Existence of measures of maximal entropy for Cr interval maps

Abstract

We show that a Cr (r>1) map of the interval f:[0,1][0,1] with topological entropy larger than logfr admits at least one measure of maximal entropy. Moreover the number of measures of maximal entropy is finite. It is a sharp improvement of the 2006 paper of Buzzi and Ruette in the case of Cr maps and solves a conjecture of J. Buzzi stated in his 1995 thesis. The proof uses a variation of a theorem of isomorphism due to J. Buzzi between the interval map and the Markovian shift associated to the Buzzi-Hofbauer diagram.

Publication
Proc. Amer. Math. Soc. 142 (2014), no. 3, 957-968
David Burguet
David Burguet
CNRS Researcher