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Tilings from non-Pisot unimodular matrices. (English) Zbl 1173.37012

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”.

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
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