General Structural Conjecture for joinings of sated $\mathbb{Z}^D$-systems with partial invariance (Austin's Conjecture 6.0.4)

A 2010 PhD dissertation of Tim Austin formulates a structural conjecture (Conjecture 6.0.4, the 'General Structural Conjecture') about joinings of probability-preserving $\mathbb{Z}^D$-systems that are only partially invariant, each under its own subgroup $\Gamma_i\le\mathbb{Z}^D$, in the regime where the subgroups $\Gamma_i$ may be linearly dependent. The conjecture predicts that after passing to systems that are sated with respect to all joins of pro-nilsystem classes, any joining of such systems decomposes with each coordinate relatively independent over a factor built from pro-nilsystem structure coming from finitely many proper supergroups $\Lambda_{i,j}\supsetneq\Gamma_i$. This would complete Austin's joining-theoretic 'interpolation' between the Host–Kra–Ziegler structure theorem for nonconventional averages and Austin's linearly independent subgroup theorem (Theorem 4.1.2). The anticipated headline application — norm convergence of polynomial nonconventional averages — was settled independently by Walsh (2012), but the joining-structure conjecture itself, and the corresponding classification of joinings for linearly dependent subgroup configurations, remain open.

Background

A probability-preserving system is a probability space together with a measure-preserving action of a countable group; here the acting group is $\mathbb{Z}^D$ for some $D$, generated by $D$ commuting invertible measure-preserving transformations. A joining of several systems is a probability measure on the product space whose marginals are the given measures and which is invariant under the product (diagonal) action; joinings encode all possible correlations among the systems, so classifying the joinings of partially related systems is a basic structural question in ergodic theory. A factor is a subsystem recovered as an invariant sub-sigma-algebra, and one factor is relatively independent from another over a third when the corresponding conditional expectations factor through the third.

If $\Gamma\le\mathbb{Z}^D$ is a subgroup, the $\Gamma$-subaction is the restriction of the action to $\Gamma$. A system is partially invariant under $\Gamma$ when this subaction is trivial on it; Austin writes $Z^\Gamma_0$ for the class of such systems, and more generally $Z^{\Lambda}_{\mathrm{nil},n}$ for the class of systems whose $\Lambda$-subaction is a direct integral of $n$-step pro-nilsystems — that is, of inverse limits of rotations on $n$-step nilmanifolds (a nilmanifold is a quotient $G/H$ of a nilpotent Lie group by a closed subgroup, acted on by translation), with the direct-integral formulation covering non-ergodic systems, where the nilmanifold fibres may vary measurably over the ergodic components.

A central tool is Austin's notion of satedness. A system $\mathbf{X}$ is sated with respect to a class $\mathcal{C}$ of systems if every joining of $\mathbf{X}$ with members of $\mathcal{C}$ (a '$\mathcal{C}$-adjoining') is relatively independent over the maximal $\mathcal{C}$-factor of $\mathbf{X}$. This is not automatic, but every system admits extensions that are sated with respect to any prescribed finite family of idempotent classes, obtained as inverse limits; passing to such sated extensions is the engine of Austin's proofs.

The motivating circle of ideas is the convergence of nonconventional ergodic averages $\frac{1}{N}\sum_{n=1}^{N}\prod_i f_i\circ T^{p_i(n)}$, which underlie Furstenberg's ergodic proofs of Szemerédi-type theorems (Szemerédi's theorem asserts that every set of integers with positive upper density contains arithmetic progressions of every finite length). Host and Kra (2005) and Ziegler (2007) proved that for a single ergodic transformation the averages are controlled by pro-nilsystem factors, a deep structure theorem equivalent to a classification of the Furstenberg self-joinings of $\mathbf{X}$ — the self-joinings obtained as weak-* limit points of these averages, which encode the asymptotic joint distributions of the iterates. Austin's dissertation proves the norm-convergence and multiple-recurrence theorems for commuting transformations by softer joining arguments: his Theorem 4.1.2 shows that when $\mathbb{Z}^D$ splits as $\Gamma_1\oplus\cdots\oplus\Gamma_r\oplus\Lambda$ and each system $\mathbf{X}_i$ is partially invariant under $\Gamma_i$ and suitably sated, any joining is relatively independent over the factors invariant under the sums $\Gamma_i+\Gamma_\ell$ (so there $\Lambda_{i,j}=\Gamma_i+\Gamma_\ell$ and $m_{i,j}=0$). Linear independence of the subgroups is essential to this argument, yet the configurations arising from polynomial iterates (the Bergelson–Leibman conjecture on norm convergence of polynomial nonconventional averages, raised in their nilpotent Roth theorem paper of 2002) force linearly dependent subgroups. This leaves the meta-question of Section 4.1 of the dissertation: what are the possible joinings of partially invariant systems when the subgroups $\Gamma_i$ are linearly dependent? The dissertation's Chapter 6 answers with a conjectural picture — the General Structural Conjecture stated below — and records that at the time of writing no proof or disproof was at hand.

