Dimension approximation for diffeomorphisms preserving hyperbolic SRB measures

TL;DR

This paper investigates how the Hausdorff dimension of hyperbolic sets approximates the dimension of the unstable manifold and SRB measure for certain diffeomorphisms, using properties of hyperbolic dynamics and Gibbs measures.

Contribution

It proves the Hausdorff dimension of hyperbolic approximations converges to the unstable manifold's dimension and extends this to SRB measures in one-dimensional stable directions.

Findings

Hausdorff dimension of \\Lambda_n on unstable manifold tends to the unstable dimension

Dimension of \\mu can be approximated by \\Lambda_n in one-dimensional stable case

Utilizes u-Gibbs property and hyperbolic system properties

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\}_{n\geq1}. 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.