On fixed points of pseudo-Anosov maps

Abstract

We give a formula to estimate the number of fixed points of a pseudo-Anosov homeomorphism of a surface. When the homeomorphism satisfies a mild property called strong irreducibility, the log of the number of fixed points is coarsely equal to the Teichmüller translation length. We also discuss several applications, including an inequality relating the hyperbolic volume of a mapping torus to the rank of its Heegaard Floer homology.