Dimension and product structure of hyperbolic measures

Luis Barreira, Yakov Pesin, Jörg Schmeling

Research output: Contribution to journalArticlepeer-review

95 Scopus citations

Abstract

We prove that every hyperbolic measure invariant under a C1 + α diffeomorphism 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.

Original languageEnglish (US)
Pages (from-to)755-783
Number of pages29
JournalAnnals of Mathematics
Volume149
Issue number3
DOIs
StatePublished - May 1999

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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