Variational principles and approximation of dynamical indicators for systems with nonuniformly hyperbolic behavior

TL;DR

This paper develops a variational principle for approximating dynamical indicators like pressure and Lyapunov exponents in nonuniformly hyperbolic systems, using invariant basic sets and horseshoes.

Contribution

It introduces a variational principle that relates pressures in nonuniformly hyperbolic systems to those in uniformly hyperbolic basic sets, enabling better approximation of dynamical indicators.

Findings

Proves that $P^*(\

Establishes equality of variational pressure over the whole system and basic sets.

Applies the principle to approximate indicators using horseshoes.

Abstract

This note is concerned with approximation of dynamical indicators as pressures, Lyapunov exponents and dimension-like quantities, in systems with nonuniformly hyperbolic behavior. For this we let $P^*(\Phi) := \sup_{\mu}\{h(\mu) + \mu(\Phi)\}$ be a variational pressure defined over a suitable class of Borel measurable potentials and prove that, for regular nonuniformly hyperbolic systems, $P^*(\Phi) = \sup_{\Omega}P^*(f|\Omega,\Phi)$, supremum taken over the family of $f$-invariant uniformly hyperbolic basic sets. We then apply this variational principle to the approximation of dynamical indicators by horseshoes.