Rotation set and Entropy

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.

15 pages, 2 figures, references added

References in corpus (1)

• Rotation set and Entropy

Cited by in corpus (4)

• Transitivity and rotation sets with nonempty interior for homeomorphisms of the 2-Torus
• Rotation set and Entropy
• Rotation sets with nonempty interior and transitivity in the universal covering
• On the rotation sets of generic homeomorphisms on the torus ${\mathbb{T}}^d$