On the self similarity of generalized Cantor sets

Abstract

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

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