# Multiple mixing for a class of conservative surface flows

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## Abstract

Arnold 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, which in turn yields 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.

## 1 Introduction

A major open problem in ergodic theory is Rokhlin’s question on whether mixing implies mixing of all orders, also called multiple mixing [26]. 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 [4], horocycle flows [23], mixing systems with singular spectrum that display multiple mixing by a celebrated theorem of Host [13], and finite rank systems since Kalikow showed that rank one and mixing implies multiple mixing [14], a result that was extended to finite rank mixing systems by Ryzhikov [30].

In the second half of the last century, it was discovered that mixing is possible for a class of smooth area preserving flows on surfaces called multi-valued Hamiltonians since they are locally given by Hamilton’s equations. The Hamiltonian \(H\) in question is associated to a closed differential \(1\)-form \(\omega =dH\), the form \(\omega \) being globally defined—but of course this is not necessarily the case for \(H\). The possibility of mixing for these flows was studied in two different cases: first, by Kochergin who obtained mixing when \(\omega \) has higher order zeros and thus the flow has degenerate saddles [17], and then by Arnold [1] who suggested that mixing is possible on a minimal component even in the case where \(\omega \) is Morse but the saddles on the minimal component appear in asymmetric configurations, for example because of a saddle loop. That mixing was indeed possible in the latter context was proved in a particular case by Khanin and Sinai [32]. Absence of mixing in the case of Morse forms was obtained by Kochergin in some particular cases [18] (see also [22]), and later generalized to a typical one form by Ulcigrai [35].

Considering a \(1\)-dimensional section of a multi-valued Hamiltonian flow allows to view the dynamics on its minimal components as special flows above interval exchange transformations (IET), which in particular situations can be circular rotations. The case of non-degenerate saddles corresponds to ceiling functions with logarithmic singularities, while the case of degenerate saddle points corresponds to ceiling functions with stronger singularities such as integrable power like singularities. In the case of power singularities [17] proved mixing above any ergodic IET, while [32] established mixing in the case of a single asymmetric logarithmic singularity above a circular rotation with a typical frequency.

The study of the mixing properties of surface flows has known a revival of interest since the beginning of the 2000s, with results such as the computation of the speed of mixing [6] or extensions of the Khanin-Sinai mixing result to include all irrational translation vectors [19, 20] (see also [21]), or advances in the study of multi-valued Hamiltonian flows on surfaces in the general case where the Poincaré section return map is an IET and not just a circular rotation [3, 35, 36, 37].

Mixing surface 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 subclass of these systems given by special flows above circular rotations, with ceiling functions having asymmetric logarithmic singularities or integrable power like singularities. For convenience, we will speak of Arnold flows in the first case, and Kochergin flows in the second. Our results will depend on the arithmetics of the frequency \(\alpha \) of the rotation on the base, that determines the slope of the unique rotation vector of the corresponding surface flow.

Loosely speaking our main result is as follows (it will be made precise at the end of this introduction, see Corollaries 1.6 and 1.9).

**Theorem**

Arnold flows are mixing of all orders for a set of \(\alpha \in (0,1)\) of full Lebesgue measure and Kochergin flows are mixing of all orders for a set of \(\alpha \in (0,1)\) of full Hausdorff dimension. This is true in the case of a single singularity and the same result holds if there are many singularities, under a non resonance condition (of full Hasudorff dimension) between the positions of the singularities and the base frequency \(\alpha \).

Similar mixing mechanisms due to orbit shear as in Kochergin and Arnold flows were observed relatively recently such as in [2, 3, 7, 35] 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 1980s, Ratner developed a rich machinery to study horocycle flows [27, 28, 29] 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 Thouvenot [33]. 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-property with other systems, and in particular with itself.

A *joining* between two dynamical systems \((T,X,\fancyscript{B},\mu )\) and \((S,Y,\fancyscript{C},\nu ), (X,\fancyscript{B},\mu )\) and \((Y,\fancyscript{C},\nu )\) being standard Borel probability spaces, is a measure \(\rho \) on \(X \times Y\) invariant by \(T \times S\) whose marginals on \(X\) and \(Y\) are \(\mu \) and \(\nu \). 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 \((T_t)_{t\in \mathbb {R}}\) is said to have *FEJ-property*, an acronym for *finite extension joining*, if for every ergodic flow \((S_t)_{t\in \mathbb {R}}\) acting on \((Y,\fancyscript{C},\nu )\) and every ergodic joining \(\rho \) of \((T_t)_{t\in \mathbb {R}}\) and \((S_t)_{t\in \mathbb {R}}\) different from the product measure \(\mu \times \nu , \rho \) yields a flow which is a finite extension of \((S_t)_{t\in \mathbb {R}}\).

It was shown in [31] that a mixing flow with FEJ-property is mixing of all orders. Moreover, it was proved in [27] 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 1990s (see p. 2 in [9]).

However, whether natural classes of special flows (not necessarily mixing) over irrational rotations may have the R-property remained open until Fra̧czek and Lemańczyk [9, 10] showed that a generalized R-property holds in some classes of special flows with roof functions of bounded variation (which, by [16], 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 Arnold and Kochergin flows, the shear may occur very abruptly as orbits approach the singularity and this may prevent them from having the weak Ratner property. Indeed, we believe that these flows do not have the WR-property (it was observed by the first author that Arnold and Kochergin flows do not have a natural *strong Ratner property* as described in the next paragraph). This 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 \(\alpha \in \mathbb {R}\) be irrational of bounded type and \(f(x)=x^\gamma +r, -1<\gamma <0, r>0\). Then the special flow \((T^t_{\alpha ,f})_{t\in \mathbb {R}}\) defined above the circle rotation \(R_\alpha \) and under the ceiling function \(f\) does not have the WR-property.

Here the circle is \(\mathbb {T}=\mathbb {R}/ \mathbb {Z}\). We recall that an irrational \(\alpha \) is said to be of bounded type or \(\alpha \in \mathrm{DC}(0)\), if the partial quotients in the continued fraction of \(\alpha \) are bounded, i.e. if \(\alpha =[a_0;a_1,a_2,\ldots ]\) and there exists \(K>0\) such that \(a_i <K\) for all \(i \geqslant 1\). We refer to Sect. 3 for the exact definition of special flows. Theorem 1 has another consequence. It is known that every horocycle flow \((h_t)_{t\in \mathbb {R}}\) is loosely Bernoulli [29]; therefore, for every irrational \(\alpha \), there exists a positive function in \(f\in L^1(\mathbb {T})\) such that \((h_t)_{t\in \mathbb {R}}\) is measurably isomorphic to the special flow \((T^t_{\alpha ,f})_{t\in \mathbb {R}}\) [25]. It follows from [16] and the fact that \((h_t)_{t\in \mathbb {R}}\) is mixing that \(f\) is of unbounded variation. Moreover, by [24], \(f\) 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 is the natural property that Ratner actually obtains for horocycle flows, and it is relatively easy to show that the Kochergin flows do not have it. What is more complicated in the proof of the absence of the general R-property for special flows, is to make sure that the slow divergence does not occur much later in the future after the nearby points have split and then came back together (see the proof of Theorem 1).

