Return Times Distribution of Expanding Maps

2025/10/16 by Haydn, Nicolai T A
#37A25 #37A50 #60F05 #60G70 #Dynamical Systems (math.DS) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2510.14298

Abstract

We consider expanding systems with invariant measures that are uniformly expanding everywhere except on a small measure set and show that the limiting statistics of hitting times for zero measure sets are compound Poisson provided the limits for the cluster size distributions exist. This extends previous results from neighbourhoods around single points to neighbourhoods around zero measure sets. The assumptions require the correlations to decay at least polynomially and the non-uniformly expanding part of the iterates of the map also has to satisfy some decay condition. We also require some regularity conditions around the limiting zero measure target set.

Citations

• Exponential Law for Random Maps on Compact Manifolds
• Extremes and Recurrence in Dynamical Systems
• Convergence of Marked Point Processes of Excesses for Dynamical Systems
• Almost sure mixing rates for non-uniformly expanding maps
• Limiting distribution and error terms for the number of visits to balls in non-uniformly hyperbolic dynamical systems
• Poisson law for some nonuniformly hyperbolic dynamical systems with polynomial rate of mixing
• Return times distribution for Markov towers with decay of correlations
• Poisson approximation for the number of visits to balls in nonuniformly hyperbolic dynamical systems
• Hitting and returning into rare events for all alpha-mixing processes
• Where to place a hole to achieve a maximal escape rate
• Where to place a hole to achieve a maximal escape rate

Related

• Stochastic-like behaviour in nonuniformly expanding maps
• On piecewise expanding maps
• Markov structures and decay of correlations for non-uniformly expanding dynamical systems
• Almost sure mixing rates for non-uniformly expanding maps
• Lower Bounds for the Decay of Correlations in Non-uniformly Expanding Maps