The weak Ratner property, introduced by Frączek and Lemańczyk in their study of special flows over two-dimensional irrational rotations, is a quantitative shearing property of parabolic flows: for most pairs of nearby points on distinct orbits, the two orbits eventually realign, one shifted by a time drawn from a fixed compact set P⊂R∖{0}P\subset\mathbb R\setminus\{0\}, and then stay close for a definite proportion of a long time window. It preserves the joining-rigidity consequences of Marina Ratner's original property for horocycle flows — mild mixing, the finite-extension-of-joinings property, and mixing of all orders when combined with mixing — while covering classes of flows for which the original property fails. However, unlike the original Ratner property and all variants with a finite shift set, which are known isomorphism invariants of measure-preserving flows, it is not known whether the weak Ratner property survives measurable isomorphism; equivalently, for an infinite compact shift set it is not known whether the property depends on the choice of compatible metric, an issue Frączek and Lemańczyk already flagged when introducing it. The question was posed explicitly by Kanigowski, Kułaga-Przymus and Ulcigrai in 2019, and it remains open. The general Rokhlin multiple mixing problem, invoked as context by the same source, is a separate, broader open problem.

Background

Ergodic theory studies measure-preserving flows (St)t∈R(S_t)_{t\in\mathbb R}, that is, one-parameter groups of transformations of a probability space (X,B,μ)(X,\mathcal B,\mu), each of which preserves μ\mu. A flow is mixing if any two measurable sets become asymptotically independent, μ(StA∩B)→μ(A)μ(B)\mu(S_t A\cap B)\to\mu(A)\mu(B) as t→∞t\to\infty, and is mixing of order k≥2k\ge 2 (has kk-fold mixing) if the analogous independence holds for kk sets as all consecutive time gaps tend to infinity; mixing of all orders means mixing of order kk for every k≥2k\ge 2. Since V. A. Rokhlin introduced these invariants in 1949, it has been a famous open problem — Rokhlin's multiple mixing problem — whether mixing alone implies mixing of all orders. It is known for systems with singular spectrum, for finite-rank transformations, for unipotent flows on homogeneous spaces, and, by Kanigowski, Kułaga-Przymus and Ulcigrai [1], for typical smooth area-preserving flows on compact surfaces of any genus having at least one periodic component.

The source of the present question, by Kanigowski, Kułaga-Przymus and Ulcigrai, established multiple mixing for such surface flows through a quantitative shearing mechanism going back to Marina Ratner's work on horocycle flows (the flows on unit tangent bundles of hyperbolic surfaces obtained by moving each tangent vector in one fixed direction). Ratner discovered that horocycle flows possess what is now called the Ratner property: a controlled form of slow divergence of nearby orbits, in which for most pairs of sufficiently close points lying on distinct orbits there is a large time MM after which the two trajectories realign — one shifted by a fixed time p∈{1,−1}p\in\{1,-1\} — and then stay close for a definite proportion of a further long time interval. This property is characteristic of parabolic systems, meaning zero-entropy systems such as horocycle flows, their smooth time changes, and typical surface flows, in which nearby orbits diverge polynomially or subpolynomially rather than exponentially. Its strength lies in rigidity of joinings: a joining of two flows is an invariant measure on the product space whose marginals are the given measures, and the Ratner property forces ergodic self-joinings to resemble finite extensions of the original flow (the finite-extension-of-joinings property), which implies mild mixing — a strengthening of weak mixing that rules out rigid factors — and, combined with mixing, mixing of all orders.

Frączek and Lemańczyk broadened this mechanism to special flows (suspensions): given a measure-preserving transformation TT of a base space and a positive roof function ff, the special flow over TT under ff moves each point vertically at unit speed under the graph of ff and applies TT when the roof is reached. Their finite Ratner property (2006) allows the realignment shift to range over an arbitrary finite set, and their weak Ratner property (2010) allows it to range over an arbitrary compact set P⊂R∖{0}P\subset\mathbb R\setminus\{0\}; all joining-rigidity consequences survive these weakenings. The weak Ratner property was verified for special flows over two-dimensional irrational rotations with roof functions of bounded variation. Kanigowski, Kułaga-Przymus and Ulcigrai, dealing with surface flows whose special-flow representations have roof functions with logarithmic singularities (hence not of bounded variation; special flows over rotations with such logarithmic roof singularities are commonly called Kochergin flows), introduced a further switchable variant that permits the controlled divergence to be observed in backward time, and proved it for typical flows on higher-genus surfaces; their multiple-mixing theorem follows because the switchable property carries the same joining-rigidity consequences.

