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Strong Borel--Cantelli Lemmas for Recurrence

Tomas Persson, Alejandro Rodriguez Sponheimer

TL;DR

This work advances dynamical recurrence theory by proving a strong Borel--Cantelli lemma with an explicit error term for recurrence events in systems with exponential 3-fold decay of correlations and a short return time condition. The authors develop a general RSBC framework for arbitrary target sets and then specialize it to recurrence targets $E_k=\{x: T^k x\in B(x,r_k(x))\}$, yielding a sharp asymptotic with error $O(\Phi(n)^{1/2}(\log \Phi(n))^{3/2+\varepsilon})$ where $\Phi(n)=\sum_{k=1}^n\mu(B_k(x))$. They prove a key large-$l$ bound ( Lemma \ref{['lem:largel']} ) and employ the Gal--Koksma lemma to convert variance control into almost-sure convergence. Applications include piecewise expanding interval maps with Gibbs measures and linear Anosov maps on the torus preserving Lebesgue measure, thereby enlarging the class of systems for which RSBC holds and accommodating critical decay rates up to and including $1/k$. The results connect recurrence rates to the underlying measure structure and correlations, with potential implications for hitting times and pointwise dimension in chaotic dynamics.

Abstract

Let $(X,T,μ,d)$ be a metric measure-preserving system for which $3$-fold correlations decay exponentially for Lipschitz continuous observables. Suppose that $(M_k)$ is a sequence satisfying some weak decay conditions and suppose there exist open balls $B_k(x)$ around $x$ such that $μ(B_k(x)) = M_k$. Under a short return time assumption, we prove a strong Borel--Cantelli lemma, including an error term, for recurrence, i.e., for $μ$-a.e. $x \in X$, \[ \sum_{k=1}^{n} \mathbf{1}_{B_k(x)} (T^k x) = Φ(n) + O \bigl( Φ(n)^{1/2} (\log Φ(n))^{3/2 + \varepsilon} \bigr), \] where $Φ(n) = \sum_{k=1}^{n} μ(B_k(x))$. Applications to systems include some non-linear piecewise expanding interval maps and hyperbolic automorphisms of $\mathbf{T}^2$.

Strong Borel--Cantelli Lemmas for Recurrence

TL;DR

This work advances dynamical recurrence theory by proving a strong Borel--Cantelli lemma with an explicit error term for recurrence events in systems with exponential 3-fold decay of correlations and a short return time condition. The authors develop a general RSBC framework for arbitrary target sets and then specialize it to recurrence targets , yielding a sharp asymptotic with error where . They prove a key large- bound ( Lemma \ref{['lem:largel']} ) and employ the Gal--Koksma lemma to convert variance control into almost-sure convergence. Applications include piecewise expanding interval maps with Gibbs measures and linear Anosov maps on the torus preserving Lebesgue measure, thereby enlarging the class of systems for which RSBC holds and accommodating critical decay rates up to and including . The results connect recurrence rates to the underlying measure structure and correlations, with potential implications for hitting times and pointwise dimension in chaotic dynamics.

Abstract

Let be a metric measure-preserving system for which -fold correlations decay exponentially for Lipschitz continuous observables. Suppose that is a sequence satisfying some weak decay conditions and suppose there exist open balls around such that . Under a short return time assumption, we prove a strong Borel--Cantelli lemma, including an error term, for recurrence, i.e., for -a.e. , where . Applications to systems include some non-linear piecewise expanding interval maps and hyperbolic automorphisms of .

Paper Structure

This paper contains 13 sections, 13 theorems, 102 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Paper structure
  3. Results
  4. Setting and notation
  5. Main results
  6. Lemmas
  7. Proof of Main Theorems
  8. Proof of the general theorem
  9. Proof of the dynamical theorem
  10. Proof of Lemma \ref{['lem:largel']}
  11. Applications
  12. Interval maps preserving a Gibbs measure
  13. Linear Anosov maps on T2 preserving the Lebesgue measure

Key Result

Theorem 1

Let $(X,T,\mu,d)$ be an m.m.p.s. and assume that, for some $\alpha>0$, the Hausdorff $\alpha$-measure, $\mathcal{H}_\alpha$, is $\sigma$-finite on $X$. Then for $\mu$-a.e. $x \in X$, Moreover, if $\mathcal{H}_\alpha(X) = 0$, then for $\mu$-a.e. $x\in X$,

Figures (3)

  • Figure 1: The position of $T^{k}a$ relative to $a$ and $p$ when $T' > 0$.
  • Figure 2: The position of $T^{k}a$ and $T^{k}b$ relative to $a$, $b$ and $p$ when $T' < 0$.
  • Figure 3: An illustration of the intersection of rectangles in $\hat{E}_k$ and $\hat{E}_{k+l}$.

Theorems & Definitions (26)

  • Theorem 1: Boshernitzan
  • Proposition 1
  • Proposition 2
  • Remark 1
  • Remark 2: Multiple mixing
  • Remark 3
  • Theorem 2: General Theorem
  • Theorem 3
  • Lemma 1: Gál--Koksma
  • Lemma 2
  • ...and 16 more