• 検索結果がありません。

Dimension and product structure of hyperbolic measures

N/A
N/A
Protected

Academic year: 2022

シェア "Dimension and product structure of hyperbolic measures"

Copied!
29
0
0

読み込み中.... (全文を見る)

全文

(1)

Dimension and product structure of hyperbolic measures

By Luis Barreira, Yakov Pesin, andorg Schmeling*

Abstract

We prove that every hyperbolic measure invariant under a C1+α diffeo- morphism of a smooth Riemannian manifold possesses asymptotically “almost”

local product structure, i.e., its density can be approximated by the product of the densities on stable and unstable manifolds up to small exponentials.

This has not been known even for measures supported on locally maximal hyperbolic sets.

Using this property of hyperbolic measures we prove the long-standing Eckmann-Ruelle conjecture in dimension theory of smooth dynamical systems:

the pointwise dimension of every hyperbolic measure invariant under a C1+α diffeomorphism exists almost everywhere. This implies the crucial fact that virtually all the characteristics of dimension type of the measure (including the Hausdorff dimension, box dimension, and information dimension) coincide.

This provides the rigorous mathematical justification of the concept of fractal dimension for hyperbolic measures.

1. Introduction

In this paper we provide an affirmative solution of the long-standing prob- lem in the interface of dimension theory and dynamical systems known as the Eckmann-Ruelle conjecture.

In the late 70’s–beginning 80’s, attention of many physicists and applied mathematicians had turned to the study of dimension of strange attractors

*This paper was written while L. B. was on leave from Instituto Superior T´ecnico, Department of Mathematics, at Lisbon, Portugal, and J. S. was visiting Penn State. L. B. was partially supported by FCT’s Pluriannual Funding Program and PRAXIS XXI grants 2/2.1/MAT/199/94, BD5236/95, and PBIC/C/MAT/2139/95. Ya. P. was partially supported by the National Science Foundation grant #DMS9403723. J. S. was supported by the Leopoldina-Forderpreis. L. B. and Ya. P. were partially supported by the NATO grant CRG970161.

1991Mathematics Subject Classification. 58F11, 28D05.

Key words and phrases. Eckmann-Ruelle conjecture, hyperbolic measures, pointwise dimen- sion, product structure

(2)

756 LUIS BARREIRA, YAKOV PESIN, AND JORG SCHMELING¨

(i.e., attracting invariant sets with some hyperbolic structure) in evolution- type systems (see, for example, [6], [7], [22]). The dimension was used to characterize a (finite) number of independent modes needed to describe the infinite-dimensional system. Several results were obtained which indicated relations between the dimension of the attractor and other invariants of the system (such as Lyapunov exponents and entropy; see, for example, [13], [14], [16], [23]). This study has become an important breakthrough in understanding the structure of systems of evolution type.

In the survey article [4], Eckmann and Ruelle summarized this activity and outlined a rigorous mathematical foundation for it. They considered dynamical systems with chaotic behavior of trajectories and described relations between persistence of chaotic motions and existence of strange attractors. They also discussed various concepts of dimension and pointed out the importance of the so-called pointwise (local) dimension of invariant measures. For a Borel measureµ on a complete metric spaceM, the latter is defined by

(1) d(x)def= lim

r0

logµ(B(x, r)) logr

whereB(x, r) is the ball centered at x of radius r (provided the limit exists).

It was introduced by Young in [23] and characterizes the local geometrical structure of an invariant measure with respect to the metric in the phase space of the system. Its crucial role in dimension theory of dynamical systems was acknowledged by many experts in the field (see, for example, the paper by Farmer, Ott, and Yorke [6], the ICM address by Young [24, p. 1232] and also [25, p. 318]).

If the limit in (1) does not exist one can consider the lower and upper limits and introduce respectivelythe lower and upper pointwise dimensionsof µatx which we denote byd(x) and d(x).

The existence of the limit in (1) for a Borel probability measure µ on M implies the crucial fact that virtuallyallthe known characteristics of dimension type of the measure coincide (this is partly described in Prop. 1 in §2). The common value is a fundamental characteristic of the fractal structure ofµ — thefractal dimension of µ.

In this paper we consider a C1+α diffeomorphism of a compact smooth Riemannian manifold without boundary. Our goal is to show the existence of the pointwise dimension in the case whenµishyperbolic, i.e., all the Lyapunov exponents off are nonzero atµ-almost every point (see Main Theorem in§3).

This statement has been an open problem in dimension theory of dynamical systems for about 15 years and is often referred to as the Eckmann-Ruelle conjecture.

(3)

Since hyperbolic measures play a crucial role in studying physical models with persistent chaotic behavior and fractal structure of invariant sets, our result provides a rigorous mathematical foundation for such a study.

The problem of the existence of the pointwise dimension has a long history.

