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Quantitative recurrence problem on some Bedford-McMullen carpets

Yu-Liang Wu, Na Yuan

TL;DR

This work computes the Hausdorff dimension of quantitative recurrent sets for the canonical endomorphism on Bedford-McMullen carpets in the regime where the carpet's Hausdorff and box dimensions coincide. The authors develop a probabilistic, self-affine framework, constructing a piecewise Bernoulli measure to capture the local dimension and employing a Falconer-type argument to obtain a sharp lower bound, then complementing it with a delicate covering argument to yield the matching upper bound. The main result expresses dim_H W_gamma(K,T,psi) as min{t1,t2}, with explicit formulas for t1 and t2 depending on scale parameters tau1, tau2, entropy terms H1(p), H2(p), and the column-count data M,N, plus a gamma-dependent reduction to simpler cases when gamma=1. The paper also demonstrates concrete applications to diagonal endomorphisms and Cantor-set products, illustrating how the general formula specializes to familiar non-conformal settings and highlighting the role of self-affine geometry in shrinking-target-type problems.

Abstract

In this paper, we study the Hausdorff dimension of the quantitative recurrent set of the canonical endomorphism on the Bedford-McMullen carpets whose Hausdorff dimension and box dimension are equal.

Quantitative recurrence problem on some Bedford-McMullen carpets

TL;DR

This work computes the Hausdorff dimension of quantitative recurrent sets for the canonical endomorphism on Bedford-McMullen carpets in the regime where the carpet's Hausdorff and box dimensions coincide. The authors develop a probabilistic, self-affine framework, constructing a piecewise Bernoulli measure to capture the local dimension and employing a Falconer-type argument to obtain a sharp lower bound, then complementing it with a delicate covering argument to yield the matching upper bound. The main result expresses dim_H W_gamma(K,T,psi) as min{t1,t2}, with explicit formulas for t1 and t2 depending on scale parameters tau1, tau2, entropy terms H1(p), H2(p), and the column-count data M,N, plus a gamma-dependent reduction to simpler cases when gamma=1. The paper also demonstrates concrete applications to diagonal endomorphisms and Cantor-set products, illustrating how the general formula specializes to familiar non-conformal settings and highlighting the role of self-affine geometry in shrinking-target-type problems.

Abstract

In this paper, we study the Hausdorff dimension of the quantitative recurrent set of the canonical endomorphism on the Bedford-McMullen carpets whose Hausdorff dimension and box dimension are equal.
Paper Structure (8 sections, 4 theorems, 83 equations, 1 table)

This paper contains 8 sections, 4 theorems, 83 equations, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. The main theorem
  4. Organization of the paper
  5. Preliminaries
  6. Proof of the lower bound
  7. Proof of the upper bound
  8. Applications

Key Result

Theorem 1.1

Let $W_{\gamma}(K,T,\psi)$ be defined in e2, $\tau_1 \ge 0$, and $0\leq \gamma\leq \log_{m_1}m_2$. Let $M$ be the number of columns containing at least one chosen rectangle and $N_i$ the number of rectangles chosen from the $i$-th non-empty column. Suppose that $\dim_{\mathrm{B}} K = \dim_{\mathrm{H In particular, if $\gamma = 1$, this is reduced to the following form. $(1)$ If $\log_{m_1} m_2 > 1

Theorems & Definitions (13)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 3 more