Dimension approximation for diffeomorphisms preserving hyperbolic SRB measures

2022/02/23 by Wang, Juan, Qu, Congcong, Cao, Yongluo
#37C45 #37D20 #37D35 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2202.11395

Abstract

For a C^1+α diffeomorphism f preserving a hyperbolic ergodic SRB measure μ, Katok's remarkable results assert that μcan be approximated by a sequence of hyperbolic sets \Λ_n\_n≥1. In this paper, we prove the Hausdorff dimension for Λ_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 μcan be approximated by the Hausdorff dimension of Λ_n. To establish these results, we utilize the u-Gibbs property of the conditional measure of the equilibrium measure of -ψ^s(⋅,fⁿ) and the properties of the uniformly hyperbolic dynamical systems.

Related

• Stable sets, hyperbolicity and dimension
• Dimension approximation in smooth dynamical systems
• Local stable and unstable sets for positive entropy C¹ dynamical systems
• Dimension, entropy and Lyapunov exponents
• Emergence for diffeomorphisms with nonzero Lyapunov exponents