On the union of homogeneous symmetric Cantor set with its translations

TL;DR

This paper investigates conditions under which unions of a symmetric Cantor set and its translations form self-similar sets, providing infinite examples and a complete characterization for certain parameters based on graph theory.

Contribution

It introduces a method to identify when unions of translated Cantor sets are self-similar, using graph cycle detection and nilpotency of adjacency matrices, with a complete characterization for specific beta values.

Findings

Existence of infinitely many translation vectors forming self-similar unions.

Complete characterization for self-similarity when 0<β<1/(2N+1).

Use of graph theory to determine self-similarity conditions.

Abstract

Fix a positive integer $N$ and a real number $0< \beta < 1/(N+1)$. Let $\Gamma$ be the homogeneous symmetric Cantor set generated by the IFS $$ \Big\{ \phi_i(x)=\beta x + i \frac{1-\beta}{N}: i=0,1,\cdots, N \Big\}. $$ For $m\in\mathbb{Z}_+$ we show that there exist infinitely many translation vectors $\mathbf t=(t_0,t_1,\cdots, t_m)$ with $0=t_0<t_1<\cdots<t_m$ such that the union $\bigcup_{j=0}^m(\Gamma+t_j)$ is a self-similar set. Furthermore, for $0< \beta < 1/(2N+1)$, we give a complete characterization on which the union $\bigcup_{j=0}^m(\Gamma+t_j)$ is a self-similar set. Our characterization relies on determining whether some related directed graph has no cycles, or whether some related adjacency matrix is nilpotent.