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Dimension and product structure of hyperbolic measures. (English) Zbl 0977.37011

The main object studied in the paper is a \(C^{1+\alpha}\) diffeomorphism \(T\) of a compact smooth Riemannian manifold without boundary. The authors prove that every \(T\)-invariant hyperbolic measure possesses asymptotically “almost” local product structure in the sense that its density can be approximated by the product of densities on stable and unstable invariant manifolds. This property is used in the paper to prove the long-standing Eckmann-Ruelle conjecture claiming that the pointwise dimension of every hyperbolic measure invariant under a \(C^{1+\alpha}\) diffeomorphism exists almost everywhere. As a corollary this result implies that a number of important dimension type characteristics of the measure, such as the Hausdorff dimension, box and information dimensions, etc., coincide.

MSC:

37C45 Dimension theory of smooth dynamical systems
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
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