Mixing of smooth time-changes of parabolic nilflows: the four open problems of Avila–Forni–Ravotti–Ulcigrai (dichotomy beyond the dense class, effective triviality, quantitative and multiple mixing)

The task is to resolve the four open problems posed in Section 1.4 of 'Mixing for smooth time-changes of general nilflows' by A. Avila, G. Forni, D. Ravotti and C. Ulcigrai (Adv. Math. 385 (2021), Paper No. 107759). That paper proved that on every compact nilmanifold of step at least 2 there exists a C-infinity-dense structured class of smooth functions (relative trigonometric polynomials) such that, for any completely irrational (uniquely ergodic) nilflow, every time-change generated by a positive element of this class is either measurably trivial or mixing. The authors explicitly posed: (1) whether every measurably non-trivial smooth time-change of a uniquely ergodic parabolic nilflow is weakly mixing, and whether it is mixing — already open for Heisenberg nilflows; (2) an effective description of the measurably trivial, non-mixing time-changes for higher-step nilflows; (3) quantitative mixing estimates (rates of decay of correlations) under a Diophantine condition on the frequencies of the toral factor; (4) mixing of all orders for mixing time-changes under similar or stronger assumptions. Known partial progress includes the weak-mixing dichotomy for all smooth time-changes of almost every Heisenberg nilflow (Forni–Kanigowski 2021), quantitative stretched-polynomial mixing for large classes of Heisenberg time-changes (Forni–Kanigowski, Astérisque 2020), multiple mixing for bounded-type Heisenberg nilflows (Forni–Kanigowski 2020), and the uniformly-shearing framework of Kanigowski–Ravotti (2024), which announces the nilflow case as forthcoming work; the core remains open: the mixing half of (1) is open even for almost every Heisenberg nilflow, and (2)–(4) are open for higher-step nilflows.

Background

A nilmanifold is a compact quotient $M=\Gamma\backslash G$ of a connected, simply connected nilpotent Lie group $G$ by a lattice $\Gamma$ (a discrete cocompact subgroup); its nilpotency step is the length of the lower central series of $G$, so step 1 is a torus and the three-dimensional Heisenberg group has step 2. A nilflow is the flow on $M$ generated by a left-invariant vector field $X$ in the Lie algebra of $G$, acting by right translations. Abelianization fibers $M$ over a torus, and every nilflow projects to a linear (constant-speed) flow on this toral factor; by a classical theorem of Auslander–Green–Hahn, the nilflow is uniquely ergodic (Haar measure is its only invariant Borel probability measure) if and only if the projected toral flow is completely irrational, meaning that no nontrivial integer vector annihilates its frequency vector. A Diophantine condition on these frequencies is a quantitative strengthening of complete irrationality: the standard requirement is $|\langle n,\alpha\rangle|\ge C\|n\|^{-\tau}$ for all non-zero integer vectors $n$ and constants $C,\tau>0$, that is, the frequencies are badly approximable by rational ones. Following the source paper, a parabolic nilflow means a nilflow on a nilmanifold of step at least 2: such flows have zero entropy and only polynomial divergence of nearby orbits, but they retain an elliptic toral factor (pullbacks of toral functions oscillate quasi-periodically), and consequently nilflows themselves are never mixing. A measure-preserving flow $(\phi_t)_{t\in\mathbb{R}}$ is mixing if correlations $\int (f\circ\phi_t)\,g\,d\mu$ converge to $(\int f\,d\mu)(\int g\,d\mu)$ for all square-integrable observables $f,g$; it is weakly mixing if it has no non-constant square-integrable eigenfunctions, a property strictly weaker than mixing; and it is mixing of all orders (multiple mixing) if correlations of every finite tuple of observables factorize asymptotically, a long-standing strengthening of mixing known as Rokhlin's problem for zero-entropy systems.

