On the self similarity of generalized Cantor sets

Abstract

We consider the self-similar structure of the class of generalized Cantor sets $$Γ_{\mathcal{D}}=\Big\{\sum_{n=1}^\infty d_nβ^{n}: d_n\in D_n, n\ge 1\Big\},$$ where $0<β<1$ and $D_n, n\ge 1,$ are nonempty and finite subsets of $\mathbb{Z}$. We give a necessary and sufficient condition for $Γ_{\mathcal{D}}$ to be a homogeneously generated self similar set. An application to the self-similarity of intersections of generalized Cantor sets will be given.