Pesin's entropy formula for C1 non-uniformly expanding maps

摘要

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 C1 generalization of Pesin's Entropy Formula: all ergodic weak-SRB-like measures satisfy Pesin's Entropy Formula for C1 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 =-log|detDf|. In particular, this holds for any C1-expanding map 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 weak-SRB-like measures.