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

Abstract

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: all ergodic weak-SRB-like measures satisfy Pesin's Entropy Formula for $C^1$ non-uniformly expanding maps. We show that for weak-expanding maps such that $\Leb$-a.e $x$ has positive frequency of hyperbolic times, then all the necessarily existing ergodic weak-SRB-like measures satisfy Pesin's Entropy Formula and are equilibrium states for the potential $\psi =-\log |\det Df|$. In particular, this holds for any $C^1$-expanding map and in this case the set of invariant probability measures that satisfy Pesin's Entropy Formula is the weak${}^{\ast }$-closed convex hull of the ergodic weak-SRB-like measures.

47 pages, 5 figures. Content updated to last accepted version to appear in Math. Z

References in corpus (3)

• Pesin Entropy Formula for C1 Diffeomorphisms with Dominated Splitting
• Equilibrium States and SRB-like measures of C1 Expanding Maps of the Circle
• Weak Pseudo-Physical Measures and Pesin's Entropy Formula for Anosov C1 diffeomorphisms