Multiple mixing for a class of conservative surface flows

Arnol'd and Kochergin mixing conservative flows on surfaces stand as the main and almost only natural class of mixing transformations for which higher order mixing has not been established, nor disproved. Under suitable arithmetic conditions on their unique rotation vector, of full Lebesgue measure in the first case and of full Hausdorff dimension in the second, we show that these flows are mixing of any order. For this, we show that they display a generalization of the so called Ratner property on slow divergence of nearby orbits, that implies strong restrictions on their joinings, that in turn yield higher order mixing. This is the first case in which the Ratner property is used to prove multiple mixing outside its original context of horocycle flows and we expect our approach will have further applications.


Introduction
A major open problem in ergodic theory is Rokhlin's question on whether mixing implies mixing of all orders, also called multiple mixing [21]. In most of the known examples of mixing dynamical systems, multiple mixing is now known to hold. Moreover, a positive answer to Rokhlin's question is actually known to generally hold within various classes of mixing dynamical systems. The most noteworthy are K-systems where multiple mixing always holds [2], horocycle flows [18], mixing systems with singular spectrum that display multiple mixing by a celebrated theorem of Host [10], and finite rank systems since Kalikow showed that rank one and mixing implies multiple mixing [11], a result that was extended to finite rank mixing systems by Ryzhikov [25].
In the second half of the last century, Arnol'd and Kochergin introduced a major class of conservative smooth mixing flows on surfaces, with non-degenerate saddle type singularities for the first and degenerate ones for the second. Mixing of these flows was proved by Kochergin in the case of degenerate power like singularities (see exact description below) in [13] and by Khanin and Sinai in a particular case of non-degenerate asymmetric saddle type singularities (see exact description below) [27]. The study of the mixing properties of these flows has known a revival of interest since the beginning of the 2000's, with results such as the computation of the speed of mixing [4] or extension of the Kahnin-Sinai mixing result to include all irrational translation vectors [17] (see also [15], [16]), or advances in the study of Arnol'd and Kochergin flows in the general case where the Poincaré section return map is an interval exchange and not just a circular rotation [30,31,32].
Arnol'd and Kochergin flows stand today as the main and almost only natural class of mixing transformations for which higher order mixing has not been established, nor disproved. Our aim here is to prove mixing of all orders for a rich subclass of these systems determined by the arithmetics of their unique rotation vector. For this, we use the representation of Arnol'd and Kochergin flows as special flows above an irrational rotation R α of the circle and under a ceiling function with asymmetric logarithmic singularities for the first, and integrable power like singularities for the second. Loosely speaking our main result is as follows (it will be made precise at the end of this introduction, see Corollaries 1.6 and 1.8).
Theorem. Arnol'd-Khanin-Sinai flows are mixing of all orders for a set of α ∈ (0, 1) of full Lebesgue measure. Kochergin flows are mixing of all orders for a set of α ∈ (0, 1) of full Hausdorff dimension.
Similar mixing mechanisms due to orbit shear as in Kochergin and Arnol'd-Khanin-Sinai flows were observed relatively recently such as in [5] or [1] and it should be possible to apply the techniques of the current paper to the study of higher order mixing for such parabolic systems.
To explain our approach we need first to make a detour by Ratner's study of horocycle flows. In the 1980's, M. Ratner developed a rich machinery to study horocycle flows [22]- [24] and, in particular, singled out a special way of controlled slow divergence of orbits of nearby points which resulted in the notion of H p -property, later called R-property by J.-P. Thouvenot [28]. This property, to which we will come back with more detail in the sequel, has important dynamical consequences, mainly expressed by a restriction on the possible joining measures of a system having the R-properties with other systems, and in particular with itself.
A joining between two dynamical systems (T, X, B, µ) and (S, Y, C , ν), (X, B, µ) and (Y, C , ν) being standard Borel probability spaces, is a measure ρ on X × Y invariant by T × S whose marginals on X and Y are µ and ν. The definition for flows is similar. An important notion in Ratner's theory is that of finite extension joinings (FEJ). Definition 1.1. An ergodic flow (T t ) t∈R is said to have FEJ-property, acronym for finite extension joining, if for every ergodic flow (S t ) t∈R acting on (Y, C , ν) and every ergodic joining ρ of (T t ) t∈R and (S t ) t∈R different from the product measure µ × ν, ρ yields a flow which is a finite extension of (S t ) t∈R .
It was shown in [26] that a mixing flow with FEJ-property is mixing of all orders. Moreover, it was proved in [22] that a flow with R-property has the FEJ-property. It follows that mixing flows with the R-property are mixing of all orders. Since the R-property for horocycle flows stemmed from polynomial shear along the orbits, and since Kochergin flows displayed a similar polynomial shear along the orbits, the idea that special flows over rotations may enjoy the R-property, and thus be multiple mixing, was then suggested by J-P. Thouvenot in the 1990's (see p. 2 in [6]).
