Atomic surfaces, tilings and coincidences II. Reducible case

Ei, Hiromi; Ito, Shunji; Rao, Hui

Annales de l'Institut Fourier, Tome 56 (2006) no. 7, pp. 2285-2313.

Les surfaces atomiques des substitutions unimodulaires de type Pisot ont été étudiées par de nombreux auteurs. Dans cet article, nous étudions les surfaces atomiques des substitutions Pisot de type réductible.

Par analogie avec le cas irréductible, nous définissons la notion de surfaces plissées et de substitution duale agissant dessus. Grâce à ces notions, nous donnons une preuve simple du fait que les surfaces atomiques forment un système de pavage auto-similaire. Nous montrons que les surfaces atomiques sont quasi-périodiques, ce qui implique qu’un recouvrement non périodique par les surfaces atomiques recouvre l’espace exactement $k$ fois.

Les surfaces atomiques ont été introduites à l’origine par Rauzy dans le but d’étudier le spectre des systèmes dynamiques substitutifs via un pavage périodique. Cependant, nous montrons qu’il n’est pas évident de savoir si les surfaces atomiques peuvent paver l’espace périodiquement ou non, en raison de la complexité du cas réductible. Il semble que la géométrie des surfaces atomiques ne peut pas être appliquée directement au problème spectral.

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.

Classification : 52C23, 37A45, 28A80, 11B85
Keywords: Atomic surfaces, Pisot substitution, tiling
Mot clés : surfaces atomiques, substitution de Pivot, pavages

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