Keplerian shear in ergodic theory

Many integrable physical systems exhibit Keplerian shear. We look at this phenomenon from the point of view of ergodic theory, where it can be seen as mixing conditionally to an invariant $\sigma$-algebra. In this context, we give a sufficient criterion for Keplerian shear to appear in a system, investigate its genericity and, in a few cases, its speed. Some additional, non-Hamiltonian, examples are discussed.

: Propagation of a wavefront at unit speed in a unit square torus. The wave starts from the corner, and propagates at unit speed. On the left: the wavefront at time 0.5. In the middle: the wavefront at time 10. On the right: the wavefront at time 500.
Other systems with Keplerian shear are the geodesic flow on a flat torus (see Fig 2), or the dynamics of a ball bouncing in a square box.
In this article, we frame Keplerian shear in the more general context of ergodic theory, as a conditional version of the notion of strong mixing.
A dynamical system (Ω, µ, (g t ) t∈R ) which preserves a probability measure is said to exhibit Keplerian shear if, for all f ∈ L 2 (Ω, µ), where I is the invariant σ-algebra and the convergence is for the weak topology on L 2 (Ω, µ).
Recall that a system (Ω, µ, (g t ) t∈R ) is mixing if and only if, for any function f ∈ L 2 (Ω, µ), where the limit is taken in the weak topology on L 2 (Ω, µ), so a system (Ω, µ, (g t ) t∈R ) is mixing if and only if it is ergodic and exhibits Keplerian shear. As such, Keplerian shear is a conditional version of the notion of strong mixing. Informally, if the system restricted to its invariant subsets is mixing, then (Ω, µ, (g t ) t∈R ) has Keplerian shear. The interesting examples occur when these restrictions are ergodic, but not mixing: that is the case, for instance, of translation flows on a torus. In this article, we give a criterion ensuring Keplerian shear for a large class of such systems; for instance, one of our result is: We also study the rate of decay of conditional covariance for the geodesic flow on T 1 T d , and give non-trivial examples of non-Hamiltonian systems with Keplerian shear.
Keplerian shear for the geodesic flow on the flat torus is related to two famous problems. The first is Landau's damping for plasma dynamics on a torus (see Landau's article [6], and [10, Theorem 3.1] for a version which follows closely our formalism), where the effect is qualitatively similar, although the underlying mechanism is different. The second is Gauss's circle problem, which consists in counting integral points in a large disc; we shall discuss it in Sub-subsection 2.4.2. The methods used to tackle these problems are either through Fourier transform (e.g. for Landau damping), or with a big arc/small arc decomposition (typical for Gauss's circle problem). While both work in our setting, we shall only use the Fourier transform.
In the context of ergodic theory, a notion closely related with Keplerian shear was used independently by F. Maucourant [7] to prove that the some hyperbolic actions on (R d SL d (R)) /(Z d SL d (Z)) are ergodic for a large class of measures. The presentation in [7] is however very different, as the phenomenon -named asynchronicity -is described as a version of unique ergodicity for measures with prescribed marginals.
Organization of the article Section 1 gives general results on the notion of Keplerian shear (including equivalences between distinct definitions), and gives us some tools to use for the remainder of the article.
Section 2 deals with a first family of systems which may exhibit Keplerian shear: fibrations by tori, where the flow acts by translation on each torus. using action-angle coordinates, this family includes integrable Hamiltonian flows. We give an explicit criterion ensuring Keplerian shear, check that it is C r -generic (r ≥ 1) and satisfied for some explicit systems, then give rates of convergence for the geodesic flow on T 1 T n . We also detail the link between Keplerian shear and the unique ergodicity as investigated in [7]. Section 3 deals with another family of dynamical systems (roughly, "fibrations by suspension flows"), which includes many non-Hamiltonian examples, and uses a different mechanism to ensure Keplerian shear.
The shorter Section 4 gives examples of systems without Keplerian shear.
A note on the terminology Given that Keplerian shear is a conditional version of the notion mixing, one could want to use a terminology such as conditional (strong) mixing. We prefer to eschew this option, and to keep the name of Keplerian shear; indeed, we think that otherwise the name of conditional (strong) mixing would be overloaded.
Indeed, in probability theory, there are already multiple notions of conditional mixing; compare for instance [11] (where it refers to conditional α-mixing) and [5], among others.
More worryingly, in ergodic theory, the notion of conditionally weakly mixing systems is wellestablished (see e.g. [13]), but if one where to conceive a notion of conditional strong mixing along this line, the resulting notion would be stronger than Keplerian shear, essentially requiring that almost every subsystem in its ergodic decomposition be mixing.

