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For any -expanding map of the circle we study the equilibrium states for the potential . We formulate a 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 -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.
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Eleonora Catsigeras, Heber Enrich, Equilibrium states and SRB-like measures of -expanding maps of the circle. Port. Math. 69 (2012), no. 3 pp. 193–212