Rotation sets of toral flows

Abstract

We consider the rotation set ρ ( Φ ) \rho (\Phi ) for a lift Φ = { Φ t } t ∈ R \Phi = {\{ {\Phi _t}\} _{t \in \mathbb {R}}} of a flow φ = { φ t : T 2 → T 2 } t ∈ R \varphi = {\{ {\varphi _t}:{\mathbb {T}^2} \to {\mathbb {T}^2}\} _{t \in \mathbb {R}}} . Our main result is that ρ ( Φ ) \rho (\Phi ) consists of either a single point, a segment of a line through 0 with rational slope, or a line segment with irrational slope and one endpoint equal to 0. Any set of one of these types is the rotation set for some flow.