Rotation set and Entropy

TL;DR

This paper establishes a converse to a known theorem, showing that for certain 2-torus diffeomorphisms with positive entropy, the rotation set's interior is non-empty, given specific dynamical conditions.

Contribution

It proves that under topological transitivity and irreducibility, positive entropy implies a non-empty interior of the rotation set for 2-torus diffeomorphisms.

Findings

Interior of rotation set is non-empty for certain diffeomorphisms with positive entropy

Topological transitivity and irreducibility are necessary conditions

Provides examples illustrating the necessity of these conditions

Abstract

In 1991 Llibre and MacKay proved that if $f$ is a 2-torus homeomorphism isotopic to identity and the rotation set of $f$ has a non empty interior then $f$ has positive topological entropy. Here, we give a converselike theorem. We show that the interior of the rotation set of a 2-torus $C^{1+ \alpha}$ diffeomorphism isotopic to identity of positive topological entropy is not empty, under the additional hypotheses that $f$ is topologically transitive and irreducible. We also give examples that show that these hypotheses are necessary.