Dimension approximation for diffeomorphisms preserving hyperbolic SRB measures

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.