Open problems
We sum up here some further leads which seem worth pursuing. The setting of Section 2 covers integrable Hamiltonian systems. However, it requires some regularity, and in particular it does not cover singular systems. A conjecture by Boshernitzan asserts that given a compact translation surface S, the geodesic flow on (T 1 S, Liouv) exhibits Keplerian shear. This question, mentioned as illumination by circles, also appears in [8], and admits a partial answer by J. Chaika and P. Hubert [1], where the convergence of Cov(f, g • g t |I) to zero is shown along a density 1 subsequence for all continuous observables f and g 1 .
In Subsection 2.5, we investigate the speed of Keplerian shear for the geodesic flow on T 1 T n . The problem is simplified by the particularities of the geometry of the sphere, more precisely the fact that its principal curvatures do not vanish. What would the speed of convergence be if the curvature vanishes (e.g. in a topologically or measure-theoretically generic setting)?
Finally, while the settings of Sections 2 and 3 are distinct, it could be that they are a special case of a more general structure. A natural candidate would be spaces fibrated by suspension tori, but we need new tools to prove Keplerian shear (or even to get a description of the invariant σ-algebra I).
Acknowledgements I would like to thank Sébastien Gouëzel, Bassam Fayad and Fraçois Maucourant for their useful comments and some of the references, as well as Jérôme Buzzi for his feedback on the presentation.

General properties of Keplerian shear
The following lemma from basic functional analysis is quite useful to prove the ergodicity and mixing of any given dynamical system, and will be instrumental in the remainder of our article.
Let B be a Banach space. Let (T t ) t≥0 be a family of operators on B, such that sup t∈R + T t B→B < +∞. Let T be an operator on B.
Let E and E * be subsets of B and B * respectively, whose span is dense in their respective space. Assume that, for all f ∈ E and g ∈ E * , lim t→+∞ g, T t f = g, T f . (1.1) Then (T t f ) t≥0 converges weakly to T f for all f ∈ B.
Since sup t∈R + T t B→B < +∞, the family of functions T t : B * × B → C is locally equicontinuous, and by the remark above, it converges to T on a dense subset. Hence, the convergence of Equation (1.1) holds for all f ∈ B and g ∈ B * .
When we use Lemma 1.1, the operator T t shall correspond to the composition by the flow g t at time t, and the operator T to the projection f → E(f |I). Since the flow is assumed to preserve the measure, for all t ≥ 0 and all p ∈ [1, +∞], the operator T t acting on L p (Ω, µ) is unitary. Lemma 1.1 implies that to prove the Keplerian shear in one of those Banach space B (potentially different from L 2 ), it is enough to restrict ourselves to subsets E of B and E * of B * whose linear span is dense. As a first consequence, in the definition of Keplerian shear, one may replace L 2 by L p for any p ∈ [1, +∞): Let (Ω, µ, (g t ) t∈R ) be a flow which preserves a probability measure. Let I be the invariant σ-algebra of the system. Then there is equivalence between: • There exists p ∈ [1, +∞) such that, for all f ∈ L p (Ω, µ), we have f • g t → E(f |I) weakly in L p .
• The system exhibits Keplerian shear.
Proof. We only prove the non-trivial implication. Let p ∈ [1, +∞). Assume such that, for all Let q ∈ [1, +∞). Since L ∞ is dense in both L q and (L q ) * , by Lemma 1.1, the convergence above occurs for all f 1 and f 2 in L q and (L q ) * respectively.
A second consequence is that Keplerian shear is not uniquely a property of the invariant measure µ, but of the class of µ.
Let (Ω, µ, (g t ) t∈R ) be a flow which preserves a probability measure and exhibits Keplerian shear. Let ν µ be a probability measure which is also (g t )-invariant. Then (Ω, ν, (g t ) t∈R ) also exhibits Keplerian shear.
Proof. Let (Ω, µ, (g t ) t∈R ) and ν be as in assumptions of the proposition.
The last lemma asserts that, in the definition of Keplerian shear, the limit object E µ (f |I) cannot be meaningfully modified.
Proof. Let g ∈ L 2 (Ω, µ). Our hypotheses imply that lim t→+∞ E µ (g · f • g t ) = E µ (gh). In addition, the function t → E µ (g · f • g t ) is measurable and bounded. By taking the Cesàro average, we get: On the other hand, by von Neumann's ergodic theorem, Since this holds for all g ∈ L 2 , we have h = E µ (f |I).

