In this talk I will talk about some new results for the escape and hitting statistics for various open dynamical systems. This includes:

1. Poisson limit laws for arbitrary slow mixing hyperbolic billiards. A connection to RH will be presented too.

2. Polynomial and exponential escape rates

3. Where orbits prefer going in the phase space of nonuniformly expanding dynamical systems.

I will outline the idea of the proof for part 3, which uses operator renewal theory and generalized Keller-Liverani perturbation theory.

Joint work with Prof. Bunimovich.