{"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/markov-partitions-for-toral-mathbb-z-2","title":"Markov partitions for toral $\\mathbb{Z}^2$-rotations featuring Jeandel-Rao Wang shift and model sets","arxiv_id":"1903.06137","date":"2019-03-14","proceeding":null,"authors":["Sébastien Labbé"],"abstract":"We define a partition $\\mathcal{P}_0$ and a $\\mathbb{Z}^2$-rotation ($\\mathbb{Z}^2$-action defined by rotations) on a 2-dimensional torus whose associated symbolic dynamical system is a minimal proper subshift of the Jeandel-Rao aperiodic Wang shift defined by 11 Wang tiles. We define another partition $\\mathcal{P}_\\mathcal{U}$ and a $\\mathbb{Z}^2$-rotation on $\\mathbb{T}^2$ whose associated symbolic dynamical system is equal to a minimal and aperiodic Wang shift defined by 19 Wang tiles. This proves that $\\mathcal{P}_\\mathcal{U}$ is a Markov partition for the $\\mathbb{Z}^2$-rotation on $\\mathbb{T}^2$. We prove in both cases that the toral $\\mathbb{Z}^2$-rotation is the maximal equicontinuous factor of the minimal subshifts and that the set of fiber cardinalities of the factor map is $\\{1,2,8\\}$. The two minimal subshifts are uniquely ergodic and are isomorphic as measure-preserving dynamical systems to the toral $\\mathbb{Z}^2$-rotations. It provides a construction of these Wang shifts as model sets of 4-to-2 cut and project schemes. A do-it-yourself puzzle is available in the appendix to illustrate the results.","url_abs":"https://arxiv.org/abs/1903.06137v3","url_pdf":"https://arxiv.org/pdf/1903.06137v3.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":"markov-partitions-for-toral-mathbb-z-2","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}