Limiting distribution and error terms for the number of visits to balls in mixing dynamical systems

This dissertation explores return statistics to metric balls in measure preserving dynamical systems which admit a Young tower with polynomial decay of the tail. The thesis opens with some background material and known results in the area. Then we proceed to analyze the recurrence to generic points of the system and estimate the measure of non-generic points. We finally apply these findings to a system equipped with an absolutely continuous measure. ❧ We show that return times are governed by an almost Poisson distribution for the generic centers of metric balls. We derive an estimate for the error between distribution describing visits to the ball and a true Poissonian and prove that the error converges to zero at a logarithmic rate. ❧ Next, we estimate the measure of the set containing the centers of balls with short return times by considering its level sets. The size of the set is inversely proportional to the logarithm of the radius of the ball. ❧ Finally we combine the two findings and apply the corollary to a system which admits a measure which is absolutely continuous with respect to Lebesgue measure. We show that absolute continuity is sufficient to satisfy the assumptions of our initial results. ❧ This thesis generalizes the paper ""Poisson approximation for the number of visits to balls in non-uniformly hyperbolic systems"" by Chazottes and Collet. Their result holds for systems which can be modeled by a Young tower with exponential decay of the tail and which are equipped with a measure absolutely continuous with respect to Lebesgue.