9.11) that

|c − c| ≤ 1

min b|PΦ(a − cb) − PΦ(a − cb)| < 2(1 + |c|)ε

minb.

Therefore,

|〈a − cb〉0| ≥ |〈a − cb〉0| − |〈a − a − (cb − cb)〉0|≥ |〈a − cb〉0| − ‖a − a‖γ − |c − c| · ‖b‖γ − |c| · ‖b − b‖γ

≥ |〈a − cb〉0| −(

1 + 2(1 + |c|)(‖b‖γ + ε)

minb+ |c|

)ε > 0,

provided that ε is sufficiently small (possibly depending on a and b). This impliesthat a is not Φ-cohomologous to cb. Hence, the ball of radius ε centered at (a, b) iscontained in G. This shows that the set G is open. �

Combining Theorems 9.1, 9.2 and 9.3 we readily obtain the following result,whose formulation has the advantage of not using the notion of cohomology.

Theorem 9.4 ([15]) Let Φ be a C1 flow with a locally maximal hyperbolic setΛ such that Φ|Λ is topologically mixing. Given γ ∈ (0,1), for (a, b) ∈ Cγ (Λ) ×C

γ+(Λ) in an open and dense set, the entropy spectrum F is analytic in the nonempty

interval (α,α) and satisfies identities (9.2) and (9.3) for α ∈ (α,α).

9.3 Examples

This section describes some applications of Theorems 9.1 and 9.2 to various spectra.In particular, we consider multifractal spectra for the local entropies, multifractalspectra for the Lyapunov exponents, and the particular case of suspension flows.


9.3.1 Multifractal Spectra for the Local Entropies

Let Φ be a continuous flow, let Λ be a Φ-invariant set, and let ν be a Φ-invariantprobability measure on Λ.

Definition 9.2 For each x ∈ Λ, we define the lower and upper ν-local entropiesof Φ at x respectively by

hν(Φ,x) = limε→0

lim inft→∞ −1

tlogν(B(x, t, ε))

and

hν(Φ,x) = limε→0

lim supt→∞

−1

tlogν(B(x, t, ε)),

with B(x, t, ε) as in (4.1).

Whenever hν(Φ,x) = hν(Φ,x), the common value is denoted by hν(Φ,x) and iscalled the ν-local entropy of Φ at x. By the Shannon–McMillan–Breiman theorem,the ν-local entropy of Φ is well defined ν-almost everywhere. In addition, if ν isergodic, then hν(Φ,x) = hν(Φ) for ν-almost every x ∈ Λ.

Definition 9.3 The entropy spectrum for the local entropies of ν is defined by

H(α) = h(Φ|Khα),

where

Khα = {

x ∈ Λ : hν(Φ,x) = hν(Φ,x) = α}.

Now let Λ be a locally maximal hyperbolic set for Φ . In this case we have

Khα =

{x ∈ Λ : lim

t→∞−1

tlogν(B(x, t, ε)) = α

}

for any sufficiently small ε > 0. Moreover, there exists a unique measure mE

of maximal entropy, that is, a Φ-invariant probability measure on Λ such thathmE

(Φ) = h(Φ). We write

αh = inf

{−∫

Λ

a dμ : μ ∈ M

}

and

αh = sup

{−∫

Λ

a dμ : μ ∈M

}.

The following result gives a conditional variational principle for the spectrum H.

9.3 Examples 121


such that Φ|Λ is topologically mixing and let ν be an equilibrium measure for aHölder continuous function a : Λ → R such that PΦ(a) = 0. Then the followingproperties hold:

1. if α ∈ [αh,αh], then Khα = ∅;

2. if α ∈ (αh,αh), then Khα = ∅ and

H(α) = max

{hμ(Φ) : −

∫

Λ

a dμ = α and μ ∈M

}

= min{PΦ(qa) + qα : q ∈R};3. if ν = mE , that is, a is Φ-cohomologous to zero, then αh = αh = c and Kh

c = Λ;4. if ν = mE , that is, a is not Φ-cohomologous to zero, then αh < αh and the func-

tion H is analytic in the interval (αh,αh).

Proof The result follows from Theorems 9.1 and 9.2 taking b = −1.

9.3.2 Multifractal Spectra for the Lyapunov Exponents

In this section we consider the multifractal spectrum for the Lyapunov exponents.Let Φ be a C1 flow with a locally maximal hyperbolic set Λ such that Φ|Λ isconformal (see Definition 5.1).

Let Zs and Zu be respectively the sets of points x ∈ Λ such that each of the limits


1

tlog‖dxϕt |Es(x)‖ and λu(x) = lim

t→+∞1

tlog‖dxϕt |Eu(x)‖

exists. As in Sect. 6.1, for any Φ-invariant probability measure ν in Λ we have

ν(Λ \ Zs) = ν(Λ \ Zu) = 0.

Definition 9.4 The stable and unstable entropy spectra for the Lyapunov exponentsare defined respectively by

Ls(α) = h(Φ|Ksα) and Lu(α) = h(Φ|Ku

α),

where

Ksα = {

x ∈ Zs : λs(x) = α}

and Kuα = {

x ∈ Zu : λu(x) = α}.

The following result gives a conditional variational principle for the spectrum Ls .We write

αs = inf

{∫

Λ

ζs dμ : μ ∈M

}


and

αs = sup

{∫

Λ

ζs dμ : μ ∈M

}.

Theorem 9.6 [15] Let Φ be a C1+δ flow with a locally maximal hyperbolic set Λ

such that Φ|Λ is conformal and topologically mixing. Then the following propertieshold:

1. if α ∈ [αs,αs], then Ksα = ∅;

2. if α ∈ (αs,αs), then Ksα = ∅ and

Ls(α) = max

{hμ(Φ) :

∫

Λ

ζs dμ = α and μ ∈M

}

= min{PΦ(qζs) − qα : q ∈ R};3. if ζs is Φ-cohomologous to zero, then αs = αs = c and Ks

c = Λ;4. if ζs is not Φ-cohomologous to zero, then αs < αs and the function Ls is analytic

in the interval (αs,αs).

Proof Since the stable and unstable distributions x → Es(x) and x → Eu(x) areHölder continuous and the flow Φ is of class C1+δ , the functions ζs and ζu are alsoHölder continuous in Λ. Hence, the result follows from Theorems 9.1 and 9.2 takinga = ζs and b = 1. �

In [82], Pesin and Sadovskaya obtained a multifractal analysis of the spec-trum Ls . One can also formulate corresponding statements for the spectrum Lu.

9.3.3 Suspension Flows

Let Ψ be a suspension flow in Y , over a homeomorphism T : X → X of the compactmetric space X, and let μ be a T -invariant probability measure on X.

Given continuous functions a, b : Y → R with b > 0 and α ∈R, let

Kα ={

x ∈ Y : limt→∞

∫ t

0 a(ψs(x)) ds∫ t

0 b(ψs(x)) ds= α

}

and consider again the spectrum F in (9.1). It follows from Proposition 7.2 that theset Kα is composed of the points (x, s) ∈ Y such that

limm→∞

∑mi=0 Ia(T

i(x))∑m

i=0 Ib(T i(x))= α

9.3 Examples 123

and s ∈ [0, τ (x)]. Let also

α = inf

{∫Y

a dν∫Y

b dν: ν ∈MΨ

}= inf

{∫X

Ia dμ∫X

Ib dμ: μ ∈ MT

}

and

α = sup

{∫Y

a dν∫Y

b dν: ν ∈MΨ

}= inf

{∫X

Ia dμ∫X

Ib dμ: μ ∈ MT

},

where MΨ (respectively MT ) is the set of all Ψ -invariant probability measures on Y

(respectively of all T -invariant probability measures on X).The following result gives a conditional variational principle for the spectrum F

in the special case when T is a topological Markov chain.

Theorem 9.7 ([15]) Let Ψ be a suspension flow over a topologically mixing two-sided topological Markov chain and let a, b : Y → R be Hölder continuous func-tions with b > 0. Then the following properties hold:

1. if α ∈ [α,α], then Kα = ∅;2. if α ∈ (α,α), then Kα = ∅ and

F(α) = max

{hμ(T )∫X

τ dμ:∫X

Ia dμ∫X

Ib dμ= α and μ ∈MT

}

= min

{

supμ∈MT

hμ(T ) + ∫X

Iqa−qαb dμ∫X

τ dμ: q ∈R

}

;

3. if a is Ψ -cohomologous to cb for some c ∈ R, that is, Ia is T -cohomologous tocIb for some c ∈ R, then α = α = c and Kc = Λ;

4. if a is not Ψ -cohomologous to a multiple of b, that is, Ia is not T -cohomologousto a multiple of Ib , then α < α and the function F is analytic in the inter-val (α,α).

Proof It follows from (4.9) that∫Y

a dν∫Y

b dν=

∫X

Ia dμ∫X

Ib dμ. (9.12)

On the other hand, by Abramov’s entropy formula, we have

hν(Ψ ) = hμ(T )∫X

τ dμ. (9.13)

By (9.12) and (9.13), using similar arguments to those in the proof of Theorem 9.1we obtain the first and second properties in the theorem. The remaining propertiesfollow from Theorem 2.1, using similar arguments to those in the proof of Theo-rem 9.2. �


9.4 Multidimensional Spectra

In this section we describe a multidimensional version of the conditional variationalprinciple in Theorem 9.1. This can be seen as a motivation for Chap. 10 where weestablish much more general results. Instead of considering Birkhoff averages (orratios of Birkhoff averages) we consider vectors of ratios of Birkhoff averages.

