Area preserving surface diffeomorphisms with polynomial decay of correlations are ubiquitous

We show that any surface admits an area preserving $C^{1+\beta}$ diffeomorphism with non-zero Lyapunov exponents which is Bernoulli and has polynomial decay of correlations. We establish both upper and lower polynomial bounds on correlations. In addition, we show that this diffeomorphism satisfies the Central Limit Theorem and has the Large Deviation Property. Finally, we show that the diffeomorphism we constructed possesses a unique hyperbolic Bernoulli measure of maximal entropy with respect to which it has exponential decay of correlations.