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 \begin{document}$ \mathbb{Z}^2 $\end{document}-rotation, i.e., \begin{document}$ \mathbb{Z}^2 $\end{document}-action defined by rotations on a 2-torus. If \begin{document}$ \mathscr{X}_{\mathscr{P}, R} $\end{document} denotes the symbolic dynamical system corresponding to a partition \begin{document}$ \mathscr{P} $\end{document} and \begin{document}$ \mathbb{Z}^2 $\end{document}-action \begin{document}$ R $\end{document} such that \begin{document}$ R $\end{document} is Cartesian on a sub-domain \begin{document}$ W $\end{document}, we express the 2-dimensional configurations in \begin{document}$ \mathscr{X}_{\mathscr{P}, R} $\end{document} as the image under a \begin{document}$ 2 $\end{document}-dimensional morphism (up to a shift) of a configuration in \begin{document}$ \mathscr{X}_{\widehat{\mathscr{P}}|_W, \widehat{R}|_W} $\end{document} where \begin{document}$ \widehat{\mathscr{P}}|_W $\end{document} is the induced partition and \begin{document}$ \widehat{R}|_W $\end{document} is the induced \begin{document}$ \mathbb{Z}^2 $\end{document}-action on \begin{document}$ W $\end{document}.

We focus on one example, \begin{document}$ \mathscr{X}_{\mathscr{P}_0, R_0} $\end{document}, for which we obtain an eventually periodic sequence of 2-dimensional morphisms. We prove that it is the same as the substitutive structure of the minimal subshift \begin{document}$ X_0 $\end{document} of the Jeandel–Rao Wang shift computed in an earlier work by the author. As a consequence, \begin{document}$ {\mathscr{P}}_0 $\end{document} is a Markov partition for the associated toral \begin{document}$ \mathbb{Z}^2 $\end{document}-rotation \begin{document}$ R_0 $\end{document}. It also implies that the subshift \begin{document}$ X_0 $\end{document} is uniquely ergodic and is isomorphic to the toral \begin{document}$ \mathbb{Z}^2 $\end{document}-rotation \begin{document}$ R_0 $\end{document} 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.