Pseudo-Anosov diffeomorphisms and the monodromy of an open book

Abstract

In this thesis, we study the proof of the fact that any mapping class on a compact oriented surface with non-empty boundary can be made pseudo-Anosov after a sequence of positive stabilizations. In the language of contact topology, it means that an abstract open book can be stabilized in order to make its monodromy isotopic to a pseudo- Anosov homeomorphism. To attain this goal we use the curve complex of the surface and the classi cation of surface di eomorphisms, the latter of which is the secondary goal of this thesis. In order to classify surface di eomorphisms, we study Thurston's compacti cation of the Teichmüller space, which uses essential curves and measured foliations.