Torus homeomorphisms whose rotation sets have empty interior

Abstract

Let $F$ be a lift of a homeomorphism $f: {\Bbb T}^{2} \to {\Bbb T}^{2}$ homotopic to the identity. We assume that the rotation set $\rho(F)$ is a line segment with irrational slope. In this paper we use the fact that ${\Bbb T}^2$ is necessarily chain transitive under $f$ if $f$ has no periodic points to show that if $v \in \rho(F)$ is a rational point, then there is a periodic point $x \in {\Bbb T}^{2}$ such that $v$ is its rotation vector.