On the Rescaled Hitting Time and Return Time Distributions to Asymptotically Small Sets

Consider a hyperbolic flow $\phi_t:M\to M$ on a smooth manifold $M$, and a sequence of open balls $(\D_n)_{n\in\N}$ with $\D_n \subset M$ and measure $m(\D_n) >0$ but also satisfying $\lim_{n \to \infty}m(\D_n)=0$. The expected time it takes for the flow to hit the set $\D_n$, known as the hitting time, or the return time if the flow started in $\D_n$, and each subsequent hit thereafter, is proportional to the measure $m(\D_n)$ of that set, provided the measure is ergodic.

In this thesis I study how the distribution of hitting times (and return times), rescaled by an appropriate sequence of constants, converges in the limit. I show conditions under which a Poisson limit law holds by considering the hitting time distributions of an associated discrete dynamical system.