Let Φ be a flow and let Λ be a Φ-invariant set. Let also a1, . . . , ad : Λ → R andb1, . . . , bd : Λ → R be continuous functions with bi > 0 for i = 1, . . . , d . We write

A ={(∫

Λa1 dμ

∫Λ

b1 dμ, . . . ,

∫Λ

ad dμ∫Λ

bd dμ

): μ ∈ M

},

where M is the set of all Φ-invariant probability measures on Λ, and we define

Kα ={

x ∈ Λ : limt→∞

(∫ t

0 a1(ϕs(x)) ds∫ t

0 b1(ϕs(x)) ds, . . . ,

∫ t

0 ad(ϕs(x)) ds∫ t

0 bd(ϕs(x)) ds

)

= α

}

for each α = (α1, . . . , αd) ∈ Rd .

Definition 9.5 The function F = F(a,b) defined by

F(α) = h(Φ|Kα)

is called the entropy spectrum for the pair (a, b) = (a1, . . . , ad, b1, . . . , bd).

The following result gives a conditional variational principle for the spectrum F.It is a multidimensional version of Theorem 9.1.


such that Φ|Λ is topologically mixing and let a1, . . . , ad and b1, . . . , bd be Höldercontinuous functions with bi > 0 for each i = 1, . . . , d . Then the following proper-ties hold:

1. if α ∈ A, then Kα = ∅;2. if α ∈ intA, then Kα = ∅ and

F(α) = max

{hμ(Φ) :

(∫Λ

a1 dμ∫Λ

b1 dμ, . . . ,

∫Λ

ad dμ∫Λ

bd dμ

)= α and μ ∈M

}

= min

{PΦ

( d∑

i=1

(qiai − qiαibi)

): (q1, . . . , qd) ∈R

d

}.

Proof The first property can be obtained in a similar manner to that in the proof ofthe first property in Theorem 9.1.

9.4 Multidimensional Spectra 125

For the second property, we briefly describe the changes that are required inthe proof of Theorem 9.1 when d > 1. For each α = (α1, . . . , αd) ∈ intA and q =(q1, . . . , qd) ∈ R

d \ {0}, take measures νq− and μ

q+ such that

d∑

i=1

qi

(∫

Λ

ai dνq− − αi

∫

Λ

bi dνq−)

< 0 <

d∑

i=1

qi

(∫

Λ

ai dνq+ − αi

∫

Λ

bi dνq+)

.

These play the role of the measures ν− and ν+ in the proof of Theorem 9.1 andsimilar arguments can be used to show that

lim inf‖q‖→∞PΦ

(d∑

i=1

(qiai − qiαibi)

)

= +∞,

where ‖·‖ is any norm in Rn. This implies that the function

F : (q1, . . . , qd) → PΦ

(d∑

i=1

(qiai − qiαibi)

)

attains its infimum at some point q(α) ∈ Rd , and hence ∂qF (q(α)) = 0. This prop-

erty allows one to use essentially the same arguments as in the proof of Theorem 9.1,replacing a and b by the vectors (a1, . . . , ad) and (b1, . . . , bd), to obtain the desiredresult. �

Theorem 9.8 is a particular case of Theorem 10.1 and for this reason we haveonly sketched the proof. Theorem 10.1 gives a conditional variational principle formultidimensional BS-dimension spectra (Theorem 9.8 considers the particular caseof multidimensional entropy spectra).

Chapter 10Multidimensional Spectra

In this chapter we present a multidimensional multifractal analysis for hyperbolicflows. More precisely, we consider multifractal spectra associated to multidimen-sional parameters, obtained from computing the entropy of the level sets associatedto several Birkhoff averages. These spectra exhibit several new phenomena that areabsent in 1-dimensional multifractal analysis. We also consider the more generalclass of flows with upper semicontinuous entropy. In this chapter the multifractalanalysis is obtained from a conditional variational principle for the topological en-tropy of the level sets.


In this section we consider a multidimensional multifractal spectrum for ratios ofBirkhoff averages of a flow and we establish a corresponding conditional variationalprinciple.

Let Φ = {ϕt }t∈R be a continuous flow in a compact metric space X. We considervectors of functions (A,B) ∈ C(X)d ×C(X)d for some d ∈ N, say with components

A = (a1, . . . , ad) and B = (b1, . . . , bd),

with bi > 0 for i = 1, . . . , d . We equip Rd with the norm ‖α‖ = |α1|+· · ·+|αd | and

C(X)d with the corresponding supremum norm. For each α = (α1, . . . , αd) ∈ Rd ,

let

Kα = Kα(A,B) =d⋂

i=1

{

x ∈ X : limt→∞

∫ t

0 ai(ϕs(x))ds∫ t

0 bi(ϕs(x))ds= αi

}

. (10.1)

Definition 10.1 Given a continuous function u : X → R+, the BS-dimension spec-

trum Fu : Rd →R of the pair (A,B) (with respect to u and Φ) is defined by

Fu(α) = dimu Kα(A,B). (10.2)


127

http://dx.doi.org/10.1007/978-3-319-00548-5_10

128 10 Multidimensional Spectra

We also consider the function P = PA,B : M →Rd defined by

P(μ) =(∫

Xa1 dμ

∫X

b1 dμ, . . . ,

∫X

ad dμ∫X

bd dμ

),

where M is the set of all Φ-invariant probability measures on X. For each α =(α1, . . . , αd) and β = (β1, . . . , βd) in R

d , we write

α ∗ β = (α1β1, . . . , αdβd) ∈ Rd and 〈α,β〉 =

d∑

i=1

αiβi ∈ R.

The following result gives a conditional variational principle for the spectrum Fu.

Theorem 10.1 ([6]) Let Φ be a continuous flow in a compact metric space X suchthat the map μ → hμ(Φ) is upper semicontinuous, and consider functions (A,B) ∈C(X)d × C(X)d such that

span{a1, b1, . . . , ad, bd, u} ⊂ D(X).

If α ∈ intP(M), then Kα = ∅ and the following properties hold:

1.

Fu(α) = max

{hμ(Φ)∫X

udμ: μ ∈M and P(μ) = α

}; (10.3)

2.

Fu(α) = min{Tu(α, q) : q ∈ Rd}, (10.4)

where Tu(α, q) is the unique real number such that

PΦ

(〈q,A − α ∗ B〉 − Tu(α, q)u)= 0; (10.5)

3. there exists an ergodic measure μα ∈ M such that P(μα) = α, μα(Kα) = 1 anddimu μα = Fu(α).

Moreover, if α ∈ P(M), then Kα = ∅.

Proof The proof follows arguments of Barreira, Saussol and Schmeling in [16] inthe case of discrete time. We use the notation μ(ψ) = ∫

Xψ dμ.

Take α ∈ Rd such that Kα = ∅. Given x ∈ Kα , we define a sequence (μn)n∈N of

probability measures on X by

μn(a) = 1

n

∫ n

0a(ϕs(x)) ds


for each a ∈ C(X). Since M is compact, this sequence has at least one accumulationpoint μ ∈M. Therefore,

α =(

limt→+∞

∫ t

0 a1(ϕs(x)) ds∫ t

0 b1(ϕs(x)) ds, . . . , lim

t→+∞

∫ t

0 ad(ϕs(x)) ds∫ t

0 bd(ϕs(x)) ds

)

=(

limn→+∞

μn(a1)

μn(b1), . . . , lim

n→+∞μn(ad)

μn(bd)

)

=(

μ(a1)

μ(b1), . . . ,

μ(ad)

μ(bd)

)= P(μ) ∈ P(M).

Now let α ∈ intP(M). The existence of the maximum in (10.3) is a consequenceof the upper semicontinuity of the map μ → hμ(Φ)/

∫X

udμ, together with thecompactness of M and the continuity of P. For each q ∈R

d , let

ϕq,α = 〈q,A − α ∗ B〉 −Fu(α)u and Fα(q) = PΦ(ϕq,α).

Let also r > 0 be the distance from α to Rd \P(M) and take q such that

‖q‖ ≥ dimu X · supu + Fα(0)

r mini infbi

= R.

For each λ ∈ (0,1) and β = (β1, . . . , βd) ∈ Rd with

βi = αi + 1

dλr sgnqi,

we have

‖β − α‖ =d∑

i=1

|βi − αi | =d∑

i=1

1

dλr sgn |qi | = λr < r.

Hence, β ∈ P(M) and there exists a μ ∈ M such that μ(A − β ∗ B) = 0. Therefore,⟨q,μ(A − α ∗ B)

⟩= ⟨q,μ((β − α) ∗ B)

⟩

=d∑

i=1

qiμ((βi − αi) ∗ bi)

=d∑

i=1

1

dλrqi sgnqi

∫

X

bi dμ

= 1

dλr

d∑

i=1

|qi |∫

X

bi dμ

≥ λr ‖q‖mini

infbi.


Since hμ(Φ) ≥ 0, it follows from Proposition 4.2 that

Fα(q) ≥ hμ(Φ) + μ(ϕq,α)

= hμ(Φ) + 〈q,μ(A − α ∗ B)〉 − Fu(α)μ(u)

≥ ‖q‖λr mini

infbi − dimu X · supu

≥ λ[dimu X · supu + Fα(0)

]− dimu X · supu.

Letting λ → 1, we obtain Fα(q) ≥ Fα(0) for every q ∈ Rd such that ‖q‖ ≥ R. By

Proposition 4.3, the function F is of class C1 and hence it reaches a minimum at apoint q = q(α) with ‖q(α)‖ ≤ R. In particular, ∂qFα(q(α)) = 0. By (4.6), we have

μα(A − α ∗ B) = ∂qFα(q(α)) = 0,

where μα is the equilibrium measure for ϕq,α . This shows that P(μα) = α. More-over,

Fα(q(α)) = hμα (Φ) − Fu(α)

∫

X

udμα. (10.6)

Now take x ∈ Kα . For i = 1, . . . , d , we have

limt→∞

∫ t

0 ai(ϕs(x)) ds∫ t

0 bi(ϕs(x)) ds= αi.

