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

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.

arXiv commentPublished in 2002. 25 pages