  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.
