On distributional limit laws for recurrence
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866910959282421760 |
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| author | Holland, Mark Todd, Mike |
| author_facet | Holland, Mark Todd, Mike |
| contents | For a probability measure preserving dynamical system $(\mathcal{X},f,μ)$, the Poincaré Recurrence Theorem asserts that $μ$-almost every orbit is recurrent with respect to its initial condition. This motivates study of the statistics of the process $X_n(x)=\text{dist}(f^n(x),x))$, and real-valued functions thereof. For a wide class of non-uniformly expanding dynamical systems, we show that the time-$n$ counting process $R_n(x)$ associated to the number recurrences below a certain radii sequence $r_n(τ)$ follows an \emph{averaged} Poisson distribution $G(τ)$. Furthermore, we obtain quantitative results on almost sure rates for the recurrence statistics of the process $X_n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_13300 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On distributional limit laws for recurrence Holland, Mark Todd, Mike Dynamical Systems Probability 37A50, 37B20, 60G55, 37E05, 60G70 For a probability measure preserving dynamical system $(\mathcal{X},f,μ)$, the Poincaré Recurrence Theorem asserts that $μ$-almost every orbit is recurrent with respect to its initial condition. This motivates study of the statistics of the process $X_n(x)=\text{dist}(f^n(x),x))$, and real-valued functions thereof. For a wide class of non-uniformly expanding dynamical systems, we show that the time-$n$ counting process $R_n(x)$ associated to the number recurrences below a certain radii sequence $r_n(τ)$ follows an \emph{averaged} Poisson distribution $G(τ)$. Furthermore, we obtain quantitative results on almost sure rates for the recurrence statistics of the process $X_n$. |
| title | On distributional limit laws for recurrence |
| topic | Dynamical Systems Probability 37A50, 37B20, 60G55, 37E05, 60G70 |
| url | https://arxiv.org/abs/2401.13300 |