Pressure and exponential rate over periodic orbits

Abstract

For a measure mu preserved by a C (1+alpha 0) (alpha(0) > 0) diffeomorphism f and a continuous function phi on the manifold, we study the relationship between the exponential growth rate of Sigma e(Sn phi(x)) over the orbits of some periodic points and the free energy h(mu)(f) + integral phi d mu (in certain cases, it is equal to the measure theoretic pressure) or the topological pressure. When mu is an ergodic hyperbolic measure, we prove that the exponential growth rate coincides with the free energy (measure theoretic pressure). And we also verify the equality of the exponential growth rate and the topological pressure when the manifold is 2-dimensional. However, for the higher-dimensional manifold, we show an inequality between the exponential growth rate and the topological pressure. For an ergodic hyperbolic measure omega, we also prove that there is a omega-full measured set (Lambda) over tilde such that for every f-invariant measure supported on (Lambda) over tilde, the exponential growth rate equals to the free energy. And moreover, we prove that there is another omega-full measured set (Delta) over tilde such that for every f-invariant measure supported on (Delta) over tilde, the exponential growth rate equals to the topological pressure. (C) 2020 Published by Elsevier Inc.