Dynamics on Surfaces

Abstract

Thurston classified surface homeomorphisms up to isotopy. Most surface homeomorphisms are so-called pseudo-Anosov. For each pseudo-Anosov homeomorphism, there is an associated number called the stretch factor which tells us how the iterations of the homeomorphism changes the length of a simple closed curve on the surface (with respect to an arbitrary metric of constant curvature). We try to find a number-theoretic characterization of these numbers, and discuss the difficulty of the problem and partial results. This talk partially represents joint work with A. Rafiqu and C. Wu.