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

## TL;DR

This paper establishes a new sufficient condition for the existence of maximal entropy measures in interval maps, linking topological entropy, local entropy, and critical point entropy, especially for non-smooth maps.

## Contribution

It introduces a novel criterion based on non-uniform hyperbolicity to guarantee maximal entropy measures for $C^1$ interval maps.

## Key findings

- Existence of maximal entropy measures when topological entropy exceeds local and critical point entropy sums.
- For $C^r$ maps, maximal entropy measures exist if topological entropy surpasses a specific bound involving the derivative.
- Provides a new perspective on entropy and measure existence in non-smooth dynamical systems.

## 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.