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

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.