Orbit closures and closed invariant sets for reducible algebraic $\mathbb{Z}^d$-actions on tori and solenoids

An algebraic $\mathbb{Z}^d$-action is a homomorphism from $\mathbb{Z}^d$ to the automorphism group of a compact abelian group such as a torus or, more generally, a solenoid. For such actions the measurable rigidity theory is essentially complete: Einsiedler and Lindenstrauss proved that a totally irreducible, non-virtually-cyclic action admits no ergodic invariant measure of positive entropy other than Haar measure, and gave a structure theorem for ergodic invariant measures of arbitrary actions without virtually cyclic factors, from which they also classified measurable factors and disjointness in the reducible case. The topological counterpart in the reducible case is open: beyond Berend's theorem, which completely describes closed invariant sets for totally irreducible actions, there is no classification of the closed invariant sets and orbit closures that can occur when total irreducibility fails. The precise expected answer in the basic arithmetic model — a higher-rank group of $S$-units of a proper subfield acting on an $S$-arithmetic solenoid — is Conjecture 5 of Lindenstrauss's 2021 survey: every orbit closure is either the whole space or is contained in the orbit itself together with finitely many cosets of proper closed subgroups. Partial results cover non-hyperbolic totally irreducible actions (Z. Wang) and self-joinings of Cartan actions of rank at least three (Lindenstrauss–Wang), while a rank-two counterexample of Lindenstrauss–Wang shows the expected classification is genuinely delicate. The problem is to prove or refute this conjectured orbit-closure classification, and more generally to classify the closed invariant sets of reducible algebraic $\mathbb{Z}^d$-actions on tori and solenoids.

Background

A solenoid is a compact connected abelian group $X$ whose Pontryagin dual — the group of continuous homomorphisms from $X$ to the circle — embeds into a finite-dimensional vector space over $\mathbb{Q}$; the simplest examples are the finite-dimensional tori $\mathbb{T}^n=(\mathbb{R}/\mathbb{Z})^n$. An algebraic $\mathbb{Z}^d$-action is a homomorphism $\alpha:\mathbb{Z}^d\to\mathrm{Aut}(X)$, i.e. a choice of $d$ commuting continuous group automorphisms of $X$. The action is irreducible if $X$ has no proper infinite closed subgroup invariant under every element of $\alpha$, and totally irreducible if this remains true for the restriction of $\alpha$ to every finite-index sublattice of $\mathbb{Z}^d$; it is virtually cyclic if some finite-index sublattice acts through powers of a single automorphism. The classical rigidity program for such actions has a topological and a measure-theoretic side.

Topologically, Furstenberg proved that a closed subset of the circle invariant under both $x\mapsto 2x$ and $x\mapsto 3x$ is finite or is the whole circle, and Berend characterized when this phenomenon persists in all dimensions: every orbit of $\alpha$ is finite or dense if and only if $\alpha$ is totally irreducible, not virtually cyclic, and some element of the action expands every common eigendirection of the automorphisms involved (an automorphism is hyperbolic if none of its eigenvalues has modulus one, and this last condition rules out isometric directions shared by the whole action). Measure-theoretically, Rudolph proved that an ergodic measure on the circle invariant under $x\mapsto px$ and $x\mapsto qx$ for multiplicatively independent $p,q$ is either Lebesgue measure or gives both maps zero entropy — the exponential rate of information production of a measure-preserving map. Einsiedler and Lindenstrauss extended this to all solenoids: if $\alpha$ is totally irreducible and not virtually cyclic, then Haar measure — the translation-invariant probability measure on the compact group — is the only $\alpha$-ergodic invariant measure giving positive entropy to any element of the action.

Crucially for the present problem, the same authors also handled the non-irreducible case on the measure side. Assuming only that $\alpha$ has no virtually cyclic factors, they proved a structure theorem: after restricting to a finite-index subgroup, every ergodic invariant measure decomposes into finitely many mutually singular ergodic components, each invariant under translations by a closed invariant subgroup, permuted by the full action, and inducing zero entropy on the quotient solenoid. From this they derived that measurable factors of such actions are algebraic — given by quotient homomorphisms — and classified disjointness: two actions are disjoint (share no joining, i.e. no invariant measure on the product projecting to the given measures) exactly when, after a finite-index restriction, they have no common algebraic factor.

