Topological pressure via saddle points

Abstract

Let $\Lambda$ 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 $P_{\rm top}(\phi)$ (with respect to the dynamical system $f|\Lambda$) for a wide class of H\"older 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 $\phi$ is entirely determined by the values of $\phi$ on the saddle points of $f$ in $\Lambda$. 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 $\Lambda$. Our results generalize several well-known results to certain non-uniformly hyperbolic systems.