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

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