ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 2, Number 1, August 1996

ON THE POINTWISE DIMENSION OF HYPERBOLIC MEASURES:

A PROOF OF THE ECKMANN–RUELLE CONJECTURE

LUIS BARREIRA, YAKOV PESIN, AND JÖRG SCHMELING

(Communicated by Svetlana Katok)

Abstract. We prove the long-standing Eckmann–Ruelle conjecture in dimension theory of smooth dynamical systems. We show that the pointwise dimension exists almost everywhere with respect to a compactly supported Borel probability measure with non-zero Lyapunov exponents, invariant under a C^{1+α} diffeomorphism of a smooth Riemannian manifold.

Let M be a smooth Riemannian manifold without boundary, and f: M → M a C^{1+α} diffeomorphism of M for some α > 0. Also let μ be an f-invariant Borel probability measure on M with a compact support. Given a set Z ⊂ M, we denote respectively by dim_{H} Z, \underline{dim}_{B}Z, and \overline{dim}_{B}Z the Hausdorff dimension of Z and the lower and upper box dimensions of Z (see for example [F]). We will be mostly interested in subsets of positive measure invariant under f. To characterize their structure we use the notions of Hausdorff dimension of μ and lower and upper box dimensions of μ. We denote them by dim_{H}μ, \underline{dim}_{B}μ, and \overline{dim}_{B}μ, respectively. We have

dim_{H}μ = inf{dim_{H} Z | μ(Z) = 1},

\underline{dim}_{B}μ = lim_{δ→0} inf\{\underline{dim}_{B}Z | μ(Z) ≥ 1 − δ\},

\overline{dim}_{B}μ = lim_{δ→0} inf\{\overline{dim}_{B}Z | μ(Z) ≥ 1 − δ\}.

It follows from the definitions that

dim_{H}μ ≤ \underline{dim}_{B}μ ≤ \overline{dim}_{B}μ.

In [Y], Young found a criterion that guarantees the coincidence of the Hausdorff dimension and lower and upper box dimensions of measures. We define the lower

Received by the editors May 13, 1996. 1991 Mathematics Subject Classification. Primary 58F11, 28D05. Key words and phrases. Eckmann–Ruelle conjecture, hyperbolic measure, pointwise

dimension.

This paper was written while L. B. was on leave from Instituto Superior Técnico, Department of Mathematics, at Lisbon, Portugal, and J. S. was visiting Penn State. L. B. was supported by Program PRAXIS XXI, Fellowship BD 5236/95, JNICT, Portugal. J. S. was supported by the Leopoldina-Forderpreis. The work of Ya. P. was partially supported by the National Science Foundation grant #DMS9403723.

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and upper pointwise dimensions of μ at a point x ∈ M by

\underline{d}_{μ}(x) = \underline{lim}_{r→0} \frac{log μ(B(x, r))}{log r},

\overline{d}_{μ}(x) = \overline{lim}_{r→0} \frac{log μ(B(x, r))}{log r},

where B(x, r) denotes the ball of radius r centered at x.

Proposition 1. Let X be a complete metric space of finite topological dimension, and μ a Borel probability measure on X. Assume that

\underline{d}_{μ}(x) = \overline{d}_{μ}(x) = d (1)

for μ-almost every x ∈ X. Then

dim_{H} μ = \underline{dim}_{B}μ = \overline{dim}_{B}μ = d.

A measure μ which satisfies (1) is called exact dimensional. Since this result was established, it has become one of the most challenging problems in the interface of dimension theory and dynamical systems to describe measures which are exact dimensional.

In [Y], Young proved that an ergodic measure with non-zero Lyapunov exponents (such measures are called hyperbolic measures), invariant under a C^{1+α} diffeomorphism of a smooth compact surface, is exact dimensional. Ledrappier [L] proved exact dimensionality of hyperbolic Bowen–Ruelle–Sinai measures. In [PY], Pesin and Yue extended his result to hyperbolic measures satisfying the so-called semi-local product structure (this class includes Gibbs measures on locally maximal hyperbolic sets).

