Recurrence to rare events for all points in countable Markov shifts

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.