Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On distributional limit laws for recurrence

Mark Holland, Mike Todd

TL;DR

This work analyzes recurrence statistics for measure-preserving dynamical systems, focusing on close returns to the initial orbit via the process $X_n(x)=d(f^n(x),x)$ and the counting process $R_n(r_n,x)$. It proves a Poisson-like distributional limit for $R_n(r_n,x)$ with $r_n=\tau/(2n)$, where the limiting law is an averaged, density-weighted Poisson mass $\int_{\mathcal{X}} \frac{\tau^k \rho(x)^{k+1}}{k!} e^{-\tau \rho(x)}\,dx$, capturing the influence of the invariant density. The results extend to asymptotically hyperbolic recurrence, with explicit examples such as Manneville-Pomeau and Misiurewicz-Thurston maps, and even intermittent cusp maps where the limit can be heavy-tailed; the authors also obtain almost-sure recurrence-rate bounds via refined correlation and short-return controls. The methodology combines decay of correlations, a hitting-time reduction, and Chen–Stein–type Poisson approximation to yield quantitative error terms and justify averaging over initial conditions, linking recurrence statistics to extreme-value-like phenomena in a rigorous dynamical-systems framework.

Abstract

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 $X_n$.

On distributional limit laws for recurrence

TL;DR

This work analyzes recurrence statistics for measure-preserving dynamical systems, focusing on close returns to the initial orbit via the process and the counting process . It proves a Poisson-like distributional limit for with , where the limiting law is an averaged, density-weighted Poisson mass , capturing the influence of the invariant density. The results extend to asymptotically hyperbolic recurrence, with explicit examples such as Manneville-Pomeau and Misiurewicz-Thurston maps, and even intermittent cusp maps where the limit can be heavy-tailed; the authors also obtain almost-sure recurrence-rate bounds via refined correlation and short-return controls. The methodology combines decay of correlations, a hitting-time reduction, and Chen–Stein–type Poisson approximation to yield quantitative error terms and justify averaging over initial conditions, linking recurrence statistics to extreme-value-like phenomena in a rigorous dynamical-systems framework.

Abstract

For a probability measure preserving dynamical system , the Poincaré Recurrence Theorem asserts that -almost every orbit is recurrent with respect to its initial condition. This motivates study of the statistics of the process , and real-valued functions thereof. For a wide class of non-uniformly expanding dynamical systems, we show that the time- counting process associated to the number recurrences below a certain radii sequence follows an \emph{averaged} Poisson distribution . Furthermore, we obtain quantitative results on almost sure rates for the recurrence statistics of the process .
Paper Structure (13 sections, 16 theorems, 108 equations, 1 figure)

This paper contains 13 sections, 16 theorems, 108 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Uniformly expanding interval maps
  3. On almost sure behaviour of recurrence statistics.
  4. On extreme statistics for recurrence
  5. Dynamical systems with asymptotic hyperbolic recurrence
  6. Dynamical systems exhibiting asymptotic hyperbolic recurrence.
  7. Proof of Theorem \ref{['thm:poisson_like']}
  8. Removing dependence on the initial condition
  9. Poisson approximation and error terms.
  10. Completing the proof of Theorem \ref{['thm:poisson_like']}.
  11. proof of Theorem \ref{['thm:almost-sure']}
  12. Proof of Theorem \ref{['thm.asym-hyp-rec']}
  13. Technical estimates concerning the invariant density

Key Result

Theorem 2.3

Suppose that $(\mathcal{X},f,\mu)$ is a measure preserving dynamical system which is hyperbolic recurrent in the sense of Definition def.hyp-rec. Consider the process with $r_n=\tau/2n$ and $\tau>0$. Then for all $k\geq 0$,

Figures (1)

  • Figure 1: Schematic picture of $\Delta, \Delta^-, \Delta_0, \Delta^+$. We will usually take $r=\frac{\tau}{2n}$ and $\delta= \frac{1}{2m}$ for $n\ll m$, for example, later in the paper we use $m= n^{1+\beta'}$ below.

Theorems & Definitions (39)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Corollary 2.7
  • Theorem 2.8
  • Example 2.1
  • Corollary 2.9
  • ...and 29 more