Pure discrete spectrum in substitution tiling spaces. (English) Zbl 1291.37024

Barge, Marcy; Štimac, Sonja; Williams, R. F.

Discrete Contin. Dyn. Syst. 33, No. 2, 579-597 (2013).

It is known that a pattern of points produces a pure point \(X\)-ray diffraction if and only if the associated tiling dynamical system has purely discrete spectrum. Consequently, a sufficient condition for having a purely discrete dynamical spectrum is of considerable interest. In this paper, a technique for establishing pure discrete spectrum for substitution systems of Pisot type is given. These results are a generalization of the one-dimensional results of the first author and J. Kwapisz [Am. J. Math. 128, No. 5, 1219–1282 (2006; Zbl 1152.37011)] for one-dimensional Pisot substitutions. This paper also clarifies the proof of the earlier result, and gives some two-dimensional examples.

Reviewer: Geoffrey R. Goodson (Towson)

MSC:

37B50	Multi-dimensional shifts of finite type, tiling dynamics (MSC2010)
52C22	Tilings in \(n\) dimensions (aspects of discrete geometry)
52C23	Quasicrystals and aperiodic tilings in discrete geometry
37B05	Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
11R06	PV-numbers and generalizations; other special algebraic numbers; Mahler measure

Keywords:

pure discrete spectrum; tiling dynamical system; substitution dynamical system of Pisot type

Citations:

Zbl 1152.37011