Large topological entropy implies existence of measures of maximal entropy: the case of interval maps

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

Discrete Contin. Dyn. Syst. Ser. A, 14 (4), 673-688, 2006.

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 C¹ 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 $h_{top}(f) > \frac{2}{r}\log \| f'\|_\infty$ possesses measures with maximal entropy.

Paper: [arXiv:1901.01073] [pdf (published paper)]