Recurrence to rare events for all points in countable Markov shifts
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918249149497344 |
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| author | Bansard-Tresse, Dylan |
| author_facet | Bansard-Tresse, Dylan |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_13536 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Recurrence to rare events for all points in countable Markov shifts Bansard-Tresse, Dylan Dynamical Systems Probability 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. |
| title | Recurrence to rare events for all points in countable Markov shifts |
| topic | Dynamical Systems Probability |
| url | https://arxiv.org/abs/2512.13536 |