Existence of measures of maximal entropy for $C^r$ interval maps

Abstract

We show that a $ \mathcal {C}^{r}$ $ (r>1)$ map of the interval $ f:[0,1]\rightarrow [0,1]$ with topological entropy larger than $ \frac {\log \Vert f'\Vert _{\infty }}{r}$ 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 $ \mathcal {C}^r$ 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.