On the union of homogeneous symmetric Cantor set with its translations

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

$$\begin{aligned} \Bigg \{ \phi _i(x)=\beta x + i \frac{1-\beta }{N}: i=0,1,\ldots , N \Bigg \}. \end{aligned}$$

For $m\in \mathbb {Z}_+$ we show that there exist infinitely many translation vectors ${\textbf{t}}=(t_0,t_1,\ldots , t_m)$ 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 finite algorithm to determine whether the union $\bigcup _{j=0}^m(\Gamma +t_j)$ is a self-similar set for any given vector ${\textbf{t}}$. Our characterization relies on determining whether some related directed graph has no cycles, or whether some related adjacency matrix is nilpotent.