Topological pressure via saddle points
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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.
19 pages, Replaced with revised version, Accepted for publication in Trans. Amer. Math. Soc
19 pages, Replaced with revised version, Accepted for publication in Trans. Amer. Math. Soc