Visiting time statistics

Many mixing dynamical systems $(X,T,μ)$ are known to satisfy the hitting time statistics result \[ \lim_{r \to 0} μ\{\, x : τ_{B(y,r)} (x) > t/μ(B(y,r))\,\} = e^{-t}, \] for $μ$-almost every $y$, where $τ_{B(y,r)} (x)$ is the first hitting time of $x$ to the ball $B(y,r)$. Taking a different point o