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

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.