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Hitting statistics for $φ$-mixing dynamical systems

Saeed Shaabanian

TL;DR

This work derives a unifying L_{α,s}(Λ) limit for hitting-time statistics and localized escape rates in φ-mixing and Gibbs-Markov dynamical systems, with Λ a nested intersection of shrinking sets of measure tending to zero. The main result expresses L_{α,s}(Λ) as 1 in the absence of short returns (π_{ess}(U_n)→∞) or as the extremal-index-based value θ_1 when short returns persist, for α in a broad range (including α=0) and under explicit decay and regularity assumptions. The authors extend these recurrence results to Gibbs-Markov systems, metric-ball observables, and provide concrete examples: a singleton, a Cantor set, and a submanifold of an Anosov map (cat map), illustrating how the extremal index and short-return structure control recurrence. Overall, the paper broadens hitting-time and escape-rate theory to zero-measure targets in polynomially φ-mixing and Gibbs-Markov settings, with implications for hyperbolic dynamics and Young tower frameworks.

Abstract

Hitting rate and escape rate are two examples of recurrence laws for a dynamical system, and a general limit connects them. We show that for both Gibbs-Markov systems or any systems with the $φ$-mixing measure, for a sequence of nested sets whose intersection is a measure zero set, this general limit equals one in the absence of short returns and less than one otherwise, which is given by an explicit formula called extremal index. One of the applications of this result is to dynamical systems on Riemannian manifolds such as hyperbolic maps and expanding maps, and it can be applied to any system with a suitable Young tower.

Hitting statistics for $φ$-mixing dynamical systems

TL;DR

This work derives a unifying L_{α,s}(Λ) limit for hitting-time statistics and localized escape rates in φ-mixing and Gibbs-Markov dynamical systems, with Λ a nested intersection of shrinking sets of measure tending to zero. The main result expresses L_{α,s}(Λ) as 1 in the absence of short returns (π_{ess}(U_n)→∞) or as the extremal-index-based value θ_1 when short returns persist, for α in a broad range (including α=0) and under explicit decay and regularity assumptions. The authors extend these recurrence results to Gibbs-Markov systems, metric-ball observables, and provide concrete examples: a singleton, a Cantor set, and a submanifold of an Anosov map (cat map), illustrating how the extremal index and short-return structure control recurrence. Overall, the paper broadens hitting-time and escape-rate theory to zero-measure targets in polynomially φ-mixing and Gibbs-Markov settings, with implications for hyperbolic dynamics and Young tower frameworks.

Abstract

Hitting rate and escape rate are two examples of recurrence laws for a dynamical system, and a general limit connects them. We show that for both Gibbs-Markov systems or any systems with the -mixing measure, for a sequence of nested sets whose intersection is a measure zero set, this general limit equals one in the absence of short returns and less than one otherwise, which is given by an explicit formula called extremal index. One of the applications of this result is to dynamical systems on Riemannian manifolds such as hyperbolic maps and expanding maps, and it can be applied to any system with a suitable Young tower.

Paper Structure

This paper contains 14 sections, 21 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Hitting times, escape rates, and their relation
  4. Escape rates for $\phi$-mixing dynamical systems
  5. Recurrence rate for an polynomially $\phi$-mixing system
  6. Main Theorem
  7. Proof of Theorem \ref{['mainTH']}: the case $\alpha=0$
  8. Proof of Theorem \ref{['mainTH']}: the case $\alpha\in(0,\infty)$
  9. recurrence rate for Gibbs-Markov systems
  10. recurrence rate and metric balls in a $\phi$-mixing system
  11. Examples
  12. $\Lambda$ as a singleton
  13. $\Lambda$ as a Cantor set
  14. $\Lambda$ as a submanifold of Anosov maps

Key Result

Theorem 1.2

BDTodd18 Let $z\in I_{cont}$ and $(U_r)_r$ be a family of intervals such that $\lim_{r\to 0}diam (U_r)=0$ and $\bigcap_r U_r=\{z\}$. Then for any $s\in \mathbb{R}^+, \alpha \in [0,\infty]$, taking $L_{\alpha,s} (z)$ with respect to $\mu_{\varphi}$, we have: where $p$-periodic means that the prime period of $z$ is $p$, i.e. the smallest positive integer $p$ such that $f^p(z)=z$.

Theorems & Definitions (66)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Definition 1.4
  • Definition 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Definition 1.9
  • Definition 1.10
  • ...and 56 more