An intersection problem of graph-directed attractors and an application

Let $(K_1,\ldots,K_n)$ be a Cantor type graph-directed attractors in $\mathbb{R}^d$. By creating an auxiliary graph, we provide an effective criterion of whether $K_i\cap K_j$ is empty for every pair of $1\leq i,j\leq n$. 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 Sierpi\'nski sponges of which the corresponding IFSs are allowed to have rotational and reflectional components.
Comment: 24 pages, 4 figures