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. For these flows, 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 \(\alpha \) is of bounded type two nearby points will accumulate sufficient stretch while staying 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 \(\alpha \).

We now introduce some notations related to the ceiling functions that will be considered and their singularities, after which we will be able to state our exact results on the SWR-property and multiple mixing.

**Definition 1.2**

*singularities of type*\(h\) at \(\{a_1,\ldots ,a_k\}\) if

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.

Definition 1.2 will allow us to state our results with some flexibility on the singularities but the reader should keep in mind that the results will target functions that are essentially of the form \(A \ln (a-x)\) or \(A (a-x)^{\gamma }, \gamma \in (-1,0)\), when \(x\) is in a left neighborhood of a singularity \(a\) and similar form on the right side of a singularity. In the case of Arnold flows all the singularities will be assumed to be essentially of logarithmic type (see more general definition in Sect. 1.1) while in the case of Kochergin flows we will be dealing with functions having at least one power like singularity while the other singularities are supposed to be of equal or weaker type (see Sect. 1.2).

Furthermore, our results can deal with functions having several singularities but require a non resonance condition between these singularities and the rotation frequency \(\alpha \) in the base of the special flow, that we now state.

**Definition 1.3**

*[Badly approximable singularities]*Given \(\alpha \in \mathbb {R}\setminus \mathbb {Q}\), we will say that \(\{a_1,\ldots ,a_k\}\) are

*badly approximable by*\(\alpha \) if there exists \(C>1\) such that for every \(x\in \mathbb {T}\) and every \(s\in \mathbb {N}\), there exists at most one \(i_0\in \{0,\ldots ,q_s-1\}\) such that

*Remark 1.4*

It was shown in [11], Lemma3] that if \(a_i-a_j\in (\mathbb {Q}+\mathbb {Q}\alpha )\setminus (\mathbb {Z}+\mathbb {Z}\alpha )\) whenever \(i\ne j\) then \(\{a_1,\ldots ,a_k\}\) are badly approximable by \(\alpha \).

Note that if there is only one singularity, that is \(k=1\), then by (2) it is always badly approximable by \(\alpha \). The following shows that for \(k \geqslant 2\) the set of singularities that are badly approximable by \(\alpha \) is a thick set in \([0,1]^k\).

**Lemma 1.5**

[34] Let \(\alpha \in \mathbb {R}\setminus \mathbb {Q}\). For any \(k \in \mathbb {N}\), the set \(E \subset [0,1]^k\) of \(k-tuples (a_1,\ldots ,a_k)\) that are badly approximable by \(\alpha \) is a product of sets of full Hausdorff dimension in \([0,1]\).

*Proof*

But it was proven in [34] (see also [5]) that the set \(B(\alpha )\) is a winning set in the sense of Schmidt (see [5, 34] and references therein). A winning set is of full Hausdorff dimension. Moreover, for a winning set \(B \in \mathbb {R}\) we have that for any \(x_1,\ldots ,x_n\) the set \(\cap _{s=1}^n (x_s+B)\) is winning. So, if \(a_1, \ldots ,a_l\) are such that \(a_i-a_j \in B(\alpha )\) for any \(i,j \in \{1,\ldots ,l\}, i\ne j\), then the set of \(a \in [0,1]\) such that \(a \in \cap _{s=1}^l (a_s + B(\alpha ))\) is winning which means that \(a_1, \ldots ,a_l,a_{l+1}\) are badly approximable by \(\alpha \) 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 \(\alpha \). \(\square \)

### 1.1 Logarithmic like singularities

In the case of logarithmic like singularities, the following theorem holds.

**Theorem 2**

*of type*\(h\) and badly approximable by \(\alpha \), with some associated constant \(C>1\). Assume that \(\sum _{i=1}^k A_i\ne \sum _{i=1}^k B_i\) and that there exists a constant \(m_0>0\) and a sequence \((x_s)\) such that for every \(s\in \mathbb {N}\), we have \(x_s < \frac{1}{q_s}\) and

- 1.
\(\lim _{s\rightarrow +\infty } \frac{h'(\frac{x_s}{4C})}{q_sh(\frac{1}{2q_s})}=0, \quad \lim _{s\rightarrow +\infty }x_sq_sh(\frac{1}{2q_s})=+\infty \);

- 2.
\(\sum _{i\notin K_\alpha }q_ix_i<+\infty \), where \(K_\alpha :=\{s\in \mathbb {N}\;:\; q_{s+1}< \frac{1}{x_s}\}\);

- 3.
\(h(\frac{1}{2q_s})/h(\frac{1}{2q_{s+1}})>m_0\).

What is \(x_s\) in Theorem 2? When we will compute the shear in the Birkhoff sums of the ceiling function \(f\), we will naturally be able to control only those points that do not go too close to the singularities under the iteration by \(R_\alpha \) on the base. Then, a controllable point will be a point that stays at distance \(x_s\) from the singularities during \(O(q_{s+1})\) iterates in the future *or* in the past. We choose \(x_s\) so that the contribution to the shear of a single visit to a singularity is negligible with respect to the accumulated shear (this, either in the future or in the past). To fix ideas, suppose the singularities are reduced to just one one-sided singularity at the origin and observe that if \(f\) at the origin is exactly \(\log \) then the choice \(x_s= {1}/{q_s \log ^{\frac{7}{8}}q_s}\) would satisfy 1. Then if \(q_{s+1}< q_s \log ^{\frac{7}{8}} q_s\) we see that for any \(x \in \mathbb {T}\) either up to \(q_{s+1}/2\) in the future or up to \(-q_{s+1}/2\) in the past the orbit of \(x\) by \(R_\alpha \) does not enter the \(x_s\) neighborhood of the origin, and this will show that the progressive accumulation in the Birkohff sums of the derivative of the ceiling function above the orbit of \(x\) in the region of time between \(O(q_s)\) and \(O(q_{s+1})\) in the future or \(O(-q_{s+1})\) and \(O(-q_s)\) in the past dominates the value of the derivative at the closest entry to the neighborhood of the origin (this is the aim of condition 1). The latter is the crucial fact that we need to show SWR. In the opposite case where for example \(q_{s+1} \geqslant q_s \log q_s\) we have to discard the points that enter the \(x_s\) neighborhood of the origin between time \(-q_s\) and \(q_s\) and then show that the remaining points stay away from the \(x_s\) neighborhood of the origin for \(O(q_{s+1})\) iterations (either in the past or in the future) and conclude as before. This is where 2. is necessary to show that the measure of the discarded points is arbitrarily small.