In [ER], Eckmann and Ruelle considered general hyperbolic measures and conjectured that they are exact dimensional. This statement has later become known as the Eckmann–Ruelle conjecture. In this paper we announce an affirmative solution of this conjecture.

We need some preliminary information on smooth dynamical systems with nonzero Lyapunov exponents. Given a point x ∈ M and a vector v ∈ T_{x}M, the Lyapunov exponent of v at x is defined by the formula

λ(x, v) = \overline{lim}_{n→∞} \frac{1}{n} log ∥d_{x}f^{n}v∥.

If x is fixed, then the function λ(x, ·) can take only finitely many values λ_{1}(x) ≥ ⋯ ≥ λ_{p}(x), where p = dim M. The functions λ_{i}(x) are measurable and invariant under f. The measure μ is said to be hyperbolic if λ_{i}(x) ≠ 0 for every i = 1, …, p, and for μ-almost every x ∈ M.

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

W^{s}(x) = \{y ∈ B(x, r(x)) | \overline{lim}_{n→+∞} \frac{1}{n} log d(f^{n}x, f^{n}y) < 0\} ,

W^{u}(x) = \{y ∈ B(x, r(x)) | \underline{lim}_{n→−∞} \frac{1}{n} log d(f^{n}x, f^{n}y) > 0\}

are immersed local manifolds called stable and unstable local manifolds at x. For each 0 < r < r(x) we consider the balls B^{s}(x, r) ⊂ W^{s}(x) and B^{u}(x, r) ⊂ W^{u}(x) centered at x with respect to the induced distances on W^{s}(x) and W^{u}(x), respectively.

In [LY], Ledrappier and Young constructed two measurable partitions ξ^{s} and ξ^{u} of M such that for μ-almost every x ∈ M:

1. ξ^{s}(x) ⊂ W^{s}(x) and ξ^{u}(x) ⊂ W^{u}(x);

2. ξ^{s}(x) and ξ^{u}(x) contain the intersection of an open neighborhood of x with

W^{s}(x) and W^{u}(x), respectively. We denote the system of conditional measures of μ with respect to the partitions ξ^{s} and ξ^{u}, respectively, by μ_{x}^{s} and μ_{x}^{u}, and for any measurable set A ⊂ M we write μ_{x}^{s}(A) = μ_{x}^{s}(A ∩ ξ^{s}(x)) and μ_{x}^{u}(A) = μ_{x}^{u}(A ∩ ξ^{u}(x)).

In [LY], Ledrappier and Young introduced the quantities

d_{μ}^{s}(x) =^{def} lim_{r→0} \frac{log μ_{x}^{s}(B^{s}(x, r))}{log r},

d_{μ}^{u}(x) =^{def} lim_{r→0} \frac{log μ_{x}^{u}(B^{u}(x, r))}{log r}

provided that the corresponding limits exist at x ∈ M. We call them, respectively, the stable and unstable pointwise dimensions of μ at x.

Proposition 2 [LY]. 1. For μ-almost every x ∈ M the limits d_{μ}^{s}(x) and d_{μ}^{u}(x) exist μ-almost everywhere.

2. If μ is a hyperbolic measure, then for μ-almost every x ∈ M,

\overline{d}_{μ}(x) ≤ d_{μ}^{s}(x) + d_{μ}^{u}(x).

In this paper we announce the following statement.

Theorem. Let f be a C^{1+α} diffeomorphism for some α > 0 on a smooth Riemannian manifold without boundary, and μ an f-invariant compactly supported Borel probability measure. If μ is hyperbolic, then it is exact dimensional and its pointwise dimension is equal to the sum of the stable and unstable pointwise dimensions, i.e., for μ-almost every x ∈ M,

\underline{d}_{μ}(x) = \overline{d}_{μ}(x) = d_{μ}^{s}(x) + d_{μ}^{u}(x).