However, whether natural classes of special flows (not necessarily mixing) over irrational rotations may have the R-property remained open until K. Fraczek and M. Lemańczyk [6,7] showed that a generalized R-property holds in some classes of special flows with roof functions of bounded variation (which, by [12], are not mixing). More precisely, they have introduced a weaker notion than the R-property, called weak Ratner or WR-property that however still implies the FEJ-property (see Definition 2.1 and the comment after it) .
Unfortunately, in the mixing examples of special flows under piecewise convex functions with singularities such as Arnol'd and Kochergin flows, the shear may occur very abruptly as orbits approach the singularity and this may prevent them from having the weak Rather property. Indeed, we believe that these flows do not have the WR-property. That this should be true is corroborated by the following result that shows that Kochergin flows, in the context of bounded type frequency in the base (that is a priori favorable to controlling the shear), do not have the WR-property. Theorem 1. Let α ∈ T be irrational with bounded partial quotients and f (x) = x γ +r, −1 < γ < 0, r > 0. Then the special flow above the circle rotation R α and under the ceiling function f does not have the WR-property.
We denote by T the circle R/Z. We refer to Section 3 for the exact definition of special flows. Theorem 1 has another consequence. It is known that every horocycle flow (h t ) t∈R is loosely Bernoulli [24]; therefore, for every irrational α, there exists a positive function in f ∈ L 1 (T) such that (h t ) t∈R is measurably isomorphic to (T f t ) t∈R [20]. It follows from [12] and the fact that (h t ) t∈R is mixing that f is of unbounded variation. Moreover, by [19], f can be made C 1 except for one point. Since the R-property implies the WR-property and the R-property is an isomorphism invariant, no special flow as in Theorem 1 is isomorphic to a horocycle flow. Actually, this line of thought can be extended to show that horocycle flows are never isomorphic to special flows above an irrational rotation and under a roof function that is convex and C 2 except at one point. For the latter result, one needs to introduce the concept of strong Ratner property, which is also an isomorphism invariant, that specifies the occurrence of slow divergence of nearby orbits to the first time when the orbits do split apart. This will be dealt with in a future work.
To bypass Theorem 1 and still use controlled divergence of orbits to show multiple mixing, our approach will be to further weaken the WR-property. Namely, we introduce the SWR-property, which stands for switchable weak Ratner property, that assumes that a pair of nearby points displays the WR-Property either under forward iteration in time or under backward iteration, and this depending on the pair of points. We show that the SWR-property is sufficient to guarantee the same FEJ consequences as the Ratner or the weak Ratner property. Consequently, a mixing flow enjoying the SWR-property is mixing of all orders.
The main idea in showing that Arnold and Kocergin special flows may have the SWR-Property is the following. The main contribution to the shear between orbits is due to the visits of the flow lines to the neighborhood of the singularities. With the representation of these flows as special flows above irrational rotations, the shear is translated into the divergence between the Birkhoff sums of the roof functions for nearby points, and this divergence is mainly due to the visits under the base rotation to the neighborhoods of the points where the roof function has its singularities. If the base rotation angle α is of bounded type two nearby points will stay sufficiently far from the singularity either when they are iterated forward or when they are iterated backward. In the case of ceiling functions with only logarithmic singularities we are also able to exploit the progressive contribution to the shear of these visits to the singularities to obtain multiple mixing for a full measure set of numbers α.
We will now describe the ceiling functions that will be considered in the sequel and state our exact results on the SWR-property and multiple mixing.
..., a k }) for some numbers a 1 , ..., a k ∈ T. We say that f has singularities of type h at {a 1 , ..., a k } if for some numbers A i , B i 0, i = 1, ..., k.
Notice that in this definition h may only reflect a domination on the singularities of f since the coefficients A i , B i may be equal to zero at some or at all i's.
In all the sequel, we will consider α ∈ R \ Q and let (q s ) be the sequence of denominators of the best rational approximations of α. Namely (q s ) is the unique increasing sequence such that q 0 = 1 and q s α < kα for any k < q s+1 , k = q s . We recall that Our results can deal with functions having several singularities but require a non resonance condition of these singularities with α.
[Badly approximable singularities] Given α ∈ R \ Q, we will say that {a 1 , ...., a k } are badly approximable by α if there exists C > 1 such that for every x ∈ T and every s ∈ N, there exists at most one i 0 ∈ {0, ..., q s − 1} such that Note that if there is only one singularity, that is k = 1, then by (2) it is always badly approximable by α. The following shows that for k 2 the set of singularities that are badly approximable by α is a thick set in [0, 1] k .
But it was proven in [29] (see also [3]) that the set B(α) is a winning set in the sense of Schmidt (see [29,3] and references therein). A winning set is of full Huasdorff dimension. Moreover, for a winning set B ∈ R we have that for any x 1 , . . . , x n the set ∩ n s=1 (x s + B) is winning. So, if a 1 , . . . , a l are such that a i − a j ∈ B(α) for any i, j ∈ {1, . . . , l}, i = j, then the set of a ∈ [0, 1] such that a ∈ ∩ l s=1 (a s + B(α)) is winning which means that a 1 , . . . , a l , a l+1 are badly approximable by α for a winning set of a l+1 . The statement of the Lemma follows then by induction and because a single a 1 is always badly approximable by α.
Our results deal with two types of singularities. Theorem 2 deals with logarithmic like singularities, while Theorem 3 deals with the case of at least one dominant power like singularities.

Logarithmic like singularities
In the case of logarithmic like singularities, the following theorem holds.
Theorem 2. Let α ∈ R \ Q and f ∈ C 2 (T \ {a 1 , ...., a k }) with the singularities {a 1 , . . . , a k } of type h and badly approximable by α, with some associated constant C > 1. Assume that and that there exist a constant m 0 > 0 and a sequence (x s ) such that for every s ∈ N, we have x s < 1 qs and Then (T f t ) t∈R has the SWR-property.
To describe a consequence of Theorem 2 (see Corollary 1.6), set h(x) = − ln(x), for x ∈ (0, 1]. For α ∈ R \ Q, let K α := {n ∈ N : q n+1 < q n log 7 8 (q n )}. We then define in view of 1. and 2. of Theorem 2 To have 3. of Theorem 2 it suffices to assume that α is Diophantine : for τ 0 define the set of Diophantine numbers α of exponent τ to be The set of Diophantine numbers is then An equivalent definition of DC(τ ) is that for any p, q ∈ Z × N * we have that |α − p q | C(α) q 2+τ for some C(α) > 0.
..., a k }) with the singularities {a 1 , . . . , a k } of type h and badly approximable by α, with some associated constant C > 1. Assume that Then (T f t ) t∈R has the SWR-Property and is mixing of all orders.
Proof. We take for x s the sequence fore (T f t ) t∈R has the SWR-property. On the other hand, it was shown in [15] that (T f t ) t∈R is mixing. Multiple mixing then follows from Theorem 4 and the FEJ-property. Corollary 1.6 covers a set of full Lebesgue measure of rotation angles α. Indeed, it is known that the set of Diophantine numbers D has full Lebesgue measure, and we will prove in Appendix A the following result. Denote by λ the Haar measure on T. Proposition 1.7. It holds that λ(E) = 1.

Power like singularities
Now, we will deal with power like singularities. We suppose f ∈ C 2 (T \ {a 1 , ...., a k }) with singularities {a 1 , ...., a k } of type h. We divide the set {a 1 , ..., a k } of singularities into two subsets : F the set of weak singularities, and E the set of strong singularities of type not less than h : namely, we suppose F = {a 1 , ..., a v } and E = {a v+1 , ..., a k } ∈ E are such that, A 2 i + B 2 i > 0 in (1) for i ∈ {v + 1, . . . , k}, while each a i ∈ F is a singularity for f of type g i with g i a positive function in C 2 (T \ {0}), decreasing on (0, 1) with g ′ i increasing and such that and lim exist and are finite, lim x→0 + gi(x) h(x) = 0 and for every s ∈ N there exists x i,s ∈ T, x i,s > 1 qsh( 1 2qs ) , such that lim s→+∞ We always assume that E is not empty.
Theorem 3. Let α be irrational with bounded partial quotients, that is, α ∈ DC(0). Assume that {a v+1 , ..., a k } are badly approximable by α with some constant C > 1. Assume that there exist constants D 1 , D 2 > 0 such that for every s ∈ N Then (T f t ) t∈R has the SWR-property.
..., a k }) with all the singularities {a 1 , ...., a k } of power-like type x γi from the left and x δi from the right, Then, if the points in E are badly approximable by α, we have that (T f t ) t∈R has the SWR-Property and is mixing of all orders.
Note that there are no combinatorial assumptions on the weak singularities a j / ∈ E.
Proof of Corollary 1.8. Take x s = 1 s 2 qs and easily check the hypothesis of Theorem 3. This gives the SWR-Property . Mixing of (T f t ) t∈R was established in [13]. Multiple mixing then follows from Theorem 4 and the FEJ-property.

Plan of the paper
In Section 2 we introduce the SWR-Property and study its joinings consequences. In Section 3 we give a criterion involving the Birkhoff sums of the ceiling function that guarantees that a special flow above an isometry has the SWR-property. The treatment of these sections is similar to [6,7]. In Section 4 we study the Birkhoff sums of logarithmic like and power like functions and prove Theorems 2 and 3. Section 5 is devoted to the proof of Theorem 1 on the absence of the SWR-Property for a subcalss of Kochergin flows. Finally Appendix A is devoted to the proof that the set of frequencies for which Theorem 2 holds has full Lebesgue measure. The second author would like to thank Professor Mariusz Lemańczyk for all his patience, help and deep insight. The authors would also like to thank Krzysztof Fraczek, Mariusz Lemańczyk and Jean-Paul Thouvenot for valuable discussions on the subject.
The results of Section 4 have been obtained by the two authors independently and the results of Section 5 by the second. The two authors decided to include Section 5 in this work because it is an integral part of the problems concerning Ratner's property for this class of special flows.

The SWR-property
Let (X, B, µ) be a probability standard Borel space. We additionally assume that X is a complete metric space with a metric d. Let (T t ) t∈R be an ergodic flow acting on (X, B, µ). Definition 2.1 (cf. [7], Definition 4). Fix t 0 ∈ R + and a compact set P ⊂ R \ {0}. One says that the flow has the switchable R(t 0 , P )-property if for every ǫ > 0 and N ∈ N there exist κ = κ(ǫ), δ = δ(ǫ, N ) and a set Z = Z(ǫ, N ) ⊂ B with µ(Z) > 1 − ǫ such that for any x, y ∈ Z with d(x, y) < δ, x not in the orbit of y there exist M = M (x, y), L = L(x, y) ∈ N with M, L > N and L M κ and p = p(x, y) ∈ P such that If the set of t 0 > 0 such that the flow (T t ) t∈R has the switchable R(t 0 , P )-property is uncountable, the flow is said to have SWR-property.
For the sake of completeness, compare the SWR-property with the definition of the WRproperty [7]. To have WR-property, we fix P ⊂ R \ {0} and t 0 ∈ R. (T f t ) t∈R has R(t 0 , P ) property if in Definition 2.1, (6) holds (the condition (7) is not taken into account) and (T f t ) t∈R has WRproperty if the set of t 0 ∈ R such that (T f t ) t∈R has R(t 0 , P ) property is uncountable. Consequently, SWR-property is weaker than WR-property (and as Theorem 1 shows, it is strictly weaker). Now, again for sake of completeness, we will present a detailed proof (using some facts proved in [7]) of the fact under the "continuty" assumption on orbits (see below) that SWR-property has FE-property as the original H p -property introduced by M. Ratner [22].
We will state a lemma which is a simple consequence of Lemma 5.2. in [7].
We will add one more natural condition on the flow (T t ) t∈R which can be viewed as "continuity" on orbits. The flow (T t ) t∈R is called almost continuous [7] if for every ǫ > 0 there exists a set For the definition and properties of joinings, we refer the reader to [28] or [9]. Our goal is now to prove the following result.
Theorem 4. Let (T t ) t∈R be a weakly mixing flow acting on a probability standard Borel space (X, B, µ). Assume that (T t ) t∈R satisfies the SWR-property. Let (S t ) t∈R be an ergodic flow acting on a probability standard Borel space (Y, C , ν) and let ρ ∈ J((T t ) t∈R , (S t ) t∈R ) be an ergodic joining. Then either ρ is equal to µ ⊗ ν or is a finite extension of the measure ν.
To prove this theorem we need some lemmas from [7]. Lemma 2.3. Let (T t ) t∈R be an ergodic almost continuous flow acting on (X, B, µ), and (S t ) t∈R be another ergodic flow acting on (Y, C , ν). Let ρ ∈ J((T t ) t∈R , (S t ) t∈R ) be such that ρ is ergodic for automorphisms T 1 × S 1 (hence, for T −1 × S −1 ). Let P ⊂ R be non-empty and compact. Let A ∈ B be such that µ(∂A) = 0 and B ∈ C . Then, for every ǫ, δ, κ > 0 there exist a natural number N = N (ǫ, δ, κ) and a set Z = Z(ǫ, δ, κ) ⊂ B ⊗ C with ρ(Z) > 1 − δ such that for any N ∋ M, L N with L M κ and any p ∈ P , we have Proof. The proof is a simple consequence of Lemma 5.4. in [7]. One uses this lemma first for the flows (T t ) t∈R and (S t ) t∈R and ergodic joining ρ ∈ J( Then, for flows (T −t ) t∈R and (S −t ) t∈R and the same ergodic joining ρ to get, for ǫ, To finish the proof one takes N := max(N + , N − ) and Next lemma is used in the proof of Theorem 3 in [22]. Lemma 2.4. Let (T t ) t∈R and (S t ) t∈R be two ergodic flows. Let ρ ∈ J e ((T t ) t∈R , (S t ) t∈R ) be an ergodic joining. Then if there exists a set V with ρ(V ) > 0 such that for any points (x, y), In what follows, we consider only (X, d) be a σ-compact metric space. Let A ∈ B. For η > 0 we denote by V η (A) := {x ∈ X : d(x, A) < η}.
Lemma 2.5. [cf. [7]] For any A ∈ B there exists R ⊂ (0, +∞) such that (0, +∞) \ R is countable and µ(∂V η (A)) = 0 for η ∈ R. It particular, there exists a dense family Proof of Theorem 4. Let ρ ∈ J((T t ) t∈R , (S t ) t∈R ) be an ergodic joining and ρ = µ × ν. Assume that (T t ) t∈R has the switchable R(t 0 , P )-property and ρ is ergodic for T t0 × S t0 (then ρ is ergodic for T −t0 × S −t0 ). Such t 0 > 0 always exists because an ergodic flow can have at most countably many non-ergodic time automorphisms. For simplicity of notation, we assume t 0 = 1. Let {B i } i 1 and {C i } i 1 be two countable dense families in the σ-algebras B and C , respectively. Consider the following real function: As in Lemma 5.4. in [7], we conclude that k : R → R is a continuous function and for any t ∈ R, k(t) > 0. Indeed, it follows by the fact that if for some r ∈ R \ {0} we have for any i, j ∈ N ρ(T r (B i ) × C j ) = ρ(B i × C j ) then ρ is product measure (recall that T t is assumed to be weak mixing hence every time r of the flow is ergodic). The set P ⊂ R \ {0} is compact, therefore there exists ǫ > 0 such that k(p) > ǫ for any p ∈ P . It follows by the definition of the function k that there exists a number R := R(ǫ) such that It follows by the fact that ρ is a joining that Next, by Lemma 2.3 applied to ǫ 8 , 1 8 , κ > 0 and the sets B i × C j , 1 i, j R, there exist N 2 ∈ N and a set U 2 ⊂ B ⊗ C with ρ(U 2 ) > 7 8 such that for every L, M N 2 with L M κ and any p ∈ P , we have and It follows that if we set N 0 := max(N 1 , N 2 ) and U 0 := U 1 ∩ U 2 , then ρ(U 0 ) > 1 2 and for every L, M N 0 with L M κ, any p ∈ P , the equations (9), (10), (11), (12) are satisfied for every (x, y) ∈ U 0 . Using the switchable R(1, P )-property with ǫ ′ > 0 and N 0 ∈ N, we obtain δ = δ(ǫ ′ , N 0 ) and Z = Z(ǫ ′ , N 0 ) with µ(Z) > 1 − ǫ ′ . Now, we will use Lemma 2.4 with the set U : Assume that the first inequality is satisfied. We will use equations (9) and (11) (in case the second one is satisfied, we use equations (10) and (12)). Let 1 δ 0 and an application of Lemma 2.4 completes the proof.

SWR-property for special flows
In this section, we will prove a sufficient condition for SWR-property in the case of special flows over an ergodic isometry We start by recalling the definition of special flows. Let T be an automorphism (X, B, µ). Let f ∈ L 1 (X, µ) such that f > 0. The special flow (T f t ) t∈R defined above T and under the ceiling function f is given by Equivalently the flow (T f t ) t∈R is defined for t + s 0 (with a similar definition for negative times) by where n is the unique integer such that and If T preserves a unique probability measure µ then the special flow will preserve a unique probability measure that is the normalized product measure of µ on the base and the Lebesgue measure on the fibers. If X is a matric space with a metric d, so is The following general lemma is a direct consequence of Birkhoff ergodic theorem.
for every x ∈ A.
Assume that additionally f is positive and bounded away from zero. Fix ǫ, κ > 0 (κ < | X f dµ| < 1/2). It follows that there are constants r 1 , or If γ > 0 is such that the automorphism T f γ is ergodic, then (T f t ) t∈R has the switchable R(γ, P )property.
Apply Remark 3.2 with the constants ǫ/4, κ to f and T , T −1 , respectively to obtain constants D 1 , D 2 > 0 such that for x ∈ A, µ(A) > 1 − ǫ/2 (the set A is the intersection of two relevant sets), we have Now, we will use ergodicity of T f γ and T f −γ . It follows that there exist N 0 := N (ǫ) and a set Z := Z(ǫ) with µ f (Z) > 1 − ǫ 2 and for every (x, s) ∈ Z and n N 0 We will consider the second case (the proof in the first case goes along the same lines).
Let us define By (19) it follows that It follows by the properties of Take Using additionally the inequality |s − s ′ | < δ ′ we hence obtain A similar reasoning shows that Therefore, by the definition of the special flow, we have (21), at least (1 − ǫ)L ′ and for any such k we get that This gives us the switchable R(γ, P )-property.
Note that if a flow (T f t ) t∈R is ergodic then the set of η ∈ R such that T f η is not ergodic, is at most countable and therefore, as a direct consequence of Proposition 3.3, we get that (T f t ) t∈R enjoys SWR-property.

SWR-property for smooth special flows with singularities
In this section we will use Proposition 3.3 to prove SWR-property for special flows given by the assumptions in Theorem 2 and Theorem 3. In all the sequel we assume {a 1 , .., a k } are badly approximable by α with a constant C > 1 (see Definition 1.3).
Lemma 4.1. Let s ∈ N be such that q s+1 > 4Cq s and x ∈ T. Then . For finite sets A, B ⊂ T, we use the notation ρ(A, B) = min a∈A,b∈B a − b .
The following lemma is a simple consequence of the Denjoy-Koksma inequality.
Thenh ∈ BV (T) and we use the Denjoy-Koksma inequality to obtain |h (qs) ri(x−ai) and increasing on (0, 1). Then there exists a constant H > 0 such that for each x ∈ T.
Proof. By assumptions, there exists a constant z 0 > 0 such that for every i = 1, ..., k and for every Moreover, since f ′ ∈ C 1 (T \ {a 1 , ..., a k }); it follows that there exists a constant R > 0 such that for every C0 } satisfies the assertion of the lemma.
In the sequel, we will assume s is an integer sufficiently large s s 1 , s 1 = s 1 (ε, N ) to be determined later, for now assume that κq s1 > N .
By assumptions 1. and 2. of Theorem 2, if s s 0 and s 0 (ε) is sufficiently large we will have and s s0,s / ∈Kα x s q s < ǫ 16k . Set v s := xs 4C . Let  . Consider x, y ∈ Z with 0 < x − y < δ (we assume that x < y).
The following proposition implies Theorem 2 due to Proposition 3.3.
We will assume that q s+1 > 2q s . If not, then in (24), m0 2qsh( 1 2qs ) < x − y and we repeat the considerations below in the time interval [q s−1 , q s ]. In other words, in this case we will see the drift between x and y before time q s . Proposition 4.5. Consider x, y ∈ Z as in (24). Part a There exists n 0 ∈ {1, ..., max( qs+1 8Cqs , 1)} satisfying Part b Let X = T n0qs x and Y = T n0qs y if (25) holds, and X = T −(n0qs+1) x and Y = T −(n0qs+1) y if (26) holds. For n = 1, ..., [κn 0 q s ] + 1 the following holds The rest of this section is devoted to the proof of For m ∈ N, we will often use the following non resonance conditions of a pair of points (x, y) with the singularities {a 1 , . . . , a k }.
Lemma 4.6. Let x, y ∈ T as in (24). Then for every m such that s 0 m s, if we have at least one of the following 1. if m / ∈ K α and (28) is satisfied 2. if m ∈ K α and q m+1 2q m , then we have at least one of (29) or (30).
Lemma 4.7. There exists s ′ ∈ N such that for every s s ′ and any points x < y ∈ T such that (28) is satisfied for x, y and m = s, then Proof. By (28) and (24), f (qs) is differantiable on [x, y]. Therefore, there exists θ ∈ [x, y] such that It is enough to show that there exist d > 0 and s ′ ∈ N such that for s s ′ It follows thatf ′ s ∈ BV (T) and where We have (if s ∈ N is sufficiently large). It follows by the assumptions on f ′ and h ′ and (28), that for every ǫ > 0 there exists s ′ = s ′ (ǫ) such that for s s ′ , we have for every i = 1, ..., k: and similarly On the other hand, by l'Hospital's rule (by xs 4C < 1 2qs ). Now, using (35)-(41), we get which allows us to conclude if we assume WLOG that ε is sufficiently small.
We will assume in the sequel that s 1 s ′ of Lemma 4.7.
We are ready now to finish the proof of Part a. of Proposition 4.5. If s / ∈ K α , then by the fact that x, y ∈ Z ⊂ B s , it follows that 1. in Lemma 4.6 is satisfied with m = s. If s ∈ K α then 2. in Lemma 4.6 is satisfied with m = s. Therefore we can use Lemma 4.6 for x, y and m = s. Now, by Lemma 4.7, if (29) holds we have (25), if (30) holds we have (26). Part a. of Proposition 4.5 is settled, we turn now to Part b.
To prove Proposition 4.5 Part b., observe first that if s 1 is sufficiently large, and up to eventually changing κ to κ ′ = κ 8C , one of two possibilities holds : 1. There exists s 0 m s, m ∈ K α , such that κn 0 q s q m 8Cκn 0 q s , or 2. There exist s 0 m s and l 1 such that lq m κn 0 q s (l + 1)q m qm+1 8C . Case 1. κn 0 q s q m 8Cκn 0 q s with s 0 m s, m ∈ K α . Lemma 4.6 implies that either (29) or (30) holds for T n0qs x, T n0qs y, m. We then apply Lemma 4.9 to T n0qs x, T n0qs y, m with l = 0, and according to whether we have (47) or (48) we will get A. or B. of Proposition 4.5 Part b. Indeed, suppose (47) holds then for n = 1, . . . , [κn 0 q s ] + 1, we have due to (46) Case 2. There exist s 0 m s and l 1 such that lq m κn 0 q s (l + 1)q m qm+1 8C . If m ∈ K α , then Lemma 4.6 implies that either (29) or (30) holds for T n0qs x, T n0qs y, m. If m / ∈ K α , then we will first prove that T n0qs x, T n0qs y, m satisfy the hypothesis of Lemma 4.9. Due to Lemma 4.6, we just have to check (28) for T n0qs x, T n0qs y, m : Indeed, let i 0 and r 1 be such that It follows by (3), that for i 0 = j ∈ {0, ..., q m − 1}, Next, by the fact that m / ∈ K α and x ∈ B m (m s 0 ), we get that x + i 0 α − a r1 4v m , and therefore (recall that n 0 qs+1 8Cqs ) and (51) is thus proved ( T n0qs x − T n0qs y (24) < v m ). We are now able to apply Lemma 4.9 to T n0qs x, T n0qs y, m with l such that lq m κn 0 q s (l + 1)q m . Now and as in case 1., if (47) holds we get A., if (48) holds we get B.

Proof of Theorem 3
We may assume WLOG that where H is from Lemma 4.3, and c is such that for every s ∈ N q s+1 cq s . Fix )). Let s 0 ∈ N be such that for +∞ s s0 q s x i,s < ǫ 8k for every i = 1, . . . , v, and for every i = 1, ...k The following proposition implies Theorem 3 due to Proposition 3.3. We can assume WLOG that x < y. Let s := s(x, y) be unique such that As in the precedent section, Proposition 4.10 follows from Proposition 4.11. Consider x, y ∈ Z as in (56). Part a. There exists i 0 ∈ {0, ..., q s−2 − 1}, such that Part b. Let X = T i0 x and Y = T i0 y if (57) holds, and X = T −i0−1 x and Y = T −i0−1 y if (58) holds, for n = 1, ..., [κi 0 ] + 1 the following holds The rest of this section is devoted to the proof of Proposition 4.11. Consider the orbit x − q s−2 α, ..., x, ..., x + (q s−2 − 1)α (the length of this orbit is smaller than q s ). It follows by (3) that there exists at most one t s ∈ [−q s−2 , q s−2 Hence at least one of the following two holds : The following Lemma directly implies the proof of Proposition 4.11. Proof. We will suppose (60) holds, the proof of the other case being analogous.

Indeed, the LHS of this inequality is equal to
. By (55), monotonicity of h ′ , (5) twice (for s and s + 1) and (56) and the claim follows. Therefore, one of the numbers As a consequence of the above lemmas, we obtain that at least one of the numbers belongs to the set P , and (57) is proved. The next result is the proof of (59).
Proof. First we show (67). Select (a unique) m ∈ N such that q m κ(i 0 + 1) q m−1 . By (3) applied to T i0 (x), by the choice of i 0 it follows that Therefore, using the same arguments which lead (62) we obtain (cf. (63)) for n = 0, ..., κ(i 0 + 1) Then for i ∈ E, again by repeating that lead to (64) we obtain , by the definition of κ. Similarly (replacing 1 2Cqm by 1 3Cqm ), we obtain x − y −h ′(qm) (a i − T i0+1 y) < ǫ 2Hk . If i ∈ F , then using monotonicity of g ′ , the choice of i 0 and m, the fact that x, y ∈ Z and (56), we get −g . We proceed, repeating what lead to (65) (with q s−1 instead of q s−2 ) and using (56) and (53) .
Using this and (69) we get |f (n) (T i0+1 x) − f (n) (T i0+1 y)| < ǫ, which yields the first case of (67). To handle the second case, notice that We now proceed as before to obtain first |f and then estimating above by We conclude exactly in the same way as in the first case.
This finishes the proof of Proposition 4.11, thus of Theorem 3.

Absence of weak Ratner's property
In this section, we will prove Theorem 1. Let f be as in Theorem 1; for simplicity we assume that T f = 1. Let c > 1 be such that for every s ∈ N, q s+1 cq s . Recall that C > 1 is a constant from Definition 1.3 (such a constant exists, since k = 1 in our case); we may assume that C > c.
Fix any compact P ⊂ R \ {0}. We will prove that for any t 0 ∈ R, (T f t ) t∈R does not have R(t 0 , P ) property. For simplicity of the notations we will assume that t 0 = 1. Let Let ǫ, κ > 0 sufficiently small, smallness that will be determined in the course of the proof. We use Lemma 3.1 for T x = x + α, to ǫ, 3κ 2 to get a set A ⊂ T, λ(A) > 1 − ǫ and N 0 ∈ N, such that (16) holds for x ∈ A and n N 0 . Let N > max 2N 0 , 1 ǫ 2 κ 2 . We will hereafter assume that (T f t ) t∈R has the R(t 0 , P ) property (see Definition 2.1) and obtain a contradiction. Thus, assume there exist a set Z ⊂ X f with λ f (Z) > 1 − ǫ and 0 < δ < ǫ such that for every (x, s), (y, s ′ ) ∈ Z with d f ((x, s), (y, s ′ )) < δ, there exist M, L N with L M κ and p ∈ P such that Consider It follows that λ f (V ) > 1 − 4ǫ. The contradiction will come from the following two Propositions, the first one of which is a consequence of (72) and (73).  Before we prove these propositions we will see how they imply Theorem 1.

Proof of Proposition 5.1
Lemma 5.5. Let x, y ∈ T and let I be an integer interval such that for every n ∈ I, |f (n) (x) − f (n) (y)| < η (where η is a sufficiently small number). Then |I| < 2cη 1+γ x − y −1 Proof. We assume that x < y. Let s ∈ N be unique such that Denote I = [a, b] ∩ Z with a, b ∈ Z. Then, by the cocycle identity, the fact that a ∈ I, for n ∈ Z, we have Let k ∈ N be unique such that We will show that there exists n 0 ∈ [0, q k+1 ] such that This, by (77), gives |f (n0+a) (x) − f (n0+a) (y)| > η and therefore n 0 + a / ∈ I. It follows that 2cη 1+γ x − y −1 1−γ which completes the proof. Now, we show (79). By (78) and η sufficiently small, we have s k. Note that there exist n 1 ∈ [0, q k+1 ) such that T a x + n 1 α ∈ [0, 1 q k+1 ]. By (76) and the fact that k + 1 s + 1, we obtain T a y + n 1 α ∈ [0, 2 q k+1 ]. Therefore for some θ ∈ [T a x + n 1 α, T a y + n 1 α] ⊂ [0, 2 q k+1 ]. Thus, by the monotonicity of f ′ and (78) the last inequality by the fact that η is small enough. Therefore at least one of the numbers, | is bigger than 2η; we set n 0 either n 1 , or n 1 + 1 to obtain (79).
The following lemma translates (72) into a property on the Birkhoff sums above R α of the ceiling function f .
Proof. Assume WLOG that x < y. Let n ∈ [M, M + L] and r n be unique such that f (rn) (x) n + s < f (rn+1) (x). We will show that Indeed, first we show that r M > N 0 . Indeed, if not, using Lemma 3.
Analogously we prove that  [a 1 , b 1 ], ..., I l = [a l , b l ] such that U = I 1 ∪ ... ∪ I l and for every i = 1, ..., l there exists m i ∈ Z such that |x − y − m i α| < ǫ and for r ∈ I i , |f (r) (x) − f (r+mi) (y) − p| < 2ǫ. Moreover we assume that for every i = 1, ..., l, I i is maximal in the sense that |f (hi) (x)−f (hi+mi) (y)−p| 2ǫ for h i = a i − 1, b i + 1. We will show that there exists i = 1, ..., l such that This will obviously finish the proof of (74) with M ′ = a i , L ′ = |I i |, and m = m i ∈ Z. Let us show (84). If l 2 there is nothing to prove. Assume l 3.
Notice that U is the set of n ′ s such that (x, s), (y, s ′ ) ∈ V are p, n-close. The next lemma implies that between any two disjoint integer intervals I j , I j+1 ⊂ U , on which (x, s), (y, s ′ ) are p, n-close, there will be an integer interval J j much longer than I j , such that for any n ∈ J j , (x, s), (y, s ′ ) are not p, n-close. (3)). Lemmata 5.7 will give (84). Indeed, by the definition of J i and I i , it follows that for i, j = 2, ..., l − 1 with j = i − 1, i, i + 1 Hence, Therefore, by the fact that |U | > aL 0 , we have |I 1 ∪ I l | > 2aL0 3 and consequently, |I w | aL0 3 for at least one of w = 0 or w = 1.
Hence to obtain (84) we just need to prove Lemma 5.7.
Lemma 5.7. Let v ∈ N be unique such that (85) such that for every N ∋ |k| ε − 1 2 , Lemmata 5.9. For every w w 0 , w 0 sufficiently large, there exists a set with λ(W w 0 ) c 0 (c 0 (d) will be specified in the proof ), such that the following holds for x ∈ W w 0 and y := x + δ w 0 : there exists i 0 ∈ {0, ..., q w−l − 1} such that for some N ∋ l 1 depending on w, to be specified later, 0 < f (n) (x) − f (n) (y) < 100c d for n i 0 and f (n) (x) − f (n) (y) > |γ|d 2 for i 0 < n < wq w . (97) Before we proof the above Lemmas, let us first show, how they imply Proposition 5.3. Let w ∈ N be such that Denote W 0 := W w 0 and δ 0 := δ w 0 . By definition, Lemma 5.10. We have that (75) holds for k = 0.
Proof. Fix any M N 2 , any p ∈ P and any k = 0 such that x − y − kα < ǫ.
This finishes the proof of Theorem 1.
Proof. We will prove that λ(A c ) = 0. To do this we will prove that µ(A c ) = 0 (λ and µ are equivalent.). Note that for k ∈ N if x = [0; a 1 , ..., ] is the continued fraction of x, then T k (x) = We will prove that µ(B c ) = 0. To do this note that where B n := x ∈ T : {k ∈ [n 2 , (n + 1) 2 ] : T k x 1 dk 7 8 2 . Moreover,