We refer to the beginning of Sect. 4.1 for an outline of the proof of Theorem 2 in which the different roles of conditions 1., 2. and 3. are all explained in detail.

We now restate Theorem 2 in the particular case of exactly logarithmic singularities, that is when \(h(x)=-\log (x),\) for \(x\in [0,1)\). To be able to choose \(x_s\) such that \(1., 2.\) and \(3.\) are satisfied we need some arithmetic restrictions on \(\alpha \), that we now introduce.

Recall first that a number \(\alpha \in \mathbb {R}\setminus \mathbb {Q}\) is said to be Diophantine if there exists \(\tau \geqslant 0\) such that for any \(p,q\in \mathbb {Z}\times \mathbb {N}^*\) we have that \(|\alpha -\frac{p}{q}|\geqslant \frac{C(\alpha )}{q^{2+\tau }}\) for some \(C(\alpha )>0\). We call \(\mathcal {D}\) the set of Diophantine numbers.

To have 3., it suffices to assume that \(\alpha \) is Diophantine since an equivalent definition of \(\alpha \in \mathbb {R}\setminus \mathbb {Q}\) being Diophantine is that its sequence of denominators \(q_n\) satisfies \(\forall _{n\in \mathbb {N}}, \ q_{n+1}<r_\alpha q^{1+\tau }_{n}\) (see (2)).

Hence we have the following

**Corollary 1.6**

Consider \(h(x)=-\log (x)\). Let \(\alpha \in \mathcal {D}\cap \mathcal {E}\). Let \(f\in C^2(\mathbb {T}\setminus \{a_1,\ldots ,a_k\})\) with the singularities \(\{a_1,\ldots ,a_k\}\)*of type *\(h\) and badly approximable by \(\alpha \). Assume that \(\sum _{i=1}^k A_i\ne \sum _{i=1}^k B_i\). Then \((T^t_{\alpha ,f})_{t\in \mathbb {R}}\) has the SWR-Property.

*Proof*

We take for \(x_s\) the sequence \(\frac{1}{q_s \log ^{\frac{7}{8}}q_s}\) and easily check the hypothesis of Theorem 2. Therefore \(( T^t_{\alpha ,f})_{t\in \mathbb {R}}\) has the SWR-property. \(\square \)

Corollary 1.6 covers a set of full Lebesgue measure of rotation angles \(\alpha \). Indeed, it is known that the set of Diophantine numbers \(\mathcal D\) has full Lebesgue measure, and we will prove in Appendix B the following result. Denote by \(\lambda \) the Haar measure on \(\mathbb {T}\).

**Proposition 1.7**

It holds that \(\lambda ({\mathcal {E}})=1\).

We now recall the following results on mixing of special flows with ceiling functions having logarithmic singularities.

**Theorem 3**

- (a)
([18]) If \(\sum _j A_j=\sum _j B_j\) then \((T_{\alpha , f}^t)_{t\in \mathbb {R}}\) is not mixing for any \(\alpha \in \mathbb {R}-\mathbb {Q}.\)

- (b)
([19, 20, 32]) If \(\sum _j A_j \ne \sum _j B_j\) then \((T_{\alpha , f}^t)_{t\in \mathbb {R}}\) is mixing for almost every \(\alpha \in \mathbb {R}-\mathbb {Q}.\)

- (c)
([19, 20]) If \(A_j-B_j \ne 0\) have the same sign for all \(j\) then \((T_{\alpha , f}^t)_{t\in \mathbb {R}}\) is mixing for each \(\alpha \in \mathbb {R}-\mathbb {Q}.\)

In Theorem 5 below, we show that the SWR property for \((T_{\alpha , f}^t)_{t\in \mathbb {R}}\) implies the FEJ-property. As an immediate consequence of this and of Theorem 3, Corollary 1.6 and the fact that mixing flows with the FEJ-property are multiple mixing [31] we get the following.

**Corollary 1.8**

Consider \(h(x)=-\log (x)\). For \(\alpha \in (0,1)\), assume \(f\in C^2(\mathbb {T}\setminus \{a_1,\ldots ,a_k\})\) has singularities \(\{a_1,\ldots ,a_k\}\)*of type *\(h\) and badly approximable by \(\alpha \). If \(A_j-B_j \ne 0\) have the same sign for all \(j\), then \((T_{\alpha , f}^t)_{t\in \mathbb {R}}\) is multiple mixing for each \(\alpha \in \mathcal {E}\cap \mathcal {D}\). If \(\sum _j A_j \ne \sum _j B_j\) then \((T_{\alpha , f}^t)_{t\in \mathbb {R}}\) is multiple mixing for almost every \(\alpha \in \mathcal {E}\cap \mathcal {D}\).

### 1.2 Power like singularities

Now, we will deal with power like singularities. We suppose \(f\in C^2(\mathbb {T}\setminus \{a_1,\ldots ,a_k\})\) with singularities \(\{a_1,\ldots ,a_k\}\) of type \(h\). We assume that there exists \(i\in \{1,\ldots ,k\}\) such that \(A_i^2+B_i^2>0\) in (1).

**Theorem 4**

The most interesting case – when \(h\) has power singularities – is discussed in the corollary below.

**Corollary 1.9**

Let \(\alpha \in DC(0)\). Let \(f\in C^2(\mathbb {T}\setminus \{a_1,\ldots ,a_k\})\) with all the singularities \(\{a_1,\ldots ,a_k\}\) of power-like type \(x^{\gamma _i}\) from the left and \(x^{\delta _i}\) from the right, \(-1<\gamma _i,\delta _i<0\). Then, if the points \(\{a_1,\ldots ,a_k\}\) are badly approximable by \(\alpha \), we have that \(( T^t_{\alpha ,f})_{t\in \mathbb {R}}\) has the SWR-Property and is mixing of all orders.

*Proof of Corollary 1.9*

It is easy to check that (4) in Theorem 4 is satisfied. This gives the SWR-Property . Mixing of \(( T^t_{\alpha ,f})_{t\in \mathbb {R}}\) was established in [17]. Multiple mixing then follows from Theorem 5 and the FEJ-property. \(\square \)

*Remark 1.10*

A stronger version of Theorem 4 actually holds : if \(\alpha \in DC(0)\) and \(f\in C^2(\mathbb {T}\setminus \{a_1,\ldots ,a_k\})\) with all the singularities \(\{a_1,\ldots ,a_k\}\) of at most power-like type \((x^{\gamma _i}\) from the left and \(x^{\delta _i}\) from the right, \(-1<\gamma _i,\delta _i<0)\) and if \(\gamma =\min _{1\leqslant i\leqslant k}\{\gamma _i,\delta _i\}\), then it is sufficient to have the singularities of maximal type badly approximable with \(\alpha \) to guarantee the SWR property, namely if the points in \(E=\{a_i\;:\; \min \{\gamma _i,\delta _i\}=\gamma \}\subset \{a_1,\ldots ,a_k\}\) are badly approximable by \(\alpha \), then \((T^t_{\alpha ,f})_{t\in \mathbb {R}}\) has the SWR-Property and is mixing of all orders. The proof in this case is similar to the proof of Theorem 4 and we omit it to avoid overloading the paper with unnecessary technicalities.

