A variational principle for systems with nonuniformly hyperbolic behavior with applications to the dimension theory

TL;DR

This paper develops a variational principle using nonadditive topological pressure for nonuniformly hyperbolic systems, providing new bounds on the dimension of stable and unstable Cantor sets.

Contribution

It introduces a nonadditive pressure framework and establishes a variational equation for nonuniformly hyperbolic diffeomorphisms, linking pressure to dimension estimates.

Findings

Proves a variational equation for nonuniformly hyperbolic systems using nonadditive pressure.

Provides lower bounds for the Cantor dimension of stable and unstable sets.

Extends thermodynamic formalism to nonuniform hyperbolicity contexts.

Abstract

Let $f$ be a $C^{1+\alpha}$ nonuniformly hyperbolic diffeomorphism. We use a a nonadditive version of the topological pressure of a class of admissible, possibly noncontinuous potentials $P^*(\Phi)$ to prove the following variational equation: $P^*(\Phi) = \sup_{\Omega \in {\mathcal H}}P^*(f|\Omega,\Phi)$ supremum taken over the set ${\mathcal H}$ of basic subsets in $M$. As a consequence we find a lower bound for the Cantor dimension of the stable and unstable Cantor sets of a non trivial conformal nonuniformly hyperbolic isolated sets.