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

TL;DR

This paper investigates the convergence of return- and hitting-time distributions for small sets in infinite measure-preserving systems, establishing conditions for their equivalence and deriving explicit limit laws in null-recurrent dynamical systems.

Contribution

It introduces a unified scaling approach for return and hitting times in infinite measure systems and characterizes their limit laws, including new results for hyperbolic periodic points.

Findings

Return and hitting time distributions converge simultaneously under certain conditions.

Explicit relation between the limit laws of return and hitting times is established.

Limit laws include a product of exponential and stable distributions, with specific cases for hyperbolic periodic points.

Abstract

We study convergence of return- and hitting-time distributions of small sets $E_{k}$ with $\mu(E_{k})\rightarrow0$ in recurrent ergodic dynamical systems preserving an infinite measure $\mu$. Some properties which are easy in finite measure situations break down in this null-recurrent setup. However, in the presence of a uniform set $Y$ with wandering rate regularly varying of index $1-\alpha$ with $\alpha\in(0,1]$, there is a scaling function suitable for all subsets of $Y$. In this case, we show that return distributions for the $E_{k}$ 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. In particular, this leads to improved sufficient conditions for convergence to $\mathcal{E}^{1/\alpha}\,\mathcal{G}_{\alpha}$, where $\mathcal{E}$ and $\mathcal{G}_{\alpha}$ are independent random variables, with $\mathcal{E}$ exponentially distributed and $\mathcal{G}% _{\alpha}$ following the one-sided stable law of order $\alpha$ (and $\mathcal{G}_{1}:=1$). The same principle also reveals the limit laws (different from the above) which occur at hyperblic periodic points of prototypical null-recurrent interval maps. We also derive similar results for the barely recurrent $\alpha=0$ case.