### 1.3 Plan of the paper

In Sect. 2 we introduce the SWR-Property and describe its joinings consequences. In Sect. 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 [9, 10]. In Sect. 4 we study the Birkhoff sums of logarithmic like and power like functions and prove Theorems 2 and 4. Appendix A is devoted to the proof of Theorem 1 on the absence of the SWR-Property for a subclass of Kochergin flows. Finally Appendix B is devoted to the proof that the set of frequencies for which Theorem 2 holds has full Lebesgue measure.

## 2 The SWR-property

Let \((X,\fancyscript{B},\mu )\) be a probability standard Borel space. We additionally assume that \(X\) is a complete metric space with a metric \(d\). Let \((T_t)_{t\in \mathbb {R}}\) be an ergodic flow acting on \((X,\fancyscript{B},\mu )\). The R-property used by Ratner in the context of horocycle flows is a property of slow divergence between the orbits of nearby points that essentially states as follows: for any \(\varepsilon >0\) there exists \(\kappa >0\) such that if \(y\) and \(y'\) are sufficiently close to each other and if they are not on the same orbit then there exists \(M(y,y')\) such that \(d(T^ty,T^{t+\iota }y')<\varepsilon \) (with \(\iota =\pm 1\)) for \(t\in [M,(1+\kappa )M]\).

It is not difficult to see that the R-property implies the FEJ property. Indeed, if a joining \(\lambda \) is not a finite extension joining then one has that there exists points \(x,y,y'\) such that \((x,y)\) and \((x,y')\) are typical for \(\lambda \) while \(y\) and \(y'\) are arbitrarily close and not in the same orbit, and by the R-property and the Birkhoff ergodic theorem one obtains an extra invariance of \(\lambda \), namely by \(\mathrm{Id} \times T^\iota \), that implies that \(\lambda \) is the product measure.

Actually, it is useful to relax the R-property by asking that the controlled divergence happens for \(x\) and \(y\) outside an exceptional set of points of measure less than \(\varepsilon \) (and for most of the times in \([M,(1+\kappa )M]\)). The proof of the FEJ property remains the same in nature but becomes a bit more technical involving some standard measure theoretical arguments (see for example [33]).

In [10], Definition 4, a slightly weaker version of the \(R\)-property is given, that is called WR or Weak Ratner property, that allows the drift to vary in some fixed compact set away from zero and infinity. There again, the FEJ consequence as well as its proof follow in practically the same way as for the R-property, with some extra standard measure theoretical arguments, and under a “continuity” assumption on orbits (see below). However, as shown in [10] the WR property is more versatile than the R-property and is adapted to nonlinear situations, where the R-property is unlikely to hold, such as in our context of reparametrizations of linear flows with singularities.

Observe now that if in the proof of the FEJ property, we used that \(d(T^{t}y,T^{t+\iota }y')<\varepsilon \) during a large interval of negative times \(t\) instead of positive times, then exactly the same conclusion of extra invariance of \(\lambda \) by \(\mathrm{Id} \times T^\iota \) would still hold. The crucial observation here is that it suffices to check one of the two slow divergences, in the future or in the past, and this possibly depending on the pair of points that is considered. This motivates the introduction of what we call the Switchable Ratner, or SWR, property that we now formally define.

**Definition 2.1**

*The switchable Ratner property*) Fix \(t_0\in \mathbb {R}_+\) and a compact set \(P\subset \mathbb {R}\setminus \{0\}\). One says that the flow has the

*switchable*\(R(t_0,P)\)-property if for every \(\epsilon >0\) and \(N\in \mathbb {N}\) there exist \(\kappa =\kappa (\epsilon ), \delta =\delta (\epsilon ,N)\) and a set \(Z=Z(\epsilon ,N)\in \fancyscript{B}\) with \(\mu (Z)>1-\epsilon \) such that for any \(x,y\in Z\) with \(d(x,y)<\delta , x\) not in the orbit of \(y\) there exist \(M=M(x,y),L=L(x,y)\in \mathbb {N}\) with \(M,L>N\) and \(\frac{L}{M}\geqslant \kappa \) and \(p=p(x,y)\in P\) such that

*SWR*-property.

For the sake of completeness, we can formally compare the SWR-property with the definition of the WR-property [10]. To have WR-property, we fix \(P\subset \mathbb {R}\setminus \{0\}\) and \(t_0\in \mathbb {R}. (T_t^f)_{t\in \mathbb {R}}\) has \(R(t_0,P)\) property if in Definition 2.1, (5) holds (the condition (6) is not taken into account) and \((T_t^f)_{t\in \mathbb {R}}\) has WR-property if the set of \(t_0\in \mathbb {R}\) such that \((T_t^f)_{t\in \mathbb {R}}\) 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).

As we just mentioned the proof of the FEJ implication from the SWR property is a direct adaptation of the proof of the same implication in the case of the R property or the WR property. For completeness, we will present a detailed proof of this fact that is stated in Theorem 5 below and that occupies the rest of the section.

As a standing assumption in all the sequel, we will add one more natural condition on the flow \((T_t)_{t\in \mathbb {R}}\) which can be viewed as “continuity” on orbits. The flow \((T_t)_{t\in \mathbb {R}}\) is called *almost continuous* [10] if for every \(\epsilon >0\) there exists a set \(X(\epsilon )\) with \(\mu (X(\epsilon ))>1-\epsilon \) such that for every \(\epsilon '>0\) there exists \(\delta '>0\) such that for every \(x\in X(\epsilon )\), we have \(d(T_t(x),T_{t'}(x))<\epsilon '\) for \(t,t'\in [-\delta ,\delta ]\).

**Theorem 5**

Let \((T_t)_{t\in \mathbb {R}}\) be a weakly mixing almost continuous flow acting on a probability standard Borel space \((X,\fancyscript{B},\mu )\). Assume that \((T_t)_{t\in \mathbb {R}}\) satisfies the SWR-property. Let \((S_t)_{t\in \mathbb {R}}\) be an ergodic flow acting on a probability standard Borel space \((Y,\fancyscript{C},\nu )\) and let \(\rho \in J((T_t)_{t\in \mathbb {R}},(S_t)_{t\in \mathbb {R}})\) be an ergodic joining. Then either \(\rho \) is equal to \(\mu \otimes \nu \) or is a finite extension of the measure \(\nu \).

For the definition and properties of joinings, we refer the reader to [33] or [12]. In the proof of Theorem 5, we will need some lemmas from [10]. But first we state a useful fact that is a simple consequence of the Birkhoff ergodic theorem.

**Lemma 2.2**

The following is a consequence of Lemma 5.4. in [10], that is itself based on the Birkhoff ergodic theorem.