In [23], Young obtained a positive answer for a hyperbolic measureµinvariant under a C1+α surface diffeomorphism f. Moreover, she showed that in this case for almost every pointx,

d(x) =d(x) =hµ(f) µ 1

λ1 1 λ2

,

where hµ(f) is the metric entropy of f and λ1 > 0 > λ2 are the Lyapunov exponents ofµ.

In [12], Ledrappier established the existence of the pointwise dimension for arbitrary SRB-measures (called so after Sinai, Ruelle, and Bowen). In [20], Pesin and Yue extended his approach and proved the existence for hyper- bolic measures satisfying the so-called semi-local product structure (this class includes, for example,Gibbs measureson locally maximal hyperbolic sets).

A substantial breakthrough in studying the pointwise dimension was made by Ledrappier and Young in [16]. They proved the existence of thestable and unstable pointwise dimensions, i.e., the pointwise dimensions along stable and unstable local manifolds for typical points (see Prop. 2 in§2). They also showed that the upper pointwise dimension at a typical point does not exceed the sum of the stable and unstable pointwise dimensions.

Our proof exploits their result in an essential way. It also uses a new and nontrivial property of hyperbolic ergodic measures that we establish in this paper. Loosely speaking, this property means that such measures have asymptotically “almost” local product structure. Let us point out that this property has not been known even for invariant measures on locally maximal hyperbolic sets (whose local topological structure is the direct product). This property also enables us to show that the pointwise dimension of a hyperbolic measure is almost everywhere the sum of the pointwise dimensions along stable and unstable local manifolds.

Acknowledgment. We would like to thank Fran¸cois Ledrappier for useful discussions and comments.

2. Preliminaries

2.1. Facts from dimension theory. We describe some most important characteristics of dimension type (see, for example, [5], [18]). Let X be a complete separable metric space. For a subsetZ ⊂Xand a numberα 0 the α-Hausdorff measure of Z is defined by

(4)

758 LUIS BARREIRA, YAKOV PESIN, AND JORG SCHMELING¨

mH(Z, α) = lim

ε0inf

G

X

U∈G

(diamU)α,

where the infimum is taken over all finite or countable coverings G of Z by open sets with diamG ≤ ε. The Hausdorff dimension ofZ (denoted dimHZ) is defined by

dimHZ = inf:mH(Z, α) = 0}= sup:mH(Z, α) =∞}.

We define thelower and upper box dimensions of Z (denoted respectively by dimBZ and dimBZ) by

dimBZ = inf:rH(Z, α) = 0}= sup:rH(Z, α) =∞}, dimBZ = inf:rH(Z, α) = 0}= sup:rH(Z, α) =∞}, where

rH(Z, α) = lim

ε0

infG

X

U∈G

εα, rH(Z, α) = lim

ε0inf

G

X

U∈G

εα

and the infimum is taken over all finite or countable coveringsG ofZ by open sets of diameterε. One can show that

dimBZ = lim

ε0

logN(Z, ε)

log(1/ε) , dimBZ = lim

ε0

logN(Z, ε) log(1/ε) ,

whereN(Z, ε) is the smallest number of balls of radius εneeded to cover the setZ.

It is easy to see that

dimHZ dimBZ dimBZ.

The coincidence of the Hausdorff dimension and lower and upper box dimension is a relatively rare phenomenon and can occur only in some “rigid” situations (see [1], [5], [19]).

In order to describe the geometric structure of a subsetZ invariant under a dynamical system f acting on X we consider a measure µ supported on Z. Its Hausdorff dimension and lower and upper box dimensions (which are denoted by dimHµ, dimBµ, and dimBµ, respectively) are

dimHµ= inf{dimHZ :µ(Z) = 1}, dimBµ= lim

δ0inf{dimBZ :µ(Z)≥1−δ}, dimBµ= lim

δ0inf{dimBZ :µ(Z)≥1−δ}.

From the definition it follows that

dimHµ≤dimBµ≤dimBµ.

(5)

Another important characteristic of dimension type ofµis itsinformation dimension. Given a partitionξ ofX, define the entropy of ξ with respect toµ by

Hµ(ξ) =X

Cξ

µ(Cξ) logµ(Cξ)

whereCξ is an element of the partition ξ. Given a number ε >0, set Hµ(ε) = inf{Hµ(ξ): diamξ ≤ε}

where diamξ = max diamCξ. We define the lower and upper information dimensions ofµby

I(µ) = lim

ε0

Hµ(ε)

log(1/ε), I(µ) = lim

ε0

Hµ(ε) log(1/ε).

There is a powerful criterion established by Young in [23] that guarantees the coincidence of the Hausdorff dimension and lower and upper box dimen- sions of measures as well as their lower and upper information dimensions.

Proposition 1 ([23]). Let X be a compact separable metric space of finite topological dimension and µ a Borel probability measure on X. Assume that

(2) d(x) =d(x) =d

for µ-almost everyx∈X. Then

dimHµ= dimBµ= dimBµ=I(µ) =I(µ) =d.

A measure µwhich satisfies (2) is called exact dimensional.

