Isogenies of minimal Cantor systems: from Sturmian to Denjoy and interval exchanges
Scott Schmieding, Christopher-Lloyd Simon
TL;DR
The paper develops a framework connecting the arithmetic of continued fractions with the dynamics of Sturmian, Denjoy, and interval exchange systems through isogenies, defined as sequences of virtual flow equivalences and infinitesimal 2‑asymptotic factors. It proves that PSL_2(Q)–equivalence of real numbers corresponds to isogeny of the associated Sturmian systems and extends the analysis to Denjoy and IES invariants, including coinvariants, states, and SAF data, yielding rational invariants that govern isogeny behavior. It further analyzes eventual flow equivalence, showing that non‑quadratic Sturmian parameters are rigid up to conjugacy, while quadratic cases exhibit total flow equivalence and arithmetic constraints, and it conjectures a full rational‑invariant classification in totally ergodic settings. Together these results weave together number theory, cohomological invariants, and dynamical systems to advance a unifying classification program for low‑complexity systems under isogeny, with potential extensions to minimal models and IETs through Rauzy data. The work thus provides a conceptual and technical bridge between arithmetic orbits and dynamical equivalence notions, offering a route to a complete classification in broad but structured settings.
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.