Transitivity and rotation sets with nonempty interior for homeomorphisms of the 2-Torus

TL;DR

This paper proves that for certain homeomorphisms of the 2-torus, the rotation set's interior contains the origin, confirming a specific case of Boyland's conjecture and linking transitivity to the rotation set's properties.

Contribution

It establishes that transitive homeomorphisms of the 2-torus have rotation sets with nonempty interior containing the origin, advancing understanding of rotation sets in dynamical systems.

Findings

The origin is in the interior of the rotation set under transitivity.

Transitivity outside elliptic islands implies the same interior property.

Confirms a particular case of Boyland's conjecture.

Abstract

We show that, if $f$ is a homeomorphism of the 2--torus isotopic to the identity, and its lift $\widetilde f$ is transitive, or even if it is transitive outside of the lift of the elliptic islands, then $(0,0)$ is in the interior of the rotation set of $\widetilde f.$ This proves a particular case of Boyland's conjecture.