Pesin's Entropy Formula for $C^1$ non-uniformly expanding maps

Vitor Araujo, Felipe Santos

Published 2017-07-13Version 1

We prove existence of equilibrium states with special properties for a class of distance expanding local homeomorphisms on compact metric spaces and continuous potentials. Moreover, we formulate a C$^1$ generalization of Pesin's Entropy Formula: for weak-expanding maps such that $\Leb$-a.e $x$ has positive frequency of hyperbolic times, then all the necessarily existing ergodic SRB-like measures satisfy Pesin's Entropy Formula. It follows that the same holds for $C^1$ non-uniformly expanding maps. Furthermore, all the necessarily existing SRB-like measures are equilibrium states for the potential $\psi=-\log|\det Df|$. In particular, this holds for any C$^1$-expanding map of finite dimensional compact Riemannian manifolds and in this case the set of invariant probability measures that satisfy Pesin's Entropy Formula is the weak$^*$-closed convex hull of the ergodic SRB-like measures. In this work no Markov structure is assumed.