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A self-similar set with non-locally connected components

Jian-Ci Xiao

Abstract

Luo, Rao and Xiong [Topol. Appl. 322 (2022), 108271] conjectured that if a planar self-similar iterated function system with the open set condition does not involve rotations or reflections, then every connected component of the attractor is locally connected. We create a homogeneous counterexample of Lalley-Gatzouras type, which disproves this conjecture.

A self-similar set with non-locally connected components

Abstract

Luo, Rao and Xiong [Topol. Appl. 322 (2022), 108271] conjectured that if a planar self-similar iterated function system with the open set condition does not involve rotations or reflections, then every connected component of the attractor is locally connected. We create a homogeneous counterexample of Lalley-Gatzouras type, which disproves this conjecture.
Paper Structure (2 sections, 4 theorems, 15 equations, 1 figure)

This paper contains 2 sections, 4 theorems, 15 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The counterexample

Key Result

Lemma 2

$\bigcup_{x\in C_1} \ell_x \subset K$.

Figures (1)

  • Figure 1: An illustration of the IFS $\Phi$ and the attractor $K$

Theorems & Definitions (9)

  • Conjecture 1: LRX22
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof