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Limit Laws for Poincaré Recurrence and the Shrinking Target Problem

Alejandro Rodriguez Sponheimer

Abstract

Let $(X,T,μ,d)$ be a metric measure-preserving system. If $B(x,r_n(x))$ is a sequence of balls such that, for each $n$, the measure of $B(x,r_n(x))$ is constant, then we obtain a self-norming CLT for recurrence for systems satisfying a multiple decorrelation property. When $μ$ is absolutely continuous, we obtain a distributional limit law for recurrence for the sequence of balls $B(x,r_n)$. In the latter case, the density of the limiting distribution is an average over Gaussian densities. An important assumption in the CLT for recurrence is that the CLT holds for the shrinking target problem. Because of this, we also prove an ASIP for expanding and Axiom A systems for non-autonomous Hölder observables and apply it to the shrinking target problem, thereby obtaining a CLT.

Limit Laws for Poincaré Recurrence and the Shrinking Target Problem

Abstract

Let be a metric measure-preserving system. If is a sequence of balls such that, for each , the measure of is constant, then we obtain a self-norming CLT for recurrence for systems satisfying a multiple decorrelation property. When is absolutely continuous, we obtain a distributional limit law for recurrence for the sequence of balls . In the latter case, the density of the limiting distribution is an average over Gaussian densities. An important assumption in the CLT for recurrence is that the CLT holds for the shrinking target problem. Because of this, we also prove an ASIP for expanding and Axiom A systems for non-autonomous Hölder observables and apply it to the shrinking target problem, thereby obtaining a CLT.
Paper Structure (25 sections, 28 theorems, 360 equations, 3 figures)

This paper contains 25 sections, 28 theorems, 360 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Structure of the Paper
  3. Main Results
  4. Assumptions
  5. Distributional Limit Laws for Recurrence
  6. Almost Sure Invariance Principles for Non-invertible Systems
  7. Almost Sure Invariance Principles for Axiom A Diffeomorphisms
  8. Variance Estimates
  9. Variance estimates for implicit radii
  10. Variance estimates for explicit radii
  11. Supporting Results
  12. Proof of CLT for Recurrence
  13. Proof of Theorem \ref{['thm:clt']}
  14. Proof of Theorem \ref{['thm:clt:constant-r']} and Corollary \ref{['cor:clt:abs']}
  15. Proof of ASIP for Non-invertible Systems
  16. ...and 10 more sections

Key Result

Theorem 2.1

Suppose that $(X,T,\mu,d)$ is an m.m.p.s. satisfying Conditions assumption:multiple-doc, assumption:frostman and assumption:thin-annuli and suppose that the sequence $(M_n)$ satisfies Condition assumption:sequence for some $\gamma \in (0,1)$ and $\upsilon > 1$. Assume that and Then converges in distribution to a standard Gaussian variable.

Figures (3)

  • Figure 1: The positions of $p,a$ and $T^{l}a$ when $(T^{l})' > 0$.
  • Figure 2: The interval $T^{l}I$ when $(T^{l})' < 0$.
  • Figure 3: The intervals $B(x,r_k(x))$ and $R_k(y)$.

Theorems & Definitions (63)

  • Remark 1: Decay of Correlations
  • Remark 2: Continuity of Measure
  • Theorem 2.1
  • Remark 3
  • Theorem 2.2
  • Remark 4
  • Corollary 2.1
  • Remark 5
  • Theorem 2.3
  • Corollary 2.2
  • ...and 53 more