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

Jérôme Buzzi; Sylvie Ruette

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

Dynamical Systems 2019-01-07 v1

Authors: Jérôme Buzzi , Sylvie Ruette

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

Keywords

interval maps maximal entropy measure topological entropy non-uniform hyperbolicity critical points

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@article{arxiv.1901.01073,
  title  = {Large entropy implies existence of a maximal entropy measure for interval maps},
  author = {Jérôme Buzzi and Sylvie Ruette},
  journal= {arXiv preprint arXiv:1901.01073},
  year   = {2019}
}

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Published in 2006

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

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