Equivalence of self-similar and pseudo-self-similar tiling spaces in \(\mathbb R^{2}\)
A derived Voronoi tiling of a given tiling \(T\) is a related tiling based on a patch of tiles in \(T\). Analogously, one may identify for a tiling space \(X\) a patch which appears regularly in all tilings in a tiling space \(X\) and forms a derived Voronoi space of tilings based on such patch. Two tilings or tiling spaces are called mutually locally derivable (MLD) if they are related by local operations. The author's main results are the following: {\parindent4mm \begin{itemize}\item[--] A tiling and its derived Voronoi tiling are MLD, as are the tiling space and its derived Voronoi space. \item[--]If \(T\) or \(X\) has high degree of hierarchical structure, then its derived Voronoi tiling or derived Voronoi space has a scaled pair. \item[--]If a derived Voronoi tiling, or derived Voronoi tiling space, contains a scaled pair, then the original tiling or tiling space has a weaker degree of hierarchical structure. \item[--]If a tiling or tiling space has a weaker degree of hierarchical structure, then it is MLD to a tiling or tiling space having high degree of hierarchical structure. \end{itemize}}
scientific article; zbMATH DE number 5906699