Since bi > 0, for each δ > 0 there exists a τ > 0 such that

∣∣∣∣∣

∫ t

0 ai(ϕs(x)) ds∫ t

0 bi(ϕs(x)) ds− αi

∣∣∣∣∣<

δ

dM

for all t > τ , where

M = maxi∈{1,...,d}

maxx∈X

bi(x).

We define

At(x) =∫ t

0A(ϕs(x)) ds and Bt(x) =

∫ t

0B(ϕs(x)) ds, (10.7)

and we let

Lδ,τ = {x ∈ X : ‖At(x) − α ∗ Bt(x)‖ < δt for all t ≥ τ

}.


Then

‖At(x) − α ∗ Bt(x)‖ =d∑

i=1

∣∣∣∣

∫ t

0ai(ϕs(x)) ds − αi

∫ t

0bi(ϕs(x)) ds

∣∣∣∣

<δ

dM

d∑

i=1

∫ t

0bi(ϕs(x)) ds < δt,

and hence,

x ∈ Lδ,τ ⊆⋃

τ∈RLδ,τ

for δ > 0. Therefore,

Kα ⊆⋂

δ>0

⋃

τ∈RLδ,τ .

Since X is compact, each function ai is uniformly continuous. Hence, there ex-ists an ε > 0 such that if (x, t) ∈ X × [0,∞) and y, z ∈ B(x, t, ε), and thus alsod(ϕs(y),ϕs(z)) < 2ε, then

|ai(ϕs(z)) − ai(ϕs(y))| < δ/d whenever 0 ≤ s ≤ t.

Let

A(x, t, ε) = (a1(x, t, ε), . . . , ad(x, t, ε))

and take y ∈ B(x, t, ε). We obtain

‖A(x, t, ε) − At(y)‖ =d∑

i=1

∣∣∣∣ai(x, t, ε) −∫ t

0ai(ϕs(y)) ds

∣∣∣∣

≤ d sup

{∫ t

0|ai(ϕs(z)) − ai(ϕs(y))|ds : z ∈ B(x, t, ε)

}

≤ d sup

{∫ t

0

δ

dds : z ∈ B(x, t, ε)

}≤ δt,

and analogously,

‖B(x, t, ε) − Bt(y)‖ ≤ δt.


Now take q ∈ Rd . Given (x, t) ∈ X × [τ,∞) with B(x, t, ε) ∩ Lδ,τ = ∅ and

y ∈ B(x, t, ε) ∩ Lδ,τ , we have

−〈q,A − α ∗ B〉(x, t, ε) ≤ |〈q,A − α ∗ B〉(x, t, ε)|≤ ‖q‖ · ‖A(x, t, ε) − α ∗ B(x, t, ε)‖≤ ‖q‖ · ‖A(x, t, ε) − At(y)‖

+ ‖q‖ · ‖α ∗ Bt(y) − α ∗ B(x, t, ε)‖+ ‖q‖ · ‖At(y) − α ∗ Bt(y)‖

≤ ‖q‖(δt + ‖α‖δt + δt) = cδt,

where c = (2 + ‖α‖)‖q‖. Hence,

exp(−Fu(α)u(x, t, ε) − βt

)= exp(ϕq,α(x, t, ε) − 〈q,A − α ∗ B〉(x, t, ε) − βt

)

≤ exp(ϕq,α(x, t, ε) − (β − cδ)t

)

for β ∈R. Let T ≥ τ and consider a finite or countable family Γ = {(xi, ti )}i∈I suchthat xi ∈ X and ti ≥ T for i ∈ I , Lδ,τ ⊂⋃

i∈I B(xi, ti , ε), and with the property thatthere exists no pair (xi, ti ) such that B(xi, ti , ε) ∩ Lδ,τ = ∅. Then

∑

(x,t)∈Γ

exp(−Fu(α)u(x, t, ε) − βt) ≤∑

(x,t)∈Γ

exp(ϕq,α(x, t, ε) − (β − cδ)t).

Taking the infimum over Γ and letting T → ∞, we obtain

M(Lδ,τ ,−Fu(α)u,β, ε

)≤ M(Lδ,τ , ϕq,α, β − cδ, ε

).

Letting ε → 0 yields the inequality

PΦ|Lδ,τ (−Fu(α)u) ≤ PΦ|Lδ,τ (ϕq,α) + cδ

for δ > 0 and q ∈ Rd . By Proposition 4.5 and the properties of the topological pres-

sure, we have

0 = PΦ|Kα (−Fu(α)u)

≤ PΦ|⋃τ∈R Lδ,τ(−Fu(α)u)

= supτ>0

PΦ|Lδ,τ (−Fu(α)u)

≤ PΦ|Lδ,τ (ϕq,α) + cδ ≤ Fα(q) + cδ

for δ > 0 and q ∈ Rd . Since δ is arbitrary, we obtain Fα(q) ≥ 0. By Proposition 4.3

and (10.6), the measure μα is ergodic and

dimu μα = hμα (Φ)∫X

udμα

≥ Fu(α).


On the other hand, since μα(A − α ∗ B) = 0, it follows from Birkhoff’s ergodictheorem that μα(Kα) = 1. This implies that

Fu(α) = dimu Kα = limε→0

dimu,ε Kα

≥ limε→0

dimu,ε μα = dimu μα,

and hence dimu μα = Fu(α). Therefore,

min{Fα(q) : q ∈ Rd} = Fα(q(α)) = hμα (Φ) − Fu(α)

∫

X

udμα

= hμα (Φ) − hμα (Φ)∫X

udμα

∫

X

udμα = 0.

Now take μ ∈ M such that P(μ) = α. Then μ(〈q,A − α ∗ B〉) = 0 and by Proposi-tion 4.2, we have

0 = min{Fα(q) : q ∈ Rd}

≥ infq∈Rd

{hμ(Φ) + μ

(〈q,A − α ∗ B〉 −Fu(α)u)}

≥ infq∈Rd

{hμ(Φ) −Fu(α)

∫

X

udμ

}

= hμ(Φ) − Fu(α)

∫

X

udμ.

Therefore, hμ(Φ)/∫X

udμ ≤ Fu(α), with equality when μ = μα . This establishesproperties 1 and 3 in the theorem.

Furthermore, since Fα(q(α)) = 0, we have

Fu(α) = Tu(α, q(α)) ≥ inf{Tu(α, q) : q ∈R

d}.

On the other hand,

Fα(q) ≥ 0 = PΦ

(〈q,A − α ∗ B〉 − Tu(α, q)u),

and hence,

Fu(α) ≤ inf{Tu(α, q) : q ∈ R

d}.


As a consequence of Proposition 4.4, the conditional variational principle in The-orem 10.1 applies in particular to a topologically mixing flow on a locally maximalhyperbolic set. In this context, the statement in Theorem 10.1 was first establishedby Barreira and Saussol in [15] in the case of the entropy (see Theorem 9.8).


It also follows from the proof of Theorem 10.1 that μα can be chosen to be anyequilibrium measure for the function 〈q(α),A−α ∗B〉−Fu(α)u, where q(α) ∈ R

d

is any vector such that

PΦ

(〈q(α),A − α ∗ B〉 −Fu(α)u)= 0.

We note that q(α) and μα need not be unique. The function Tu is implicitly definedby (10.5) and thus, by Proposition 4.3, the function

(p,α, q) → PΦ

(〈q,A − α ∗ B〉 − pu)

is of class C1. Moreover,

∂

∂pPΦ(〈q,A − α ∗ B〉 − pu)

∣∣∣(p,q)=(Tu(α,q),q)

= −∫

X

udμq < 0,

where μq is the equilibrium measure for 〈q,A−α ∗B〉−Tu(α, q)u. It follows fromthe Implicit function theorem that Tu is of class C1 in R

d ×Rd . This implies that for

each α the minimum in (10.4) is attained at a point q ∈Rd such that ∂qTu(α, q) = 0.

10.2 Finer Structure

In this section we study in greater detail the structure of a class of level sets Kα

in (10.1).Let Φ be a continuous flow in a compact metric space X. Take A,B ∈ C(X)d

and a continuous function u : X →R+. We define

ut (x) =∫ t

0u(ϕs(x)) ds

for each t > 0. Given a continuous function F : Rd ×Rd → R

d and α ∈Rd , let

Lα ={x ∈ X : lim

t→∞F

(At(x)

ut (x),Bt (x)

ut (x)

)= α

},

with At(x) and Bt(x) as in (10.7). When F(X,Y ) = X ∗ Y−1 this is simply the setKα in (10.1). We also consider the multifractal spectrum Gu defined by

Gu(α) = dimu Lα

for each α ∈Rd .

We want to establish a relation between the BS-dimension of a set Lα and theBS-dimension of the sets

Kβ,γ ={x ∈ X : lim

t→∞

(At(x)

ut (x),Bt (x)

ut (x)

)= (β, γ )

},

10.2 Finer Structure 135

with β,γ ∈Rd . We write

Hu(β, γ ) = dimu Kβ,γ .

For each q ∈ R2d , let Su(q) be the unique real number such that

PΦ

(〈q, (A,B)〉 − Su(q)u)= 0

and let μq be the equilibrium measure for 〈q, (A,B)〉 − Su(q)u (this measure willbe unique in our context).