2.2. Hyperbolic measures. Let M be a smooth Riemannian manifold without boundary, andf:M M a C1+α diffeomorphism on M. Let also µ be anf-invariant ergodic Borel probability measure on M.

Givenx∈M andv∈TxM define theLyapunov exponent ofvatxby the formula

λ(x, v) = lim

n→∞

1

nlogkdxfnvk.

If x is fixed then the function λ(x,·) can take on only finitely many values λ1(x) < · · · < λk(x)(x). The functions λi(x) are measurable and f-invariant.

Since µ is ergodic, these functions are constant µ-almost everywhere. We denote these constants byλ1<· · ·< λk. The measureµis said to behyperbolic ifλi 6= 0 for everyi= 1, . . .,k.

(6)

760 LUIS BARREIRA, YAKOV PESIN, AND JORG SCHMELING¨

There exists a measurable function r(x)>0 such that for µ-almost every x∈M the sets

Ws(x) =

½

y ∈B(x, r(x)) : lim

n+

1

nlogd(fnx, fny)<0

¾ ,

Wu(x) =

½

y ∈B(x, r(x)) : lim

n→−∞

1

nlogd(fnx, fny)>0

¾

are immersed local manifolds called stable and unstable local manifolds at x (see [17] for details). For each r (0, r(x)) we consider the balls Bs(x, r) Ws(x) and Bu(x, r) Wu(x) centered at x with respect to the induced distances onWs(x) and Wu(x) respectively.

Let ξ be a measurable partition of M. It has a canonical system of con- ditional measures: forµ-almost every x there is a probability measure µx de- fined on the element ξ(x) of ξ containing x. The conditional measuresµx are uniquely characterized by the following property: ifBξis theσ-subalgebra (of the Borelσ-algebra) whose elements are unions of elements of ξ, and A⊂M is a measurable set, thenx7→µx(A∩ξ(x)) isBξ-measurable and

µ(A) = Z

µx(A∩ξ(x))dµ(x).

In [16], Ledrappier and Young constructed two measurable partitions ξs andξu of M such that forµ-almost everyx∈M:

1. ξs(x)⊂Ws(x) and ξu(x)⊂Wu(x);

2. ξs(x) and ξu(x) contain the intersection of an open neighborhood of x with Ws(x) andWu(x) respectively.

We denote the system of conditional measures of µ with respect to the partitions ξs and ξu, respectively by µsx and µux, and for any measurable set A⊂M we write µsx(A) =µsx(A∩ξs(x)) andµux(A) =µux(A∩ξu(x)).

Given x M, consider the lower and upper pointwise dimensions of µ at x, d(x) and d(x). Since these functions are measurable and f-invariant they are constant µ-almost everywhere. We denote these constants by dand drespectively.

In [16], Ledrappier and Young introduced the quantities ds(x)def= lim

r0

logµsx(Bs(x, r)) logr , du(x)def= lim

r0

logµux(Bu(x, r)) logr

provided that the corresponding limits exist atx∈M. We call them, respec- tively,stable and unstable pointwise dimensions of µ.

(7)

Proposition 2 ([16]).

1. For µ-almost every x ∈M the limits ds(x) and du(x) exist and are con- stant µ-almost everywhere; we denote these constants by ds and du. 2. If µis a hyperbolic measure then

d≤ds+du.

When the entropy of f is zero it follows from [16] that ds =du = 0 and henced=d=ds+du = 0.

Let us point out that in [16] the authors consider a class of measures more general than hyperbolic measures (some of the Lyapunov exponents may be zero). They prove Proposition 2 under the assumption that the diffeomor- phismf is of class C2. The main ingredient of their proof is the existence of intermediatepointwise dimensions, i.e., the pointwise dimensions of the condi- tional measures generated by the invariant measure on the intermediate stable and unstable leaves. This, in turn, relies on the Lipschitz continuity of the holonomy map generated by these intermediate leaves (see the Appendix for definitions and precise statements) and is where the assumption that f is of classC2 is used (see§(4.2) in [16]; other arguments in [16] do not use the Lip- schitz continuity of the holonomy map and go well forC1+αdiffeomorphisms).

In the Appendix to the paper we provide a proof of the fact that for hy- perbolic measures the Lipschitz property of the holonomy map holds for diffeo- morphisms of classC1+α. Indeed, we prove a slightly more general statement:

the Lipschitz property holds for intermediate stable and unstable foliations even if some of the Lyapunov exponents are zero. Our approach gives a new proof of this property even in the case of diffeomorphisms of class C2. As a consequence we obtain that Proposition 2 holds for diffeomorphisms of class C1+α and so does our Main Theorem.

Let us point out that the Lipschitz continuity of the holonomy map pre- sumably fails if the map is generated by stable leaves inside the stable-neutral foliation.

3. Main Theorem

In this paper we prove the following statement.

