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Quantitative recurrence and the shrinking target problem for overlapping iterated function systems

Simon Baker, Henna Koivusalo

Abstract

In this paper we study quantitative recurrence and the shrinking target problem for dynamical systems coming from overlapping iterated function systems. Such iterated function systems have the important property that a point often has several distinct choices of forward orbit. As is demonstrated in this paper, this non-uniqueness leads to different behaviour to that observed in the traditional setting where every point has a unique forward orbit. We prove several almost sure results on the Lebesgue measure of the set of points satisfying a given recurrence rate, and on the Lebesgue measure of the set of points returning to a shrinking target infinitely often. In certain cases, when the Lebesgue measure is zero, we also obtain Hausdorff dimension bounds. One interesting aspect of our approach is that it allows us to handle targets that are not simply balls, but may have a more exotic geometry.

Quantitative recurrence and the shrinking target problem for overlapping iterated function systems

Abstract

In this paper we study quantitative recurrence and the shrinking target problem for dynamical systems coming from overlapping iterated function systems. Such iterated function systems have the important property that a point often has several distinct choices of forward orbit. As is demonstrated in this paper, this non-uniqueness leads to different behaviour to that observed in the traditional setting where every point has a unique forward orbit. We prove several almost sure results on the Lebesgue measure of the set of points satisfying a given recurrence rate, and on the Lebesgue measure of the set of points returning to a shrinking target infinitely often. In certain cases, when the Lebesgue measure is zero, we also obtain Hausdorff dimension bounds. One interesting aspect of our approach is that it allows us to handle targets that are not simply balls, but may have a more exotic geometry.
Paper Structure (19 sections, 30 theorems, 286 equations)

This paper contains 19 sections, 30 theorems, 286 equations.

Table of Contents

  1. Introduction
  2. Dynamical systems with choice
  3. Fractal Geometry
  4. Shrinking targets
  5. Recurrence
  6. Statements of results
  7. Preliminaries
  8. Notation
  9. Technical results
  10. Basic results
  11. Proof of Theorem \ref{['Main thm']}
  12. Proof of Theorem \ref{['Main thm']}.1
  13. Proofs of Theorem \ref{['Main thm']}.2 and Theorem \ref{['Main thm']}.3
  14. Proof of Theorem \ref{['Main thm']}.4
  15. Proof of Theorem \ref{['exponential theorem']}
  16. ...and 4 more sections

Key Result

Theorem 2.1

Let $\mathcal{A}=\{A_i\}_{i\in \mathcal{I}}$ be a finite set of invertible $d\times d$ matrices satisfying the following properties: Then the following statements are true:

Theorems & Definitions (58)

  • Theorem 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Remark 2.7
  • Remark 2.8
  • Remark 2.9
  • Lemma 3.1
  • ...and 48 more