Applying Theorem 10.1 to the spectrum Hu, we obtain the following result.

Theorem 10.2 Let Φ be a continuous flow in a compact metric space X suchthat the map μ → hμ(Φ) is upper semicontinuous. If span{a1, b1, . . . , ad, bd, u} ⊂D(X), then

Hu(∂qSu(q)) = Su(q) − 〈q, ∂qSu(q)〉and μq(K∂qSu(q)) = 1 for every q ∈R

2d .

Now we describe a general relation between the spectra Gu and Hu.

Proposition 10.1 ([6]) Let Φ be a continuous flow in a compact metric space X

and let F : Rd ×Rd → R

d be a continuous function. Then

Gu(α) ≥ sup{Hu(β, γ ) : (β, γ ) ∈ F−1(α)

}

for every α in the image of F .

Proof Take (β, γ ) ∈Rd ×R

d and x ∈ Kβ,γ . By the continuity of F , we have

limt→∞F

(At(x)

ut (x),Bt (x)

ut (x)

)= F

(lim

t→∞At(x)

ut (x), limt→∞

Bt(x)

ut (x)

)= F(β,γ ).

This implies that⋃

(β,γ )∈F−1(α)

Kβ,γ ⊆ Lα.

Since Kβ,γ ⊂ Lα for every (β, γ ) ∈ F−1(α), we have

dimu Kβ,γ ≤ dimu Lα.

This yields the desired inequality. �

Barreira, Saussol and Schmeling in [16] made a corresponding study in the caseof discrete time.


10.3 Hyperbolic Flows: Analyticity of the Spectrum

In this section we consider the particular case of hyperbolic flows and we establishthe analyticity of the spectrum Fu in (10.2). The proof is based on property 2 ofTheorem 10.1 saying that the spectrum is equal to the minimum of a certain functiondefined implicitly in terms of the topological pressure.

Theorem 10.3 ([6]) Let Φ be a C1 flow with a compact locally maximal hyperbolicset Λ such that Φ|Λ is topologically mixing. If the functions ai, bi : Λ → R for i =1, . . . , d and u : Λ → R

+ are Hölder continuous, then Fu is analytic in intP(M).

Proof By Proposition 4.4, the map μ → hμ(Φ) is upper semicontinuous in M.Thus, by Theorem 10.1, we have

Fu(α) = min{Tu(α, q) : q ∈ R

d},

where Tu(α, q) is the unique real number satisfying (10.5). Hence,

0 = ∂qPΦ(〈q,A − α ∗ B〉 − Tu(α, q)u)

= ∂qPΦ(〈q,A − α ∗ B〉 − pu)|p=Tu(α,q)

+ ∂pPΦ(〈q,A − α ∗ B〉 − pu)|p=Tu(α,q)∂qTu(α, q).

Now take q(α) ∈ Rd such that Fu(α) = Tu(α, q(α)). Since Tu is of class C1 (see

the discussion at the end of Sect. 10.1), we have ∂qTu(α, q(α)) = 0 and thus,

∂qPΦ

(〈q,A − α ∗ B〉 − pu)= 0

for q = q(α) and p = Tu(α, q(α)). Hence, (α, q,p) = (α, q(α),Fu(α)) is a solutionof the system

{PΦ

(〈q,A − α ∗ B〉 − pu)= 0,

∂qPΦ

(〈q,A − α ∗ B〉 − pu)= 0.

(10.8)

By Proposition 4.4, the function t → PΦ(a + tb) is analytic. We want to showthat

det

(∂[PΦ

(〈q,A − α ∗ B〉 − pu), ∂qPΦ

(〈q,A − α ∗ B〉 − pu)]

∂(q,p)

)

= 0 (10.9)

for (α, q,p) = (α, q(α),Fu(α)). The first line of the matrix in (10.9) is(

∂q

(PΦ

(〈q,A − α ∗ B〉 − pu))

,−∫

Λ

udμα

),

where μα ∈ M is the equilibrium measure for⟨q(α),A − α ∗ B

⟩ − Fu(α)u. Nowwe observe that in the last d equations of system (10.8) all values of the first line

10.3 Hyperbolic Flows: Analyticity of the Spectrum 137

vanish at (α, q(α),Fu(α)), except for the last one, which is negative. Therefore, thedeterminant in (10.9) is nonzero provided that

det[∂2qPΦ

(〈q,A − α ∗ B〉 − pu)] = 0 (10.10)

for (α, q,p) = (α, q(α),Fu(α)).

Lemma 10.1 The matrix

∂2qPΦ

(〈q,A − α ∗ B〉 − pu)

(10.11)

is positive definite for every q ∈Rd , p ∈ R and α ∈ intP(M).

Proof of the lemma If the determinant of the matrix in (10.11) is zero, then thereexists a vector v ∈R

d \ {0} such that

v∗∂2qPΦ

(〈q,A − α ∗ B〉 − pu)v = 0,

where v∗ is the transpose of v. Then

∂2t PΦ

(〈q − tv,A − α ∗ B〉 − pu)|t=0 = 0

and by Proposition 4.4, the function 〈v,A−α∗B〉 is Φ-cohomologous to a constant,say c. Therefore,

∫

Λ

〈v,A − α ∗ B〉dμ =⟨∫

Λ

Adμ − α ∗∫

Λ

B dμ

⟩= cμ(Λ)

for μ ∈ M. Since α ∈ P(M), we obtain c = 0. Hence, the function 〈v,A − α ∗ B〉 isΦ-cohomologous to 0 and

PΦ(0) = PΦ(t〈v,A − α ∗ B〉) for t ∈ R.

Since α ∈ intP(M), there exist s = 0 and μs ∈ M such that sv + α ∈ P(M) and

∫

Λ

Adμs =∫

Λ

(sv + α) ∗ B dμs.

For each t ∈R, we obtain

PΦ(0) = PΦ(t〈sv,A − α ∗ B〉)

≥ hμs (Φ) + t

⟨sv, (sv + α − α) ∗

∫

Λ

B dμs

⟩

≥ ts2|v|2 infi∈{1,...,d} infbi.


But letting t → ∞, we find that this is impossible, and hence, the matrix in (10.11)has nonzero determinant. Now we show that it is positive definite. By the continuityof the map

v → v∗∂2qPΦ

(〈q,A − α ∗ B〉 − pu)v,

if there exist vectors v = (v1, . . . , vd) and w = (w1, . . . ,wd) in Rd \ {0} such that

v∗∂2qPΦ

(〈q,A − α ∗ B〉 − pu)v < 0

and

w∗∂2qPΦ

(〈q,A − α ∗ B〉 − pu)w > 0,

then one can find t1, . . . , td ∈ (0,1) such that

x = (t1v1 + (1 − t1)w1, . . . , tdvd + (1 − td )wd

) = 0

and

x∗∂2qPΦ

(〈q,A − α ∗ B〉 − pu)x = 0.

But it was shown above that this is impossible. Therefore, the matrix in (10.11) iseither positive definite or negative definite.

Let e1 be the first element of the canonical base of Rd . By Proposition 4.4, wehave

e∗1∂2

qPΦ

(〈q,A − α ∗ B〉 − pu)e1 = ∂2

∂q21

PΦ

(〈q,A − α ∗ B〉 − pu)≥ 0.

This shows that the matrix in (10.11) is positive definite. �

By Lemma 10.1, condition (10.10) holds. Thus, by the Implicit function theo-rem, system (10.8) defines q and p as analytic functions of α in a neighborhood of(α, q(α),Fu(α)). In particular, the spectrum Fu is analytic in intP(M). �

Chapter 11Dimension Spectra

In this chapter, for conformal flows with a hyperbolic set, we establish a condi-tional variational principle for the dimension spectra of Birkhoff averages. The mainnovelty in comparison to the former chapters is that we consider simultaneouslyBirkhoff averages into the future and into the past. The main difficulty is that eventhough the local product structure is bi-Lipschitz, the level sets of the Birkhoff av-erages are not compact. Our proof is based on the use of Markov systems and isinspired by earlier arguments in the case of discrete time.

11.1 Future and Past

In this section we consider Birkhoff averages both into the future and into the past,and we compute the Hausdorff dimension of the corresponding level sets on locallymaximal hyperbolic sets for a conformal flow.

Let Φ = (ϕt )t∈R be a C1 flow with a locally maximal hyperbolic set Λ such thatΦ|Λ is conformal. We denote by Cγ (Λ) the space of Hölder continuous functionsin Λ with Hölder exponent γ ∈ (0,1). Given d ∈ N, let F = Cγ (Λ)d × Cγ (Λ)d .Moreover, given functions (A±,B±) ∈ F , we write

A+ = (a+1 , . . . , a+

d ), B+ = (b+1 , . . . , b+

d ) (11.1)

and

A− = (a−1 , . . . , a−

d ), B− = (b−1 , . . . , b−

d ), (11.2)

and we assume that all components of A− and B− are positive functions. For eachα = (α1, . . . , αd) and β = (β1, . . . , βd) in R

d , let

K+α =

d⋂

i=1

{

x ∈ Λ : limt→+∞

∫ t

0 a+i (ϕs(x)) ds

∫ t

0 b+i (ϕs(x)) ds

= αi

}


139

http://dx.doi.org/10.1007/978-3-319-00548-5_11

140 11 Dimension Spectra

and

K−β =

d⋂

i=1

{

x ∈ Λ : limt→−∞

∫ t

0 a−i (ϕs(x)) ds

∫ t

0 b−i (ϕs(x)) ds

= βi

}

.

