On distributional limit laws for recurrence

Mark Holland; Mike Todd

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arXiv · 2401.13300

On distributional limit laws for recurrence

Mark Holland·Mike Todd

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Abstract

For a probability measure preserving dynamical system $(\mathcal{X},f,\mu)$, the Poincar\'e Recurrence Theorem asserts that $\mu$-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 $r_n(\tau)$ follows an \emph{averaged} Poisson distribution $G(\tau)$. Furthermore, we obtain quantitative results on almost sure rates for the recurrence statistics of the process $X_n$.

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Mark Holland, Mike Todd. 2024-01-24. On distributional limit laws for recurrence. https://arxiv.org/abs/2401.13300

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