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

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 $P^{\ast }(\Phi ):={\sup }_{\mu }\{h(\mu )+\mu (\Phi )\}$ be a variational pressure defined over a suitable class of Borel measurable potentials and prove that, for regular nonuniformly hyperbolic systems, $P^{\ast }(\Phi )={\sup }_{\Omega }{P}^{\ast }(f|\Omega ,\Phi )$, 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.

arXiv commentThis paper has been withdrawn by the author because a crucial gap in the proof of Theorem 1

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