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

We propose a method for proving that a toral partition into polygons is a Markov partition for a given toral $mathbb{Z}^2$-rotation, i.e., $mathbb{Z}^2$-action defined by rotations on a 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 $mathcal{X}_{widehat{mathcal{P}}|_W,widehat{R}|_W}$ where $widehat{mathcal{P}}|_W$ is the induced partition and $widehat{R}|_W$ is the induced $mathbb{Z}^2$-action on the sub-domain $W$. The induced $mathbb{Z}^2$-action extends the notion of Rauzy induction of IETs to the case of $mathbb{Z}^2$-actions where subactions are polytope exchange transformations. This allows to describe $mathcal{X}_{mathcal{P},R}$ by a $S$-adic sequence of 2-dimensional morphisms. We apply the method on one example and we obtain a sequence of 2-dimensional morphisms which is eventually periodic leading to a self-induced partition. We prove that its substitutive structure 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, we deduce the equality of the two subshifts and it implies that the partition is a Markov partition for the associated toral $mathbb{Z}^2$-rotation since $X_0$ is a shift of finite type. It also implies that $X_0$ 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 the code to reproduce the proofs are provided.