**Lemma 2.3**

*Proof*

The proof is a simple consequence of the following lemma:

**Lemma 2.4**

One uses the above lemma first for the flows \((T_t)_{t\in \mathbb {R}}\) and \((S_t)_{t\in \mathbb {R}}\) and ergodic joining \(\rho \in J((T_t)_{t\in \mathbb {R}},(S_t)_{t\in \mathbb {R}})\) to get, for \(\epsilon ,\frac{\delta }{2}, \kappa >0\), a natural number \(N_+\in \mathbb {N}\) and a set \(Z_+\subset \fancyscript{B}\otimes \fancyscript{C}\) with \(\rho (Z_+)>1-\frac{\delta }{2}\). Then, for flows \((T_{-t})_{t\in \mathbb {R}}\) and \((S_{-t})_{t\in \mathbb {R}}\) and the same ergodic joining \(\rho \) to get, for \(\epsilon ,\frac{\delta }{2},\kappa >0\), a natural number \(N_-\in \mathbb {N}\) and a set \(Z_-\subset \fancyscript{B}\otimes \fancyscript{C}\) with \(\rho (Z_-)>1-\frac{\delta }{2}\). To finish the proof one takes \(N:=\max (N_+,N_-)\) and \(Z=Z_+\cap Z_-\). \(\square \)

The next lemma is used in the proof of Theorem 3 in [27].

**Lemma 2.5**

Let \((T_t)_{t\in \mathbb {R}}\) and \((S_t)_{t\in \mathbb {R}}\) be two ergodic flows. Let \(\rho \in J^e((T_t)_{t\in \mathbb {R}},(S_t)_{t\in \mathbb {R}})\) be an ergodic joining. Then if there exists a set \(V\) with \(\rho (V)>0\) such that for any points \((x,y),(x',y)\in V\) either \(x\) is in the orbit of \(x'\) or \(d(x,x')>c_0\) for some constant \(c_0>0\), then \(\rho \) is a finite extension of \(\nu \).

In what follows, we consider only \((X,d)\) be a \(\sigma \)-compact metric space. Let \(A\in \fancyscript{B}\). For \(\eta >0\) we denote by \(V_\eta (A):=\{x\in X:\; d(x,A)<\eta \}\).

**Lemma 2.6**

(cf. [10]) For any \(A\in \fancyscript{B}\) there exists \(R\subset (0,+\infty )\) such that \((0,+\infty )\setminus R\) is countable and \(\mu (\partial V_{\eta }(A))=0\) for \(\eta \in R\). It particular, there exists a dense family \((B_i)_{i\geqslant 1}\) in \(\fancyscript{B}\) with the property \(\mu (\partial B_i)=0\) for every \(i\in \mathbb {N}\).

*Remark 2.7*

Since \((X,d)\) is a Polish space, by Lemma 2.6 and regularity of \(\mu \), there exists a dense family \(\{B_i\}_{i\geqslant 1}\in \fancyscript{B}\), such that \(\mu (\partial B_i)=0\) for \(i\geqslant 1\).

*Proof of Theorem 5*

## 3 SWR-property for special flows

*s*pecial flow \((T_t^f)_{t\in \mathbb {R}}\) defined above \(T\) and under the ceiling function \(f\) is given by

The following general lemma is a direct consequence of Birkhoff ergodic theorem.

**Lemma 3.1**

*Remark 3.2*

We now state the main result of this section. It is similar to Lemma 6 of [9].

**Proposition 3.3**

*Proof*

Let \(\delta ':=\delta (\epsilon ', 2\gamma \frac{\max (N_0,N)}{D_1})\). Take two points \((x,s),(x',s')\in Z'\), such that \(x\ne x'\) and \(d^f((x,s),(x',s'))<\delta '\). It follows by definition of \(d^f\) that \(d(x,x')<\delta '\) and therefore by our assumptions there exist \(M,L\geqslant 2\gamma \frac{\max (N_0,N)}{D_1}\) with \(\frac{L}{M}\geqslant \kappa , p\in P\) and such that for all \(n\in [M,M+L]\) either \(|f^{(n)}(x)-f^{(n)}(y)-p|<\epsilon '\) or for all \(n\in [M,M+L], |f^{(-n)}(x)-f^{(-n)}(y)-p|<\epsilon '\). We will consider the second case (the proof in the first case goes along the same lines).

## 4 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 4. In all the sequel we assume \(\{a_1,\ldots ,a_k\}\) are badly approximable by \(\alpha \) with a constant \(C>1\) (see Definition 1.3). We start with an easy combinatorial fact about the visits of an orbit by the rotation \(R_\alpha \) to the neighborhood of the singularities.

**Lemma 4.1**

*Proof*

We will often use the Denjoy-Koksma inequality to control the growth of the Birkhoff sums. For a reference, see for example [4].

**Proposition 4.2**

The following lemma is a simple consequence of the Denjoy-Koksma inequality. It will be very useful in separating the contribution to the shear of the visits to the neighborhood to the singularities from the rest of the orbit.

**Lemma 4.3**

*Proof*

**Lemma 4.4**

*Proof*

By assumptions, there exists a constant \(z_0>0\) such that for every \(i=1,\ldots ,k\) and for every \(x\in [-z_0+a_i,a_i) |f'(x)|<-Kr_i'(a_i-x)\) and for every \(x\in (a_i,a_i+z_0]; |f'(x)|<-Kr_i'(x-a_i)\) for some constant \(K\geqslant 0\).

Moreover, since \(f'\in C^1(\mathbb {T}\setminus \{a_1,\ldots ,a_k\})\); it follows that there exists a constant \(R>0\) such that for every \(x\in \mathbb {T}\setminus \bigcup _{i=1}^k[-z_0+a_i,a_i+z_0], |f'(x)|<R\). Denote by \(C_0:=\min _{i=1,\ldots ,k}|\sup _{\mathbb {T}}r_i'|\) Now, the constant \(H:=2\max _{i=1,\ldots ,k}\{K,\frac{R}{C_0}\}\) satisfies the assertion of the lemma. \(\square \)

### 4.1 Proof of Theorem 2

#### 4.1.1 Outline of the proof.

We first give an outline of the proof in which we suppose for simplicity that the ceiling function has just one singularity that is exactly logarithmic. The SWR-property (in the same way as the original Ratner’s property) consists of two parts. First, for two “nearby” points, we need to show that their orbits drift apart by a controlled amount at some time \(M (M\) may be positive or negative in the case of the SWR property). Second, we need to make sure that their orbits keep essentially the same drift during an interval of time comparable to \(M\). For special flows over rotations (or IET’s) we gave in Proposition 3.3 a characterization of the SWR property based on the divergence for nearby points between the corresponding Birkhoff sums of the ceiling function \(f\). The latter is controlled by the Birkhoff sums of the derivative of the ceiling function that were the subject of investigation of several other works (see e.g. [8, 17, 19, 20, 32, 35]).

