The connectedness of Sierpiński sponges with rotational and reflectional components and associated graph-directed systems

Abstract

We provide two methods to characterize the connectedness of all $d$-dimensional generalized Sierpiński sponges whose corresponding IFSs are allowed to have rotational and reflectional components. Our approach is to reduce it to an intersection problem between the coordinates of graph-directed attractors. More precisely, let $(K_1,\dots ,K_n)$ be a Cantor-type graph-directed attractor in ${\mathbb{R}}^d$. By creating an auxiliary graph, we provide an effective criterion for whether $K_i\cap K_j$ is empty for every pair of $1\le i,j\le n$. Moreover, the emptiness can be checked by examining only a finite number of geometric approximations of the attractor. The approach is also applicable to more general graph-directed systems.

24 pages, 4 figures