Invariant partitions for homeomorphisms isotopic to a pseudo-Anosov homeomorphism

Pseudo-Anosov homeomorphisms are a generalization of Anosov automorphisms of the 2-torus to higher  genus surfaces. In this talk, we will discuss how much the dynamics of a homeomorphism isotopic to a pseudo-Anosov homeomorphism looks like the dynamics of the actual pseudo-Anosov homeomorphism. More precisely, we will discuss the connections between a semi-conjugacy result by Handel, a more precise result by Fathi which gives invariant stable and unstable partitions of the surface and a new way to obtain these partitions.