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 ±3−n±3−n, n∈Nn∈N. In particular, a simple criterion is formulated to characterize self-similar subsets of C with equal contraction ratio in modulus.