- a.
There exists \(c>0\) such that for any point \(x \in \mathbb {T}\) either \(R^i_\alpha x\) is disjoint from \([-c/q_s,c/q_s]\) for every \(i=0,\ldots ,l< \frac{q_s}{2}\) or \(R^{-i}_\alpha x\) is disjoint from \([-c/q_s,c/q_s]\) for every \(i=1,\ldots ,l< \frac{q_s}{2}\).

- b.
By the Denjoy-Koksma inequality, if the forward (backward) orbit of length \(q_s\) a point \(x\) by \(R_\alpha \) is disjoint with the \(1/(q_s \ln q_s)\) neighborhood of the origin we have that \(|f'^{(q_s)}(x)|\) (or \(|f'^{(-q_s)}(x)|\)) is of order \(q_s \ln q_s\) (this is easy to understand if we rearrange the orbit of \(x\) almost like \(c/q_s, (c+1)/q_s,(c+2)/q_s...\) with \(c \in (0,1)\) far from \(0\) and from \(1\)). Hence for any point \(x\) either \(|f'^{(q_s)}(x)|\) or \(|f'^{(-q_s)}(x)|\) is of order \(q_s \ln q_s\). This is the content of Lemma 4.8. In case \(q_{s+1} \leqslant C q_s\), this would give the macroscopic yet controlled drift between the points \(x,y\) such that \(1/(q_{s+1} \ln q_{s+1}) \leqslant d(x,y) \leqslant 1/(q_{s} \ln q_{s})\) either in the future at time \(q_s\) or in the past at time \(-q_s\). So, if \(\alpha \) is of constant type, we would be done with the proof of Proposition 4.6 Part a.

- c.
In the case \(q_{s+1} \gg q_s\) but \(s \in K_\alpha \), we may have to consider the Birkhoff sums beyond \(q_s\) to see the drift between the orbits of \(x\) and \(y\) such that \(1/(q_{s+1} \ln q_{s+1}) \leqslant d(x,y) \ll 1/(q_{s} \ln q_{s})\). Since \(s \in K_\alpha \), we have that \(1/(q_s \ln q_s) \ll c/q_{s+1}\) hence a. applied to \(s+1\) implies that either up to \(q_{s+1}/2\) in the future or up to \(-q_{s+1}/2\) in the past the orbit of \(x\) by \(R_\alpha \) does not enter the \(1/(q_s \ln q_s)\) neighborhood of the origin (see Lemma 4.7, case \(m \in K_\alpha \)). Using this, the estimate of b. and the Denjoy-Koksma inequality we get by the cocycle identity that \(f'^{(\iota kq_s)}(x)\) behaves like \(kq_s \ln q_s\) for \(s\) sufficiently large, \(k \in [1,O(q_{s+1}/q_s)]\) and \(\iota =1\) or \(-1\) (see Lemma 4.9). As a consequence there exists a time of the form \(n_0q_s, n_0 \in [1,O(q_{s+1}/q_s)]\) such that \(f^{(\iota n_0q_s)}(x)-f^{(\iota n_0q_s)}(y)\) is in some compact set \(P\) away from \(0\), which finishes the proof of Proposition 4.6 Part a in the case \(K_\alpha \) contains all sufficiently large integers.

- d.
In the case \(s \notin K_\alpha \), we define \(x_s := 1/(q_s (\ln q_s)^{7/8})\) and use our arithmetic condition \(\alpha \in \mathcal {E}\) to define a set \(Z\) of almost full measure (see definition of \(Z\) in Sect. 4.1.2) such that \(R^i_\alpha x\) does not enter the \(x_s\) neighborhood of the origin (see Definition (25)) for every \(i\in \{-q_s,\ldots , 0,\ldots , q_s-1\}\) for every \(s \notin K_\alpha \) sufficiently large. This actually implies that either up to \(\frac{q_{s+1}}{4C}\) in the future or up to \(-\frac{q_{s+1}}{4C}\) (\(C\) is a constant coming from Definition 1.3; in case there is only one singularity \(C=1\)) in the past the orbit of \(x\) by \(R_\alpha \) does not enter the \(x_s\) neighborhood of the origin (see Lemma 4.7, case \(m \notin K_\alpha \)). From there the proof of Proposition 4.6 Part a is similar to case 3. above except that the condition 3 of Theorem 2, namely \(h(\frac{1}{2q_s})/h(\frac{1}{2q_{s+1}})>m_0\) is used to show that a stretch of order \( n_0q_s \ln q_s\) where \(n_0\) can be taken to be as large as \(O(q_{s+1}/q_s)\) is sufficient to produce a drift between the point \(x,y\) such that \(d(x,y)\) is comparable to \(1/(q_{s+1} \ln q_{s+1})\). This is where the Diophantine condition 3. of Theorem 2 is crucial.

Note however that in all the cases above, it is crucial to use the possibility to control the drift in the future or in the past depending on the pair of points.

The second part (keeping the drift) is proved in Proposition 4.6 Part b. We need to consider the points \(R_\alpha ^{n_0q_s}x\) and \(R_\alpha ^{n_0q_s}y\) and apply similar arguments as in Part a to bound the drift during time \(\kappa n_0q_s\). The main ingredient is Lemma 4.10 which is another lemma that allows us to bound the drift between the Birkhoff sums in the future (or in the past) up to a time comparable to \(q_{m+1}\) for points that stay away from the singularities in the future (or in the past) during this time. We then have to “situate” \(\kappa n_0 q_s\) relatively to the denominators of \(\alpha \) and check that the conditions of Lemma 4.10 are satisfied by \(R_\alpha ^{n_0q_s}x\) and \(R_\alpha ^{n_0q_s}y\). Of course if \(s\) is such that \(q_{s+1}\gg q_s\) (for example if \(s \notin K_\alpha \)) and if \(1/(q_{s+1} \ln q_{s+1}) \ll d(x,y) \ll 1/(q_s \ln q_s)\), then the same argument of Part a would allow to keep the drift under control for additional \(\kappa n_0q_s\) time. But in the other cases where we have in particular to interpolate between the constant type and non-constant type behavior, our proof gets a bit technical and treats different cases separately.

#### 4.1.2 Notations and standing assumptions

Observe that \(\lambda (Z)\geqslant 1- \epsilon (\lambda (W_s)\geqslant 1-16kv_sq_s )\).

Set \(\delta :=\frac{1}{q_{s_0}h(\frac{1}{2q_{s_0}})}\). Consider \(x,y\in Z\) with \(0<\Vert x-y\Vert <\delta \). We will assume WLOG that \(x<y\) (we consider the trigonometric order on \(\mathbb {T}\)).

#### 4.1.3 Controlling the drift

The following proposition implies Theorem 2 due to Proposition 3.3.

