On the self similarity of generalized Cantor sets

TL;DR

This paper investigates the conditions under which generalized Cantor sets exhibit self-similarity, providing a complete characterization and exploring implications for intersections of such sets.

Contribution

It offers a necessary and sufficient condition for generalized Cantor sets to be homogeneously self-similar, advancing understanding of their structural properties.

Findings

Characterization of self-similarity in generalized Cantor sets

Conditions for homogeneously generated self-similar sets

Application to intersections of generalized Cantor sets

Abstract

We consider the self-similar structure of the class of generalized Cantor sets $$\Gamma_{\mathcal{D}}=\Big\{\sum_{n=1}^\infty d_n\beta^{n}: d_n\in D_n, n\ge 1\Big\},$$ 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.