A characterization of annularity for area-preserving toral   homeomorphisms
  We prove that if an area-preserving homeomorphism of the torus in the homotopy class of the identity has a rotation set which is a nondegenerate vertical segment containing the origin, then there exists an essential invariant annulus. In particular, some lift to the universal covering has uniformly bounded displacement in the horizontal direction. 
Nancy Guelman, Andres Koropecki, Fabio Armando Tal
math.DS