On the structure of λ-Cantor set with overlaps

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