Setting and main theorem
We generalize our introductory examples to a class of flows on fibre bundles by tori which leave the basis invariant. More specifically, the spaces on which we work are the following: 1. An affine tori bundle is a C 1 manifold Ω which is a fiber bundle by d-dimensional tori, with group structure T d GL d (Z). In other words, there exist: • two integers n, d ≥ 1; • a n-dimensional C 1 real manifold M ; • a C 1 projection π : Ω → M ; • a maximal atlas A on M , such that, for all U ∈ A, we have a diffeomorphism ψ U : π −1 (U ) → U × T d such that π 1 • ψ U = π, and the change of charts are given by: The notions of "subset of zero Lebesgue measure" or "subset of full Lebesgue measure" are welldefined on C 1 manifolds (as they are invariant by diffeomorphisms), and thus so is the notion of "probability measure absolutely continuous with respect to the Lebesgue measure". We will abuse notations and write Leb(A) = 0 for a measurable subset of zero Lebesgue measure A, and µ Leb for an absolutely continuous measure.

Definition 2.2.
Let Ω be an affine tori bundle. A flow (g t ) t∈R on Ω is said to be compatible on a chart ψ U : A σ-finite measure µ on Ω is said to be compatible on a chart ψ U : A flow or a measure is said to be compatible if it is compatible on all charts.
A compatible measure is always invariant under a compatible flow. In addition, this notion behaves well with respect to the affine structure on the manifolds we work with. If a flow or a measure is compatible on some chart ψ U : U ∩ V → π(U ∩ V ) × T d and if ψ U,V is a change of charts, then the flow or the measure is compatible on the chart ψ V |U ∩V : In what follows, we are working mostly with absolutely continuous measures. In this case, what happens on a subset of zero Lebesgue measure does not matter: the assumption that M be a manifold can be weakened to account for singularities or boundaries.
In light of the previous paragraph, the introduction of the structure group T d GL d (Z) might look gratuitous: one can always cut out the manifold M along a set of zero Lebesgue measure to get a disjoint union of simply connected domains, on which there is no holonomy. However, this structure appears naturally in many examples. For instance, for all n ≥ 1, we can work with the geodesic flow on T S n : if we ignore the set of null tangent vectors, which is negligible, we get a fibre bundle over R * + × Gr(2, n + 1) with fibre S 1 . With the same adaptation, our setting also includes billiards in ellipsoids or the geodesic flow on ellipsoids (see C. Jacobi [4] for the geodesic flow on ellipsoids, J. Moser [9] for similar examples, and S. Tabachnikov [12] for the relation between the geodesic flow and the billiard). Let us also mention the study of the geodesic flow on [7], in which the same structure appears.
Another important remark is that, when we change charts from chart U to chart V , we have We are now ready to state our main theorem.
Let π : Ω → M be an affine d-dimensional tori bundle over a manifold M . Let (g t ) t∈R be a compatible flow, and µ be an absolutely continuous compatible probability measure.
Hence, up to completion by the measure µ, the invariant σ-algebra of the flow is I : Our goal is to find a family of observables which is large enough to generate a dense subset of L 2 (Ω, µ), and specific enough to make our computations manageable. Roughly, we choose a specific frequency in the direction of the torus T d . Under the hypothesis of the theorem, we can rectify the differential form ξ, v so that it has a very simple expression. Then we choose observables which split into an observable a in the direction of ξ, v , and another observable b in the direction of the kernel. The later observable b does not see the shearing at all, so the shearing only affects a.
Let (U i , ϕ i ) i∈I be a countable cover of M by disjoint open charts 2 , up to a Lebesgue-negligible set, with ϕ i : Using the local normal form of submersions, we can find a finite or countable family (V ξ ij ) j∈J(i,ξ) of open sets which are pairwise disjoint, cover V ξ i up to a Lebesgue-negligible set, and with charts ϕ ξ ij : to be a singleton and take V 0 ij := V i . Given a point p ∈ R n , we write p x its first coordinate in R n , and p y for its remaining n − 1 coordinates in R n . Given a point p ∈ M × T d , we write p z for its coordinate in T d . We apply Lemma 1.1, with the Banach space B = B * = L 2 (Ω, µ), and: Let f j = a j b j e ξ j ,· , with j ∈ {1, 2}, be in E. If the corresponding indices i ∈ I are different, then We can thus assume without loss of generality that they are supported by the same open set π −1 (V i ).
If the corresponding frequencies ξ j ∈ 2πZ d are different, then the integral of f 1 · f 2 • g t on each torus T d vanishes, and a least one of E µ (f 2 |I) or E µ (f 2 |I) vanishes, so for all t ∈ R: We can thus assume without loss of generality that their frequencies ξ j are the same; let us denote it by ξ. If ξ = 0, then f 1 and f 2 are invariant under the flow, so there is nothing more to prove. We further assume that ξ = 0.
If the corresponding indices j ∈ J(i, ξ) are different, then the supports of f 1 and f 2 •g t are disjoint for all t, so then again there is nothing more to prove. We thus fruther assume that these indices are the same. Write Then, for all t ∈ R: is integrable for almost every y. By the Riemann-Lebesgue lemma, the inner integral decay to 0 as t → ±∞. The inner integral is bounded by: which is integrable as a function of y. Hence, by the dominated convergence theorem,

