Rauzy induction of polygon partitions and toral ${\mathbb{Z}}^2$-rotations

Abstract

We extend the notion of Rauzy induction of interval exchange transformations to the case of toral ${\mathbb{Z}}^2$-rotation, i.e., ${\mathbb{Z}}^2$-action defined by rotations on a 2-torus. If ${\mathcal{X}}_{\mathcal{P},R}$ denotes the symbolic dynamical system corresponding to a partition $\mathcal{P}$ and ${\mathbb{Z}}^2$-action $R$ such that $R$ is Cartesian on a sub-domain $W$, we express the 2-dimensional configurations in ${\mathcal{X}}_{\mathcal{P},R}$ as the image under a $2$-dimensional morphism (up to a shift) of a configuration in ${\mathcal{X}}_{\hat{\mathcal{P}}{|}_W,\hat{R}{|}_W}$ where $\hat{\mathcal{P}}{|}_W$ is the induced partition and $\hat{R}{|}_W$ is the induced ${\mathbb{Z}}^2$-action on $W$. We focus on one example ${\mathcal{X}}_{{\mathcal{P}}_0,R_0}$ for which we obtain an eventually periodic sequence of 2-dimensional morphisms. We prove that it is the same as the substitutive structure of the minimal subshift $X_0$ of the Jeandel-Rao Wang shift computed in an earlier work by the author. As a consequence, ${\mathcal{P}}_0$ is a Markov partition for the associated toral ${\mathbb{Z}}^2$-rotation $R_0$. It also implies that the subshift $X_0$ is uniquely ergodic and is isomorphic to the toral ${\mathbb{Z}}^2$-rotation $R_0$ which can be seen as a generalization for 2-dimensional subshifts of the relation between Sturmian sequences and irrational rotations on a circle. Batteries included: the algorithms and code to reproduce the proofs are provided.

v1:36 p, 11 fig; v2:40 p, 12 fig, rewritten before submission; v3:after reviews; v4:typos and updated references; v5:typos and abstract; v6: added a paragraph commenting that Algo 1 may not halt. Jupyter notebook available at https://nbviewer.jupyter.org/url/www.slabbe.org/Publications/arXiv_1906_01104.ipynb

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• A self-similar aperiodic set of 19 Wang tiles
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• Markov partitions for toral ${\mathbb{Z}}^2$-rotations featuring Jeandel-Rao Wang shift and model sets
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Cited by in corpus (2)

• Nonexpansive directions in the Jeandel-Rao Wang shift
• Metallic mean Wang tiles II: the dynamics of an aperiodic computer chip