All of these properties are defined in terms of a metric dd on the phase space compatible with the measure algebra, and this is where the present question arises. For the original Ratner property and every variant with a finite shift set, Frączek–Lemańczyk and Kanigowski–Kułaga-Przymus–Ulcigrai showed, by a Luzin/Egorov argument combined with Birkhoff ergodic averages, that the property does not depend on the compatible metric and is preserved under measurable isomorphism of flows (a measure-space isomorphism ψ\psi with ψ∘St=Tt∘ψ\psi\circ S_t=T_t\circ\psi for all tt); such invariance is what lets these properties serve in classification statements, for example that certain Kochergin flows are not measurably isomorphic to horocycle flows. For the weak Ratner property, whose shift set PP is an infinite compact set, the same argument breaks down: Frączek–Lemańczyk explicitly could not decide whether the property depends on the metric (they expected that it does), and Kanigowski–Kułaga-Przymus–Ulcigrai remark that it is not known whether it is an isomorphism invariant.

Problem Statement

Let (X,d)(X,d) be a σ\sigma-compact metric space with Borel σ\sigma-algebra B\mathcal B and Borel probability measure μ\mu, and let (St)t∈R(S_t)_{t\in\mathbb R} be an ergodic measure-preserving flow on (X,B,μ)(X,\mathcal B,\mu). The metric is required to be compatible with B\mathcal B and to satisfy the Frączek–Lemańczyk almost-continuity condition: for every ε>0\varepsilon>0 there is a set X(ε)∈BX(\varepsilon)\in\mathcal B with μ(X(ε))>1−ε\mu(X(\varepsilon))>1-\varepsilon such that for every ε′>0\varepsilon'>0 there is ε1>0\varepsilon_1>0 with d(Stx,St′x)<ε′d(S_t x,S_{t'}x)<\varepsilon' for all x∈X(ε)x\in X(\varepsilon) and all t,t′∈[−ε1,ε1]t,t'\in[-\varepsilon_1,\varepsilon_1]. Ergodicity is the standing assumption of the Frączek–Lemańczyk and Kanigowski–Kułaga-Przymus–Ulcigrai framework: the known metric-independence proofs for finite shift sets run via Birkhoff averages of the ergodic time maps.

For a compact set P⊂R∖{0}P\subset\mathbb R\setminus\{0\} and t0∈R∖{0}t_0\in\mathbb R\setminus\{0\}, the flow has property R(t0,P)R(t_0,P) if for every ε>0\varepsilon>0 and N∈NN\in\mathbb N there exist κ=κ(ε)>0\kappa=\kappa(\varepsilon)>0, δ=δ(ε,N)>0\delta=\delta(\varepsilon,N)>0 and Z=Z(ε,N)∈BZ=Z(\varepsilon,N)\in\mathcal B with μ(Z)>1−ε\mu(Z)>1-\varepsilon such that whenever x,x′∈Zx,x'\in Z, x′x' is not on the orbit of xx, and d(x,x′)<δd(x,x')<\delta, there exist M,L≥NM,L\ge N with L/M≥κL/M\ge\kappa and a time shift ρ∈P\rho\in P such that

1L#{n∈Z∩[M,M+L] : d(Snt0x, Snt0+ρx′)<ε}>1−ε.\frac{1}{L}\#\{n\in\mathbb Z\cap[M,M+L]\ :\ d(S_{nt_0}x,\,S_{nt_0+\rho}x')<\varepsilon\}>1-\varepsilon.

The flow has the weak Ratner property if it has property R(P)R(P) for some compact P⊂R∖{0}P\subset\mathbb R\setminus\{0\}, where having property R(P)R(P) means that the set of s∈Rs\in\mathbb R for which the flow has property R(s,P)R(s,P) is uncountable.

Question (posed by Kanigowski–Kułaga-Przymus–Ulcigrai, 2019, after Frączek–Lemańczyk, 2010): is the weak Ratner property an isomorphism invariant of ergodic measure-preserving flows? That is, if (St)(S_t) on (X,μ)(X,\mu) has the weak Ratner property with respect to some compatible almost-continuous metric, and ψ:X→Y\psi:X\to Y is a measure-space isomorphism onto a flow (Tt)(T_t) on (Y,ν)(Y,\nu) satisfying ψ∘St=Tt∘ψ\psi\circ S_t=T_t\circ\psi for every t∈Rt\in\mathbb R, must (Tt)(T_t) also have the weak Ratner property with respect to every compatible almost-continuous metric on YY? Equivalently, for infinite compact PP, does property R(s,P)R(s,P) of a given flow depend on the choice of the compatible metric? For shift sets PP that are finite, both isomorphism invariance and metric-independence are known; the question concerns precisely the compact infinite case.

References

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