Genericity
We check in this subsection that the sufficient condition in Theorem 2.3 is C r -generic for all r ∈ [1, +∞]. Given a C r affine tori bundle Ω, we begin by endowing the space of C r compatible flows with a topology. Let r ∈ [1, +∞], and π : Ω → M be a C r affine d-dimensional tori bundle over a manifold M . Let (U i ) i∈I be a locally finite open cover of M with trivializing charts ϕ i : if, for all i ∈ I, all the derivatives of (v n,i ) n≥0 (up to order r) converge to those of v uniformly on each K i . This topology does not depends on the choice of the charts (U i ) i∈I nor on that of the compacts (K i ) i∈I , and makes F r (M, R d ) a Baire space.

Proposition 2.4.
Let r ∈ [1, +∞]. Let π : Ω → M be a C r affine d-dimensional tori bundle over a manifold M . For a Baire generic subset of compatibles flows in F r (M, R d ), the dynamical system (Ω, µ, (g t )) exhibits Keplerian shear for all absolutely continuous compatible measures µ.

Proof.
We use the criterion of Theorem 2.3. It is enough to prove that, for all ξ ∈ Z d \ {0} and all i ∈ I: All is left is to prove that m≥1 B ξ,i,n,m is meager. Note that: Let v ∈ B c ξ,i,n,m . By inner regularity of the Lebesgue measure on (1, 0, . . . , 0). By the pigeonhole principle, for all m ≥ 1, at least one of the functions v(2k/( ξ 2 m)), with 0 ≤ k ≤ n Leb(ϕ i (K i )) , belongs to B c ξ,i,n,m . Thus there exists a sequence (t m ) m≥1 such that v(t m ) ∈ B c ξ,i,n,m and lim m→+∞ t m = 0. This finishes the proof. Remark 2.5.
If Ω = M × T d and r ≥ 2, we can conclude using the (well known, but more difficult to prove) fact that a generic function in C r (M, R) is Morse.

Examples
The simplest non-trivial example of Keplerian shear is given by the map x, y ∈ T}. This transformation preserves the Lebesgue measure, as well as all the circles T × {y}. Keplerian shear is rather easy to prove 3 , as there is no need to play with charts; one can use directly the Fourier basis on L 2 (T 2 , Leb), which behaves well under T . A slightly more sophisticated version of this argument is used in Sub-subsection 2.5.1 to compute the speed of decay of correlations.
All systems are not that simple. Besides genericity, Theorem 2.3 provides a useful criterion to prove that a given dynamical system exhibits Keplerian shear. We now use it to prove Keplerian shear for two dynamical systems: the billiard in the unit ball B n ⊂ R n , and the unit speed geodesic flow on T n (with the flat metric).

Billiard in a ball
Let B n be the unit ball in R n , with n ≥ 2. Consider a particle moving with unit speed in B n , which reflects specularly on the boundary S n−1 . The phase state is an orbifold T 1 B n , and the flow (g t ) t∈R preserves the Liouville measure µ n (which here is essentially the Lebesgue measure on B n × S n−1 ).
The dynamical system (T 1 B n , µ n , (g t ) t∈R ) exhibits Keplerian shear. Proof.
If we exclude trajectories which go through the origin, then any given trajectory lie in the unique plane generated by the position and the speed at any given time. Restricted to any such plane, the billiard is isomorphic to the billiard in B 2 . Since a disjoint union of systems with Keplerian shear still has Keplerian shear, it is enough to prove that (T 1 B 2 , µ 2 , (g t ) t∈R ) has Keplerian shear.
For all ξ ∈ Z 2 \ {0}, the function ξ, v is analytic and non-zero, and thus its zero set is discrete. By Theorem 2.3, the system (T 1 B 2 , µ 2 , (g t ) t∈R ) has Keplerian shear.
A similar proof applies to the billiard in an ellipsoid, or the geodesic flow on an ellipsoid.

