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Finite type, open set conditions and weak separation conditions. (English) Zbl 1226.28004

Summary: We study graph-directed iterated function systems of finite type. We show that such an IFS of finite type induces another graph-directed IFS of finite type where every strongly connected component satisfies the open set condition. We introduce the notions of topological and geometric weak separation properties, and summarize the relationship between the different separation conditions. For the induced IFS, similarity, growth, box and Hausdorff dimension coincide. Finally, we prove that the generalized finite-type condition for graphs implies the geometric weak separation property.

MSC:

28A78 Hausdorff and packing measures
28A80 Fractals
05C10 Planar graphs; geometric and topological aspects of graph theory
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