Preprint C58/2007
On the Dynamics of Torus Homeomorphisms

Andres Koropecki

**Keywords: **
torus homeomorphisms | rotation set | periodic points

This work is divided into four parts. In the first two chapters we focus on the rotation set of torus homeomorphisms in the isotopy class of the identity and the existence of free (essential, simple, closed) curves. The first result we obtain is a topological version of a theorem of Franks, replacing the area preserving hypothesis by the condition of having no free curves: If a torus homeomorphism is homotopic to the identity, has no free curves, and its rotation set has empty interior, then every rational point in the rotation set is realized by a periodic orbit. Our second result is inspired by a result of Guillou on the annulus: If a torus homeomorphism is isotopic to the identity has no free curves, then either it has a fixed point or its rotation set has nonempty interior. As a consequence (by results of Franks and Llibre-McKay), either there is a fixed point or the homeomorphism has positive topological entropy and periodic points of arbitrarily large periods. In the third chapter we consider an opposite situation: what happens when a torus homeomorphism has a curve that is always free, i.e. such that its iterates are pairwise disjoint? With the assumption that the orbit of this curve is dense, we prove that the homeomorphism is semi-conjugate to an irrational rotation on the circle. In the last chapter, we study the (C^\infty) closure of the conjugancy class of translations of the torus. In that space, Fathi and Herman showed that a generic diffeomorphism is minimal and uniquely ergodic. We prove that in fact for a generic diffeomorphism in that space, its associated dynamic cocycle (the map induced on the unit tangent bundle) is minimal. This allows us to prove partially the following claim, which was announced by Herman but whose proof never appeared (to our knowledge): For a generic diffeomorphism in the closure of the conjugancy class of translations, there is no invariant codimension 1 topological foliation.