Recurrence to rare events for all points in countable Markov shifts

TL;DR

This paper investigates the statistical behavior of return times to rare events in countable Markov shifts, revealing diverse limit laws depending on recurrence type and point properties, with comprehensive classifications and explicit examples.

Contribution

It provides a complete classification of return-time limit laws in countable Markov shifts, including explicit constructions for each possible behavior.

Findings

Positive recurrent case yields Poisson and compound Poisson limits.

Null-recurrent case exhibits fractional and Pareto-dominated limit laws.

Complete description of return-time statistics for canonical null-recurrent systems.

Abstract

We study quantitative recurrence to rare events in Countable Markov Shifts with recurrent potentials, focusing on return-time statistics to natural target sets for every point. In the positive recurrent case, return-time processes associated with non-periodic points converge to a standard Poisson process, while those for periodic points converge to a compound Poisson limit. In the null-recurrent regime, three distinct behaviors arise. Points satisfying certain combinatorial conditions (in particular, this class is generic in the measure-theoretic sense) exhibit fractional Poisson limits, whereas periodic points yield compound fractional Poisson limits. For all remaining points, we describe the full family of possible limit laws, each Pareto-dominated. This classification is sharp: every limit behavior in this family can be realized by an explicit system. For canonical families of null-recurrent CMS, we identify the limit process for every point, providing a complete description of return-time statistics in these systems.