Visiting time statistics

Abstract

Many mixing dynamical systems $(X,T,\mu)$ are known to satisfy the hitting time statistics result \[ \lim_{r \to 0} \mu \{\, x : \tau_{B(y,r)} (x) > t/\mu(B(y,r))\,\} = e^{-t}, \] for $\mu$-almost every $y$, where $\tau_{B(y,r)} (x)$ is the first hitting time of $x$ to the ball $B(y,r)$. Taking a different point of view, we fix $x$ and consider $\tau_{B(y,r)} (x)$ as a function of $y$. We call this the visiting time of $y$ from $x$, i.e. the time it takes for $y$ to get a visit from $x$ within a neighbourhood of radius $r$. We prove that \[ \lim_{r \to 0} \mu \{\, y : \tau_{B(y,r)} (x) > t/\mu(B(y,r)) \,\} = e^{-t}, \] for $\mu$-almost every $x$. As a byproduct we obtain a new method of proof for hitting time statistics.