Return- and hitting-time distributions of small sets in infinite measure preserving systems

Simon Rechberger, Roland Zweimüller

Published 2015-03-17Version 1

We study convergence of return- and hitting-time distributions of small sets in recurrent dynamical systems preserving an infinite measure $\mu$. In the presence of a Darling-Kac set with regularly varying wandering rate there is a scaling function suitable for all its subsets. In this case, we show that return distributions for a sequence $(E_{k})$ of sets with $\mu(E_{k})\rightarrow 0$ converge iff the corresponding hitting time distributions do, and we derive an explicit relation between the two limit laws. Some consequences of this result are discussed.