Furukado, Maki Tilings from non-Pisot unimodular matrices. (English) Zbl 1173.37012 Hiroshima Math. J. 36, No. 2, 289-329 (2006). Summary: Using the unimodular Pisot substitution of the free monoid on \(d\) letters, the existence of graph-directed self-similar sets \(\{X_i\}_{i=1,2,\dots,d}\) satisfying the set equation \[ A^{-1}X_i= \bigcup_{j=1}^{l_i} (V-j^{(i)}+X_j) \quad\text{(non-overlapping)} \]with the positive measure on the \(A\)-invariant contracting plane \(P\) is well-known, where \(A\) is the incidence matrix of the substitution. Moreover, under some conditions, the set \(\{X_i\}_{i=1,2,\dots,d}\) is the prototile of the quasi-periodic tiling of \(P\). In this paper, even in the case of non-Pisot matrix A, the generating method of graphdirected self-similar sets and quasi-periodic tilings is proposed under the “blocking condition”. Cited in 4 Documents MSC: 37B50 Multi-dimensional shifts of finite type, tiling dynamics (MSC2010) 52C23 Quasicrystals and aperiodic tilings in discrete geometry 37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010) 28A80 Fractals Keywords:Non-Pisot unimodular substitutions; graph-directed self-affine tilings PDFBibTeX XMLCite \textit{M. Furukado}, Hiroshima Math. J. 36, No. 2, 289--329 (2006; Zbl 1173.37012)