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We extend the notion of Rauzy induction of interval exchange transformationsto the case of toral $\mathbb{Z}^2$-rotation, i.e., $\mathbb{Z}^2$-actiondefined by rotations on a 2-torus. If $\mathcal{X}_{\mathcal{P},R}$ denotes thesymbolic dynamical system corresponding to a partition $\mathcal{P}$ and$\mathbb{Z}^2$-action $R$ such that $R$ is Cartesian on a sub-domain $W$, weexpress the 2-dimensional configurations in $\mathcal{X}_{\mathcal{P},R}$ asthe image under a $2$-dimensional morphism (up to a shift) of a configurationin $\mathcal{X}_{\widehat{\mathcal{P}}|_W,\widehat{R}|_W}$ where$\widehat{\mathcal{P}}|_W$ is the induced partition and $\widehat{R}|_W$ is theinduced $\mathbb{Z}^2$-action on $W$.  We focus on one example $\mathcal{X}_{\mathcal{P}_0,R_0}$ for which we obtainan eventually periodic sequence of 2-dimensional morphisms. We prove that it isthe same as the substitutive structure of the minimal subshift $X_0$ of theJeandel-Rao Wang shift computed in an earlier work by the author. As aconsequence, $\mathcal{P}_0$ is a Markov partition for the associated toral$\mathbb{Z}^2$-rotation $R_0$. It also implies that the subshift $X_0$ isuniquely 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 therelation between Sturmian sequences and irrational rotations on a circle.Batteries included: the algorithms and code to reproduce the proofs areprovided.

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