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Hausdorff dimension of the set of eventually always hitting points on a self-conformal set

Xintian Zhang

Abstract

Recurrence problems are fundamental in dynamics, and for example, sizes of the set of points recurring infinitely often to a target have been studied extensively in many contexts. For example, the problem of finding the dimension for shrinking target set in an iterated function system is an active research area. In the current work, we consider a set with a finer recurrence quality, the eventually always hitting set. In a sense, the points in the intersection of an eventually always hitting set and a shrinking target set not only return infinitely often but also at a bounded rate. We study this set in the context of self-conformal iterated function systems, and compute upper and lower bounds for its Hausdorff dimension. Additionally, as an intermediate theorem, we obtain a Hausdorff dimension result for the intersection of eventually always hitting and shrinking target sets.

Hausdorff dimension of the set of eventually always hitting points on a self-conformal set

Abstract

Recurrence problems are fundamental in dynamics, and for example, sizes of the set of points recurring infinitely often to a target have been studied extensively in many contexts. For example, the problem of finding the dimension for shrinking target set in an iterated function system is an active research area. In the current work, we consider a set with a finer recurrence quality, the eventually always hitting set. In a sense, the points in the intersection of an eventually always hitting set and a shrinking target set not only return infinitely often but also at a bounded rate. We study this set in the context of self-conformal iterated function systems, and compute upper and lower bounds for its Hausdorff dimension. Additionally, as an intermediate theorem, we obtain a Hausdorff dimension result for the intersection of eventually always hitting and shrinking target sets.
Paper Structure (11 sections, 15 theorems, 116 equations, 2 figures)

This paper contains 11 sections, 15 theorems, 116 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background and notations
  3. Main results
  4. The influence of shrinking speed
  5. Possible values of $ve$ and $vs$
  6. Relation between ve and vs
  7. Hausdorff dimension of the intersections: Proof of Theorem 2
  8. Ideal covering: upper bound
  9. A proper measure and the local dimension: lower bound
  10. Hausdorff dimension of eventually always hitting set: Proof of Theorem 1
  11. Applications and explicit results

Key Result

Lemma 1.1

(Lemma 6.1 in Sascha) For a conformal iterated function system $\{f_i\}_1^S$ defined on $U$, there exists a bounded open convex set $V\subset \mathbb{R}^d$ such that $f_i(V)\subset V\subset \Bar{V} \subset U$ for all $i\in \{1,\cdots,S\}$. Furthermore, if $\Lambda\subset V$ is the associated self-co for all $x,y\in V$ and $\mathbf{i}\in \Sigma_*$, for all $\mathbf{i}\in \Sigma_*$, and for all $\

Figures (2)

  • Figure 1: Illustration of the relation between $s^+$ and $\Bar{s}$.
  • Figure 2: Illustration of ${\dim_\mathscr{H}(\pi(R_e((a_n)_n,\mathbf{t})) )}$.

Theorems & Definitions (37)

  • Lemma 1.1
  • Definition 1.1
  • Remark 1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1
  • Theorem 2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 27 more