Towards Nielsen-Thurston classification for surfaces of infinite type

Abstract

We introduce and study tame homeomorphisms of surfaces of infinite type. These are maps for which curves under iterations do not accumulate onto geodesic laminations with non-proper leaves, but rather just a union of possibly intersecting curves or proper lines. Assuming an additional finiteness condition on the accumulation set, we prove a Nielsen-Thurston type classification theorem. We prove that for such maps there is a canonical decomposition of the surface into invariant subsurfaces on which the first return is either periodic or a translation.