Topological pressure via saddle points

Let $\mathrm{\Lambda }$ 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 $P_{\mathrm{t}\mathrm{o}\mathrm{p}}(\varphi )$ (with respect to the dynamical system $f\mathrm{\mid }\mathrm{\Lambda }$) 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 $\varphi$ is entirely determined by the values of $\varphi$ on the saddle points of $f$ in $\mathrm{\Lambda }$. 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 $\mathrm{\Lambda }$. Our results generalize several well-known results to certain non-uniformly hyperbolic systems.