Self-similar subsets of the Cantor set

In this paper, we study the following question raised by Mattila in 1998: what are the self-similar subsets of the middle-third Cantor set $\C$? We give criteria for a complete classification of all such subsets. We show that for any self-similar subset $\F$ of $\C$ containing more than one point every linear generating IFS of $\F$ must consist of similitudes with contraction ratios $\pm 3^{-n}$, $n\in \N$. In particular, a simple criterion is formulated to characterize self-similar subsets of $\C$ with equal contraction ratio in modulus.