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 multiplicative action by

and the projective action of

PGL 2 ( Z ) $\operatorname {\mathrm {PGL}}_2(\mathbb {Z})$ upper P upper G upper L 2 left parenthesis double struck upper Z right parenthesis

; hence, of

PGL 2 ( Q ) $\operatorname {\mathrm {PGL}}_2(\mathbb {Q})$ upper P upper G upper L 2 left parenthesis double struck upper Q right parenthesis

. A real number gives rise to a Sturmian system encoding a rotation of the circle, which is a particular case of an interval exchange system. It is well known that

PGL 2 ( Z ) $\operatorname {\mathrm {PGL}}_2(\mathbb {Z})$ upper P upper G upper L 2 left parenthesis double struck upper Z right parenthesis

-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 ∈ Z $m\in \mathbb {Z}$ m element of double struck upper 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 flow equivalences, virtual conjugacies 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

PGL 2 ( Q ) $\operatorname {\mathrm {PGL}}_2(\mathbb {Q})$ upper P upper G upper L 2 left parenthesis double struck upper Q right parenthesis

-equivalence of real numbers in terms of Sturmian systems. Considering powers of Sturmian systems leads naturally to the realm of Denjoy systems, and we classify Denjoy systems up to isogenies preserving this class via the action of

PGL 2 ( Q ) $\operatorname {\mathrm {PGL}}_{2}(\mathbb {Q})$ upper P upper G upper L 2 left parenthesis double struck upper Q right parenthesis

on their invariants. We also investigate eventual flow equivalence and show that it implies topological conjugacy for Sturmian and Denjoy systems. For this, we prove the following novel arithmetic result that we believe is of independent interest. Consider

α , β ∈ R ∖ Q $\alpha ,\beta \in \mathbb {R}\setminus \mathbb {Q}$ alpha comma beta element of double struck upper R minus double struck upper Q

; if for all but finitely many prime integers

n ∈ P $n\in \mathbb {P}$ n element of double struck upper P

the numbers

n α , n β $n\alpha ,n\beta $ n alpha comma n beta

are

PGL 2 ( Z ) $\operatorname {\mathrm {PGL}}_2(\mathbb {Z})$ upper P upper G upper L 2 left parenthesis double struck upper Z right parenthesis

-equivalent, then

α ≡ ± β mod Z $\alpha \equiv \pm \beta \bmod {\mathbb {Z}}$ alpha identical to plus or minus beta mod double struck upper Z

. 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). We describe their 2-AI factors in terms of minimal models. 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.