Joining classification for higher-rank diagonalizable actions beyond the no-local-rank-one-factor hypothesis, and the equivalent measure classification for commuting toral endomorphism actions without virtually cyclic factors

Einsiedler and Lindenstrauss (Duke Math. J. 2007) proved that ergodic joinings between higher-rank diagonalizable actions on locally homogeneous spaces are algebraic under a no-local-rank-one-factor hypothesis, and posed two extensions: (1) remove that hypothesis, keeping only the absence of global rank-one factors, which they said would require a general low-entropy argument they were writing; (2) classify joinings — equivalently, for these actions, all invariant measures — for actions of nonvirtually cyclic groups of commuting toral endomorphisms with no virtually cyclic factor, where only descriptions up to zero-entropy components were available. Direction (1) was subsequently carried out by the same authors: the general low-entropy method appeared in 2008, and the joining classification for higher-rank torus actions on S-arithmetic quotients, now admitting ambient groups that are products of forms of $\mathrm{SL}_2$ (the local-rank-one-factor cases), was published in 2019, with positive-characteristic analogues in 2012 and 2020. The surviving open core is direction (2): the classification, without entropy hypotheses, of ergodic invariant measures (equivalently ergodic joinings) for commuting toral endomorphism actions without virtually cyclic factors, whose base case is Furstenberg's 1967 conjecture that every ergodic $\times p,\times q$-invariant measure on the circle is Lebesgue or atomic. All known techniques — high-entropy, low-entropy, and entropy-symmetry arguments — require positive entropy, so the zero-entropy regime is the open frontier; recent work achieves only conditional statements.

Background

Ergodic theory studies measure-preserving systems: a probability space together with transformations or flows preserving the measure. A system is ergodic when every invariant measurable set is null or conull. A joining of two ergodic systems carrying actions of the same group is an invariant probability measure on the product space whose push-forwards to the two factors are the given measures; the product measure is the trivial joining, and when it is the only joining the systems are called disjoint. Joinings measure common structure: any measurable factor (a quotient system of both) yields a nontrivial joining, so classifying joinings is a strong rigidity statement. Measure-theoretic entropy quantifies the exponential orbit-complexity of a measure-preserving map; a joining or invariant measure is said to have positive entropy for an action when some element of the action has positive entropy on it.

The systems of interest are algebraic. Let $G$ be a connected Lie group and $\Gamma\subset G$ a lattice (a discrete subgroup of finite covolume); quotients $G/\Gamma$ are locally homogeneous spaces. An injective homomorphism $\alpha:\mathbb{R}^n\to G$ into an $\mathbb{R}$-split torus (a conjugate of a group of diagonal matrices, with diagonalizable adjoint action on the Lie algebra) realizes by translation a higher-rank diagonalizable action on $G/\Gamma$ preserving Haar measure (the unique translation-invariant probability measure on a compact group). For $n\ge2$ such actions are strikingly rigid: although each one-parameter subaction has an abundance of invariant measures, the full action is expected to admit only algebraic invariant measures — Haar measure on a single closed orbit of a closed subgroup. Furstenberg introduced both joinings and this rigidity phenomenon in 1967 through the simplest example, multiplication by multiplicatively independent integers $p$ and $q$ (no powers of $p$ and $q$ coincide) on the circle $\mathbb{R}/\mathbb{Z}$; he conjectured that every ergodic $\times p,\times q$-invariant measure is Lebesgue or atomic, and this conjecture remains open. For homogeneous spaces, Katok and Spatzier formulated analogous measure-classification conjectures for higher-rank abelian hyperbolic actions. All known progress requires positive entropy: Rudolph (1990) proved the $\times p,\times q$ statement under positive entropy (the coprimality restriction was later removed and alternative proofs found, all still entropy-dependent), and Katok–Spatzier, Kalinin–Katok, and Kalinin–Spatzier established positive-entropy rigidity for higher-rank actions by automorphisms of tori and homogeneous spaces under additional irreducibility or total non-symplecticness assumptions (total non-symplecticness excluding invariant symplectic subspaces). The entropy-based tools are the high-entropy method, which derives invariance from maximal entropy contributions along the Lyapunov decomposition (the splitting of the Lie algebra into the weight spaces of the action), and the low-entropy method (developed in Lindenstrauss's work on arithmetic quantum unique ergodicity, i.e. the equidistribution of Laplacian eigenfunctions on arithmetic surfaces, and in the Einsiedler–Katok–Lindenstrauss theorem on Littlewood's conjecture in simultaneous Diophantine approximation), which derives unipotent invariance (invariance under one-parameter subgroups whose adjoint action has all eigenvalues equal to one) from small-but-positive entropy along conditional measures on foliations.

