Entry times statistics on metric spaces

In this thesis we investigate the hitting and return times statistics for maps on metric spaces. In Section 2 we look at the asymptotics of the recurrence time for Bowen balls for ergodic systems. We prove the exponential law for the first hitting and return times distribution of Bowen balls in Section 3 and the Poisson law for the higher order hitting times in Section 4. In Section 5 we study random dynamical systems and prove a exponential law for geometric balls under some assumption on the geometrical structure of the system. Finally the dimension of a measure is studied in Section 6. For toral automorphism, a central limit theorem is proven. This generalizes a previous result of Leplaideur and Saussol [38] where they requires the measure to be equilibrium state.