On non-contractible periodic orbits for surface homeomorphisms

In this work we study homeomorphisms of closed orientable surfaces homotopic to the identity, focusing on the existence of non-contractible periodic orbits. We show that, if g is such a homeomorphism, and if (g) over cap is its lift to the universal covering of S that commutes with the deck transformations, then one of the following three conditions must be satisfied: (1) the set of fixed points for (g) over cap projects to a closed subset F which contains an essential continuum; (2) g has non-contractible periodic points of every sufficiently large period; or (3) there exists a uniform bound M > 0 such that, if (x) over cap projects to a contractible periodic point, then the (g) over cap orbit of (x) over cap has diameter less than or equal to M. Some consequences for homeomorphisms of surfaces whose rotation set is a singleton are derived.