{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/rauzy-induction-of-polygon-partitions-and","title":"Rauzy induction of polygon partitions and toral $\\mathbb{Z}^2$-rotations","arxiv_id":"1906.01104","date":"2019-06-03","proceeding":null,"authors":["Sébastien Labbé"],"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 $\\mathcal{X}_{\\widehat{\\mathcal{P}}|_W,\\widehat{R}|_W}$ where $\\widehat{\\mathcal{P}}|_W$ is the induced partition and $\\widehat{R}|_W$ is the induced $\\mathbb{Z}^2$-action on $W$. 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.","url_abs":"https://arxiv.org/abs/1906.01104v6","url_pdf":"https://arxiv.org/pdf/1906.01104v6.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"rauzy-induction-of-polygon-partitions-and","repo_url":"https://github.com/seblabbe/slabbe","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}