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 $\mathbb{Z}^2$-periodic extension $H=F+\mathbb{Z}^2$ of a fractal square $F$ and of its complement $H^c=\mathbb{R}^2\setminus H$. A fractal square $F\subset\mathbb{R}^2$ 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 $\mathbb R^3$ for which the set $H^c=\mathbb{R}^3\setminus H$ 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 $C(\mathbb R^3)$, it is bi-Lipschitz isomorphic to the Cantor set $C_{1/5}$, all the sets $K_\alpha+\mathbb{Z}^3$ 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 $[a,b]$.