Geodesic flow on the torus
The second example we discuss is the unit speed geodesic flow on the torus T n , with n ≥ 1. This flow, again, preserves the Liouville measure.

Proof.
The manifold T 1 T n is trivializable, and thus isomorphic to T n × S n−1 . The geodesic flow (g t ) t∈R acts on T 1 T n by: Let ξ ∈ Z n \ {0}. Then d ξ, v vanishes at only two points, which are ±ξ/ ξ . By Theorem 2.3, the system (T 1 T n , Liouv, (g t ) t∈R ) has Keplerian shear.

Unique ergodicity
In this subsection, we describe the relation between Keplerian shear and the unique ergodicity of a transformation acting on spaces of probability measures, as introduced by F. Maucourant [7]. We drop the assumption that the function v generating the flow be C 1 : here, continuity is enough.

Definition and relation with Keplerian shear
Let π : Ω → M be a compact affine tori bundle, (g t ) a compatible flow on Ω, and ν ∈ P(M ). Denote by P ν ⊂ P(Ω) the subspace of probability measuresμ such that π * μ = ν, and by ν ⊗ Leb the unique compatible measure on Ω such that π * (ν ⊗ Leb) = ν. Let G t := g t, * act continuously on P(Ω), which is compact when endowed with the weak convergence. Since the flow is compatible, (G t ) preserves P ν , which is also compact. Note that ν ⊗ Leb is a fixed point of (G t ), so δ ν⊗Leb is (G t )-invariant.
Let π : Ω → M be a compact affine tori bundle. Let (g t ) be a compatible flow on Ω. Let ν ∈ P(M ).

Proof.
Let π : Ω → M , (g t ) and ν be as in the hypotheses of the theorem. First, we assume that (Ω, ν ⊗ Leb, (g t )) exhibits Keplerian shear. We can find a countable cover of M by disjoint open charts (U i ) i∈I , up to a ν-negligible subset. Then all (U i × T d , ν |U i ⊗ Leb, (g t )) exhibit Keplerian shear.
Let µ be in M(U i × T d ) with π * µ = ν |U i . Endow U i with any bounded Riemannian metric, and T d with a flat metric. This yields a Riemannian metric on U i × T d (e.g. the product metric), from which we get a Wasserstein distance d W , which metrizes the weak convergence.
Assume now that G t (µ) → ν ⊗ Leb for all µ ∈ P ν . By [7, Theorem 1], g 1 is asynchronuous, so the set of points x of M such that (g t ) acts on {x} × T d by an irrational translation has full ν-measure. Hence, the invariant σ-algebra is π * B M .
Let (U i ) i∈I be an open cover of M by charts. Let f ∈ C(Ω, C). Let i ∈ I and ρ(x, y) = a(x)b(y) Then µ ∈ P ν , and, for all t: By assumption, G t (µ) converges weakly to ν ⊗ Leb, so the quantity above converges to: By Lemma 1.1, (Ω, ν ⊗ Leb, (g t )) exhibits Keplerian shear.
Remark 2.9 (Keplerian shear is stronger than unique ergodicity). F. Maucourant gives an example [7] of a compatible flow and a measure ν such that (P ν , (G t )) is uniquely ergodic, but the fixed point ν ⊗ Leb behaves like an indifferent fixed point: there are exceptional sequences of times (t i ) for which G t i (ν ⊗ δ 0 ) is far from ν ⊗ Leb. As a corollary, the unique ergodicity of (P ν , (G t )) does not imply that (Ω, ν ⊗ Leb, (g t )) has Keplerian shear.

An application : Gauss' circle problem
The alternative characterization of Keplerian shear given by Theorem 2.8 is also useful in settings which use non-absolutely continuous measures. Let us give an elementary application to a variation on Gauss' circle problem. Let S(x, r) be the sphere of center x and radius r in R n , with n ≥ 2. Let ε ∈ (0, 1/2). What is the number of integer points in an ε-neighborhood of S(x, r)?
Let σ x,r be the uniform measure on S(x, r), and the canonical projection from R n to T n . Take Ω := S n−1 × T n , with g t (v, y) = (v, y + tv) and ν the uniform measure on S n−1 . Let f (y) := 1 |y|≤ε on T n . Then: The system (Ω, ν ⊗ Leb, (g t )) has Keplerian shear by Proposition 2.7, so that: In addition, S(x, r) ∩ B(Z n , ε) consists of finitely many caps, which get flatter and flatter as r increases; the number of integer points ε-close to S(x, r) is the number of such caps. Let us direct there caps by the outward normal at their center. Since the measure supported by the projection on S n−1 × T n of these caps equidistributes in S n−1 × B(0, ε), we get that the average area (for * σ x,r ) of each cap converges to: Hence, the number of integer points in an ε-neighborhood of S(x, r) converges, as r goes to infinity, to: ε n Leb(B R n (0, 1)) · 2r n−1 Leb n−1 (S n−1 ) ε n−1 Leb(B R n (0, 1)) = 2εr n−1 Leb n−1 (S n−1 ).
This stays true if the sphere is replaced by any compact manifold, under non-resonancy conditions which ensure Keplerian shear for the relevant dynamical system. Note also that for the sphere, by integrating over r, one recovers the more elementary fact that the number of integral points at distance r from the origin is equivalent to r n Leb(B R n (0, 1)).
This result is not optimal. For instance, the best known bounds for Gauss' circle problem [3] imply that:

