Dimension approximation for diffeomorphisms preserving hyperbolic SRB measures

DOI10.1016/J.JDE.2022.07.041arXiv2202.11395MaRDI QIDQ6391934

Publication date: 23 February 2022

Abstract: For a C^{1+alpha} diffeomorphism f preserving a hyperbolic ergodic SRB measure mu, Katok's remarkable results assert that mu can be approximated by a sequence of hyperbolic sets {Lambda_n}_{ngeq1}. In this paper, we prove the Hausdorff dimension for Lambda_n on the unstable manifold tends to the dimension of the unstable manifold. Furthermore, if the stable direction is one dimension, then the Hausdorff dimension of mu can be approximated by the Hausdorff dimension of Lambda_n. To establish these results, we utilize the u-Gibbs property of the conditional measure of the equilibrium measure of -psi^{s}(cdot,f^n) and the properties of the uniformly hyperbolic dynamical systems.

Mathematics Subject Classification ID

Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) (37D20) Thermodynamic formalism, variational principles, equilibrium states for dynamical systems (37D35) Dynamical systems with hyperbolic orbits and sets (37D05) Dimension theory of smooth dynamical systems (37C45)

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