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

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 $P^{\ast }(\Phi )$ to prove the following variational equation: $P^{\ast }(\Phi )={\sup }_{\Omega \in \mathcal{H}}{P}^{\ast }(f|\Omega ,\Phi )$ 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.