Isogenies of minimal Cantor systems: from Sturmian to Denjoy and interval exchanges

[2503.02168] Isogenies of minimal Cantor systems: from Sturmian to Denjoy and interval exchanges Skip to main content Search Submit Donate Log in Mathematics > Dynamical Systems arXiv:2503.02168 (math) [Submitted on 4 Mar 2025 ( v1 ), last revised 26 Aug 2026 (this version, v2)] Title: Isogenies of minimal Cantor systems: from Sturmian to Denjoy and interval exchanges Authors: Scott Schmieding , Christopher-Lloyd Simon View PDF HTML (experimental) Abstract: This work is motivated by the study of continued fraction expansions of real numbers: we describe in dynamical terms their orbits under the action of $\mathrm{PGL}_2(\mathbb{Q})$. A real number gives rise to a Sturmian system encoding a rotation of the circle. It is well known that $\mathrm{PGL}_2(\mathbb{Z})$-equivalence of real numbers, characterized by the tails of their continued fraction expansions, amounts to flow equivalence of Sturmian systems. We show that the multiplicative action of $m\in \mathbb{Z}$ on a real number corresponds to taking the $m$th-power followed by what we call an infinitesimal 2-asymptotic factor of its Sturmian system. This leads us to introduce the notion of isogeny between zero-dimensional systems: it combines virtual flow equivalences and infinitesimal asymptotic equivalences. We develop tools for classifying systems up to isogeny involving cohomological invariants and states. We then use this to give a complete description of $\mathrm{PSL}_2(\mathbb{Q})$-equivalence of real numbers in terms of Sturmian systems. We classify Denjoy systems up to isogenies within this class via the action of $\mathrm{PGL}_{2}(\mathbb{Q})$ on their invariants. We also investigate eventual flow equivalence of Sturmian systems: we show that for non-quadratic parameters it amounts to topological conjugacy and for quadratic parameters it implies total flow equivalence and other arithmetic constraints. In another direction, we consider interval exchanges satisfying Keane's condition. We characterize flow equivalence in terms of interval-induced subsystems (or the tails of their paths in the bilateral Rauzy induction diagram). Finally we find rational invariants for isogeny involving the length modules and SAF invariants of the associated ergodic measures. This leads to a conjecture for their classification up to isogeny, which we prove in the totally ergodic case. Comments: New version includes proofs of two conjectures from Version 1: Conjecture A.6, that eventual PGL(2,Z)-equivalence of irrationals implies equality up to sign and addition of an integer, and Conjecture 7.7, that eventual flow equivalence implies topological conjugacy for Sturmian subshifts. These are both part of an expanded Section 7 on virtual and eventual flow equivalence Subjects: Dynamical Systems (math.DS) Cite as: arXiv:2503.02168 [math.DS] (or arXiv:2503.02168v2 [math.DS] for this version) https://doi.org/10.48550/arXiv.2503.02168 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Scott Schmieding [ view email ] [v1] Tue, 4 Mar 2025 01:15:32 UTC (55 KB) [v2] Wed, 26 Aug 2026 23:49:45 UTC (80 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.DS < prev | next > new | recent | 2025-03 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. 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