The mildest perturbation of a flow that preserves its orbits is a time-change: replacing the generator $X$ by $f^{-1}X$ for a smooth positive function $f$ on $M$, which reparametrizes the speed along each orbit. A time-change is measurably trivial if $f$ is measurably cohomologous to a constant, that is, $f=c+Xu$ for a measurable function $u$ and a constant $c$; then the reparametrized flow is measurably conjugate to the original one (up to a constant rescaling) and inherits its failure to mix. Deciding such triviality is a question about the cohomological equation $Xu=g$: its obstructions are the invariant distributions, linear functionals on Sobolev spaces that commute with the flow and hence vanish on all coboundaries $g=Xu$; these are explicitly computable only in renormalizable settings. Renormalization here means an action of a larger group (for horocycle flows — the flows along the horocycle circles of compact hyperbolic surfaces — the geodesic flow) that maps rescaled time-changes back to time-changes, a tool available for horocycle flows and Heisenberg nilflows but with no known counterpart for higher-step nilflows.

For step-1 (toral) flows, smooth time-changes are typically not mixing, by classical perturbative rigidity results; for horocycle flows it is classical that all smooth time-changes are mixing. For nilflows, Avila–Forni–Ulcigrai (2011) proved that Heisenberg nilflows admit a $C^\infty$-dense class of smooth time-changes within which mixing holds exactly when the time-change is not smoothly cohomologous to a constant; Ravotti (2019) proved the analogous dichotomy for quasi-abelian filiform nilflows, a class isomorphic to suspensions of unipotent toral skew-translations. Forni–Kanigowski then strengthened the Heisenberg picture in two directions: quantitative mixing, with stretched-polynomial decay of correlations for a large residual class (one containing a countable intersection of open dense sets) of time-changes of almost every three-dimensional Heisenberg nilflow, via renormalization-invariant asymptotics of ergodic integrals (the integrals $\int_0^t (f\circ\phi_s)\,ds$ of an observable along flow trajectories); and multiple mixing for time-changes of bounded-type Heisenberg nilflows (bounded type being a Diophantine, that is, quantitatively irrational, condition on the flow generator), proved through the Ratner property — a quantitative shearing property for time-changes first exploited by Ratner in her work on horocycle flows, and which combined with mixing implies mixing of all orders. The source paper (Avila–Forni–Ravotti–Ulcigrai 2021) generalized the qualitative dichotomy to every nilmanifold of step at least 2, with no Diophantine condition: there is a $C^\infty$-dense class $\mathscr{P}$ of smooth functions, the relative trigonometric polynomials, consisting of functions that at each level of a tower of toral central extensions of the nilmanifold behave like trigonometric polynomials in the fibers, such that for any completely irrational nilflow every time-change generated by a positive element of $\mathscr{P}$ is either measurably trivial or mixing. The authors warn that Problems 2, 3 and 4 seem hard mainly because fine quantitative pointwise estimates on ergodic integrals of higher-step nilflows — in particular pointwise lower bounds on sets of large measure — are not available, and because renormalization is so far available only for translation flows, horocycle flows, and Heisenberg nilflows.

Problem Statement

The problem is to complete the theory of mixing for smooth time-changes of parabolic nilflows along the four directions explicitly posed in Section 1.4 (Problems 1-4) of Avila–Forni–Ravotti–Ulcigrai (2021). The common objective is to remove the restriction of the known dichotomy to the dense structured class $\mathscr{P}$ of relative trigonometric polynomials, to make the trivial side of the dichotomy effective, and to upgrade it quantitatively and to higher order. Concretely:

1. (Generality of the dichotomy) Let $\phi^{\mathbb{R}}$ be a uniquely ergodic nilflow on a compact nilmanifold of step at least 2 (a parabolic nilflow; the question is already open for Heisenberg nilflows). Is every measurably non-trivial (that is, not cohomologous to a constant with a measurable transfer function) smooth time-change of $\phi^{\mathbb{R}}$ weakly mixing? Is every such time-change mixing?

2. (Effective triviality) Give an effective description of the class of measurably trivial, non-mixing time-changes for higher-step nilflows, that is, characterize in concrete algebraic or analytic terms the smooth positive functions that are measurably cohomologous to a constant.

3. (Quantitative mixing) Under a Diophantine condition on the frequencies of the toral (Abelianized) factor of an ergodic parabolic nilflow, prove quantitative mixing estimates, i.e. rates of decay of correlations, for mixing time-changes in a suitably large class.

4. (Higher-order mixing) Under similar (or stronger) Diophantine or arithmetic assumptions, prove that mixing time-changes are mixing of all orders (multiple mixing).

The four components are posed at the stated generality: uniquely ergodic or ergodic parabolic nilflows, the higher-step cases being the essential open territory.

References

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