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

We show that a C^r (r > 1) map of the interval f: [0, 1] → [0, 1] with topological entropy larger than 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 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.