Coloring quasicrystals with prescribed symmetries and frequencies (Q1807618)

The authors study tilings \({\mathcal T}\) of \(\mathbb{R}^n\) by tiles congruent to a finite number of tile types. They consider the following additional properties of the tilings. \(\bullet\) The tiling is repetitive: Given any patch of \({\mathcal T}\) occurring in any bounded region of \(\mathbb{R}^n\), this patch is repeated in any other sufficiently large region. \(\bullet\) The tiling is hierarchical: There are rules to pass from any given tiling \({\mathcal T}_0\) by tiles congruent to the given tile types to a tiling \({\mathcal T}_1\) with the same set of tiles on a larger scale \(\tau\). Moreover, when \({\mathcal T}_1\) is superimposed over \({\mathcal T}_0\), the tiles of \({\mathcal T}_1\) are unions of tiles of \({\mathcal T}_0\), and two congruent tiles of \({\mathcal T}_1\) are decomposed into tiles of \({\mathcal T}_0\) in the same way. Also, \({\mathcal T}_1\) gives rise to \({\mathcal T}_2\), \({\mathcal T}_2\) to \({\mathcal T}_3\) etc. \(\bullet\) The tiling has local \((G,\sigma,M)\)-symmetry: Let the tiles be coloured by a finite set \(M\) and let \(G\) be a finite group acting on \(M\) via \(\sigma:G\times M\to M\). Then, given \(g\in G\) and given a patch \(\Sigma\) of the tiles of \({\mathcal T}\) in any bounded region, there exists \(R>0\) such that in any disk of radius \(R\) one can find a copy \(\Sigma'\) of the patch \(\Sigma\), except that the colouring of \(\Sigma'\) differs from that of \(\Sigma\) by the permutation \(\sigma(g)\). Now the authors prove: Let \({\mathcal T}\) be a hierarchical repetitive tiling. Then for any \((G,\sigma,M)\) there always exists a colouring of \({\mathcal T}\) with local \((G,\sigma,M)\)-symmetry. They redefine this result by imposing some extra conditions on the colouring, such as colour isolation and frequency as defined in the paper.