Parametrization of a two-dimensional quasiperiodic Rauzy tiling

Abstract:

With the help of an affine inflation $B$, a two-dimensional quasiperiodic Rauzy tiling $\mathcal {R}^\infty$ is constructed, together with a parametrization of its tiles by algebraic integers $\mathbb {Z}[\zeta ] \subset [0,1)$, where $\zeta$ is a certain Pisot number (specifically, the real root of the polynomial $x^3+x^2+x-1$). The coronas (clusters) of the tiling $\mathcal {R}^\infty$ are classified by disjoint half-intervals in $[0,1)$ the lengths of which are proportional to the frequencies of the corresponding corona types. It is proved that, for each of the three basis tiles, there exists an odd number of corona types of an arbitrary level. The parametrization obtained describes local rules (tile adjacency conditions) for $\mathcal {R}^\infty$, and it conjugates the action of the affine rotation $B$ of the plane $\mathbb {R}^2$ by an irrational angle with a shift of the coding sequences.