Problem Statement

Prove or refute Austin's General Structural Conjecture (Conjecture 6.0.4 of arXiv:1006.0491). Let $\Gamma_1,\dots,\Gamma_r\le\mathbb{Z}^D$ be subgroups among which there are no pairwise inclusions, and let $n_1,\dots,n_r\ge0$ be integers. The conjecture asserts that these data determine finite families of pairs $(\Lambda_{i,1},m_{i,1}),\dots,(\Lambda_{i,k_i},m_{i,k_i})$, $i=1,\dots,r$, where each $m_{i,j}\ge0$ is an integer and each $\Lambda_{i,j}\le\mathbb{Z}^D$ is a subgroup properly containing $\Gamma_i$, such that the following holds: whenever each $\mathbf{X}_i$ is a $\mathbb{Z}^D$-system whose $\Gamma_i$-subaction is a direct integral of $n_i$-step pro-nilsystems ($\mathbf{X}_i\in Z^{\Gamma_i}_{\mathrm{nil},n_i}$) and each $\mathbf{X}_i$ is sated with respect to every class that is a join of classes of the form $Z^{\Gamma}_{\mathrm{nil},n}$ for $\Gamma\le\mathbb{Z}^D$ and $n\ge0$, then for any joining of the $\mathbf{X}_i$, realized by factor maps $\pi_i:\mathbf{Y}\to\mathbf{X}_i$, the lifted sigma-algebras $\pi_i^{-1}(\Sigma_i)$, where $\Sigma_i$ is the full sigma-algebra of $\mathbf{X}_i$ (so $\pi_i^{-1}(\Sigma_i)$ is the $i$-th coordinate factor of the joining), are relatively independent over their further factors

$$\pi_i^{-1}\Big(\bigvee_{j=1}^{k_i}\Phi_{i,j}\Big),$$

where $\Phi_{i,j}$ is the factor of $\mathbf{X}_i$ generated by the map to $(Z^{\Gamma_i}_0\cap Z^{\Lambda_{i,j}}_{\mathrm{nil},m_{i,j}})\mathbf{X}_i$. The conjecture as formulated asserts only the existence of such finite families; the known special cases fix what they reduce to there: for linearly independent subgroups, $\Lambda_{i,j}=\Gamma_i+\Gamma_\ell$ for $\ell\ne i$ and $m_{i,j}=0$ (Theorem 4.1.2), and the three-directions-in-general-position case settled by the 'Pleasant extensions' papers. The conjecture fails if, for some configuration of subgroups and every candidate family of proper supergroups, there are partially invariant, suitably sated systems and a joining violating the asserted relative independence.

References

1. Tim Austin, "Multiple recurrence and the structure of probability-preserving systems", PhD dissertation (UCLA), 2010. arXiv:1006.0491, https://arxiv.org/abs/1006.0491 (primary source of Conjecture 6.0.4 and the Section 4.1 meta-question)
2. Tim Austin, "On the norm convergence of non-conventional ergodic averages", Ergodic Theory and Dynamical Systems 30(2) (2009) 321–338. DOI 10.1017/S014338570900011X, https://doi.org/10.1017/S014338570900011X
3. Bernard Host and Bryna Kra, "Nonconventional ergodic averages and nilmanifolds", Annals of Mathematics (2) 161(1) (2005) 397–488. DOI 10.4007/annals.2005.161.397, https://doi.org/10.4007/annals.2005.161.397
4. Tamar Ziegler, "Universal characteristic factors and Furstenberg averages", Journal of the American Mathematical Society 20(1) (2007) 53–97. DOI 10.1090/S0894-0347-06-00532-7, https://doi.org/10.1090/S0894-0347-06-00532-7
5. V. Bergelson and A. Leibman, "A nilpotent Roth theorem", Inventiones Mathematicae 147(2) (2002) 429–470. DOI 10.1007/s002220100179, https://doi.org/10.1007/s002220100179
6. Tim Austin, "Extensions of probability-preserving systems by measurably-varying homogeneous spaces and applications", preprint (2009). arXiv:0905.0516, https://arxiv.org/abs/0905.0516
7. Qing Chu, Nikos Frantzikinakis and Bernard Host, "Ergodic averages of commuting transformations with distinct degree polynomial iterates", Proceedings of the London Mathematical Society (3) 102(5) (2011) 801–842. DOI 10.1112/plms/pdq037, https://doi.org/10.1112/plms/pdq037 (preprint arXiv:0912.2641)
8. Tim Austin, "Pleasant extensions retaining algebraic structure, I", Journal d'Analyse Mathématique 125 (2015) 1–36. DOI 10.1007/s11854-015-0001-9, https://doi.org/10.1007/s11854-015-0001-9 (preprint arXiv:0905.0518)
9. Tim Austin, "Pleasant extensions retaining algebraic structure, II", Journal d'Analyse Mathématique 126 (2015) 1–111. DOI 10.1007/s11854-015-0013-5, https://doi.org/10.1007/s11854-015-0013-5 (preprint arXiv:0910.0907)
10. Miguel Walsh, "Norm convergence of nilpotent ergodic averages", Annals of Mathematics (2) 175(3) (2012) 1667–1688. DOI 10.4007/annals.2012.175.3.15, https://doi.org/10.4007/annals.2012.175.3.15 (preprint arXiv:1109.2922, https://arxiv.org/abs/1109.2922)
11. Tim Austin, "Equidistribution of joinings under off-diagonal polynomial flows of nilpotent Lie groups", Ergodic Theory and Dynamical Systems 33(6) (2013) 1667–1708. DOI 10.1017/etds.2012.113, https://doi.org/10.1017/etds.2012.113 (preprint arXiv:1105.5612)
12. Tim Austin, "Non-conventional ergodic averages for several commuting actions of an amenable group", Journal d'Analyse Mathématique 130(1) (2016) 243–274. DOI 10.1007/s11854-016-0036-6, https://doi.org/10.1007/s11854-016-0036-6 (preprint arXiv:1309.4315, https://arxiv.org/abs/1309.4315)
13. Nikos Frantzikinakis, "Some open problems on multiple ergodic averages", preprint (2011). arXiv:1103.3808, https://arxiv.org/abs/1103.3808