turkmath.org

Thurston's theory of homeomorphisms from a compact surface of genus greater than one to itself, classifies such maps (and their mapping classes) into three types: periodic, pseudo-Anosov, or reducible - where the surface decomposes into pieces on which the restrictions of the map are either periodic or pseudo-Anosov. The pseudo-Anosov case is thus of great interest in studying these maps. In this case, an integrable quadratic holomorphic differential exists on the surface w.r.t. a Riemann surface structure. When the foliations of the quadratic differential are orientable, namely when there is an Abelian differential preserved by the map, we will characterize the structure of the surface and the pseudo-Anosov map in terms of a permutation of integers.