Main Theorem. Letf be aC1+α diffeomorphism on a smooth Riemann- ian manifoldM without boundary, and µ an f-invariant compactly supported ergodic Borel probability measure. If µis hyperbolic then the following proper- ties hold:

(8)

762 LUIS BARREIRA, YAKOV PESIN, AND JORG SCHMELING¨

1. for every δ >0,there exist a setΛ⊂M withµ(Λ)>1−δ and a constant κ≥1 such that for every x∈Λ and every sufficiently small r (depending on x),

rδµsx(Bs(x, r

κ))µux(Bu(x,r

κ))≤µ(B(x, r)) (3)

≤rδµsx(Bs(x, κr))µux(Bu(x, κr));

2. µ is exact dimensional and its pointwise dimension is equal to the sum of the stable and unstable pointwise dimensions,i.e.,

d=d=ds+du.

This statement provides an affirmative solution to the Eckmann-Ruelle conjecture and describes the most broad class of measures invariant under smooth dynamical systems which are exact dimensional.

Note that an SRB-measure is locally equivalent on Λ to the direct product of an absolutely continuous measure on an unstable leaf and a measure on a stable leaf (see [12]). Hence, statement 1 holds in this case automatically. One can also show that Gibbs measures on a locally maximal hyperbolic set Λ are locally equivalent to the direct product of a measure on an unstable leaf and a measure on a stable leaf (see [9]). Therefore, statement 1 holds in this case as well.

Let us also point out that neither of the assumptions of the Main Theorem can be omitted. Ledrappier and Misiurewicz [15] constructed an example of a smooth map of a circle preserving an ergodic measure with zero Lyapunov exponent which is not exact dimensional. In [19], Pesin and Weiss presented an example of a H¨older homeomorphism with H¨older constant arbitrarily close to 1 whose measure of maximal entropy is not exact dimensional.

Remarks. 1. Statement 1 of the Main Theorem establishes a new and nontrivial property of an arbitrary hyperbolic measure. Loosely speaking, it means that every hyperbolic invariant measure possesses asymptotically “al- most” local product structure. This statement has not been known even for measures supported on (uniformly) hyperbolic locally maximal invariant sets.

The lower bound in (3) can be easily obtained from results in [16] while the upper bound is one of the main ingredients of our proof. Note that statement 2 follows from statement 1. The proof of statement 2 exploits the existence of stable and unstable pointwise dimensions and the argument in [20] (see §6).

In order to illustrate the property of having asymptotically “almost” local product structure, let us consider an ergodic measureµinvariant under the full shift σ on the space Σp of all two-sided infinite sequences of pnumbers. This space is endowed with the usual “symbolic” metric dβ, for each fixed number

(9)

β≥2, defined by

dβ1, ω2) =X

i∈Z

β−|i|i1−ωi2|,

whereω1= (ωi1) andω2 = (ω2i). Fix ω= (ωi)Σp. The cylinder Cn(ω) =¯ = (¯ωi) : ¯ωi =ωi fori=−n, . . . , n}

can be identified with the direct product Cn+(ω)×Cn(ω) where Cn+(ω) =¯ = (¯ωi) : ¯ωi =ωi fori= 0, . . . , n}

and

Cn(ω) =¯ = (¯ωi) : ¯ωi=ωi fori=−n, . . . ,0}

are the “positive” and “negative” cylinders at ω of “size”n. Define measures µ+n(ω) =µ|Cn+(ω) and µn(ω) =µ|Cn(ω).

The measureµis said to have local product structure if the measureµ|Cn(ω) is equivalent to the product µ+n(ω)×µn(ω) uniformly over ω Σp and n >

0. It is known that Gibbs measures have local product structure (see, for example, [18]). For an arbitraryσ-invariant ergodic measureµit follows from statement 1 of the Main Theorem (see (3)) that for everyδ >0 there exist a set ΛΣp withµ(Λ)>1−δ and an integerm≥1 such that for everyω Λ and every sufficiently largen(depending on ω),

βδ|n|µ+n+m(ω)×µn+m(ω)≤µ|Cn(ω)≤βδ|n|µ+nm(ω)×µnm(ω).

2. It follows immediately from the Main Theorem that the pointwise di- mension of an ergodic invariant measure supported on a (uniformly) hyperbolic locally maximal invariant set is exact dimensional. This result has not been known before. We emphasize that in this situation the stable and unstable foliations need not be Lipschitz (in fact, they are “generically” not Lipschitz;

see [21]), and, in general, the measure need not have a local product struc- ture despite the fact that the set itself does. Therefore, both statements of the Main Theorem are nontrivial even for measures supported on hyperbolic locally maximal invariant sets.

3. The role of the Eckmann-Ruelle conjecture in dimension theory of dynamical systems is similar to the role of the Shannon-McMillan-Breiman theorem in the entropy theory. In order to illustrate this, consider the full shift σ on the space Σp.

