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Atomic surfaces, tilings and coincidences. II: Reducible case. (English) Zbl 1119.52013
Summary: The atomic surfaces of unimodular Pisot substitutions of irreducible type have been studied by many authors. In this article, we study the atomic surfaces of Pisot substitutions of reducible type. As an analogue of the irreducible case, we define the stepped-surface and the dual substitution over it. Using these notions, we give a simple proof to the fact that atomic surfaces form a self-similar tiling system. We show that the stepped-surface possesses the quasi-periodic property, which implies that a non-periodic covering by the atomic surfaces covers the space exactly \(k\)-times. The atomic surfaces are originally designed by Rauzy to study the spectrum of the substitution dynamical system via a periodic tiling. However, we show that, since the stepped-surface is complicated in the reducible case, it is not clear whether the atomic surfaces can tile the space periodically or not. It seems that the geometry of the atomic surfaces can not applied directly to the spectral problem.
[For Part I of this paper see S. Ito and H. Rao, “Atomic surfaces, tilings and coincidences. I: Irreducible case”, Isr. J. Math. 153, 129–155 (2006; Zbl 1143.37013); Part III: \(\beta\)-tiling and super-coincidence” by H. Ei, S. Ito and H. Rao is in prepration.]

MSC:
52C23 Quasicrystals and aperiodic tilings in discrete geometry
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
28A80 Fractals
11B85 Automata sequences
Citations:
Zbl 1143.37013
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