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Projections of four corner Cantor set: total self-similarity, spectrum and unique codings

Derong Kong, Beibei Sun

Abstract

Given $ρ\in (0,1/4]$, the four corner Cantor set $E\subset \mathbb{R}^{2}$ is a self-similar set generated by the iterated function system \[ \left\{(ρx, ρy), \quad(ρx, ρy+1-ρ),\quad (ρx+1-ρ, ρy),\quad(ρx+1-ρ,ρy+1-ρ)\right\}. \] For $θ\in[0,π)$ let $E_θ$ be the orthogonal projection of $E$ onto a line with an angle $θ$ to the $x$-axis. In this paper we give a complete characterization on which the projection $E_θ$ is totally self-similar. We also study the spectrum of $E_θ$, which turns out that the spectrum of $E_θ$ achieves its maximum value if and only if $E_θ$ is totally self-similar. Furthermore, when $E_θ$ is totally self-similar, we calculate its Hausdorff dimension and study the subset $U_θ$ which consists of all $x\in E_θ$ having a unique coding. In particular, we show that $\dim_H U_θ=\dim_H E_θ$ for Lebesgue almost every $θ\in[0,π)$. Finally, for $ρ=1/4$ we describe the distribution of $θ$ in which $E_θ$ contains an interval. It turns out that the possibility for $E_θ$ to contain an interval is smaller than that for $E_θ$ to have an exact overlap.

Projections of four corner Cantor set: total self-similarity, spectrum and unique codings

Abstract

Given , the four corner Cantor set is a self-similar set generated by the iterated function system For let be the orthogonal projection of onto a line with an angle to the -axis. In this paper we give a complete characterization on which the projection is totally self-similar. We also study the spectrum of , which turns out that the spectrum of achieves its maximum value if and only if is totally self-similar. Furthermore, when is totally self-similar, we calculate its Hausdorff dimension and study the subset which consists of all having a unique coding. In particular, we show that for Lebesgue almost every . Finally, for we describe the distribution of in which contains an interval. It turns out that the possibility for to contain an interval is smaller than that for to have an exact overlap.
Paper Structure (8 sections, 25 theorems, 182 equations, 3 figures)

This paper contains 8 sections, 25 theorems, 182 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Total self-similarity of $E_\lambda$
  3. Total self-similarity of $E_\lambda$ for $\lambda\in(0,\rho)$
  4. Total self-similarity of $E_\lambda$ for $\lambda\in \left(\frac{1-2\rho}{2},\frac{1-\rho}{2}\right)$
  5. The spectrum of $E_\lambda$
  6. Unique codings of $E_\lambda$ when $E_\lambda$ is totally self-similar
  7. Typical result for the Hausdorff dimension of $U_\lambda$
  8. The possibility for $E_{\lambda}$ to contain an interval

Key Result

Theorem 1.2

Let $\lambda \in(0,\rho)\cup \left(\frac{1-2\rho}{2},\frac{1-\rho}{2}\right)$.

Figures (3)

  • Figure 1: The first two levels for the geometric construction of $E_\lambda$ with $\lambda\in (0,\rho)$ (left) and $\lambda\in \left(\frac{1-2\rho}{2},\frac{1-\rho}{2}\right)$ (right).
  • Figure 2: The overlapping structure of $E_\lambda$ with $\lambda=\lambda_k$ for some $k\in\mathbb{N}$. Then $f_0(E_\lambda)\cap f_\lambda(E_\lambda)=f_{ {\mathbf{i}}_1}(E_\lambda)\cup f_{ {\mathbf{i}}_2}(E_\lambda)$ and $f_{1-\rho-\lambda}(E_\lambda)\cap f_{1-\rho}(E_\lambda)=f_{ {\mathbf{i}}_3}(E_\lambda)\cup f_{ {\mathbf{i}}_4}(E_\lambda)$ with $f_{ {\mathbf{i}}_\ell}=f_{ {\mathbf{j}}_\ell}$ for all $\ell\in\left\{1,2,3,4\right\}$.
  • Figure 3: The graph of $\hat{W}\cap[1,N]^2$ with $N=100$.

Theorems & Definitions (55)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Remark 1.9
  • Definition 1.10
  • ...and 45 more