MULTIPLE MIXING AND DISJOINTNESS FOR TIME CHANGES OF BOUNDED-TYPE HEISENBERG NILFLOWS

. — We study time changes of bounded type Heisenberg nilﬂows ( φ t ) acting on the Heisenberg nilmanifold M . We show that for every positive τ ∈ W s ( M ) , s > 7 / 2 , every non-trivial time change ( φ τt ) enjoys the Ratner property . As a consequence, every mixing time change is mixing of all orders. Moreover, we show that for every τ ∈ W s ( M ) , s > 9 / 2 and every p,q ∈ N , p (cid:54) = q , ( φ τpt ) and ( φ τqt ) are disjoint. As a consequence, Sarnak conjecture on Möbius disjointness holds for all such time changes.


MULTIPLE MIXING AND DISJOINTNESS FOR TIME CHANGES OF BOUNDED-TYPE HEISENBERG NILFLOWS by Giovanni Forni & Adam Kanigowski
Abstract.-We study time changes of bounded type Heisenberg nilflows (φt) acting on the Heisenberg nilmanifold M .We show that for every positive τ ∈ W s (M ), s > 7/2, every nontrivial time change (φ τ t ) enjoys the Ratner property.As a consequence, every mixing time change is mixing of all orders.Moreover, we show that for every τ ∈ W s (M ), s > 9/2 and every p, q ∈ N, p = q, (φ τ pt ) and (φ τ qt ) are disjoint.As a consequence, Sarnak conjecture on Möbius disjointness holds for all such time changes.

Introduction
In this paper we study ergodic properties of time changes of Heisenberg nilflows.Nilsystems on (non-abelian) nilmanifolds are classical examples of systems which share some features from both the elliptic and the parabolic world.They always have a nontrivial Kronecker factor which is responsible for the elliptic behavior (in particular they are never weakly mixing).On the other hand, orthogonally to the elliptic factor they exhibit polynomial speed of divergence of nearby trajectories and are polynomially mixing, which are properties characteristic of parabolic systems.
We are interested in the lowest dimensional (non-abelian) situation, i.e., nilflows on 3-dimensional Heisenberg nilmanifolds.In [3] it was shown that, for every (ergodic) Heisenberg nilflow, there exists a dense set of smooth time changes which are mixing.This result was strengthened in [13], where it was shown that for a full measure set of Heisenberg nilflows a generic time change is mixing, and moreover one has a "stretched-polynomial" decay of correlations for any pair of sufficiently smooth observables.For Heisenberg nilflows of bounded type the decay of correlations is estimated in [13] to be polynomial, as expected according to the "parabolic paradigm" (see [18, §8.2.f]).The mixing result of [3] was generalized in [27] to a class of nilflows on higher step nilmanifolds, called quasi-Abelian, which includes suspension flows over toral skew-shifts, and then recently to all non-Abelian nilflows in [4].These general results reach no conclusion about the speed of mixing.
For time changes of horocycle flows, polynomial decay of correlations, as well as the property that the maximal spectral type is Lebesgue, were proved in [14].A different proof of the absolute continuity of the spectrum was given simultaneously and independently by R. Tiedra de Aldecoa [29] under the so-called Kushnirenko condition (a positivity condition which holds for small time changes) and later in general [30].More recently, B. Fayad and the authors [6] have proved that the spectrum is Lebesgue of countable multiplicity, also for a class of Kochergin flows on the two-torus.Nothing is known about the spectrum of smooth-time changes of nilflows beyond the mixing results of [3] and [13].In particular, even for Heisenberg nilflows of bounded type, it is unclear whether the spectrum has an absolutely continuous component.
It follows from [3] and [13] that by a time change one can alter the dynamical features of Heisenberg nilflows, i.e., the elliptic factor becomes trivial for the timechanged flow and the mixing property holds (with polynomial decay of correlations for bounded type nilflows).It is therefore natural to ask to what extent the time-changed flow can behave, roughly speaking, as a "prototypical" parabolic flow (there is no widely accepted formal definition of a parabolic system).One of the characteristic features of parabolic systems is the Ratner property which quantifies the polynomial speed of divergence of nearby trajectories.It was first established by M. Ratner in [25] in the class of horocycle flows and was applied to prove Ratner's rigidity phenomena in this class.Moreover, in [26], M. Ratner showed that the Ratner property survives under C 1 smooth time changes of horocycle flows, hence similar rigidity phenomena hold for time changes.One of the most important consequences of this property is that a mixing system with the Ratner property is mixing of all orders.Indeed, Ratner showed that the Ratner property implies so-called PID property of the set of joinings which in turn implies higher order mixing, as observed by del Junco and Rudolph.
Recently Ratner's property (or its variants) was observed in a new class of (nonhomogeneous) systems, that of smooth flows on surfaces with finitely many (saddlelike) singularities.In [7], the authors studied the case of smooth mixing flows on the two-torus and established the SWR-property. (1)This property allows to establish the Ratner-type divergence of orbits, either in the future or in the past (depending on points), and moreover it has the same dynamical consequences as the original Ratner property.Then the authors showed in particular that the SWR-property holds for a full measure set of mixing flows with logarithmic singularities (Arnol'd flows) thereby proving higher order mixing in this class.The result in [7] was strengthened in [20], where the authors showed that the SWR-property holds for a full measure set of Arnol'd flows on surfaces of higher genus.
It is therefore natural to ask whether a Ratner property holds in the class of Heisenberg nilflows.In [21] it is shown that Ratner's property implies in particular that the Kronecker factor is trivial and hence no nilflow can enjoy it.The situation is very different for non-trivial time changes of Heisenberg nilflows.
Let H denote the 3-dimensional Heisenberg group and let h denote its Lie algebra.Let M := Γ\H denote a Heisenberg nilmanifold, that is, the quotient of H over a (co-compact) lattice Γ < H.For any W ∈ h, the flow (φ W t ) generated by the projection to M of the left-invariant vector field W on H, is called a Heisenberg nilflow (see Section 2.4 for the definition).
For any W ∈ h and any positive function τ ∈ C 1 (M ), let (φ W,τ t ) denote the time change of the nilflow (φ W t ), that is, the flow generated by the vector field τ W on M .The flow (φ W,τ t ) is a reparametrization of the nilflow (φ W t ) in the sense that there exists a function T : One can verify that the group property of the flow (φ W,τ t ) implies that the function T is a cocycle over (φ W,τ t ), in the sense that and it is uniquely determined by the above cocycle condition and by the identity ) is called trivial if the cocycle T is cohomologous to a constant, that is, there exist a constant c ∈ R and a measurable function u : (1) Acronym for switchable weak Ratner.It was also shown that the original Ratner property does not hold.
J.É.P. -M., 2020, tome 7 All trivial time changes (φ W,τ t ) are measurably conjugate to the nilflow (φ W t ) and the regularity of the conjugacy is equal to the regularity of the so-called transfer function u on M .
A vector field W ∈ h and the corresponding flow (φ W t ) are called of bounded type if their projections on the Kronecker factor (which are respectively a constant coefficients vector field and the corresponding linear flow on a 2-dimensional torus) are of bounded type.
Our first main result establishes the Ratner property (see Section 3 for the definition) for time changes of bounded type Heisenberg nilflows in a very strong sense.In fact our first main result is the following.For every s > 0, let W s (M ) denote the standard Sobolev space.By the Sobolev embedding theorem we have that W s (M ) ⊂ C k (M ), for every s > 3/2 + k.Recall that the famous Rokhlin problem asks whether mixing implies mixing of all orders.The above result implies that the answer to the Rokhlin problem is positive for smooth time changes of bounded type Heisenberg nilflows: Corollary 1.1.-Let W ∈ h be of bounded type and let Then, for any s 7/2, every element of D s (W ) is mixing if and only if it is mixing of all orders.As proved in [3] and [13], for s 7/2 and W of bounded type, mixing is generic in the set D s (W ).Moreover, by [21], we have the following strong dichotomy for time changes of Heisenberg nilflows: Our second main result deals with disjointness properties of time changes of Heisenberg nilflows.It is based on a variant of a parabolic disjointness criterion from [22].We have: Theorem 2. -Let W ∈ h be of bounded type.For any positive function τ ∈ W s (M ) with s > 9/2, if the time change is non-trivial, then the flows (φ W,τ pt ) and (φ W,τ qt ) are disjoint for all p, q ∈ N, p = q.
As mentioned above, if the time-change τ is trivial, then (φ W pt ) and (φ W qt ) are not disjoint (since (φ W t ) has a non-trivial Kronecker factor).However, L. Flaminio, K. Fraczek, M. Lemańczyk and J. Kułaga-Przymus [11] have recently proved that any ergodic nilsystem satisfies the AOP-property: as p, q → +∞, the set of joinings between (φ W pt ) and (φ W qt ) weakly converges to the product measure.The above theorem should be compared with analogous disjointness results for other flows with Ratner's property.It follows from the renormalization equation for the horocycle flow, which states that g s h t = h e −2s t g s for all s, t ∈ R, that h pt and h qt are isomorphic (and hence not disjoint) for any p, q ∈ R {0}.In [26], joinings of time changes of horocycle flows were completely characterized by M. Ratner.From Ratner's work one can derive that distinct powers of the same time change are disjoint unless the time change is trivial [10].A different proof of this result, based on a new disjointness criterion for parabolic flows, was given recently in [22].Moreover, in [22] it is proved that for almost every Arnol'd flow on T 2 the same assertion as in Theorem 2 holds.Therefore, among known flows with Ratner's property, the horocycle flow is the only one for which the conclusion of Theorem 2 does not hold.The heuristic reason for that is that the Ratner property for the horocycle flow depends only on the distance between points, and not on their position in space (since the space is homogeneous).In all other examples (for flows as in Theorem 2 in particular) the divergence depends also on position which allows to get stronger consequences (see Section 4).
Let us now briefly discuss the connection between Theorem 2 and Sarnak's conjecture on Möbius disjointness [28], which is recently under extensive study, see e.g.[8].We say that a continuous flow (T t ) on a compact metric space (X, d) is Möbius disjoint, if for every F ∈ C(X) and every x ∈ X and every t ∈ R we have (1) lim here µ denotes the classical Möbius function. (2) Möbius disjointness for horocycle flows was proved by J. Bourgain, P. Sarnak and T. Ziegler [5].Moreover, as explained in [10], it follows from Ratner's work [26] that Möbius disjointness also holds for non-trivial time changes of horocycle flows.Moreover, in view of a criterion due to I. Kátai [23] and to Bourgain, Sarnak and Ziegler [5], for non-trivial time changes of horocycle flows the convergence in (1) is uniform in x ∈ X.It is an open question whether for horocycle flows the convergence in (1) (for fixed F and t ∈ R {0}) is uniform with respect to x ∈ X.A corollary of Theorem 2, again by the Kátai-Bourgain-Sarnak-Ziegler (KBSZ) criterion [23], [5], is the following: ) is Möbius disjoint.Moreover, the convergence in formula (1) is uniform with respect to x ∈ M .
(2) Sarnak's conjecture states that every system of zero topological entropy is Möbius disjoint.
J.É.P. -M., 2020, tome 7 It follows from the work of B. Green and T. Tao [17] that, if the time change is trivial, then (φ W,τ t ) is Möbius disjoint and the convergence in formula ( 1) is also uniform (the uniform convergence in (1) follows also from the AOP property, [11]).
The structure of the paper is as follows.In Section 2 we recall some basic definitions of the theory of joinings and recall the definition Heisenberg nilflows and their special flow representations over skew-shifts of the 2-torus.In Section 3 we recall the Ratner property and then formulate a version of it for special flows.In Section 4 we state a disjointness criterion (Proposition 4.1) and then formulate a version of it for special flows (Lemma 4.3).In Section 5 we derive from results of [9] (see also [12]) estimates on Birkhoff sums for smooth functions over skew-shifts of the 2-torus.Finally, in Sections 6 and 7 we prove our main theorems by applying the estimates from Section 5.

Heisenberg nilflows. -The (three-dimensional) Heisenberg group H is given by
where K is a positive integer.Notice that M has a (normalized) volume element vol given by the projection of the (bi-invariant) Haar measure on H. Since the Abelianized Lie group Let W be any element of the Lie algebra h of H.The Heisenberg nilflow for W is given by φ W t (x) = x exp(tW ), for all (x, t) ∈ M × R. Every Heisenberg nilflow (φ W t ) on M preserves the volume element vol on M .The classical ergodic theory of nilflows (see [2]) implies that a Heisenberg nilflow (φ W t ) is uniquely ergodic iff it is ergodic iff it is minimal iff the projected flow on M (which is isomorphic to its Kronecker factor) has rationally independent frequencies.More generally, the Diophantine properties of a vector W ∈ h, and of the corresponding nilflow (φ X t ), under the renormalization dynamics introduced in [9] can be entirely read from the Diophantine properties of the projection W of W onto the Abelianized Lie algebra h := h/[h, h], which is also isomorphic to R 2 (as a Lie algebra).In particular, a vector where the (φ W,τ t )-cocycle T (x, t) is uniquely defined by the condition that The above condition is equivalent to either of the following integral conditions By definition the time change is trivial if the (φ W,τ t )-cocycle T is cohomologous to a constant.It follows from the above formula that the time change is trivial if and only if the function τ is (φ W,τ t )-cohomologous to constant, in the sense that there exist a constant c ∈ R and a function u : or, infinitesimally, if and only if τ W u = τ − c.Finally we see that this condition is equivalent to the property that the function We have thus reduced the condition that a time change of a nilflow be trivial to the existence of solutions of cohomological equation for the nilflow itself, a question that can be analyzed by Fourier analysis [24], [9], [3], [12].µ, d) be an ergodic automorphism of a compact metric probability space and let f : X → R be strictly positive.We recall that the special flow (Φ t ) := (Φ f t ) constructed above Φ and under f acts on where N (x, s, t) is the unique integer such that Notice that the flow (Φ t ) preserves the measure µ f = µ⊗λ R restricted to X f , where λ R denotes the Lebesgue measure on R. Let moreover d f denote the product metric (of d and the absolute value metric on R).
2.4.Special flow representation of nilflows.-It is classical that an ergodic nilflow (φ W t ) can be represented as a special flow, where the base automorphism Let (q n ) +∞ n=1 denote the sequence of denominators of α ∈ [0, 1) Q.The vector field W ∈ h is of bounded type if and only if α is of bounded type, i.e., there exists C α > 0 such that q n+1 C α q n for every n ∈ N.
For any function ) denote the special flow over Φ α,β and under f .Then every time change φ W,τ t is isomorphic to a special flow (Φ fτ ,α,β t ), where the roof function f τ is as smooth as τ .In view of the above representation, Theorems 1 and 2 are respectively equivalent to the following two theorems: ) are disjoint for all p, q ∈ N with p = q.
Remark 2.1.-It seems to the authors that a necessary condition for the Ratner property (or any of its variants) to hold in
In [19], Proposition 3.2 was proved for the SR-property, which is a modification of Ratner's property in which one allows for divergence either in the future or in the past.However the proof in [19] immediately extends to a proof of Proposition 3.2.
We will use Proposition 3.2 to prove Theorem 3.
The proof of the above proposition follows similar lines to the proof of [22,Th. 3].We provide a proof for completeness.
Proof.-Let ρ ∈ J((φ t ) t∈R , (ψ t ) t∈R ) be an ergodic joining with ρ = µ × ν.Since (φ t ) is weakly mixing it follows that, for v ∈ {−v , v }, the map φ v is ergodic and hence disjoint from Id.Therefore, there exist Since ρ is a joining, by the triangle inequality, for each t ∈ R, we have ( 4) By applying Birkhoff point-wise ergodic theorem to the joining flow (φ t × ψ t , ρ) and to the characteristic functions of the sets , and for all (x, y) ∈ U 1 , we have Let U 2 := U 1 ∩ (X × Z), where Z = Z(ε, N 0 ) comes from our assumptions.Then ρ(U 2 ) > (1 − c/50)ρ(X × Y ).Note also that since X × Y is σ-compact, the measure ρ is regular, and hence we can additionally assume that U 2 is compact.Define proj : X × Y → X, proj(x, y) = x.Then the fibers of proj are σ-compact, and since U 2 is compact, the fibers of the map proj | U2 : U 2 → proj(U 2 ) ⊂ X are also σ-compact and proj(U 2 ) is also compact.Thus, by Kunugui's selection theorem (see e.g.[15,Th. 4.1]), it follows that there exists a measurable (selection) . By Luzin's theorem there exists X cont ⊂ proj(U 2 ), with µ(X cont ) (1 − c/50)µ(X), such that s Y is uniformly continuous on X cont .Finally, we set Hence, by the definitions of sequences (A k ) and Let δ = δ(ε, N 0 ) come from the assumptions of our theorem.By the uniform continuity of s Y : X cont → Y it follows that there exists 0 < δ < δ such that . By definition, (x, y), (x , y ) ∈ U and d 2 (y, y ) < δ and all other assumptions of our theorem are satisfied for (x, y), (x , y ) (so that we obtain M, L, V, v depending on (x, y) and (x , y ) satisfying ( 2)).
We claim that ( 8) Indeed, in view of (4), the estimate (8) follows if we can prove that Hence to complete the proof of Claim (8), it is enough to show that Notice however that hence the estimate in (9) follows from (6) with M = M + V and L = L.By a similar reasoning, we get so putting together ( 8) and (10) we derive the estimate This however contradicts (3), hence the argument is complete.

4.1.
Disjointness criterion for special flows.-In this section we assume that act on X f with metric d f 1 and (Ψ g t ) act on X g with metric d g 2 .For (x, s) ∈ X f and t ∈ R we denote by n(x, s, t) ∈ Z the unique number for which We define m(y, r, t) analogously for (y, r) ∈ Y g .We tacitly assume that f and g are bounded away from zero.
Before we state a disjointness criterion for special flows, we need the following general lemma: The statement follows by the definition of V (t) since d f 1 is the product metric and we have d 1 (Φ n(x,s,t) x, Φ n(x,s,t) x ) < ε 2 and |s − s | ε 2 .Lemma 4.3.-Let V ∈ R and P = {−p , p } for p = 0 and ζ := X f dµ/ Y gdν.Let A k ∈ Aut(X, B, µ), A k → Id uniformly.Assume moreover that for every ε > 0 and N ∈ N there exist 0 < κ = κ (ε ) < ε , δ = δ (ε , N ) > 0, such that for all y, y ∈ Y satisfying d 2 (y, y ) < δ , every k such that d 1 (A k , Id) < δ and every x, x := A k x ∈ X there are M N , L 1, L /M κ and p ∈ P satisfying: The proof of the above proposition follows similar lines (although is simpler) than the proof of [22,Prop. 4.1].We provide a proof here for completeness.By the ergodic theorem for Φ f t there exist N ∈ R and a set E = E(ε) ⊂ X f , µ f (E) > 1 − ε, such that for every M, L N , L/M κ and every (x, s) ∈ E, we have Similarly, by the ergodic theorem for Ψ g t there exist N ∈ R and Z = Z(ε) ⊂ Y f , ν g (Z) > 1 − ε such that for every M, L N , L/M κ and every (y, r) ∈ Z, we have and for t ∈ R such that m(y, r, t) ∈ [M, M + L], we have (18) |t − m(y, r, t) 1 ((x, s), (x , s)) < δ and (y, r), (y , r ) ∈ Z with d g 2 ((y, r), (y , r )) < δ.Let M , L , p come from Lemma 4.3 for x, x and y, y .Define M, L by n(x, s, M ) X f dµ = M and n(x, s, M + L) X f dµ = M + L .It follows by (16) that L/M κ and M N .
We will use Lemma 4.3 to prove Theorem 4.

Birkhoff sums over toral skew-shifts
In what follows Φ α,β (x, y) = (x + α, y + x + β) is a (linear) skew-shift on T 2 , with α of bounded type and g ∈ W s (T 2 ), with s > 7/2 and T 2 gdλ T 2 = 0, where λ T 2 denotes the normalized (Haar) Lebesgue measure on T 2 .We also assume that g is not a coboundary (although some lemmas below are true also for coboundaries).

The solution of the cohomological equation, for any function
, is given by the following formula.If g = j∈Z g j e m+jn,n , the solution J.É.P. -M., 2020, tome 7 u = j∈Z u j e m+jn,n has Fourier coefficients: If g ∈ W s (H m,n ) for any s > 1 and D m,n (g) = 0, then the above solution u belongs to W t (H m,n ) for all t < s − 1 and there exists a constant C s,t > 0 such that The results below establish the quantitative behavior of the square mean of ergodic averages for smooth functions under the skew-shift.
lim sup

General estimates
Lemma 5.2.-There exists a constant C α,g > 0 such that, for every N ∈ N and every (x, y) ∈ T 2 , we have Proof.-Since α ∈ R Q is of bounded type, the statement follows from [12,Lem. 1.4.9] or from [13, Lem.6.1 & Th.6.2].In fact, since any constant roof suspension of Φ α,β is smoothly isomorphic to a Heisenberg nilflow (φ W t ) on a nilmanifold M , generated by a bounded type vector field W ∈ h, for any g ∈ W s (T 2 ) and every (x, y) ∈ T 2 , there exist a function G ∈ W s (M ) and p ∈ M such that By [12, Lem.1.4.9], for any Heisenberg triple F := (X, Y, Z) and for any σ > 2, there exists a function B σ (F , T ) (defined [12, Eq. (1.71)]) such that, for any function f ∈ W σ (M ) and for all (p, T ) ∈ M × R, For X = W of bounded type, by definition there exists a constant C > 0 such that B σ (F , T ) C σ T 1/2 , hence we derive (see also the comments after the proof of [12, Lem.1.4.9]).Alternatively, from [13, Th. 6.2] we derive that for a = (X, Y, Z) satisfying an explicit Diophantine condition (depending only on Y ) and for any f ∈ W s (M ), there exists a Hölder cocycle and [13, Lem.6.1] implies that whenever X = W is of bounded type, the cocycle β f (a, p, T ) satisfies the upper bound which again implies our statement.
Lemma 5.3.-There exists a constant c g > 0 such that for every N ∈ N, Proof.-Since g has zero average, but it is not a coboundary, and α ∈ R Q has bounded type, we can assume that g ∈ W s (H m,n ), for some (m, n) with n = 0.In fact, otherwise g is the pull back of a function on the circle T, which belongs to W s (T) with s > 7/2, and since α is of constant type, it follows by Fourier series that g is a coboundary with transfer function u ∈ W t (T) for all t < s − 1 (in particular u ∈ C 2 (T)).
By orthogonality of the decomposition W s (T 2 ) as a direct sum of components W s (H m,n ), for all s ∈ R, we can assume that g ∈ W s (H m,n ), for some (m, n) with n = 0, hence by Lemma 5.1 there exists c g > 0 such that, for all N ∈ N, This finishes the proof.
By contradiction, if the statement is not true, then This contradicts the choice of k.The proof of the second inequality follows the same lines.
Lemma 5.5.-There exists C = C α,g > 0 such that, for every (x, y), (x , y ) ∈ T 2 and for every n ∈ N, we have Proof.
-By the mean value theorem for S n (g), we have for some By the chain rule and Lemma 5.2 Moreover, by the chain rule, we have and by summation by parts By Lemma 5.2, for some C > 0, Using the above estimates in (19) finishes the proof.
Lemma 5.6.-Fix q ∈ N.For every η > 0 there exists D η > 1 such that for every n, m ∈ N and every (x, y) ∈ T 2 , we have Proof.-Since α is of bounded type, there exists D η > 1 such that for every (x, y) ∈ T 2 and every n ∈ N, the orbit , which together with Lemma 5.3 finishes the proof is η > 0 is small enough.
We have by the above Since K T and by the definition of (x, y), (z, w), this finishes the proof of ( 21) with K = K, W = W + 1,n q n and Q = n q n , hence the proof of Lemma 5.9 is complete.
Therefore, it only remains to show (29).By ( 28) and ( 27) and the assumptions of (A), we have This finishes the proof of (29) and hence also the proof of case (A).
(B) |(w − w )q 1/2 /(y − y )p 1/2 | − 1 χ.In this case the LHS in (26) is larger than By Lemma 5.9 for T = T and the definition of T , there exists an Moreover, by Lemma 5.2 and the definition of T for h it follows that if χ > 0 is sufficiently small.This finishes the proof of Lemma 5.10.

Ratner's property: proof of Theorem 3
In this section we will use the estimates from Section 5 to prove Theorem 3. We will use Proposition 3.2.Before we do that, we will prove a crucial proposition: For (x, y), (x , y ) ∈ T This property, together with (37), finishes the proof of the hypotheses of Proposition 3.2 and hence also the proof of Theorem 3.

Theorem 1 .
-Let W ∈ h be of bounded type.For any positive function τ ∈ W s (M ) with s > 7/2, either the time change is trivial or the time-changed flow (φ W,τ t ) enjoys the Ratner property.

Corollary 1 . 2 .
-Let W ∈ h be of bounded type.Then for every positive function τ ∈ W s (M ) with s > 7/2, either the time change is trivial or (φ W,τ t ) is mildly mixing (no non-trivial rigid factors).It turns out that Heisenberg nilflows of bounded type (as in Theorem 1) are the only known examples, beyond horocycle flows and their time changes, for which the original Ratner property holds.