Martingale Central Limit Theorem and Nonuniformly Hyperbolic Systems

MARTINGALE CENTRAL LIMIT THEOREM FOR NONUNIFORMLY HYPERBOLIC SYSTEMS SEPTEMBER 2013 LUKE MOHR, B.S., UNIVERSITY AT BUFFALO M.S., UNIVERSITY OF MASSACHUSETTS AMHERST Ph.D., UNIVERSITY OF MASSACHUSETTS AMHERST Directed by: Professor HONG-KUN ZHANG In this thesis we study the central limit theorem (CLT) for nonuniformly hyperbolic dynamical systems. We examine cases in which polynomial decay of correlations leads to a CLT with a non-standard scaling factor of √ n lnn. We also formulate an explicit expression for the the diffusion constant σ in situations where a return time function on the system is a certain class of supermartingale. We then demonstrate applications by exhibiting the CLT for the return time function in four classes of dynamical billiards, including one previously unproven case, the skewed stadium, as well as for the linked twist map. Finally, we introduce a new class of billiards which we conjecture are ergodic, and we provide numerical evidence to support that claim.