The following result expresses the dimensions of the level sets K+α and K−

β interms of the topological pressure.


such that Φ|Λ is conformal and topologically mixing and let (A±,B±) ∈ F . Foreach α,β ∈R

d , x+ ∈ K+α and x− ∈ K−

β , we have

dimH K+α = dimH (K+

α ∩ V u(x+)) + ts + 1

= dimζuK+α + ts + 1 (11.3)

and

dimH K−β = dimH (K−

β ∩ V s(x−)) + tu + 1

= dim−ζs K−β + tu + 1, (11.4)

with ts and tu as in (5.6).

Proof By (3.1) and the uniform continuity of a±i and b±

i in Λ, we have

Λ ∩ V s(x) ⊂ K+α for x ∈ K+

α ,

and thus,

Λ ∩⋃

t∈Rϕt (V

s(x)) ⊂ K+α

for x ∈ K+α , since the set K+

α is Φ-invariant.On the other hand, since Φ is conformal on Λ, it follows from results of Hassel-

blatt in [53] that the distributions x → Es(x) ⊕ E0(x) and x → Eu(x) ⊕ E0(x) areLipschitz. Therefore, on a sufficiently small open neighborhood of a point x ∈ K+

α

there exists a Lipschitz map with Lipschitz inverse from the set K+α to the product

⋃

t∈I

ϕt (Vs(x)) × V u(x),

where I is some open interval containing zero. Therefore,

dimH K+α = dimH

((K+

α ∩ V u(x)) ×(

Λ ∩⋃

t∈I

ϕt (Vs(x))

)). (11.5)

11.2 Conditional Variational Principle 141

On the other hand, by Theorem 5.1, we have

dimH

(Λ ∩

⋃

t∈I

ϕt (Vs(x))

)= dimB

(Λ ∩

⋃

t∈I

ϕt (Vs(x))

)= ts + 1. (11.6)

Since

dimH E + dimH F ≤ dimH (E × F) ≤ dimH E + dimBF

for any sets E,F ⊂ Rm (see for example [41]), it follows from (11.5) and (11.6)

that

dimH K+α = dimH (K+

α ∩ V u(x)) + ts + 1.

For the second equality in (11.3), we note that∫ t

0ζu(ϕs(x)) ds = log‖dxϕt |Eu(x)‖.

Since the distribution x → Eu(x) ⊕ E0(x) is Lipschitz and Φ is of class C1+δ , thefunction ζu is Hölder continuous and for each ε > 0 there exist constants c1, c2 > 0such that

c1 exp(−αζu(x, t, ε)) ≤ [diam

(B(x, t, ε) ∩ V u(x)

)]α ≤ c2 exp(−αζu(x, t, ε))

for every x ∈ Λ and t > 0. Hence, it follows from the definition of Hausdorff di-mension that

dimH (Z ∩ V u(x)) = dimζuZ

for every set Z ⊂ Λ. The second equality in (11.3) is obtained by taking Z = K+α .

The arguments for the set K−β and (11.4) are entirely analogous. �

11.2 Conditional Variational Principle

In this section we establish a conditional variational principle for the dimensionspectrum obtained from the level sets K+

α ∩ K−β .

Definition 11.1 The dimension spectrum D : Rd ×Rd →R associated to the func-

tions in (11.1) and (11.2) is defined by

D(α,β) = dimH (K+α ∩ K−

β ).

The following result is a conditional variational principle for the spectrum D.


such that Φ|Λ is conformal and topologically mixing and let (A±,B±) ∈ F . Thenthe following properties hold:


1. if

α ∈ intP+(M) and β ∈ intP−(M), (11.7)

then

D(α,β) = dimH K+α + dimH K−

β − dimH Λ

= max

{hμ(Φ)∫Λ

ζu dμ: μ ∈ M and P+(μ) = α

}

+ max

{hμ(Φ)

− ∫Λ

ζs dμ: μ ∈ M and P−(μ) = β

}+ 1;

2. the function D is analytic in intP+(M) × intP−(M).

Proof The proof is based on arguments of Barreira and Valls in [18], using alsoresults of Barreira and Saussol in [12]. We separate the argument into several steps.

Step 1. Construction of auxiliary measures

Consider a Markov system R1, . . . ,Rk for Φ on Λ and the associated symbolicdynamics (see Sect. 3.3). The following statement is a consequence of a constructiondescribed by Bowen in [28].

Lemma 11.1 For i, j = 1, . . . , d there exist Hölder continuous functions

aui , bu

i , du : Σ+A →R and as

j , bsj , d

s : Σ−A → R,

and continuous functions g+i , h+

i , g−j , h−

j , ρ± : ΣA →R such that

Ia+i

◦ π = aui ◦ π+ + g+

i − g+i ◦ σ,

Ib+i

◦ π = bui ◦ π+ + h+

i − h+i ◦ σ,

Iζu ◦ π = du ◦ π+ + ρ+ − ρ+ ◦ σ

and

Ia−j

◦ π = asj ◦ π− + g−

j − g−j ◦ σ−1,

Ib−j

◦ π = bsj ◦ π− + h−

j − h−j ◦ σ−1,

I−ζs ◦ π = ds ◦ π− + ρ− − ρ− ◦ σ−1.


We write

Au = (au

1 , . . . , aud

), Bu = (

bu1 , . . . , bu

d

)

and

As = (as

1, . . . , asd

), Bs = (

bs1, . . . , b

sd

).

Given q± ∈Rd , we define Hölder continuous functions U : Σ+

A →R and S : Σ−A →

R by

U = 〈q+,Au − α ∗ Bu〉 − d+du,

S = 〈q−,As − β ∗ Bs〉 − d−ds,(11.8)

where

d+ = dimH K+α − ts − 1 and d− = dimH K−

β − tu − 1. (11.9)

Now let μu be the equilibrium measure for U in Σ+A (with respect to σ+) and let μs

be the equilibrium measure for S in Σ−A (with respect to σ−). The following result

is a simple consequence of Theorem 10.1.

Lemma 11.2 For each α and β as in (11.7), there exist q± ∈Rd such that

Pσ+(U) = Pσ−(S) = 0,

∫

Σ+A

Au dμu = α ∗∫

Σ+A

Bu dμu

and∫

Σ−A

Au dμs = β ∗∫

Σ−A

Bs dμs.

Given x ∈ Z =⋃ki=1 Ri , let R(x) be a rectangle of the Markov system that con-

tains x. We define measures νu and νs on R(x) by

νu = μu ◦ π+ ◦ π−1 and νs = μs ◦ π− ◦ π−1,

taking the vectors q± given by Lemma 11.2. Finally, we define a measure ν on R(x)

by ν = νu × νs . Since μu and μs are Gibbs measures (see (7.8)), we have

ν(R(x)) = μu(C+

i0

)μs(C−

i0

)> 0,

with C+i0

and C−i0

as in (3.15) and (3.16).

Step 2. Lower pointwise dimension

Here and in the following steps we establish several properties of the measure ν.


Lemma 11.3 For ν-almost every x ∈ Z, we have

lim infr→0

logν(B(x, r))

log r≥ dimH K+

α + dimH K−β − dimH Λ − 1.

Proof of the lemma We follow arguments in the proof of Lemma 4 in [18]. By thevariational principle for the topological pressure applied to the functions U and S

in (11.8) together with Lemma 11.2, we obtain

hμu(σ+)∫Σ+

Adu dμu

= d+ andhμs (σ−)

∫Σ−

Ads dμs

= d−.

By the Shannon–McMillan–Breiman theorem and Birkhoff’s ergodic theorem,given ε > 0, for μs -almost every ω+ ∈ C+

i0and μu-almost every ω− ∈ C−

i0there

exists an s(ω) ∈ N, with ω+ = π+(ω) and ω− = π−(ω), such that

d+ − ε < − logμu(C+i0···in )∑n

k=0 du(σ k+(ω+))< d+ + ε

and

d− − ε < − logμs(C−i−m···i0)∑m

k=0 ds(σ k−(ω−))< d− + ε

for n,m > s(ω). For any sufficiently small r > 0, let n = n(ω, r) and m = m(ω, r)

be the unique positive integers such that

−n∑

k=0

du(σ k+(ω+)) > log r, −n+1∑

k=0

du(σ k+(ω+)) ≤ log r (11.10)

and

−m∑

k=0

ds(σ k−(ω−)) > log r, −m+1∑

k=0

ds(σ k−(ω−)) ≤ log r. (11.11)

On the other hand, as in the proof of Theorem 8.3 (see (8.21)), there exists aρ > 1 (independent of x = π(ω) and r) such that

B(y, r/ρ) ∩ Z ⊂ π(Ci−m···in) ⊂ B(x,ρr) ∩ Z (11.12)

for some point y ∈ π(Ci−m···in ), where ω = (· · · i−1i0i1 · · · ). Now we recall a resultof Barreira and Saussol in [13] (see also Lemma 15.2.2 in [3]).

Lemma 11.4 Given a probability measure ν on a set Z ⊂ Rm, there exists a con-

stant η > 1 such that for ν-almost every y ∈ Z and every ε > 0 there exists ac = c(y, ε) such that

ν(B(y,ηr)) ≤ ν(B(y, r))r−ε for r < c.


