Towards the Nielsen-Thurston classification for surfaces of infinite type

The fundamental theorem of Thurston states that any homeomorphism of a
surface of finite type can be isotoped so that some multi-curve is
invariant and so that for every complementary component the first
return map is either periodic or pseudo-Anosov. Homeomorphisms of
infinite type surfaces are much more complicated. In this work we
focus on the class of tempered homeomorphisms -- these are the ones
that do not have any mixing behavior. We show that up to isotopy
there is an invariant geodesic lamination so that the first return maps
display one of three qualitatively different behaviors. This work is in progress and it is
joint with Federica Fanoni and Jing Tao.