Existence of measures of maximal entropy for 𝒞^{𝓇} interval maps

Abstract

We show that a C r \mathcal {C}^{r} ( r > 1 ) (r>1) map of the interval f : [ 0 , 1 ] → [ 0 , 1 ] f:[0,1]\rightarrow [0,1] with topological entropy larger than log ⁡ ‖ f ′ ‖ ∞ r \frac {\log \|f’\|_{\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 C r \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.