Atomic surfaces, tilings and coincidences II. Reducible case

Hiromi Ei[1]; Shunji Ito[2]; Hui Rao[3]

  • [1] Chuo University Kasuga, Bunkyo-ku Department of Information and System Engineering Tokyo (Japan)
  • [2] Kanazawa University Department of Mathematical Kanazawa (Japan)
  • [3] Tsinghua University Department of Mathematics Beijing (China)

Annales de l’institut Fourier (2006)

  • Volume: 56, Issue: 7, page 2285-2313
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Ei, Hiromi, Ito, Shunji, and Rao, Hui. "Atomic surfaces, tilings and coincidences II. Reducible case." Annales de l’institut Fourier 56.7 (2006): 2285-2313. <http://eudml.org/doc/10205>.

@article{Ei2006,
abstract = {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.},
affiliation = {Chuo University Kasuga, Bunkyo-ku Department of Information and System Engineering Tokyo (Japan); Kanazawa University Department of Mathematical Kanazawa (Japan); Tsinghua University Department of Mathematics Beijing (China)},
author = {Ei, Hiromi, Ito, Shunji, Rao, Hui},
journal = {Annales de l’institut Fourier},
keywords = {Atomic surfaces; Pisot substitution; tiling; atomic surfaces},
language = {eng},
number = {7},
pages = {2285-2313},
publisher = {Association des Annales de l’institut Fourier},
title = {Atomic surfaces, tilings and coincidences II. Reducible case},
url = {http://eudml.org/doc/10205},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Ei, Hiromi
AU - Ito, Shunji
AU - Rao, Hui
TI - Atomic surfaces, tilings and coincidences II. Reducible case
JO - Annales de l’institut Fourier
PY - 2006
PB - Association des Annales de l’institut Fourier
VL - 56
IS - 7
SP - 2285
EP - 2313
AB - 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.
LA - eng
KW - Atomic surfaces; Pisot substitution; tiling; atomic surfaces
UR - http://eudml.org/doc/10205
ER -

References

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