Speed of mixing
Keplerian shear is a qualitative property of a measure-preserving dynamical system, which asserts the convergence to zero on average of the conditional correlations: As with the notion of mixing, one cannot expect a rate of convergence for all observables f 1 , f 2 ∈ L 2 . However, we may get a rate of convergence if f 1 and f 2 are regular enough. We may also need assumptions of the measure µ and the critical points of the functions ξ, v .
In the examples we discuss below, f 1 and f 2 shall belong to anisotropic Sobolev spaces (or, more precisely, weighted anisotropic Sobolev spaces). The regularity of such observables depends on the direction. We refer the reader to the monography by H. Triebel for additional information [15, 5 .
In our setting, we need relatively little regularity in the direction of the invariant tori: what matters most is the regularity transversaly to the invariant tori. This is not surprising in view of Theorem 2.8, which asserts roughly that E(Cov t (f 1 , f 2 |I)) vanishes, where f 1 is Lipschitz and f 2 is e.g. Leb ⊗δ 0 on M × T d . In this case, f 2 is a distribution which is more regular transversaly to the invariant tori than in the direction of the invariant tori.
Instead of working out a general statement, we discuss two simple systems: the parabolic automorphism of T 2 at the beginning of Subsection 2.3, and the unit speed geodesic flow on T n .

Transvection on T 2
Consider the map acting on T 2 , endowed with the Lebesgue measure. Let us define suitable anisotropic Sobolev spaces. For ξ ∈ R 2 , let: For any real number s ≥ 0, let: The following proposition gives decay bounds on the correlation coefficients for Sobolev or analytic observables.

Proposition 2.10.
Let f 1 , f 2 be in H s,0 (T 2 , R). Then: If f 1 and f 2 are analytic, then there exist constants c, C > 0 (depending on f 1 and f 2 ) such that, for all n ∈ Z, |E(Cov n (f 1 , f 2 |I))| ≤ Ce −c|n| . 5 A small difference is that our spaces H s,0 and H s, n−1 2 below do not fit exactly in the framework of Triebel, because the weights do not satisfy the assumptions at the beginning of [15,Chapters 6]. However, one can write for instance H s,0 (T 2 ) = L 2 (T 1 ) ⊕ H s,0 (T 2 ), where L 2 (T 1 ) has no effect on the correlations and H s,0 (T 2 ) fits into Triebel's framework.

Proof.
Let f 1 , f 2 be in H s,0 (T 2 ). By Plancherel's theorem, so that: is maximal for ξ 1 = −nξ 2 /2, where its value is (1 + n 2 /4) −s , so that: The proof for analytic functions is essentially the same. The only remark needed is that, if f is analytic on the torus, then there exist constants c , C > 0 such that |f |(ξ) ≤ C e −c |ξ| .
The map T is especially well-behaved: not only does it acts nicely on Fourier series, but its shearing (the derivative of v) does not vanish. The estimates of Proposition 2.10 are thus a best case behaviour, that we do not expect to hold for more general systems.

Speed for the geodesic flow on the torus
The geodesic flow is harder to analyse than the previous example: not only does it lack its algebraic structure, but the functions ξ, v have vanishing gradient at two points for any non-zero ξ. Hence, we cannot expect the same rate of convergence. We use the stationary phase method to compute the speed of convergence. This yields a polynomial rate of decay for a large space of observables belonging again to some anisotropic Sobolev spaces (Proposition 2.11).
The definition of these anisotropic Sobolev spaces is however slightly more delicate. Let n ≥ 2 and s > (n − 1)/2. For (k, ξ) ∈ R n−1 × 2πZ n , let: We see T 1 T n as S n−1 × T n . Fix a finite open cover by charts (U i , ϕ i ) of S n−1 , and a smooth partition of the unit (χ i ) subordinated to (U i ). Then define: and denote by · 2 H s, n−1 2 the norm appearing in this definition. In the same way, we define the Sobolev space H s (S n−1 ). These spaces do not depend on the choice of the family of charts and of the partition of the unit.
The following proposition gives decay bounds on the correlation coefficients for observables in H s, n−1 2 .

