Topological pressure via saddle points

Abstract

Let $Λ$ be a compact locally maximal invariant set of a $C^2$-diffeomorphism $f:M\to M$ on a smooth Riemannian manifold $M$. In this paper we study the topological pressure $P_{\rm top}(ϕ)$ (with respect to the dynamical system $f|Λ$) for a wide class of Hölder continuous potentials and analyze its relation to dynamical, as well as geometrical, properties of the system. We show that under a mild nonuniform hyperbolicity assumption the topological pressure of $ϕ$ is entirely determined by the values of $ϕ$ on the saddle points of $f$ in $Λ$. Moreover, it is enough to consider saddle points with ``large'' Lyapunov exponents. We also introduce a version of the pressure for certain non-continuous potentials and establish several variational inequalities for it. Finally, we deduce relations between expansion and escape rates and the dimension of $Λ$. Our results generalize several well-known results to certain non-uniformly hyperbolic systems.