Equilibrium states and SRB-like measures of $C^1$-expanding maps of the circle

Abstract

For any C^1 -expanding map f of the circle we study the equilibrium states for the potential \psi=-\log |f'| . We formulate a C^1 generalization of Pesin’s Entropy Formula that holds for all the Sinai–Ruelle–Bowen (SRB) measures if they exist, and for all the (necessarily existing) SRB-like measures. In the C^1 -generic case Pesin’s Entropy Formula holds for a unique SRB measure which is not absolutely continuous with respect to Lebesgue. The result also stands in the non-generic case for which no SRB measure exists.