Recurrence and fixed points of surface homeomorphisms

We prove that if f is a homeomorphism of the annulus which is homotopic to the identity and has a compact invariant chain transitive set L , then either f has a fixed point or every point of L moves uniformly in one direction: clockwise or counterclockwise. If f is area-preserving, then the annulus itself is a chain transitive set, so, in the presence of a boundary twist condition, one obtains a fixed point. The same techniques apply to homeomorphisms of the torus T 2 . In this setting we show that if f is homotopic to the identity, preserves Lebesgue measure and has mean translation 0, then it has at least one fixed point.