Mixing C r maps of the interval without maximal measure

Israel Journal of Math., 127, 253-277, 2002.


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.

Paper: [arXiv:1901.00325] [pdf]