We notice that using the ergodic decomposition of the measure one can reduce the proof to the case when μ is an ergodic measure. In this case the proof of the theorem is based upon a special countable partition 𝔓 of M constructed by Ledrappier and Young in [LY]. This partition simulates Markov partitions for hyperbolic sets. Let 𝔓(x) be the element of the partition 𝔓 containing x ∈ M. The elements of the “shift” partition 𝔓_{n} = ⋁_{k=−n}^{n} f^{−k}𝔓 decompose 𝔓(x) into finitely many subsets called rectangles, since they have the “direct product structure”: for any z, y ∈ 𝔓_{n}(x) the intersection W^{s}(y) ∩ W^{u}(z) consists of a unique point which belongs to 𝔓_{n}(x). The number of rectangles which have positive measure is asymptotically equal to exp(nh), where h is the measure-theoretic entropy of f with respect to μ. We thoroughly study the distribution of these rectangles inside 𝔓(x). In general, it does not reproduce the direct product structure. For “typical” points y ∈ 𝔓(x) the number of rectangles intersecting W^{s}(y) (respectively, W^{u}(y)) is “asymptotically” the same up to a subexponential factor, but their distribution along W^{s}(y) (respectively, W^{u}(y)) may differ from point to point, causing a deviation from direct product structure. A simple combinatorial argument is used to show that the deviation can grow at most with a subexponential rate in n. We then estimate the measure of a ball by the product of its stable and unstable measures.

Remarks. 1. Let us point out that neither of the assumptions of the theorem can be omitted. Ledrappier and Misiurewicz [LM] constructed an example of a smooth map of a circle preserving an ergodic measure with zero Lyapunov exponent which is not exact dimensional. In [PW], Pesin and Weiss presented an example of a Hölder homeomorphism whose measure of maximal entropy is not exact dimensional.

2. It follows immediately from the theorem that the pointwise dimension of an ergodic invariant measure supported on a (uniformly) hyperbolic locally maximal 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, and in general the measure need not have a local product structure despite the fact that the set itself does. This illustrates that the theorem is non-trivial even for measures supported on hyperbolic locally maximal sets.

References

[ER] J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Modern Phys. 57 (3) (1985), 617–656. MR 87d:58083a

[F] K. Falconer, Fractal geometry. Mathematical foundations and applications, John Wiley & Sons, 1990. MR 92j:28008

[L] F. Ledrappier, Dimension of invariant measures, Proceedings of the conference on ergodic theory and related topics, II (Georgenthal, 1986), Teubner-Texte Math., vol. 94, Leipzig, 1987, pp. 116–124. MR 89b:58120

[LM] F. Ledrappier and M. Misiurewicz, Dimension of invariant measures for maps with exponent zero, Ergod. Theory and Dyn. Syst. 5 (1985), 595–610. MR 87j:58058

[LY] F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin’s entropy formula, Ann. of Math. (2) 122 (3) (1985), 509– 539; The metric entropy of diffeomorphisms. II. Relations between entropy, exponents and dimension, Ann. of Math. (2) 122 (3) (1985), 540–574. MR 87i:58101a, b

[PW] Ya. Pesin and H. Weiss, On the dimension of deterministic and random Cantor-like sets, symbolic dynamics, and the Eckmann–Ruelle conjecture, Comm. Math. Phys. (to appear).

[PY] Ya. Pesin and C. Yue, The Hausdorff dimension of measures with non-zero Lyapunov exponents and local product structure, PSU preprint.

[Y] L.-S. Young, Dimension, entropy and Lyapunov exponents, Ergodic Theory Dynam. Systems 2 (1982), 109–124. MR 84h:58087

Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, U.S.A.

   E-mail address: luis@math.psu.edu

Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, U.S.A.

   E-mail address: pesin@math.psu.edu

Weierstrass Institute of Applied Analysis and Stochastics, Mohrenstrasse 39, D– 10117 Berlin, Germany

   E-mail address: schmeling@wias-berlin.de