Improved estimates of statistical properties in some non-uniformly hyperbolic dynamical systems

Building upon previous works by Young, Chernov–Zhang and Bruin–Melbourne–Terhesiu, we present a general scheme to improve bounds on the statistical properties (in particular, decay of correlations, and rates in the almost sure invariant principle) for a class of non-uniformly hyperbolic dynamical systems. Specifically, for systems with polynomial, yet summable mixing rates, our method removes logarithmic factors of earlier arguments, resulting in essentially optimal bounds. Applications include Wojtkowski’s system of two falling balls, dispersing billiards with flat points and Bunimovich’s flower-shaped billiard tables.