Realizing rotation vectors for torus homeomorphisms

Abstract

We consider the rotation set ρ ( F ) \rho (F) for a lift F F of a homeomorphism f : T 2 → T 2 f:{T^2} \to {T^2} , which is homotopic to the identity. Our main result is that if a vector v v lies in the interior of ρ ( F ) \rho (F) and has both coordinates rational, then there is a periodic point x ∈ T 2 x \in {T^2} with the property that \[ F q ( x 0 ) − x 0 q = v \frac {{{F^q}({x_0}) - {x_0}}}{q} = v \] where x 0 ∈ R 2 {x_0} \in {R^2} is any lift of x x and q q is the least period of x x .