**Proposition 4.5**

Consider \(x,y\in Z\) with \(0<\Vert x-y\Vert <\delta \). Then there exists \(p \in P, M,L\geqslant \kappa M \geqslant N\) such that either (16) holds for \(n \in [M,M+L]\) or (17) holds for \(n \in [M,M+L]\).

**Proposition 4.6**

Consider \(x,y\in Z\) as in (26).

**Part a**There exists \(n_0\in \{1,\ldots ,\max (\frac{q_{s+1}}{8Cq_s},1)\}\) satisfying

**Part b**Let \(X=T^{n_0q_s}x\) and \(Y=T^{n_0q_s}y\) if (27) holds, and \(X=T^{-(n_0q_s+1)}x\) and \(Y=T^{-(n_0q_s+1)}y\) if (28) holds. For \(n=1,\ldots ,[\kappa n_0q_s]+1\) we have

The rest of this section is devoted to the proof of Proposition 4.6. But before this we show how it implies Proposition 4.5 and thus Theorem 2.

*Proof of Proposition 4.5*

Suppose (27) holds, the other case being similar. If A. from (29) holds, set \(M:=n_0q_s, L:=[\kappa M]+1\) and \(p:=f^{(n_0q_s)}(x)-f^{(n_0q_s)}(y) \in P\). If B. holds, we set \(M:=[\frac{n_0q_s}{1+\kappa }], L:=[\kappa M]+1\) and \(p:=f^{(n_0q_s)}(x)-f^{(n_0q_s)}(y)\in P\). Notice that in both cases \(M,L\geqslant \frac{1}{2} \kappa n_0q_s\geqslant \frac{1}{2}\kappa q_s\geqslant \frac{1}{2} \kappa q_{s_0}\geqslant N\). Finally, using \(A.\) or \(B.\) and the cocycle identity for the Birkhoff sums and the triangular inequality shows that for \(n\in [M,M+L], |f^{(n)}(x)-f^{(n)}(y)-p|< \varepsilon \) for some \(p\in P\). \(\square \)

#### 4.1.4 Proof of Proposition 4.6 Part a

**Lemma 4.7**

*Proof*

Observe first that since by (26), \(\Vert x-y\Vert \leqslant v_s\leqslant v_m\), then it suffices to prove (31) or (32) with just \(x\) instead of \([x,y]\) on the LHS and \( 2v_m\) instead of \(v_m\) on the RHS.

Assume \(m\notin K_\alpha \). Since \(m\geqslant s_0\) we may assume that \(q_{m+1}\geqslant 16Cq_m\).

In the following lemma we control the drift between the Birkhoff sums up to \(q_s\) or \(-q_s\) between nearby points that do not go too close to the singularities. Recall that we have assumed that \(d=\sum _{i=1}^k (A_i-B_i)>0\).

**Lemma 4.8**

*Proof*

We now concatenate the inequalities of Lemma 4.8 if (31) or (32) are satisfied.

**Lemma 4.9**

*Proof*

We are ready now to finish the proof of Part a. of Proposition 4.6. If \(s\notin K_\alpha \), then by the fact that \(x,y\in Z\subset W_s\), it follows that 1. in Lemma 4.7 is satisfied with \(m=s\). If \(s\in K_\alpha \) then 2. in Lemma 4.7 is satisfied with \(m=s\). Therefore we can use Lemma 4.7 for \(x,y\) and \(m=s\). Now, by Lemma 4.9, if (31) holds we have (27), if (32) holds we have (28). Part a. of Proposition 4.6 is settled, we turn now to Part b.

#### 4.1.5 Proof of Proposition 4.6 Part b

**Lemma 4.10**

*Proof*

This finishes the proof. \(\square \)

To prove Proposition 4.6 Part b., observe first that if \(s_0\) is sufficiently large, and up to eventually changing \(\kappa \) to \(\kappa ' = \frac{\kappa }{8C}\), one of two possibilities holds : \(\mathbf{1.}\) There exists \(s_0\leqslant m\leqslant s, m \in K_\alpha \), such that \( \kappa n_0 q_s < q_{m} \leqslant 8 C \kappa n_0 q_s\), or \(\mathbf{2.}\) There exist \(s_0\leqslant m\leqslant s\) and \(l \geqslant 1\) such that \( l q_m \leqslant \kappa n_0 q_s < (l+1) q_m \leqslant \frac{q_{m+1}}{8C}\).

**Case 1.**\( \kappa n_0 q_s < q_{m} \leqslant 8 C \kappa n_0 q_s\) with \(s_0\leqslant m\leqslant s, m \in K_\alpha \). Lemma 4.7 implies that either (31) or (32) holds for \(T^{n_0 q_s}x,T^{n_0 q_s}y,m\). Therefore, (48) or (49) holds for \(m\) and \(l=0\). We then apply Lemma 4.10 to \(T^{n_0 q_s}x,T^{n_0 q_s}y,m\) with \(l=0\), and according to whether we have (50) or (51) we will get A. or B. of Proposition 4.6 Part b. Indeed, suppose (50) holds. Then, since \(\kappa n_0 q_s < q_{m}\), for \(n =1, \ldots , [\kappa n_0 q_s]+1\), we have due to (47)

**Case 2.**There exist \(s_0\leqslant m\leqslant s\) and \(l \geqslant 1\) such that \( l q_m \leqslant \kappa n_0 q_s <(l+1) q_m \leqslant \frac{q_{m+1}}{8C}\). We will first prove that \(T^{n_0 q_s}x,T^{n_0 q_s}y,m,l\) satisfy the hypothesis of Lemma 4.10. If \(m\in K_\alpha \), then Lemma 4.7 implies that either (31) or (32) holds for \(T^{n_0 q_s}x,T^{n_0 q_s}y,m\). Therefore, since \(l\leqslant \frac{q_{m+1}}{8Cq_m}-1\), either (48) or (49) holds for \(T^{n_0 q_s}x,T^{n_0 q_s}y,m,l\). If \(m\notin K_\alpha \), then we consider two cases:

**I.**\(m=s\). In this case \(n_0>\frac{1}{\kappa }\) and therefore \(q_{s+1}>32Cq_s\). Since \(l\leqslant [\kappa n_0]+1, n_0\leqslant \frac{q_{s+1}}{8Cq_s}\) and \(\kappa \ll 1\), we get that

**II.**\(m<s\). Since \(l\leqslant \frac{q_{m+1}}{8Cq_m}-1\), it is enough to show that \(T^{n_0 q_s}x,T^{n_0 q_s}y,m\) satisfy (31) or (32). Due to Lemma 4.7, we just have to check (30) for \(T^{n_0 q_s}x,T^{n_0 q_s}y,m\):

**Case 2.**at least one of (48) or (49) is satisfied for \(T^{n_0q_s}x, T^{n_0q_s}y,m,l\).

**Case 1.**, if (50) holds we get A., if (51) holds we get B. Indeed, assume WLOG that \(T^{n_0q_s}x, T^{n_0q_s}y,m,l\) satisfy (50) (the proof in the other case is analogous). Using (47) and the fact that \([\kappa n_0 q_s]+1\leqslant (l+1)q_m\leqslant 2\kappa n_0q_s\), we get for \(n =1, \ldots , (l+1)q_m\)

The proof of Proposition 4.6 is thus completed and Theorem 2 follows. \(\square \)

### 4.2 Proof of Theorem 4

#### 4.2.1 Outline of the proof

- a.
If for some \(c\in (0,1), R^i_\alpha I\) is disjoint from \([-c/q_n,c/q_n]\) for every \(i=0,\ldots ,l \leqslant q_n\) then \(f'^{(l)}(\theta )\leqslant C q_n^{1+\gamma }\) for any \(\theta \in I\) for some \(C\) that depends on \(c\) (with a similar statement for negative iterates).

