C^r surface diffeomorphisms with no maximal entropy measure

TL;DR

This paper constructs specific smooth surface diffeomorphisms that lack any measure of maximal entropy, challenging assumptions about the existence of such measures in dynamical systems.

Contribution

It provides explicit examples of $C^r$-diffeomorphisms on surfaces with no measure of maximal entropy, extending the understanding of entropy measures in smooth dynamics.

Findings

Existence of $C^r$-diffeomorphisms with no maximal entropy measure

Construction on the disk extends to any surface manifold

Advances the understanding of entropy measures in smooth dynamical systems

Abstract

For any $1\leq r<\infty$, we build on the disk and therefore on any manifold, a $C^r$-diffeomorphism with no measure of maximal entropy.