Polynomial decay of correlations of pseudo-Anosov diffeomorphisms

Abstract

We give a construction of a smooth realization of a pseudo-Anosov diffeomorphism of a Riemannian surface, and show that it admits a unique SRB measure with polynomial decay of correlations, large deviations, and the central limit theorem. The construction begins with a linear pseudo-Anosov diffeomorphism whose singularities are fixed points. Near the singularities, the trajectories are slowed down, and then the map is conjugated with a homeomorphism that pushes mass away from the origin. The resulting map is a $C^{2+\epsilon}$ diffeomorphism topologically conjugate to the original pseudo-Anosov map. To prove that this map has polynomial decay of correlations, our main technique is to use the fact that this map has a Young tower, and study the decay of the tail of the first return time to the base of the tower.