The connectedness of Sierpi\'nski sponges with rotational and reflectional components and associated graph-directed systems

TL;DR

This paper introduces two methods to determine the connectedness of generalized Sierpiński sponges with rotational and reflectional symmetries by reducing the problem to intersection checks of graph-directed attractors.

Contribution

It provides an effective, finite-approximation-based criterion for connectedness in high-dimensional Sierpiński sponges with symmetries, extending to general graph-directed systems.

Findings

Connectedness can be characterized via intersection criteria.

Finite geometric approximations suffice for checking intersections.

The methods are applicable to broader graph-directed systems.

Abstract

We provide two methods to characterize the connectedness of all $d$-dimensional generalized Sierpi\'nski 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 $(K_1,\ldots,K_n)$ 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 $1\leq i,j\leq n$. 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.