Proof.
In this proof, the letter C shall denote a constant which may change from line to line, but which depends only on the dimension n and on the parameter s.
That finishes the proof for smooth observables f 1 and f 2 . But, for fixed t, the correlation function E(Cov t (·, ·|I)) is bilinear and continuous from L 2 to C. Since the H s, n−1 2 norm is stronger than the L 2 norm, E(Cov t (|I)) is also continuous from H s, n−1 2 to C. But C ∞ is dense in H s, n−1 2 , so the bound (2.1) actually holds for any two observables in H s, n−1 2 .
Assuming that the observables f 1 and f 2 have higher regularity, standard formulations of the stationary phase method yield a higher order development of E(Cov t (f 1 , f 2 |I)) as t goes to infinity.
Assume now that we change the flow on S n−1 × T n , for instance by making the velocity depend on the direction. Then the rates we got in Proposition 2.11 may not be generic. We shall sketch the difficulties encountered with more general systems. Let n ≥ 3 and M be a compact connected (n − 1)-dimensional smooth manifold, and let v : M → R n be smooth. Consider the flow g t (x, y) = (x, y + tv(x)) on M × T n . If Dv is never degenerate (which is a C 1 -open condition on v), then v is an immersion. If in addition the extrinsic curvature of the immersed manifold is never degenerate, then we get rates of convergence as in Proposition 2.11. However, if the extrinsic curvature is never degenerate, then the Gauss map M → S n−1 is a local diffeomorphism, so a diffeomorphism (since n ≥ 3), and thus M is a sphere.
In other words, if M is not a sphere, then we have to deal with degenerescences of the extrinsic curvature of v(M ). If such a degenerescence happens in a rational direction of R n , then we would get a speed of convergence in O(t − n−1−r 2 ), where r is the corank of the Hessian in the given direction. If this degenerescence happens in a direction u which is not rational, then this bound could be improved, although any improvement would depend on the Diophantine properties of u (the bound getting better if u is badly approximable by rationals). In particular, one cannot hope to get a significantly better bound than O(t − n−1−r 2 ) in a Baire generic setting, as Baire generic directions are Liouville.
For n ≥ 2, the same kind of obstruction may happen for v : S n−1 → R n . For a C 3 -open set of such functions v, the map v has non-degenerate inflexion points. Without further argument about the directions these inflexion points occur, this would for instance yield a rate of decay of only O(t − 1 3 ) if n = 2.

Stretched Birkhoff sums
We present in this sub-section another class of systems which may exhibit Keplerian shear. The examples of Subsection 2.1 are based on translations on the torus, which are a family of non-mixing dynamical systems. In this section, the elementary brick will be given by suspension flows with constant roof function. The family of examples we get includes many non-Hamiltonian systems.
Such a suspension flow is ergodic, but cannot be mixing, as it has the rotation on the circle as a factor. Now, we give ourselves: • a n-dimensional C 1 manifold M , with n ≥ 1; • a measure-preserving ergodic dynamical system (A, ν, T ); • a measurable function v : M → R * + .
With this data we construct a new semi-flow (Ω, (g t ) t≥0 ) with Ω := A×M and g t (x, y) := (g v(y) t (x), y). A measure µ ∈ P(Ω) is said to be compatible if it is equal toν ⊗μ for someμ ∈ P(M ). Compatible measures are preserved by (g t ).
If (A, ν, T ) is invertible, the suspension semi-flow can be extended to a flow, in which case v may take negative values. The following theorem also holds in this alternative setting.

