An intersection problem of graph-directed attractors and an application

Abstract

Let (K1, . . . , Kn) be a Cantor type graph-directed attractors in R d . By creating an auxiliary graph, we provide an effective criterion of whether Ki ∩ Kj is empty for every pair of 1 ≤ i, j ≤ n. Moreover, the emptiness can be checked by examining only a finite number of the attractor's geometric approximations. The method is also applicable for more general graph-directed systems. As an application, we are able to determine the connectedness of all d-dimensional generalized Sierpiński sponges of which the corresponding IFSs are allowed to have rotational and reflectional components.