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

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