The source paper (Einsiedler–Lindenstrauss, Duke Math. J. 2007) united the two themes of Furstenberg's work: it classified ergodic joinings between two higher-rank diagonalizable actions $\alpha_i$ on $G_i/\Gamma_i$ under the hypothesis that the actions have no local rank-one factors — for every proper Lie ideal $\mathfrak h$ of the Lie algebra of $G_i$, the adjoint weights of $\alpha_i$ on the quotient Lie algebra do not all lie in a one-dimensional subspace of $(\mathbb{R}^n)^*$, where a weight is the logarithm of the absolute value of an adjoint eigenvalue, viewed as a linear functional of the acting parameter. Their theorem shows that every ergodic joining is Haar measure on a single closed orbit of a closed subgroup $H\le G_1\times G_2$ projecting onto both factors (an algebraic joining), with consequences for measurable factors and equidistribution. The hypothesis excludes ambient groups built from rank-one pieces; for example, for $G=\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})$ with an irreducible lattice, each factor has rank-one Lie algebra so local rank-one factors exist, even though the induced action on the quotient has no global rank-one factor (no quotient system on which the action degenerates to rank one). In the introduction the authors state that they expect similar results even when local rank-one factors are present, provided the actions on the quotients have no global rank-one factors, but that this requires a totally different proof — a general version of the low-entropy argument — which they were in the process of writing. They also contrast the homogeneous setting with Furstenberg's: for actions of nonvirtually cyclic groups of commuting toral endomorphisms (a group is nonvirtually cyclic when no finite-index subgroup is cyclic; the $\times p,\times q$ action is the basic example, and these are generally non-invertible maps), no full classification of joinings exists — the available results describe ergodic joinings only up to zero-entropy components — and for such actions the classification of joinings is equivalent to the classification of all invariant measures, a well-known open problem. The standard formulation of that open problem — also the standing hypothesis of the positive-entropy theorems — concerns actions with no virtually cyclic factor, meaning that no quotient by a closed invariant subgroup carries an induced action generated, up to finite index, by a single element of the acting group; the acting group being nonvirtually cyclic alone does not exclude such degenerate rank-one factors, on which arbitrary invariant measures of a single element persist. The natural invertible models of these endomorphism actions live on solenoids: compact connected abelian groups, such as inverse limits of circles under multiplication maps, whose Pontryagin dual (the group of continuous characters) embeds in a finite-dimensional $\mathbb{Q}$-vector space; a non-invertible endomorphism is studied through its natural extension (the inverse-limit system).

Problem Statement

The problem is the source paper's stated extension of its joining classification beyond the no-local-rank-one-factor hypothesis, under the common objective of completing the algebraic classification of joinings — equivalently, where applicable, of invariant measures — for higher-rank algebraic actions. It has two linked levels: (1) rank-one ambient factors, and (2) commuting toral endomorphisms. The second is the surviving open core, and the first remains open only in its residual generality.

