A characterization of pseudo-Anosov foliations

Abstract

Let M be a closed oriented smooth surface of genus g > 2, and let Jf ^ denote the space of equivalence classes of measured foliations on M. The importance of measured foliations began with Thurston's work on diffeomorphisms of surfaces: he defined the space Jt'J** and recognized the natural action of the mapping class group on J3P as an extension of the action of this group on the Teichmijller space of M. In these investigations, there arose the concept of a pseudo-Anosov map which fixes a pair of transverse projective measured foliation classes on M, and the question evolves of recognizing the foliation classes fixed by some pseudo-Anosov map. Our main result provides a solution to this problem: we give a combinatorial characterization of these projective measured foliation classes. The combinatorial formulation of this problem uses the theory of train tracks.