On the self similarity of generalized Cantor sets

We consider the self-similar structure of the class of generalized Cantor sets $${\mathrm{\Gamma }}_{\mathcal{D}}=\{\sum _{n=1}^{\mathrm{\infty }}{d}_n{\beta }^n:d_n\in D_n,n\ge 1\},$$ 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 ${\mathrm{\Gamma }}_{\mathcal{D}}$ to be a homogeneously generated self similar set. An application to the self-similarity of intersections of generalized Cantor sets will be given.