Letµbe aσ-invariant ergodic measure on Σp. By the Shannon-McMillan- Breiman theorem, forµ-almost everyω Σp,

(4) lim

n→∞ 1

2n+ 1logµ(Cn(ω)) =h

(10)

764 LUIS BARREIRA, YAKOV PESIN, AND JORG SCHMELING¨

whereh=hµ(σ). Since the cylinderCn(ω) is the ball (in the symbolic metric dβ) centered at ω of radius n (for some c > 0), the quantity in the right- hand side in (4) divided by logβ is the pointwise dimension of µ at ω. The Shannon-McMillan-Breiman theorem (applied to the shift map) thus claims that the pointwise dimension ofµexists almost everywhere; it is clearly almost everywhere constant; furthermore, the common value is the measure-theoretic entropy ofµdivided by logβ.

As an important consequence of this theorem one obtains that various definitions of the entropy (due to Kolmogorov and Sinai [11], Katok [8], Brin and Katok [2], etc.) coincide (see [18] for details).

4. Let µbe an arbitrary (not necessarily ergodic) invariant measure for a C1+α diffeomorphismf ofM. The measureµis said to be hyperbolicif on al- most every ergodic component the induced measure is a hyperbolic measure on M. The Main Theorem remains true for anyf-invariant compactly supported hyperbolic Borel probability measure, i.e., the pointwise dimension of such a measure exists almost everywhere (but may not be any longer a constant; see

§7).

4. Description of a special partition

We use the following notation. Let η be a partition. For every integers k,l 1, we define the partition ηkl =Wl

n=kfnη. We observe that η0k(x) ηl0(x) =ηlk(x).

From now on we assume that µ is hyperbolic. In [16], Ledrappier and Young constructed a special countable partition Pof M of finite entropy sat- isfying the following properties. Given 0< ε <1, there exists a set Γ ⊂M of measureµ(Γ)>1−ε/2, an integern01, and a numberC >1 such that for everyx∈Γ and any integer n≥n0, the following statements hold:

a. For all integersk,l≥1 we have

C1e(l+k)h(l+k)ε≤µ(Plk(x))≤Ce(l+k)h+(l+k)ε, (5)

C1ekh≤µsx(P0k(x))≤Cekh+kε, (6)

C1elh≤µux(Pl0(x))≤Celh+lε, (7)

whereh is the Kolmogorov-Sinai entropy of f with respect to µ.

b.

ξs(x) \

n0

Pn0(x)⊃Bs(x, en0), (8)

ξu(x) \

n0

P0n(x)⊃Bu(x, en0).

(9)

(11)

c.

edsn≤µsx(Bs(x, en))≤edsn+nε, (10)

edun≤µux(Bu(x, en))≤edun+nε. (11)

d.

Panan(x)⊂B(x, en)P(x), (12)

P0an(x)∩ξs(x)⊂Bs(x, en)P(x)∩ξs(x), (13)

Pan0 (x)∩ξu(x)⊂Bu(x, en)P(x)∩ξu(x), (14)

whereais the integer part of 2(1 +ε) max{−λ1, λk,1}. e. Define Qn(x) by

(15) Qn(x) =[

Panan(y) where the union is taken over all y∈Γ for which

Pan0 (y)∩Bu(x,2en)6=6°and P0an(y)∩Bs(x,2en)6=6°; then

(16) B(x, en)Γ⊂Qn(x)⊂B(x,4en), and for each y∈Qn(x),

Panan(y)⊂Qn(x).

Increasing n0 if necessary we may also assume that f. For everyx∈Γ andn≥n0,

Bs(x, en)Γ⊂Qn(x)∩ξs(x)⊂Bs(x,4en), (17)

Bu(x, en)Γ⊂Qn(x)∩ξu(x)⊂Bu(x,4en).

(18)

The above statements are slightly different versions of statements in [16].

Property (5) essentially follows from the Shannon-McMillan-Breiman theorem applied to the partitionPwhile properties (6) and (7) follow from “leaf-wise”

versions of this theorem. The inequalities in (10) and (11) are easy conse- quences of the existence of the stable and unstable pointwise dimensions ds and du (see Prop. 2). Since the Lyapunov exponents at µ-almost every point are constant and equal toλ1,. . .,λk, the properties (12), (13), and (14) follow from (8), (9), and the choice ofa indicated above. The inclusions in (16) are based upon the continuous dependence of stable and unstable manifolds in the C1+α topology on the base point (in each Pesin set). We need the following well-known result.

(12)

766 LUIS BARREIRA, YAKOV PESIN, AND JORG SCHMELING¨

Proposition 3 (Borel density lemma). Let µ be a finite Borel measure andA⊂M a measurable set. Then for µ-almost everyx∈A,

rlim0

µ(B(x, r)∩A) µ(B(x, r)) = 1.

Furthermore, if µ(A) > 0 then, for each δ > 0, there is a set A with µ(∆)> µ(A)−δ,and a numberr0 >0 such that for allx∈and0< r < r0, we have

µ(B(x, r)∩A)≥ 1

2µ(B(x, r)).

