Large entropy implies existence of a maximal entropy measure for interval maps

Abstract

We give a new type of sufficient condition for the existence of measures with maximal entropy for an interval map $f$, using some non-uniform hyperbolicity to compensate for a lack of smoothness of $f$. More precisely, if the topological entropy of a $C^1$ interval map is greater than the sum of the local entropy and the entropy of the critical points, then there exists at least one measure with maximal entropy. As a corollary, we obtain that any $C^r$ interval map $f$ such that $h_{\mathrm{top}}(f)>2\log \Vert f^{\prime }{\Vert }_{\infty }/r$ possesses measures with maximal entropy.

Published in 2006