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

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.

28 pages, 1 figure