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 $\psi =-\log |f^{\prime }|$. 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.