Without loss of generality, we take η = 2ρ. By (11.12) and Lemma 11.4, weobtain

ν(B(x, r)) ≤ ν

(B(y,2ρ

r

ρ)

)≤ ν(B(y, r/ρ))

(r

ρ

)−ε

≤ ν(π(Ci−m···in ))(

r

ρ

)−ε

= μu(C+i0···in )μ

s(C−i−m···i0)

(r

ρ

)−ε

≤ exp

[

(−d+ + ε)

n∑

k=0

du(σ k+(ω+))

]

× exp

[

(−d− + ε)

m∑

k=0

ds(σ k−(ω−))

](r

ρ

)−ε

≤ exp[(log r + ‖du‖∞)(d+ − ε)] exp[(log r + ‖ds‖∞)(d− − ε)](

r

ρ

)−ε

for r < c, and hence,

lim infr→0

logν(B(x, r))

log r≥ d+ + d− − 2ε (11.13)

for ν-almost every x ∈ Z.On the other hand, by Theorem 5.2, we have

dimH Λ = ts + tu + 1. (11.14)

Thus, by (11.9) and (11.13), we obtain

lim infr→0

logν(B(x, r))

log r≥ dimH K+

α + dimH K−β − dimH Λ − 1 − 2ε

and the desired result follows from the arbitrariness of ε. �

Step 3. Upper pointwise dimension

Now we obtain an upper bound for the upper pointwise dimension.

Lemma 11.5 For each x ∈ K+α ∩ K−

β ∩ Z, we have

lim supr→0

logν(B(x, r))

log r≤ dimH K+

α + dimH K−β − dimH Λ − 1.

Proof of the lemma We follow arguments in the proofs of Lemmas 5 and 6 in [18].Take x ∈ K+

α ∩ K−β ∩ Z and ω ∈ ΣA such that π(ω) = x, and let ω± = π±(ω).


It follows from Lemma 11.1 that

Ia+i(T k(π(ω))) = Ia+

i(π(σ k+(ω)))

= aui (π+(σ k(ω))) + g+

i (σ k(ω)) − g+i (σ k+1(ω))

= aui (σ k+(ω+)) + g+

i (σ k(ω)) − g+i (σ k+1(ω)),

with analogous identities for the functions Ib+i

, Ia−j

and Ib−j

. Therefore,

∑n−1k=0 Ia+

i(T k(x))

∑n−1k=0 Ib+

i(T k(x))

=∑n−1

k=0 aui (σ k+(ω+)) + g+

i (ω) − g+i (σ n(ω))

∑n−1k=0 bu

i (σ k+(ω+)) + h+i (ω) − h+

i (σ n(ω))

and∑n−1

k=0 Ia−j(T −k(x))

∑n−1k=0 Ib−

j(T −k(x))

=∑n−1

k=0 asj (σ

k−(ω−)) + g−j (ω) − g−

j (σ−n(ω))∑n−1

k=0 bsj (σ

k−(ω−)) + h−j (ω) − h−

j (σ−n(ω)).

On the other hand,

n−1∑

k=0

bui (σ k+(ω+)) ≥ n infb+

i inf τ − 2‖h+i ‖∞

andn−1∑

k=0

bsj (σ

k−(ω−)) ≥ n infb−j inf τ − 2‖h−

j ‖∞.

Since b+i , b−

j > 0 and inf τ > 0, this ensures that the limits

limn→∞

∑n−1k=0 Ia+

i(T k(x))

∑n−1k=0 Ib+

i(T k(x))

, limn→∞

∑n−1k=0 Ia−

j(T −k(x))

∑n−1k=0 Ib−

j(T −k(x))

exist respectively if and only if the limits

limn→∞

∑n−1k=0 au

i (σ k+(ω+))∑n−1

k=0 bui (σ k+(ω+))

, limn→∞

∑n−1k=0 as

j (σk−(ω−))

∑n−1k=0 bs

j (σk−(ω−))

exist, in which case they (respectively) coincide.By Theorem 2.3, if x ∈ K+

α ∩ K−β ∩ Z and ω ∈ ΣA are such that π(ω) = x, then

given ε > 0, there exists an r(ω) ∈N such that∥∥∥∥∥

⟨

q+,

n∑

k=0

(Au − α ∗ Bu)(σ k+(ω+))

⟩∥∥∥∥∥< εn‖〈q+,Bu〉‖∞


and∥∥∥∥∥

⟨

q−,

n∑

k=0

(As − β ∗ Bs)(σ k−(ω−))

⟩∥∥∥∥∥

< εn‖〈q−,Bs〉‖∞

for n > r(ω). By Lemma 11.2, we have Pσ+(U) = 0 and since μu is a Gibbs mea-sure (see (7.8)), there exists a D > 0 such that

D−1 <μu(C+

i0···in )exp

∑nk=0 U(σk+(ω+))

< D

for i0 = 1, . . . , p and n ∈ N. Therefore,

μu(C+i0···in ) > D−1 exp

[

−d+n∑

k=0

du(σ k+(ω+)) − εn‖〈q+,Bu〉‖∞

]

. (11.15)

Similarly, for every i0 = 1, . . . , p and n ∈ N we have

μs(C−i−m···i0) > D−1 exp

[

−d−m∑

k=0

ds(σ k−(ω−)) − εm‖〈q−,Bs〉‖∞

]

. (11.16)

Since infx∈Λ τ > 0 (see (3.7)), it follows from the hyperbolicity of Φ on Λ thatthere exists an r > 0 such that n(ω, r) > r(ω) and m(ω, r) > r(ω) (see (11.10)and (11.11)). Moreover, by (11.12), there exists a ρ > 0 (independent of x = π(ω)

and r) such that

B(x,ρr) ∩ Z ⊃ π(Ci−m···in ),

where n = n(ω, r) and m = m(ω, r). Combining (11.15) and (11.16) with (11.10)and (11.11), we obtain

ν(B(x,ρr)) ≥ ν(π(Ci−m···in ))

= μu(C+i0···in )μ

s(C−i−m···i0)

≥ D−2rd++d−exp

(−εn‖〈q+,Bu〉‖∞ − εm‖〈q−,Bs〉‖∞)

for any sufficiently small r > 0. On the other hand, it follows from (11.10)and (11.11) that

−n infdu > log r and − m infds > log r.

Therefore, for each x ∈ K+α ∩ K−

β ∩ Z we have

lim supr→∞

logν(B(x, r))

log r≤ d+ + d− + ε

(‖〈q+,Bu〉‖∞infdu

+ ‖〈q−,Bs〉‖∞infds

).


Since ε can be made arbitrarily small, we obtain

lim supr→∞

logν(B(x, r))

log r≤ d+ + d−.

Together with (11.9) and (11.14) this yields the desired result. �

Step 4. Conclusion

Combining Lemmas 11.3 and 11.5 yields the following result.

Lemma 11.6 For each α and β as in (11.7), there exists a probability measure ν inZ such that ν(K+

α ∩ K−β ) = 1,

limr→∞

logν(B(x, r))

log r= dimH K+

α + dimH K−β − dimH Λ − 1 (11.17)

for ν-almost every x ∈ Z, and

lim supr→∞

logν(B(x, r))

log r≤ dimH K+

α + dimH K−β − dimH Λ − 1 (11.18)

for every x ∈ K+α ∩ K−

β ∩ Z.

We proceed with the proof of the theorem. It follows from (11.17) (see for exam-ple [3, Theorem 2.1.5]) that

dimH ν = dimH K+α + dimH K−

β − dimH Λ − 1,

where

dimH ν = inf{dimH Z : ν(Z) = 1

}.

Since ν(K+α ∩ K−

β ) = 1, we obtain

dimH (K+α ∩ K−

β ∩ Z) ≥ dimH K+α + dimH K−

β − dimH Λ − 1.

On the other hand, it follows from (11.18) (see for example [3, Theorem 2.1.5]) that

dimH (K+α ∩ K−

β ∩ Z) ≤ dimH K+α + dimH K−

β − dimH Λ − 1,

and thus,

dimH (K+α ∩ K−

β ∩ Z) = dimH K+α + dimH K−

β − dimH Λ − 1.

Since K+α ∩ K−

β is locally diffeomorphic to a product of K+α ∩ K−

β ∩ Z and aninterval, we obtain

D(α,β) = dimH K+α + dimH K−

β − dimH Λ.


By Theorems 10.1 and 11.1 together with (11.14), we conclude that

D(α,β) = dimH (K+α ∩ V u(x)) + dimH (K−

β ∩ V s(x)) + 1

= dimζuK+α + dim−ζs K

−β + 1

= max

{hμ(Φ)∫Λ

ζu dμ: μ ∈ M and P+(μ) = α

}

+ max

{hμ(Φ)

− ∫Λ

ζs dμ: μ ∈M and P−(μ) = β

}+ 1.

The second statement is now a simple consequence of Theorem 10.1. �

References

1. L. Abramov, On the entropy of a flow, Dokl. Akad. Nauk SSSR 128 (1959), 873–875.2. L. Barreira, A non-additive thermodynamic formalism and applications to dimension theory

of hyperbolic dynamical systems, Ergod. Theory Dyn. Syst. 16 (1996), 871–927.3. L. Barreira, Dimension and Recurrence in Hyperbolic Dynamics, Progress in Mathematics

272, Birkhäuser, Basel, 2008.4. L. Barreira, Thermodynamic Formalism and Applications to Dimension Theory, Progress in

Mathematics 294, Birkhäuser, Basel, 2011.5. L. Barreira, Ergodic Theory, Hyperbolic Dynamics and Dimension Theory, Universitext,

Springer, Berlin, 2012.6. L. Barreira and P. Doutor, Birkhoff averages for hyperbolic flows: variational principles and

applications, J. Stat. Phys. 115 (2004), 1567–1603.7. L. Barreira and P. Doutor, Dimension spectra of hyperbolic flows, J. Stat. Phys. 136 (2009),

505–525.8. L. Barreira and K. Gelfert, Multifractal analysis for Lyapunov exponents on nonconformal

repellers, Commun. Math. Phys. 267 (2006), 393–418.9. L. Barreira and G. Iommi, Suspension flows over countable Markov shifts, J. Stat. Phys. 124

(2006), 207–230.10. L. Barreira and Ya. Pesin, Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero

Lyapunov Exponents, Encyclopedia of Mathematics and Its Applications 115, CambridgeUniversity Press, Cambridge, 2007.

