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

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.