# Mixing $C^r$ maps of the interval without maximal measure

## TL;DR

This paper constructs a $C^r$ interval map that is topologically mixing but lacks a measure of maximal entropy, demonstrating the necessity of high smoothness for such measures.

## Contribution

It provides a $C^r$ example of a mixing map without a maximal measure, highlighting the importance of smoothness in entropy measure existence.

## Key findings

- Constructed a $C^r$ map with no maximal measure
- Showed smoothness is crucial for measure existence
- Computed the local entropy of the example

## Abstract

We construct a $C^r$ transformation of the interval (or the torus) which is topologically mixing but has no invariant measure of maximal entropy. Whereas the assumption of $C^{\infty}$ ensures existence of maximal measures for an interval map, it shows we cannot weaken the smoothness assumption. We also compute the local entropy of the example.