Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The shrinking target and recurrence problem for non-autonomous systems

Ayesha Bennett

TL;DR

The paper develops a unified framework for shrinking target and recurrence problems in non-autonomous measure-preserving systems on compact spaces. It proves a quantitative shrinking target theorem under a uniform mixing condition, yielding optimal error terms and a clear zero-one criterion based on the convergence/divergence of the sum of target measures. It then extends these ideas to non-autonomous recurrence, establishing zero-measure laws, and provides autonomous zero-one criteria for recurrence in the special class of centred, one-component inner functions, using Markov partitions and distortion control. Applications include zero-one laws for patterns in Cantor-series (multibase) expansions and a cohesive treatment that bridges non-autonomous dynamics with inner-function dynamics. The results unify shrinking-target and recurrence phenomena across autonomous and non-autonomous dynamics, highlighting the roles of uniform mixing, Ahlfors regularity, and partition regularity in determining measure-theoretic outcomes.

Abstract

We investigate the shrinking target and recurrence set associated to non-autonomous measure-preserving systems on compact metric spaces, establishing zero-one criteria in the spirit of classical Borel-Cantelli results. Our first main theorem gives a quantitative shrinking target result for non-autonomous systems under a uniform mixing condition, providing asymptotics with an optimal error term. This general result is applicable to certain families of inner functions, yielding concrete applications such as patterns of zeros in the multibase expansion. Turning to recurrence, we establish new zero-measure laws for non-autonomous systems. In the autonomous case, we prove a zero-one criterion for recurrence sets of centred, one-component inner functions via Markov partitions and distortion estimates. Together, these results provide a unified framework for shrinking target and recurrence problems in both autonomous and non-autonomous dynamics.

The shrinking target and recurrence problem for non-autonomous systems

TL;DR

The paper develops a unified framework for shrinking target and recurrence problems in non-autonomous measure-preserving systems on compact spaces. It proves a quantitative shrinking target theorem under a uniform mixing condition, yielding optimal error terms and a clear zero-one criterion based on the convergence/divergence of the sum of target measures. It then extends these ideas to non-autonomous recurrence, establishing zero-measure laws, and provides autonomous zero-one criteria for recurrence in the special class of centred, one-component inner functions, using Markov partitions and distortion control. Applications include zero-one laws for patterns in Cantor-series (multibase) expansions and a cohesive treatment that bridges non-autonomous dynamics with inner-function dynamics. The results unify shrinking-target and recurrence phenomena across autonomous and non-autonomous dynamics, highlighting the roles of uniform mixing, Ahlfors regularity, and partition regularity in determining measure-theoretic outcomes.

Abstract

We investigate the shrinking target and recurrence set associated to non-autonomous measure-preserving systems on compact metric spaces, establishing zero-one criteria in the spirit of classical Borel-Cantelli results. Our first main theorem gives a quantitative shrinking target result for non-autonomous systems under a uniform mixing condition, providing asymptotics with an optimal error term. This general result is applicable to certain families of inner functions, yielding concrete applications such as patterns of zeros in the multibase expansion. Turning to recurrence, we establish new zero-measure laws for non-autonomous systems. In the autonomous case, we prove a zero-one criterion for recurrence sets of centred, one-component inner functions via Markov partitions and distortion estimates. Together, these results provide a unified framework for shrinking target and recurrence problems in both autonomous and non-autonomous dynamics.

Paper Structure

This paper contains 17 sections, 35 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Shrinking target background and proofs
  3. Borel-Cantelli Lemmas
  4. Borel-Cantelli Lemmas
  5. Proof of Theorem \ref{['ABthA']}
  6. Background on inner functions
  7. Proof of Theorem \ref{['ABthB']}
  8. Background on base expansions
  9. Proof of Theorem \ref{['ABcf']}
  10. Proof of non-autonomous recurrence laws
  11. Proof of Theorem \ref{['ABconv']}
  12. Proof of Theorem \ref{['ABconvin']}
  13. Autonomous recurrence background and proof
  14. Background on one-component inner functions
  15. Markov Partition for one-component inner functions
  16. ...and 2 more sections

Key Result

Theorem A

Let $(X,\mathcal{B},\mu,T_{\Bbb N},d)$ be a dynamical system. Let $\{B(x_n,r_n)\}_{n \in {\Bbb N}}$ be a sequence of balls in $X$ with centres $x_n$ and decreasing radii $r_n$. Suppose that $T_{\Bbb N}$ is uniformly mixing at $\{x_n\}_{n \in {\Bbb N}}$. Then, for any given $\varepsilon>0$, we have t where $\Phi(N):=\sum\limits_{n=1}^N \mu (B(x_n,r_n))$. In particular,

Theorems & Definitions (72)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Theorem F
  • Lemma 2.1: First Borel-Cantelli Lemma
  • ...and 62 more