It immediately follows from the Borel density lemma that one can choose an integer n1 n0 and a set Γb Γ of measure µ(Γ)b > 1−ε such that for everyn≥n1 and x∈Γ,b

µ(B(x, en)Γ) 1

2µ(B(x, en));

(19)

µsx(Bs(x, en)Γ) 1

2µsx(Bs(x, en));

(20)

µux(Bu(x, en)Γ) 1

2µux(Bu(x, en)).

(21)

We establish two additional properties of the partitions Pk0 and P0k. Proposition 4. There exists a positive constantD=D(Γ)b <1such that for everyk≥1 and x∈Γ,

µsx(Pk0(x)Γ)≥D;

µux(P0k(x)Γ)≥D.

Proof. By (8), for every k≥1 and x Γ, the set Pk0(x)Γ contains the setBs(x, en0)Γ. It follows from (20) and (10) that

µsx(Pk0(x)Γ) 1

2µsx(Bs(x, en0)) 1

2edsn0n0εdef=D.

The second inequality in the proposition can be proved in a similar fashion using the properties (9), (21), and (11).

The next statement establishes the property of the partitionPwhich sim- ulates the well-known Markov property.

Proposition 5. For every x∈Γ andn≥n0, Panan(x)∩ξs(x) =P0an(x)∩ξs(x);

Panan(x)∩ξu(x) =Pan0 (x)∩ξu(x).

(13)

Proof. It follows from (13) and (8) that

P0an(x)∩ξs(x)P0an(x)∩Bs(x, en)P0an(x)∩Bs(x, en0)

P0an(x)Pan0 (x)∩ξs(x) =Panan(x)∩ξs(x).

SincePanan(x)P0an(x), this completes the proof of the first identity. The proof of the other identity is similar.

5. Preparatory lemmata

Fix x Γ and an integerb n≥ n1. We consider the following two classes R(n) and F(n) of elements of the partition Panan (we call these elements “rect- angles”):

R(n) ={Panan(y)P(x) :Panan(y)Γ6=6°};

F(n) ={Panan(y)P(x) :P0an(y)6=6°and Pan0 (y)6=6°}.

The rectangles inR(n) carry all the measure of the setP(x)Γ, i.e., X

RR(n)

µ(R∩Γ) =µ(P(x)Γ).

Obviously, the rectangles in R(n) that intersect Γ belong tob F(n). If these were the only ones in F(n), the measure µ|P(x)Γ would have the “direct product structure” at the “level” n. One could then use the approach in [12], [20] to estimate the measure of a ball by the product of its stable and unstable measures. In the general case, the rectangles in the class F(n) are obtained from the rectangles in R(n) (that intersect Γ) by “filling in” the gaps in theb

“product structure” (see Fig. 1).

We wish to compare the number of rectangles inR(n) andF(n) intersect- ing a given set. This will allow us to evaluate the deviation of the measure µ from the direct product structure at the level n. Our main observation is that for “typical” points y Γ the number of rectangles from the classb R(n) intersectingWs(y) (respectivelyWu(y)) is “asymptotically” the same up to a factor that grows at most subexponentially withn.

However, in general, the distribution of these rectangles along Ws(y) (re- spectivelyWu(y)) may be different for different points y. This causes a devi- ation from the direct product structure. We will use a simple combinatorial argument to show that this deviation grows at most subexponentially withn.

One can then say that the measureµhas an “almost direct product structure.”

(14)

768 LUIS BARREIRA, YAKOV PESIN, AND JORG SCHMELING¨

R( )n F( )n R( )n F( )n F( )n R( )n ( )

W x

x

s

( ) W xu

Figure 1. The procedure of “filling in” rectangles.

To effect this, for each set A⊂P(x), we define N(n, A) = card{R R(n) :R∩A6=6°},

Ns(n, y, A) = card{R R(n) :R∩ξs(y)Γ∩A6=6°}, Nu(n, y, A) = card{R R(n) :R∩ξu(y)Γ∩A6=6°}, Nbs(n, y, A) = card{R F(n) :R∩ξs(y)∩A6=6°}, Nbu(n, y, A) = card{R F(n) :R∩ξu(y)∩A6=6°}.

Note thatN(n,P(x)) is the cardinality of the setR(n), andNs(n, y,P(x)) (respectivelyNu(n, y,P(x))) is the number of rectangles inR(n) that intersect Γ and the stable (respectively unstable) local manifold at y. The product Nbs(n, y,P(x))×Nbu(n, y,P(x)) is the cardinality of the setF(n) for a “typical”

pointy∈P(x).

Let Qn(y) be the set defined by (15).

Lemma 1. For each y∈P(x)Γ and integer n≥n0,we have:

Ns(n, y, Qn(y))≤µsy(Bs(y,4en))·Ceanh+anε; Nu(n, y, Qn(y))≤µuy(Bu(y,4en))·Ceanh+anε.

