Equilibrium States and SRB-like measures of C1 Expanding Maps of the Circle

TL;DR

This paper investigates equilibrium states and SRB-like measures for C1 expanding maps on the circle, generalizing Pesin's Entropy Formula and analyzing measure uniqueness and existence in various generic and non-generic scenarios.

Contribution

It introduces a C1 generalization of Pesin's Entropy Formula applicable to all SRB-like measures and characterizes measure uniqueness in generic cases.

Findings

Pesin's Entropy Formula holds for unique SRB measures in generic cases.

The formula applies to all SRB-like measures regardless of their existence.

In non-generic cases, no SRB measure exists, but the formula still applies.

Abstract

For any C1 expanding map f of the circle we study the equilibrium states for the potential -log |f'|. We formulate a C1 generalization of Pesin's Entropy Formula that holds for all the SRB measures if they exist, and for all the (necessarily existing) SRB-like measures. In the C1-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.