Visiting time statistics

2026/08/02 by Maxim Kirsebom, Philipp Kunde, Tomas Persson
Mathematics · #math.DS #msc:37A50 #msc:37B20 #msc:37E05

paper · pdf

26 pages, 1 figure

arxiv created 2026/08/02 · arxiv updated 2026/08/04

Abstract

Many mixing dynamical systems (X,T,μ) are known to satisfy the hitting time statistics result lim_{r → 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 of view, we fix x and consider τ_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 → 0} μ\ y : τ_B(y,r) (x) > t/μ(B(y,r)) \ = e^-t, for μ-almost every x. As a byproduct we obtain a new method of proof for hitting time statistics.

Citations

• On distributional limit laws for recurrence
• Dichotomy results for eventually always hitting time statistics and almost sure growth of extremes
• Entry and Return times distribution
• Poisson approximation for the number of visits to balls in nonuniformly hyperbolic dynamical systems
• A spectral gap for transer operators of piecewise expanding maps
• Return time statistics via inducing