11. L. Barreira, Ya. Pesin and J. Schmeling, Dimension and product structure of hyperbolic mea-sures, Ann. Math. (2) 149 (1999), 755–783.

12. L. Barreira and B. Saussol, Multifractal analysis of hyperbolic flows, Commun. Math. Phys.214 (2000), 339–371.

13. L. Barreira and B. Saussol, Hausdorff dimension of measures via Poincaré recurrence, Com-mun. Math. Phys. 219 (2001), 443–463.

14. L. Barreira and B. Saussol, Variational principles and mixed multifractal spectra, Trans. Am.Math. Soc. 353 (2001), 3919–3944.

15. L. Barreira and B. Saussol, Variational principles for hyperbolic flows, Fields Inst. Commun.31 (2002), 43–63.

16. L. Barreira, B. Saussol and J. Schmeling, Higher-dimensional multifractal analysis, J. Math.Pures Appl. 81 (2002), 67–91.

17. L. Barreira and J. Schmeling, Sets of “non-typical” points have full topological entropy andfull Hausdorff dimension, Isr. J. Math. 116 (2000), 29–70.

18. L. Barreira and C. Valls, Multifractal structure of two-dimensional horseshoes, Commun.Math. Phys. 266 (2006), 455–470.

L. Barreira, Dimension Theory of Hyperbolic Flows,Springer Monographs in Mathematics, DOI 10.1007/978-3-319-00548-5,© Springer International Publishing Switzerland 2013

151

http://dx.doi.org/10.1007/978-3-319-00548-5

152 References

19. L. Barreira and C. Wolf, Measures of maximal dimension for hyperbolic diffeomorphisms,Commun. Math. Phys. 239 (2003), 93–113.

20. L. Barreira and C. Wolf, Pointwise dimension and ergodic decompositions, Ergod. TheoryDyn. Syst. 26 (2006), 653–671.

21. L. Barreira and C. Wolf, Dimension and ergodic decompositions for hyperbolic flows, Dis-crete Contin. Dyn. Syst. 17 (2007), 201–212.

22. T. Bedford, Crinkly curves, Markov partitions and box dimension of self-similar sets, Ph.D.Thesis, University of Warwick, 1984.

23. T. Bedford, The box dimension of self-affine graphs and repellers, Nonlinearity 2 (1989),53–71.

24. T. Bedford and M. Urbanski, The box and Hausdorff dimension of self-affine sets, Ergod.Theory Dyn. Syst. 10 (1990), 627–644.

25. C. Bonatti, L. Díaz and M. Viana, Discontinuity of the Hausdorff dimension of hyperbolicsets, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), 713–718.

26. H. Bothe, The Hausdorff dimension of certain solenoids, Ergod. Theory Dyn. Syst. 15 (1995),449–474.

27. R. Bowen, Symbolic dynamics for hyperbolic flows, Am. J. Math. 95 (1973), 429–460.28. R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphism, Lect.

Notes in Math. 470, Springer, Berlin, 1975.29. R. Bowen, Hausdorff dimension of quasi-circles, Inst. Hautes Études Sci. Publ. Math. 50

(1979), 259–273.30. R. Bowen and D. Ruelle, The ergodic theory of axiom A flows, Invent. Math. 29 (1975),

181–202.31. R. Bowen and P. Walters, Expansive one-parameter flows, J. Differ. Equ. 12 (1972), 180–

193.32. M. Brin and A. Katok, On local entropy, in Geometric Dynamics (Rio de Janeiro, 1981),

edited by J. Palis, Lect. Notes in Math. 1007, Springer, Berlin, 1983, pp. 30–38.33. P. Collet, J. Lebowitz and A. Porzio, The dimension spectrum of some dynamical systems,

J. Stat. Phys. 47 (1987), 609–644.34. L. Díaz and M. Viana, Discontinuity of Hausdorff dimension and limit capacity on arcs of

diffeomorphisms, Ergod. Theory Dyn. Syst. 9 (1989), 403–425.35. A. Douady and J. Oesterlé, Dimension de Hausdorff des attracteurs, C. R. Acad. Sci. Paris

290 (1980), 1135–1138.36. M. Dysman, Fractal dimension for repellers of maps with holes, J. Stat. Phys. 120 (2005),

479–509.37. K. Falconer, The Hausdorff dimension of self-affine fractals, Math. Proc. Camb. Philos. Soc.

103 (1988), 339–350.38. K. Falconer, A subadditive thermodynamic formalism for mixing repellers, J. Phys. A, Math.

Gen. 21 (1988), 1737–1742.39. K. Falconer, Dimensions and measures of quasi self-similar sets, Proc. Am. Math. Soc. 106

(1989), 543–554.40. K. Falconer, Bounded distortion and dimension for non-conformal repellers, Math. Proc.

Camb. Philos. Soc. 115 (1994), 315–334.41. K. Falconer, Fractal Geometry. Mathematical Foundations and Applications, Wiley, New

York, 2003.42. H. Federer, Geometric Measure Theory, Grundlehren der mathematischen Wissenschaften

153, Springer, Berlin, 1969.43. A. Fan, Y. Jiang and J. Wu, Asymptotic Hausdorff dimensions of Cantor sets associated with

an asymptotically non-hyperbolic family, Ergod. Theory Dyn. Syst. 25 (2005), 1799–1808.44. D. Feng, Lyapunov exponents for products of matrices and multifractal analysis. I. Positive

matrices, Isr. J. Math. 138 (2003), 353–376.45. D. Feng, The variational principle for products of non-negative matrices, Nonlinearity 17

(2004), 447–457.

References 153

46. D. Feng and K. Lau, The pressure function for products of non-negative matrices, Math. Res.Lett. 9 (2002), 363–378.

47. D. Gatzouras and S. Lalley, Hausdorff and box dimensions of certain self-affine fractals,Indiana Univ. Math. J. 41 (1992), 533–568.

48. D. Gatzouras and Y. Peres, Invariant measures of full dimension for some expanding maps,Ergod. Theory Dyn. Syst. 17 (1997), 147–167.

49. K. Gelfert, Dimension estimates beyond conformal and hyperbolic dynamics, Dyn. Syst. 20(2005), 267–280.

50. B. Gurevic, Topological entropy of a countable Markov chain, Sov. Math. Dokl. 10 (1969),911–915.

51. T. Halsey, M. Jensen, L. Kadanoff, I. Procaccia and B. Shraiman, Fractal measures and theirsingularities: the characterization of strange sets, Phys. Rev. A 34 (1986), 1141–1151; erratain 34 (1986), 1601.

52. P. Hanus, R. Mauldin and M. Urbanski, Thermodynamic formalism and multifractal analysisof conformal infinite iterated function systems, Acta Math. Hung. 96 (2002), 27–98.

53. B. Hasselblatt, Regularity of the Anosov splitting and of horospheric foliations, Ergod. The-ory Dyn. Syst. 14 (1994), 645–666.

54. B. Hasselblatt and J. Schmeling, Dimension product structure of hyperbolic sets, Electron.Res. Announc. Am. Math. Soc. 10 (2004), 88–96.

55. B. Hasselblatt and J. Schmeling, Dimension product structure of hyperbolic sets, in Mod-ern Dynamical Systems and Applications, Cambridge University Press, Cambridge, 2004,pp. 331–345.

56. M. Hirayama, An upper estimate of the Hausdorff dimension of stable sets, Ergod. TheoryDyn. Syst. 24 (2004), 1109–1125.

57. V. Horita and M. Viana, Hausdorff dimension of non-hyperbolic repellers. I. Maps with holes,J. Stat. Phys. 105 (2001), 835–862.

58. V. Horita and M. Viana, Hausdorff dimension for non-hyperbolic repellers. II. DA diffeomor-phisms, Discrete Contin. Dyn. Syst. 13 (2005), 1125–1152.

59. H. Hu, Box dimensions and topological pressure for some expanding maps, Commun. Math.Phys. 191 (1998), 397–407.

60. G. Iommi, Multifractal analysis for countable Markov shifts, Ergod. Theory Dyn. Syst. 25(2005), 1881–1907.

61. T. Jordan and K. Simon, Multifractal analysis of Birkhoff averages for some self-affine IFS,Dyn. Syst. 22 (2007), 469–483.

62. A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, En-cyclopedia of Mathematics and Its Applications 54, Cambridge University Press, Cambridge,1995.

63. G. Keller, Equilibrium States in Ergodic Theory, London Mathematical Society StudentTexts 42, Cambridge University Press, Cambridge, 1998.

64. R. Kenyon and Y. Peres, Measures of full dimension on affine-invariant sets, Ergod. TheoryDyn. Syst. 16 (1996), 307–323.

65. M. Kesseböhmer and B. Stratmann, A multifractal formalism for growth rates and applica-tions to geometrically finite Kleinian groups, Ergod. Theory Dyn. Syst. 24 (2004), 141–170.

66. M. Kesseböhmer and B. Stratmann, A multifractal analysis for Stern-Brocot intervals, con-tinued fractions and Diophantine growth rates, J. Reine Angew. Math. 605 (2007), 133–163.

67. F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms I. Characterization ofmeasures satisfying Pesin’s entropy formula, Ann. Math. (2) 122 (1985), 509–539.