(1) Rank-one ambient factors (now established in the S-arithmetic setting). Classify ergodic joinings between higher-rank diagonalizable actions on quotients $G_i/\Gamma_i$ when the ambient groups $G_i$ may have local rank-one factors, assuming only that the actions on the quotients have no global rank-one factors. This was the extension the source paper said would require a general low-entropy argument; it has since been established for S-arithmetic quotients (quotients of products of real and $p$-adic points of algebraic $\mathbb{Q}$-groups by lattices) of perfect algebraic $\mathbb{Q}$-groups and $\mathbb{Z}^d$-actions (Einsiedler–Lindenstrauss 2019; reference 13). What may still be asked here is the same statement in the original Lie-group generality — arbitrary lattices and $\mathbb{R}^n$-flows rather than arithmetic lattices and $\mathbb{Z}^d$-actions — to the extent it is not already covered by the established S-arithmetic theorem via the Margulis arithmeticity theorem (which asserts that irreducible lattices in semisimple Lie groups of real rank at least two are arithmetic) and straightforward modification of the acting group.

(2) Commuting toral endomorphisms (the surviving open core). For an action with no virtually cyclic factor of a group of commuting toral endomorphisms — that is, no quotient by a closed invariant subgroup carries an induced action generated, up to finite index, by a single element of the acting group; in particular the acting group is nonvirtually cyclic — e.g. the semigroup generated by $\times p$ and $\times q$ with $p,q$ multiplicatively independent on the circle $\mathbb{R}/\mathbb{Z}$, and the analogous actions by endomorphisms or automorphisms of higher-dimensional tori and solenoids — classify, with no entropy hypothesis, the ergodic joinings and, equivalently per the source paper, all ergodic invariant measures. The expected resolution is that every ergodic invariant measure, and every ergodic joining, is algebraic: Haar/Lebesgue measure on the whole space or on a translate of a closed invariant subgroup, or an atomic measure supported on a finite invariant set; the base case is Furstenberg's conjecture that an ergodic $\times p,\times q$-invariant measure on the circle is Lebesgue or atomic. The exclusion of virtually cyclic factors is the standing hypothesis of the positive-entropy theorems and is necessary: on a quotient whose induced action is generated up to finite index by a single element, arbitrary invariant measures of that element lift to invariant measures of the whole action and need not be algebraic.

