The limiting distribution and error terms for return times of dynamical systems

We develop a new framework that allows to prove that the limiting distribution of return times for a large class of mixing dynamical systems are Poisson distributed. We demonstrate our technique in several settings and obtain more general results than previously has been proven. We also obtain error estimates. For $\phi$-mixing maps we obtain a close to exhausting description of return times. For $(\phi,f)$-mixing maps it is shown how the separation function affects error estimates for the limiting distribution. As examples of $(\phi,f)$-mixing we prove that for piecewise invertible maps and for rational maps return times are in the limit Poisson distributed.