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A Recurrence-type Strong Borel--Cantelli Lemma for Axiom A Diffeomorphisms

Alejandro Rodriguez Sponheimer

TL;DR

The paper addresses strong recurrence laws for shrinking targets in dynamical systems with dependence, focusing on Axiom A diffeomorphisms. It develops a framework based on exponential decay of $3$-fold correlations for Lipschitz observables to bound recurrence sets $E_n=\{x: T^{n}x \u203a B(x,r_n(x)) ext{ for } bmu ext{-a.e. }x ext{} ight ight ext{, and applies Persson’s strong Borel–Cantelli criterion to obtain } \lim_{n\to ty} rac{ sum_{k=1}^{n} oldsymbol{1}_{B_k(x)}(T^{k}x)}{ sum_{k=1}^{n} bmu(B_k(x))} = 1$. The main result holds under natural geometric assumptions on balls, annuli, and packings, with $(M_n) o0$ at a controlled rate; this yields a general recurrence theorem (Theorem general) and, for Axiom A diffeomorphisms with Hölder potentials, the specific corollaries (Theorems main) applying to equilibrium states. The work also derives a hitting-time consequence $ lim_{r o0} rac{ log au_{B(x,r)}(x)}{- log bmu(B(x,r))}=1$ and connects these recurrence rates to pointwise dimensions, providing a significant extension of recurrence results from one-dimensional or specially structured systems to broad hyperbolic settings with meaningful geometric measures.

Abstract

Let $(X,μ,T,d)$ be a metric measure-preserving dynamical system such that $3$-fold correlations decay exponentially for Lipschitz continuous observables. Given a sequence $(M_k)$ that converges to $0$ slowly enough, we obtain a strong dynamical Borel--Cantelli result for recurrence, i.e., for $μ$-a.e. $x\in X$ \[ \lim_{n \to \infty}\frac{\sum_{k=1}^{n} \mathbf{1}_{B_k(x)}(T^{k}x)} {\sum_{k=1}^{n} μ(B_k(x))} = 1, \] where $μ(B_k(x)) = M_k$. In particular, we show that this result holds for Axiom A diffeomorphisms and equilibrium states under certain assumptions.

A Recurrence-type Strong Borel--Cantelli Lemma for Axiom A Diffeomorphisms

TL;DR

The paper addresses strong recurrence laws for shrinking targets in dynamical systems with dependence, focusing on Axiom A diffeomorphisms. It develops a framework based on exponential decay of -fold correlations for Lipschitz observables to bound recurrence sets . The main result holds under natural geometric assumptions on balls, annuli, and packings, with at a controlled rate; this yields a general recurrence theorem (Theorem general) and, for Axiom A diffeomorphisms with Hölder potentials, the specific corollaries (Theorems main) applying to equilibrium states. The work also derives a hitting-time consequence and connects these recurrence rates to pointwise dimensions, providing a significant extension of recurrence results from one-dimensional or specially structured systems to broad hyperbolic settings with meaningful geometric measures.

Abstract

Let be a metric measure-preserving dynamical system such that -fold correlations decay exponentially for Lipschitz continuous observables. Given a sequence that converges to slowly enough, we obtain a strong dynamical Borel--Cantelli result for recurrence, i.e., for -a.e. where . In particular, we show that this result holds for Axiom A diffeomorphisms and equilibrium states under certain assumptions.
Paper Structure (9 sections, 10 theorems, 115 equations)

This paper contains 9 sections, 10 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Setting and Notation
  4. Main Result
  5. Return Times Corollary
  6. Properties of m and r_n
  7. Measure and Correlations of E_n
  8. Proof of Theorems \ref{['thm:main']} and \ref{['thm:general']}
  9. Final Remarks

Key Result

Theorem 1.1

Consider a compact smooth $N$-dimensional manifold $M$. Suppose $f \colon M \to M$ is an Axiom A diffeomorphism that is topologically mixing on a basic set and $\mu$ is an equilibrium state corresponding to a Hölder continuous potential on the basic set. Suppose that there exist $c_1>0$ and $s > N - for all $x \in M$ and $r \geq 0$. Assume further that $(M_n)$ is a sequence converging to $0$ satis

Theorems & Definitions (20)

  • Theorem 1.1
  • Definition 2.1: $r$-fold Decay of Correlations
  • Theorem 2.1: (Persson persson2023Strong)
  • Theorem 2.2
  • Remark 2.3
  • Corollary 2.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 10 more