(15)

Proof. It follows from (17) that

µsy(Bs(y,4en))≥µsy(Qn(y))≥Ns(n, y, Qn(y))

·minsy(R) :R∈R(n) andR∩ξs(y)∩Qn(y)Γ6=6°}.

(Note that the conditionR∩ξs(y)∩Qn(y)6=6°implies that R R(n).) Let z R∩ξs(y)∩Qn(y)Γ for some R R(n). By Proposition 5 we obtain µsy(R) = µsy(P0an(z)) = µsz(P0an(z)). The first inequality in the lemma follows now from (6). The proof of the second inequality is similar.

Lemma 2. For each y∈P(x)Γb and integer n≥n1, µ(B(y, en))≤N(n, Qn(y))·2Ce2anh+2anε. Proof. It follows from (19) and (16) that

1

2µ(B(y, en))≤µ(B(y, en)Γ)≤µ(Qn(y)Γ)

≤N(n, Qn(y))·max{µ(R) :R∈R(n) and R∩Qn(y)6=6°}.

(Note that the conditionR∩Qn(y)6=6°implies thatR∈R(n).) The desired inequality follows from (5).

We now estimate the number of rectangles in the classes R(n) andF(n).

Lemma 3. Forµ-almost everyy P(x)there is an integern2(y)≥n1

such that for eachn≥n2(y), we have:

N(n+ 2, Qn+2(y))≤Nbs(n, y, Qn(y))·Nbu(n, y, Qn(y))·2C2e4a(h+ε)e4anε. Proof. By the Borel density lemma (withA=bΓ), forµ-almost everyy∈Γb there is an integern2(y)≥n1 such that for alln≥n2(y),

2µ(B(y, en)Γ)b ≥µ(B(y, en)).

SinceΓbΓ, it follows from (16) that for all n≥n2(y),

2µ(Qn(y)Γ)b 2µ(B(y, en)Γ)b ≥µ(B(y, en)) (22)

≥µ(B(y,4en2))≥µ(Qn+2(y)).

For anym≥n2(y), by (5) and property (e), we have µ(Qm(y)) = X

Pamam(z)Qm(y)

µ(Pamam(z))≥N(m, Qm(y))·C1e2amh2amε. Similarly, for everyn≥n2(y), we obtain

µ(Qn(y)bΓ) = X Panan(z)Qn(y)

µ(Panan(z)Γ)b ≤Nn·Ce2anh+2anε,

(16)

770 LUIS BARREIRA, YAKOV PESIN, AND JORG SCHMELING¨

where Nn is the number of rectangles Panan(z) R(n) that have nonempty intersection withbΓ.

Set m=n+ 2. The last two inequalities together with (22) imply that (23) N(n+ 2, Qn+2(y))≤Nn·2C2e4a(h+ε)+4anε

.

On the other hand, sincey Γ the intersectionsb Pan0 (y) ξu(y) Γ andb P0an(y) ξs(y) Γ are nonempty.b

Consider a rectangle Panan(v) Qn(y) that has nonempty intersection withbΓ. Then the rectangles P0an(v)Pan0 (y) and P0an(y)Pan0 (v) are in F(n) and intersect respectively the stable and unstable local manifolds aty. Hence, to any rectanglePanan(v) ⊂Qn(y) with nonempty intersection with Γ, one canb associate the pair of rectangles (P0an(v)Pan0 (y),P0an(y)Pan0 (v)) in

{R F(n) :R∩ξs(y)∩Qn(y)6=6°} × {R∈F(n) :R∩ξu(y)∩Qn(y)6=6°}.

Clearly this correspondence is injective. Therefore,

Nbs(n, y, Qn(y))·Nbu(n, y, Qn(y))≥Nn. The desired inequality follows from (23).

Our next goal is to compare the growth rate in nof the number of rectan- gles inF(n) with the number of rectangles inR(n). We start with an auxiliary result.

Lemma 4. For each x∈and integer n≥n1, we have:

Nbs(n, x,P(x))≤D1C2eanh+3anε; Nbu(n, x,P(x))≤D1C2eanh+3anε.

Proof. Since the partition Pis countable we can find pointsyi such that the union of the rectanglesPan0 (yi) isP(x), and these rectangles are mutually disjoint. Without loss of generality we can assume that yi bΓ whenever Pan0 (yi)6=6°. We have

(24) N(n,P(x))X

i

Ns(n, yi,Pan0 (yi)) X

i:Pan0 (yi)bΓ6=6°

Ns(n, yi,Pan0 (yi)).

