On the approximation of dynamical indicators in systems with nonuniformly hyperbolic behavior

TL;DR

This paper demonstrates how to approximate dynamical indicators like topological pressure for systems with nonuniform hyperbolic behavior using sequences of basic sets, bridging the gap between hyperbolic measures and topological invariants.

Contribution

It introduces a method to approximate topological pressure for nonuniformly hyperbolic systems via basic sets, extending the understanding of dynamical indicators in complex systems.

Findings

Approximation of pressure for hyperbolic measures using basic sets

Existence of sequences of basic sets converging to free energy

Identification of potential classes with converging pressure sequences

Abstract

Let $f$ be a $C^{1+\alpha}$ diffeomorphism of a compact Riemannian manifold and $\mu$ an ergodic hyperbolic measure with positive entropy. We prove that for every continuous potential $\phi$ there exists a sequence of basic sets $\Omega_n$ such that the topological pressure $P(f|\Omega_n,\phi)$ converges to the free energy $P_{\mu}(\phi) = h(\mu) + \int\phi{d\mu}$. Then we introduce a class of potentials $\phi$ for which there exists sequence of basic sets $\Omega_n$ such that $P(f|\Omega_n,\phi) \to P(\phi)$.