Rotation sets with nonempty interior and transitivity in the universal covering

TL;DR

This paper characterizes when a lift of a transitive torus homeomorphism is transitive based on the rotation set, showing that the lift is transitive if and only if the rotation set contains the origin in its interior.

Contribution

It establishes a precise criterion linking the transitivity of a lift to the interior of its rotation set for torus homeomorphisms.

Findings

Lift of a transitive homeomorphism is transitive iff the rotation set contains the origin.

Provides a characterization of rotation sets with nonempty interior.

Connects topological transitivity with geometric properties of rotation sets.

Abstract

Let $f$ be a transitive homeomorphism of the two-dimensional torus in the homotopy class of the identity. We show that a lift of $f$ to the universal covering is transitive if and only if the rotation set of the lift contains the origin in its interior.