- b.
If for some \(c'\in (0,1), R^{i_0}_\alpha I\) is included in \([-c'/q_n,0)\) for some \(i_0\geqslant 0\) then \(f'^{(i_0+1)}(\theta )-f'^{(i_0)}(\theta )\geqslant C' q_n^{1+\gamma }\) for any \(\theta \in I\) for some \(C'\) that depends on \(c'\) (with a similar statement for negative iterates).

- c.
If \(\alpha \) is of constant type then there exists \(c'>c>0\) such that one of the following holds if \(n\) is sufficiently large : 1. there exists \(i_0\geqslant 0\) such that \(R^{i_0}_\alpha I\) is included in \([-c'/q_n,0)\) and \(R^i_\alpha I\) is disjoint from \([-c/q_n,c/q_n]\) for every \(i=0,\ldots ,i_0\) or 2. there exists \(i_0<0\) such that \(R^{i_0}_\alpha I\) is included in \([-c'/q_n,0)\) and \(R^i_\alpha I\) is disjoint from \([-c/q_n,c/q_n]\) for every \(i=-1,\ldots ,-i_0\).

- d.
If \(\alpha \) is not of constant type and if \(q_{n+1}\gg q_n\) and \(y-x=\varepsilon _n/q_n^{1+\gamma }\) while \(|x-p/q_n|\leqslant \varepsilon _n^2/q_n\) with \(\varepsilon _n\rightarrow 0\), then as long as for \(i \in [0,l]\) (or for \(i \in [-l,-1]\)) \(R^i_\alpha I\) is disjoint from \([-1/(2q_n),1/(2q_n)]\) we have that \(f'^{(l)}(\theta )\leqslant C q_n^{1+\gamma }\) for any \(\theta \in I\), while if \(l\) is the first integer such that \(R^l_\alpha I\) intersects \([-1/(2q_n),1/(2q_n)]\) then \(f'^{(l)}(\theta )\geqslant q_n^{1+\gamma }/\varepsilon _n \).

In Sublemma 4.14 we essentially prove a. and in Sublemma 4.15 we essentially prove b.

We now explain why the constant type condition is necessary in our proof. Indeed, if \(\alpha \) is not of constant type, d. gives an example of pairs \(x,y\) for which the drift between the forward orbits jumps from \(\varepsilon _n\) to \(1/\varepsilon _n\) and the same happens for backward orbits, which contradict the SWR property for this pair. Furthermore, if \(\varepsilon _n\) is taken to converge very slowly to \(0\) such pairs can be produced with \(x \in Z\) for any \(Z\) having positive measure. Observe that this does not imply that the SWR property would not reappear much later in time but this is very unlikely as demonstrated for the absence of the WR property in the particular case of Theorem 1.

Observe finally that the same type of pairs \((x,y)\) described in d. show that it is necessary to use the SWR property instead of the WR property. Indeed, only one of the alternatives c.1 or c.2 holds for such pairs and we are obliged, if we want to see a controlled drift, to iterate in the future or in the past.

#### 4.2.2 Notations and standing assumptions

Fix \(\varepsilon \ll 1\) and \(N\in \mathbb {N}\). We will assume that \(\epsilon <\frac{C_kD_1^2}{8c}\). Let \(\kappa :=\kappa (\epsilon )=\frac{\epsilon }{2(3D_2+2)kCH}\).

#### 4.2.3 Controlling the drift

The following proposition implies Theorem 4 due to Proposition 3.3.

**Proposition 4.11**

Consider \(x,y\in \mathbb {T}\) with \(0<\Vert x-y\Vert <\delta \). Then there exists \(p \in P, M,L\geqslant \kappa M \geqslant N\) such that either (16) or (17) holds for \(n \in [M,M+L]\).

**Proposition 4.12**

Consider \(x,y\in \mathbb {T}\) as in (58).

**Part a.**There exists \(i_0 \in \{0,\ldots ,q_{s-2}-1\}\), such that

**Part b.**Let \(X=T^{i_0}x\) and \(Y=T^{i_0}y\) if (59) holds, and \(X=T^{-i_0-1}x\) and \(Y=T^{-i_0-1}y\) if (60) holds, for \(n=1,\ldots ,[\kappa i_0]+1\) the following holds

The rest of Sect. 4.2 is devoted to the proof of Proposition 4.12.

#### 4.2.4 Proof of Lemma 4.13

We will suppose (62) holds, the proof of the other case being analogous. We will need some lemmas.

**Sublemma 4.14**

*Proof*

**Sublemma 4.15**

There exists \(i_0 \in \{0,\ldots ,q_{s-2}-1\}\), such that \(|f^{(i_0)}(x)-f^{(i_0)}(y)| \geqslant \frac{A_kD_1^2}{4c}\).

*Proof*

As a consequence of the above lemmas, we obtain that at least one of the numbers \(f^{(i_0+1)}(x)-f^{(i_0+1)}(y)\) or \(f^{(i_0)}(x)-f^{(i_0)}(y)\) belongs to the set \(P\), and (59) is proved. The next result will give the proof of (61).

**Sublemma 4.16**

*Proof*

We proceed to the proof of Lemma 4.13 in the case (62) is satisfied. If \(f^{(i_0+1)}(x)-f^{(i_0+1)}(y)\in P\), then (69) gives A. in (61); if \(f^{(i_0)}(x)-f^{(i_0)}(y)\in P\), then (70) gives B. in (61). The proof of Lemma 4.13 is thus completed since the case where (63) is satisfied is analogous. \(\square \)

This finishes the proof of Proposition 4.12, thus of Theorem 4. \(\square \)

## Notes

### Acknowledgments

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 Fra̧czek, Mariusz Lemańczyk and Jean-Paul Thouvenot for valuable discussions on the subject. The results of Sect. 4 have been obtained by the two authors independently and the results of Appendix A by the second. The two authors decided to include Appendix A in this work because it is an integral part of the problems concerning Ratner’s property for this class of special flows.

The authors are grateful to the referee for constructive criticisms that helped us improve the text.

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