Polynomial Decay of Correlations for Pseudo-Anosov Diffeomorphisms

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 C2+β C 2 + β diffeomorphism with nonzero Lyapunov exponents, which is topologically conjugate to the original pseudo-Anosov map. We also show that with respect to its SRB measure, the map has polynomial upper and lower large deviation bounds and the central limit theorem; and has a unique measure of maximal entropy with respect to which the map is Bernoulli, has nonzero Lyapunov exponents, and has both exponential decay of correlations and the central limit theorem with respect to Hölder observables.