68. F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms II. Relations betweenentropy, exponents and dimension, Ann. Math. (2) 122 (1985), 540–574.

69. A. Lopes, The dimension spectrum of the maximal measure, SIAM J. Math. Anal. 20 (1989),1243–1254.

70. N. Luzia, A variational principle for the dimension for a class of non-conformal repellers,Ergod. Theory Dyn. Syst. 26 (2006), 821–845.

154 References

71. N. Luzia, Measure of full dimension for some nonconformal repellers, Discrete Contin. Dyn.Syst. 26 (2010), 291–302.

72. R. Mañé, The Hausdorff dimension of horseshoes of diffeomorphisms of surfaces, Bol. Soc.Bras. Mat. (N.S.) 20 (1990), 1–24.

73. R. Mauldin and M. Urbanski, Dimensions and measures in infinite iterated function systems,Proc. Lond. Math. Soc. (3) 73 (1996), 105–154.

74. R. Mauldin and M. Urbanski, Conformal iterated function systems with applications to thegeometry of continued fractions, Trans. Am. Math. Soc. 351 (1999), 4995–5025.

75. R. Mauldin and M. Urbanski, Parabolic iterated function systems, Ergod. Theory Dyn. Syst.20 (2000), 1423–1447.

76. H. McCluskey and A. Manning, Hausdorff dimension for horseshoes, Ergod. Theory Dyn.Syst. 3 (1983), 251–260.

77. C. McMullen, The Hausdorff dimension of general Sierpinski carpets, Nagoya Math. J. 96(1984), 1–9.

78. K. Nakaishi, Multifractal formalism for some parabolic maps, Ergod. Theory Dyn. Syst. 20(2000), 843–857.

79. J. Palis and M. Viana, On the continuity of Hausdorff dimension and limit capacity forhorseshoes, in Dynamical Systems (Valparaiso, 1986), edited by R. Bamón, R. Labarca andJ. Palis, Lecture Notes in Mathematics 1331, Springer, Berlin, 1988, pp. 150–160.

80. W. Parry and M. Pollicott, Zeta Functions and the Periodic Orbit Structure of HyperbolicDynamics, Astérisque 187–188, 1990.

81. Ya. Pesin, Dimension Theory in Dynamical Systems: Contemporary Views and Applications,Chicago Lectures in Mathematics, Chicago University Press, Chicago, 1997.

82. Ya. Pesin and V. Sadovskaya, Multifractal analysis of conformal axiom A flows, Commun.Math. Phys. 216 (2001), 277–312.

83. Ya. Pesin and H. Weiss, A multifractal analysis of equilibrium measures for conformal ex-panding maps and Markov Moran geometric constructions, J. Stat. Phys. 86 (1997), 233–275.

84. Ya. Pesin and H. Weiss, The multifractal analysis of Gibbs measures: motivation, mathemat-ical foundation, and examples, Chaos 7 (1997), 89–106.

85. M. Piacquadio and M. Rosen, Multifractal spectrum of an experimental (video feedback)Farey tree, J. Stat. Phys. 127 (2007), 783–804.

86. M. Pollicott and H. Weiss, The dimensions of some self-affine limit sets in the plane andhyperbolic sets, J. Stat. Phys. 77 (1994), 841–866.

87. M. Pollicott and H. Weiss, Multifractal analysis of Lyapunov exponent for continued fractionand Manneville-Pomeau transformations and applications to Diophantine approximation,Commun. Math. Phys. 207 (1999), 145–171.

88. F. Przytycki and M. Urbanski, On the Hausdorff dimension of some fractal sets, Stud. Math.93 (1989), 155–186.

89. D. Rand, The singularity spectrum f (α) for cookie-cutters, Ergod. Theory Dyn. Syst. 9(1989), 527–541.

90. M. Ratner, Markov partitions for Anosov flows on n-dimensional manifolds, Isr. J. Math. 15(1973), 92–114.

91. D. Ruelle, Statistical mechanics on a compact set with Zν action satisfying expansiveness

and specification, Trans. Am. Math. Soc. 185 (1973), 237–251.92. D. Ruelle, Thermodynamic Formalism, Encyclopedia of Mathematics and Its Applications 5,

Addison-Wesley, Reading, 1978.93. D. Ruelle, Repellers for real analytic maps, Ergod. Theory Dyn. Syst. 2 (1982), 99–107.94. O. Sarig, Thermodynamic formalism for countable Markov shifts, Ergod. Theory Dyn. Syst.

19 (1999), 1565–1593.95. S. Savchenko, Special flows constructed from countable topological Markov chains, Funct.

Anal. Appl. 32 (1998), 32–41.96. J. Schmeling, Symbolic dynamics for β-shifts and self-normal numbers, Ergod. Theory Dyn.

Syst. 17 (1997), 675–694.

References 155

97. J. Schmeling, On the completeness of multifractal spectra, Ergod. Theory Dyn. Syst. 19(1999), 1595–1616.

98. J. Schmeling, Entropy preservation under Markov coding, J. Stat. Phys. 104 (2001), 799–815.

99. K. Simon, Hausdorff dimension for noninvertible maps, Ergod. Theory Dyn. Syst. 13 (1993),199–212.

100. K. Simon, The Hausdorff dimension of the Smale–Williams solenoid with different contrac-tion coefficients, Proc. Am. Math. Soc. 125 (1997), 1221–1228.

101. K. Simon and B. Solomyak, Hausdorff dimension for horseshoes in R3, Ergod. Theory Dyn.

Syst. 19 (1999), 1343–1363.102. B. Solomyak, Measure and dimension for some fractal families, Math. Proc. Camb. Philos.

Soc. 124 (1998), 531–546.103. F. Takens, Limit capacity and Hausdorff dimension of dynamically defined Cantor sets, in

Dynamical Systems (Valparaiso 1986), edited by R. Bamón, R. Labarca and J. Palis, LectureNotes in Mathematics 1331, Springer, Berlin, 1988, 196–212.

104. M. Urbanski and C. Wolf, Ergodic theory of parabolic horseshoes, Commun. Math. Phys.281 (2008), 711–751.

105. P. Walters, Equilibrium states for β-transformations and related transformations, Math. Z.159 (1978), 65–88.

106. P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics 79, Springer,Berlin, 1982.

107. Y. Yayama, Dimensions of compact invariant sets of some expanding maps, Ergod. TheoryDyn. Syst. 29 (2009), 281–315.

108. L.-S. Young, Dimension, entropy and Lyapunov exponents, Ergod. Theory Dyn. Syst. 2(1982), 109–124.

109. M. Yuri, Multifractal analysis of weak Gibbs measures for intermittent systems, Commun.Math. Phys. 230 (2002), 365–388.

110. Y. Zhang, Dynamical upper bounds for Hausdorff dimension of invariant sets, Ergod. TheoryDyn. Syst. 17 (1997), 739–756.

Index

Bbase, 20bounded variation, 30, 115Bowen–Walters distance, 25Bowen’s equation, 2box dimension, 45

lower –, 46upper –, 46

BS-dimension, 43, 44spectrum, 127

Ccoding map, 37cohomologous functions, 20cohomology, 19

class, 20conditional variational principle, 111, 115conformal

flow, 51map, 91

Ddiameter, 45dimension

box –, 45Hausdorff –, 45, 46, 67lower box –, 45, 46lower pointwise –, 46, 82pointwise –, 61, 84spectrum, 127, 139, 141

for the pointwise dimensions, 84, 92, 96upper box –, 45, 46upper pointwise –, 46, 82

Eentropy, 39

local –, 62, 120

lower local –, 120spectrum, 111, 124

for the Birkhoff averages, 89, 105for the local entropies, 87, 120for the Lyapunov exponents, 121

topological –, 40upper local –, 120

equilibrium measure, 41ergodic

decomposition, 47, 67measure, 40

expansive flow, 42

Fflow

conformal –, 51expansive –, 42hyperbolic –, 33, 91suspension –, 19topologically mixing –, 42topologically transitive –, 107

functionheight –, 19, 25transfer –, 35

HHausdorff

dimension, 45, 46, 67measure, 45

height function, 19, 25horizontal segment, 25hyperbolic

flow, 33, 91set, 33

Iinvariant measure, 40irregular set, 87, 106

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158 Index

Llocal entropy, 62, 120locally maximal set, 34lower

box dimension, 45, 46local entropy, 120pointwise dimension, 46, 82

Mmap

coding –, 37conformal –, 91transfer –, 35

Markovchain, 36system, 35, 36

measureequilibrium –, 41ergodic –, 40Hausdorff –, 45invariant –, 40of maximal dimension, 70

multidimensional spectrum, 124, 127multifractal analysis, 91, 127

Ppointwise dimension, 61, 84

Rrectangle, 35, 63repeller, 91

Ssegment

horizontal –, 25vertical –, 26

semiflow, 25suspension –, 25

sethyperbolic –, 33irregular –, 87, 106locally maximal –, 34

spectrumBS-dimension –, 127dimension –, 84, 92, 96, 139, 141entropy –, 87, 89, 105, 111, 120, 121, 124

stable manifold, 34suspension

flow, 19over expanding map, 91semiflow, 25

symbolic dynamics, 36, 81

Ttopological

entropy, 40Markov chain, 36pressure, 39, 40

topologicallymixing flow, 42transitive flow, 107

transferfunction, 35map, 35

transition matrix, 36

Uu-dimension spectrum, 84unstable manifold, 34upper

box dimension, 45, 46local entropy, 120pointwise dimension, 46, 82

Vvariational principle, 111, 115vertical segment, 26

dimension theory of hyperbolic flows--luis barreira

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