Abstract
The well-known plastic number substitution gives rise to a ternary inflation tiling of the real line whose inflation factor is the smallest Pisot-Vijayaraghavan number. The corresponding dynamical system has pure point spectrum, and the associated control point sets can be described as regular model sets whose windows in two-dimensional internal space are Rauzy fractals with a complicated structure. Here, we calculate the resulting pure point diffraction measure via a Fourier matrix cocycle, which admits a closed formula for the Fourier transform of the Rauzy fractals.
Highlights
The algebraic integer β is a unit, and the smallest Pisot–Vijayaraghavan (PV) number, known as the plastic number ; compare [4, p. 50 and Ex. 2.17]
Which is the dynamical spectrum of the tiling dynamical system as well as the model set dynamical system defined by the control point sets; see [6] for background
It is known that we obtain a regular model set [10], with three topologically regular windows that have disjoint interiors
Summary
The algebraic integer β is a unit, and the smallest Pisot–Vijayaraghavan (PV) number, known as the plastic number ; compare [4, p. 50 and Ex. 2.17]. To turn the symbolic sequences into tilings, we choose intervals of natural length, namely 1, β and β2, with control points on their left endpoints. For each sequence in the symbolic hull of, this leads to a typed point set, Λ = Λa ∪ ̇ Λb ∪ ̇ Λc, from the three types of control
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