Graph-directed structures of self-similar sets with overlaps
This paper shows that a certain family of self-similar sets of the real line (i.e., satisfying the set-theoretic equation \(K= \bigcup_{i= 1} ^{m} \varphi_i(K)\), each \(\varphi_i\) being a contractive similarity on \({\mathbf R}\)) with non-trivial overlapping can be obtained as invariant sets associated with a graph-directed self-similar construction satisfying the standard (``open set'') separation condition. The class under consideration includes as particular cases some \(\lambda\)-Cantor sets (i.e., \(m=3\) and \(\varphi_i(x)= x/3+a_i\), \(a_i=0,\lambda,2/3\) with \(\lambda\in (0,1/3)\)) and some sets associated with the so-called \(\{0,1,3\}\)-problem (i.e., \(m=3\) and \(\varphi_i(x)=\beta(x+i)\) with \(i=0,1,3\) and \(\beta\in (1/3,2/5)\)). Applying dimension theory for graph-directed self-similar sets, the fractional dimensions of some of these sets are computed.
scientific article; zbMATH DE number 1547186