Self-similar subsets of the Cantor set

Abstract

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.