Surface diffeomorphisms via train-tracks

In this paper we describe an algorithm to classify a given surface diffeomorphism in the sense of Thurston. We show that the algorithm can be put into a computer with exponential time in terms of the length of the word representing the diffeomorphism. The algorithm is based on the piecewise linear action of the mapping class group on the space of measured train-tracks (measured geodesic laminations), which was first noticed by Thurston. We use a new coordinate system for the space of measured train-tracks, which is based on the action of the mapping class group on the fundamental group of the surface, first used by Nielsen. We first discuss the algorithm for a punctured surface, and then describe the necessary modifications in the case of a closed surface. We will also indicate the relation with the Birman-Series conditions for linear action of a diffeomorphism on a set of simple closed curves. We also present some open questions.