The open set conditions in graph-directed constructions (Q1978842)

The article is devoted to questions connected with Hausdorff dimension of self-similar fractals, especially to the role of the Open Set Condition (OSC) and the strong OSC. The setting is known as mixed self-similar sets, graph-directed constructions or Mauldin-Williams graphs. Under certain conditions it is known (see \textit{A. Schief} [Proc. Am. Math. Soc. 122, No.~1, 111-115 (1994; Zbl 0807.28005)]) that the strong OSC and the OSC are equivalent. A partial extension to the setting used in the article is given: the necessity of the strong OSC is given. The article contains six theorems (without proofs), too technical to state them here in details.