(17)

We now estimateNs(n, yi,Pan0 (yi)) from below for yi Γ. By Propositions 4b and 5, and (6),

Ns(n, yi,Pan0 (yi)) µsyi(Pan0 (yi)Γ)

maxsz(Panan(z)) :z∈ξs(yi)P(x)Γ} (25)

D

maxsz(Panan(z)) :z∈ξs(yi)P(x)Γ}

= D

maxsz(P0an(z)) :z∈ξs(yi)P(x)Γ}

≥DC1eanhanε. Similarly (5) implies that

(26) N(n,P(x)) µ(P(x))

min{µ(Panan(z)) :z∈P(x)Γ} ≤Ce2anh+2anε. We now observe that

(27) Nbu(n, x,P(x)) = card{i:Pan0 (yi)6=6°}. Putting (24), (25), (26), and (27) together we conclude that

Ce2anh+2anε ≥N(n,P(x))

X

i:Pan0 (yi)bΓ6=6°

Ns(n, yi,Pan0 (yi))

≥Nbu(n, x,P(x))·DC1eanhanε.

This yields Nbu(n, x,P(x)) D1C2eanh+3anε. The other inequality can be proved in a similar way.

We emphasize that the procedure of “filling in” rectangles to obtain the class F(n) may substantially increase the number of rectangles in the neigh- borhood of some points. However, the next lemma shows that this procedure of “filling in” does not add too many rectangles at almost every point.

Lemma 5. For µ-almost every y∈P(x)bΓ,we have:

nlim+

Nbs(n, y, Qn(y))

Ns(n, y, Qn(y))e7anε<1;

nlim+

Nbu(n, y, Qn(y))

Nu(n, y, Qn(y))e7anε<1.

Proof. By (17) and (20), for eachn≥n1 andy Γ,b µsy(Qn(y))≥µsy(Bs(y, en)Γ) 1

2µsy(Bs(y, en)).

(18)

772 LUIS BARREIRA, YAKOV PESIN, AND JORG SCHMELING¨

SincePanan(z)P0an(z) for everyz, by virtue of (6) and (10) we obtain Ns(n, y, Qn(y)) µsy(Qn(y))

maxsz(Panan(z)) :z∈ξs(y)P(x)Γ} (28)

1 2

µsy(Bs(y, en))

maxsz(P0an(z)) :z∈ξs(y)P(x)Γ}

1 2C

edsn eanh+anε. Let us consider the set

F = (

y∈bΓ : lim

n+

Nbs(n, y, Qn(y))

Ns(n, y, Qn(y))e7anε 1 )

.

For each y ∈F there exists an increasing sequence {mj}j=1 ={mj(y)}j=1 of positive integers such that

Nbs(mj, y, Qmj(y)) 1

2Ns(mj, y, Qmj(y))e7amjε (29)

1

4Cedsmj+amjh+5amjε for allj (note thata >1).

We wish to show that µ(F) = 0. Assume on the contrary thatµ(F)>0.

LetF0⊂F be the set of points y∈F for which there exists the limit

rlim0

logµsy(Bs(y, r)) logr =ds.

Clearlyµ(F0) =µ(F)>0. Then we can find y∈F such that µsy(F) =µsy(F0) =µsy(F0P(y)∩ξs(y))>0.

It follows from Frostman’s lemma that

(30) dimH(F0∩ξs(y)) =ds. Let us consider the countable collection of balls

B={B(z,4emj(z)) :z∈F0∩ξs(y);j= 1, 2, . . .}.

By the Besicovitch covering lemma (see, for example, [3]) one can find a sub- cover C B of F0∩ξs(y) of arbitrarily small diameter and finite multiplic- ity ρ = ρ(dimM). This means that for any L > 0 one can choose a se- quence of points{zi ∈F0∩ξs(y)}i=1 and a sequence of integers{ti}i=1, where ti∈ {mj(zi)}j=1 andti > L for each i, such that the collection of balls

C={B(zi,4eti) :i= 1, 2, . . .}

参照

関連したドキュメント

This result allows us to [partially] generalize combinato- rial cuspidalization results obtained in previous papers to the case of outer automorphisms of pro-Σ fundamental groups

We consider the Cauchy problem periodic in the spatial variable for the usual cubic nonlinear Schrödinger equation and construct an infinite sequence of invariant mea- sures

We consider the Cauchy problem periodic in the spatial variable for the usual cubic nonlinear Schrödinger equation and construct an infinite sequence of invariant mea- sures

We study the classical invariant theory of the B´ ezoutiant R(A, B) of a pair of binary forms A, B.. We also describe a ‘generic reduc- tion formula’ which recovers B from R(A, B)

Our basic strategy for the construction of an invariant measure is to show that the following “one force, one solution ” principle holds for (1.4): For almost all ω, there exists

In secgion 3 we prove a general theorem which ensures ghe exisgence and uniqueness of invarian measures for McKean-Vlasov nonlinear stochastic differential equations.. In section 4

In particular, each path in B(γ, ) is nonconstant. Hence it is enough to show that S has positive Q–dimensional Hausdorff measure.. According to Lemma 2.8 we can choose L ≥ 2 such

The case when the space has atoms can easily be reduced to the nonatomic case by “putting” suitable mea- surable sets into the atoms, keeping the values of f inside the atoms