References

1. Harry Furstenberg, 'Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation', Mathematical Systems Theory 1 (1967), 1–49. DOI 10.1007/BF01692494. https://doi.org/10.1007/BF01692494
2. A. Katok and R. J. Spatzier, 'Invariant measures for higher-rank hyperbolic abelian actions', Ergodic Theory and Dynamical Systems 16 (1996), 751–778. DOI 10.1017/S0143385700009081. https://doi.org/10.1017/S0143385700009081
3. Daniel J. Rudolph, '×2 and ×3 invariant measures and entropy', Ergodic Theory and Dynamical Systems 10 (1990), 395–406. DOI 10.1017/S0143385700005629. https://doi.org/10.1017/S0143385700005629
4. Boris Kalinin and Anatole Katok, 'Measurable rigidity and disjointness for Z^k actions by toral automorphisms', Ergodic Theory and Dynamical Systems 22 (2002), 507–523. DOI 10.1017/S014338570200024X. https://doi.org/10.1017/S014338570200024X
5. Boris Kalinin and Ralf Spatzier, 'Rigidity of the measurable structure for algebraic actions of higher-rank Abelian groups', Ergodic Theory and Dynamical Systems 25 (2005), 175–200. DOI 10.1017/S014338570400046X. https://doi.org/10.1017/S014338570400046X
6. Elon Lindenstrauss, 'Invariant measures and arithmetic quantum unique ergodicity', Annals of Mathematics (2) 163 (2006), 165–219. DOI 10.4007/annals.2006.163.165. https://doi.org/10.4007/annals.2006.163.165
7. Manfred Einsiedler, Anatole Katok, and Elon Lindenstrauss, 'Invariant measures and the set of exceptions to Littlewood's conjecture', Annals of Mathematics (2) 164 (2006), 513–560. DOI 10.4007/annals.2006.164.513. https://doi.org/10.4007/annals.2006.164.513
8. Manfred Einsiedler and Elon Lindenstrauss, 'Rigidity properties of Z^d-actions on tori and solenoids', Electronic Research Announcements of the American Mathematical Society 9 (2003), 99–110. DOI 10.1090/S1079-6762-03-00117-3. https://doi.org/10.1090/S1079-6762-03-00117-3
9. Manfred Einsiedler and Elon Lindenstrauss, 'Joinings of higher-rank diagonalizable actions on locally homogeneous spaces', Duke Mathematical Journal 138 (2007), no. 2, 203–232. DOI 10.1215/s0012-7094-07-13822-5. https://doi.org/10.1215/s0012-7094-07-13822-5
10. Manfred Einsiedler and Elon Lindenstrauss, 'On measures invariant under diagonalizable actions: the rank-one case and the general low-entropy method', Journal of Modern Dynamics 2 (2008), no. 1, 83–128. DOI 10.3934/jmd.2008.2.83. https://doi.org/10.3934/jmd.2008.2.83
11. Manfred Einsiedler and Amir Mohammadi, 'A joinings classification and a special case of Raghunathan's conjecture in positive characteristic (with an appendix by Kevin Wortman)', Journal d'Analyse Mathématique 116 (2012), 299–334. DOI 10.1007/s11854-012-0008-4. https://doi.org/10.1007/s11854-012-0008-4
12. Manfred Einsiedler and Elon Lindenstrauss, 'Symmetry of entropy in higher rank diagonalizable actions and measure classification', Journal of Modern Dynamics 13 (2018), 163–185. DOI 10.3934/jmd.2018016. https://arxiv.org/abs/1803.07762
13. Manfred Einsiedler and Elon Lindenstrauss, 'Joinings of higher rank torus actions on homogeneous spaces', Publications Mathématiques de l'IHÉS 129 (2019), 83–127. DOI 10.1007/s10240-019-00103-y. https://arxiv.org/abs/1502.05133
14. Manfred Einsiedler, Elon Lindenstrauss, and Amir Mohammadi, 'Diagonal actions in positive characteristic', Duke Mathematical Journal 169 (2020), no. 1, 117–175. DOI 10.1215/00127094-2019-0038. https://arxiv.org/abs/1705.10418
15. Manfred Einsiedler and Elon Lindenstrauss, 'Rigidity properties for commuting automorphisms on tori and solenoids', Ergodic Theory and Dynamical Systems 42 (2022), no. 2, 691–736. DOI 10.1017/etds.2021.74. https://arxiv.org/abs/2101.11120
16. Menny Aka, 'Joinings classification and applications (after Einsiedler and Lindenstrauss)', Séminaire Bourbaki, volume 2021/2022, exposé 1185, Astérisque 438, Société Mathématique de France (2022), 181–245. DOI 10.24033/ast.1186. https://arxiv.org/abs/2207.10132
17. Manfred Einsiedler and Elon Lindenstrauss, 'Rigidity of non-maximal torus actions, unipotent quantitative recurrence, and Diophantine approximations', preprint (2023). arXiv:2307.04163. https://arxiv.org/abs/2307.04163
18. Huichi Huang, 'Equidistribution and measure rigidity under ×p,×q', preprint (2014). arXiv:1411.4522. https://arxiv.org/abs/1411.4522
19. Yunping Jiang, 'Geometry in the Furstenberg Conjecture', preprint (2022). arXiv:2206.13569. https://arxiv.org/abs/2206.13569
20. Rotem Yaari, 'Density of finitely supported invariant measures for automorphisms of compact abelian groups', preprint (2025). arXiv:2507.14113. https://arxiv.org/abs/2507.14113
21. Douglas Finamore, André Magalhães de Sá Gomes, and Christian S. Rodrigues, 'Pushforward dynamics on Wasserstein spaces and measure rigidity', preprint (2026). arXiv:2609.00451. https://arxiv.org/abs/2609.00451