On the union of homogeneous symmetric Cantor set with its translations
Fix a positive integer $N$ and a real number $0< β< 1/(N+1)$. Let $Γ$ be the homogeneous symmetric Cantor set generated by the IFS $$ \Big\{ ϕ_i(x)=βx + i \frac{1-β}{N}: i=0,1,\cdots, N \Big\}. $$ 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 $\bigcup_{j=0}^m(Γ+t_j)$ is a self-similar set. Furthermore, for $0< β< 1/(2N+1)$, we give a complete characterization on which the union $\bigcup_{j=0}^m(Γ+t_j)$ 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.
math.DS↗