On connected components of fractal cubes

The paper shows an essential difference between fractal squares and fractal cubes. The topological classification of fractal squares proposed in 2013 by K.-S. Lau et al. was based on analyzing the properties of the Z(2)-periodic extension H = F + Z(2) of a fractal square F and of its complement H-c = R-2\H. A fractal square F subset of R-2 contains a connected component different from a line segment or a point if and only if the set H-c contains a bounded connected component. We show the existence of a fractal cube F in R-3 for which the set F = R-3\H is connected whereas the set Q of connected components K-alpha of F possesses the following properties: Q is a totally disconnected self-similar subset of the hyperspace C(R-3), it is bi-Lipschitz isomorphic to the Cantor set C-1/5, all the sets K-alpha + Z(3) are connected and pairwise disjoint, and the Hausdorff dimensions dim(H)(K-alpha) of the components K-alpha assume all values from some closed interval [a, b].