On the structure of $\lambda$-Cantor set with overlaps

TL;DR

This paper characterizes when the self-similar set $E_\lambda$ with overlaps is totally self-similar, investigates its generating systems, and explores the properties of its spectrum, including conditions for spectrum vanishing.

Contribution

It provides a complete characterization of total self-similarity for $E_\lambda$, analyzes all generating IFSs, and links spectrum properties to the irrationality of $\lambda$.

Findings

Necessary and sufficient condition for total self-similarity of $E_\lambda$.

Determination of the size of points with finite triadic codings.

Spectrum of $E_\lambda$ vanishes iff $\lambda$ is irrational.

Abstract

Given $\lambda\in(0, 1)$, let $E_\lambda$ be the self-similar set generated by the iterated function system $\{x/3,(x+\lambda)/3,(x+2)/3\}$. Then $E_\lambda$ is a self-similar set with overlaps. We obtain the necessary and sufficient condition for $E_\lambda$ to be totally self-similar, which is a concept first introduced by Broomhead, Montaldi, and Sidorov in 2004. When $E_\lambda$ is totally self-similar, all its generating IFSs are investigated, and the size of the set of points having finite triadic codings is determined. Besides, we give some properties of the spectrum of $E_\lambda$ and show that the spectrum of $E_\lambda$ vanishes if and only if $\lambda$ is irrational.