Linearization of conservative toral homeomorphisms

We give an equivalent condition for the existence of a semi-conjugacy to an irrational rotation for conservative homeomorphisms of the two-torus. This leads to an analogue of Poincaré’s classification of circle homeomorphisms for conservative toral homeomorphisms with unique rotation vector and a certain bounded mean motion property. For minimal toral homeomorphisms, the result extends to arbitrary dimensions. Further, we provide a basic classification for the dynamics of toral homeomorphisms with all points non-wandering.