In the topological category the situation is the reverse. For totally irreducible actions Berend's theorem (extended by Z. Wang, who removed the expansion condition) gives a complete description of closed invariant sets. Without total irreducibility little is known: beyond the totally irreducible case, where Berend gave definitive results, no classification of the closed invariant sets or orbit closures is available. The basic arithmetic model is as follows. A place of a number field $K$ is an equivalence class of absolute values arising from a real or complex embedding or a $p$-adic valuation; fixing a finite set $S$ of places containing all infinite ones, the ring of $S$-integers $\mathcal{O}_S$ consists of the elements of $K$ that are integral outside $S$, and $X=\prod_{v\in S}K_v/\mathcal{O}_S$ is a solenoid on which the units of $\mathcal{O}_S$ (the $S$-units) act by multiplication. Irreducible actions correspond to unit groups not contained in any proper subfield; the reducible case already includes the actions of a higher-rank subgroup $\Sigma$ of the $S_L$-units of a proper subfield $L<K$.

For this setting Lindenstrauss's 2021 survey proposes the expected answer (Conjecture 5): if no finite-index subgroup of $\Sigma$ lies in a proper subfield of $L$, then for every $x\in X$ the orbit closure $Y=\overline{\Sigma.x}$ is either all of $X$, or is contained in the orbit $\Sigma.x$ together with finitely many cosets $X_i+p_i$ of proper closed subgroups $X_i$ of $X$ at torsion points $p_i$ (points of finite order). The survey identifies the key obstacle: although the supports of invariant measures are completely classified by the measure rigidity above, the correspondence between invariant measures and individual orbits is weak in the diagonalizable setting. The best partial results are the classification of topological self-joinings of Cartan actions — maximal-rank diagonal $\mathbb{Z}^r$-actions on tori — by Lindenstrauss and Z. Wang: for rank at least three every closed invariant topologically transitive set (a set that is the closure of one orbit) is homogeneous, a finite union of translates of an invariant subtorus, while for rank two they constructed a counterexample whose orbit closure is the disjoint union of the orbit and three transverse invariant subtori, proving the naive homogeneous classification is false. No general theorem or counterexample for the reducible case is known.

Problem Statement

Let $\alpha$ be a $\mathbb{Z}^d$-action by automorphisms of a solenoid $X$ (in particular of a finite-dimensional torus), with $d\ge 2$, such that $\alpha$ is not totally irreducible but has no virtually cyclic factors (i.e., no quotient of $\alpha$ by a closed invariant subgroup is virtually cyclic). Understand the closed $\alpha$-invariant subsets of $X$; equivalently, classify the possible orbit closures $\overline{\alpha(\mathbb{Z}^d).x}$, since a closed invariant set is topologically transitive precisely when it is one orbit closure, and general closed invariant sets are built from these.

The pinned authoritative form of this problem is Conjecture 5 of Lindenstrauss's 2021 survey (arXiv:2101.11114). Let $K$ be a number field, $S$ a finite set of places of $K$ containing all infinite places, and $\mathcal{O}_S$ the ring of $S$-integers. Let $L<K$ be a proper subfield, $S_L$ the set of valuations of $L$ corresponding to places in $S$, and $\Sigma$ a higher-rank subgroup (that is, a subgroup of rank at least two) of the group of $S_L$-units of $L$, such that no finite-index subgroup of $\Sigma$ is contained in a proper subfield of $L$. Let $\Sigma$ act by multiplication on the solenoid $X=\prod_{v\in S}K_v/\mathcal{O}_S$, and for $x\in X$ set $Y=\overline{\Sigma.x}$. Prove or refute: either $Y=X$, or there exist finitely many closed proper subgroups $X_i<X$ and torsion points $p_i\in X$ such that $Y\subseteq\Sigma.x\cup\bigcup_i(X_i+p_i)$.

More generally, establish the analogous structure theorem for orbit closures and closed invariant sets of all algebraic $\mathbb{Z}^d$-actions by automorphisms of tori and solenoids without virtually cyclic factors that are not totally irreducible, the conjecture above being its $S$-unit model case. A single explicit counterexample to the conjecture above — arithmetic data for $K$, $S$, $L$, $\Sigma$ and a point $x$ with a rigorous determination of $\overline{\Sigma.x}$ violating both stated alternatives — also constitutes a resolution.

References

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