Proof.
Let I A be the invariant σ-algebra of (A, T ), and B M the Borel σ-algebra of M . As a measured space, we can see Ω as A × S 1 × M . Up to completion with respect to µ, the invariant σ-algebra of (Ω, (g t ) t≥0 ) is I : Let U := {dv = 0} ⊂ M . Let (U i , ψ i ) i∈I be a countable cover of U by charts, with ϕ i : U i → W i ⊂ R n and W i bounded. Using the local normal form of submersions, we assume that v•ϕ −1 i (z) = z 1 > 0. We write z = (z 2 , . . . , z n ). Let (V i ) i∈I be a partition of U by open sets, up to a Lebesgue negligible subset of U , such that V i ⊂ U i for all i. We write W i := ϕ i (V i ).
We apply Lemma 1.1, with the Banach space B = B * = L 2 (Ω, µ), and: Let us write d(z) := b(ϕ i (z) 1 )c(ϕ i (z) ) for z ∈ U i . Let (p i ) i∈I be a sequence of positive numbers such that i∈I p i Leb(W i ) = 1. By Proposition 1.3, without loss of generality, we replaceμ byμ := i∈I p i ϕ * i Leb |W i . Let f j = a j e iξ j · d j , with j ∈ {1, 2}, be in E. If the d j have disjoint support, then E(f 1 · f 2 • g t ) = 0 = E(f 1 E(f 2 |I)) for all t, and there is nothing more to prove. We assume without loss of generality that the h j are supported by the same open set V i . Let h(z 1 ) : Then, for all t ≥ 0: If ξ 1 = ξ 2 , there is nothing more to prove. Assume that ξ 1 = ξ 2 =: ξ. Then: We now distinguish between two cases, depending on whether ξ = 0 or not.
In the spirit of Riemann-Lebesgue's lemma, we use an integration by parts to show that the oscillations make the integral decay.
In this case, where: By von Neumann's ergodic theorem, where the convergence is in L 2 norm. Hence, so that: Since the sufficient criterion in Theorem 3.1 is the same as in Theorem 2.3, genericity follows (as for Proposition 2.4): Let (A, ν, T ) be a system preserving a probability measure, M a n-dimensional manifold (with n ≥ 1). Let r ∈ [1, +∞]. For v ∈ C r (M, R * + ), let (Ω, (g v t ) t≥0 ) be defined as above. For C r generic roof functions v, the system (Ω, µ, (g v t ) t≥0 ) exhibits Keplerian shear for any absolutely continuous compatible measure µ.
We shall not discuss the speed of decay of correlations for such systems: not only do the critical points of v matter, so do the decay of correlations on (A, ν, T ).

Systems without Keplerian shear
While systems with Keplerian shear are abundant in the classes we discussed -since the conditions in Theorems 2.3 and 3.1 are generic -, we shall finish with a couple of examples of non-ergodic systems without Keplerian shear. The first is the geodesic flow on the sphere, which falls in the setting of Section 2 but lacks asynchronicity; the second is given by a large class of p-adic translations.

Geodesic flow on a sphere
Let n ≥ 2. The manifold T 1 S n is a fibre bundle over the oriented Grassmannian Gr(2, n + 1) with fibre S 1 . This comes from the fact that the orbits of the geodesic flow on this manifold are oriented grand circles, and the space of oriented grand circle is isomorphic to the space of oriented 2-planes in R n+1 . The geodesic flow acts by translations on the grand circles. Hence the dynamical system (T 1 S n , Liouv, (g t )) belongs to the class of examples discussed in Section 2. The invariant σ-algebra I is isomorphic to B Gr(2,n+1) , and thus non trivial.
However, all grand circles are of the same length, so g t+2π = g t . In particular, given any integrable function h which is not I-measurable, the sequence of functions (h • g t ) t cannot converge to a (g t )invariant function.
Finally, the geodesic flow on T 1 S 1 is isomorphic to the disjoint union of two rotations on S 1 , which are ergodic but not mixing. Hence, the system (T 1 S n , Liouv, (g t )) does not have Keplerian shear for any n ≥ 1.

p-adic translations
Until now, we have seen classes of dynamical systems for which Keplerian shear is generic, with the geodesic flow on T 1 S n being an exception rather than the rule. As we shall see now, the situation is completely different for p-adic translations. Recall that, for p a prime number, the ring Z p is the completion of Z for the p-adic norm. It is compact, and thus supports an invariant probability, which we shall denote Leb.
We shall see that, when one replaces translations on a torus by translations on Z p , the system they get typically does not exhibit Keplerian shear. The reason is that, on Z p , errors do not accumulate: if we change a translation on Z p by a small quantity, the iterates of the two translations still stay close one to another at all times. Proof. If v ≡ 0 almost everywhere, then T is essentially the identity, which has Keplerian shear. Assume that this is not the case. Then one can find A ⊂ M , N ≥ 0, i ∈ {1, . . . , d} and k ∈ {1, . . . , p − 1} such that ν(A) > 0 and v i (x) = kp N + (x)p N +1 for all x ∈ A. Let χ be a non-trivial character on Z /pZ . Let Then, for (x, y) ∈ A × (Z p ) d , f • T n (x, y) = χ(y N,i + nk) = χ(y N,i )χ(k) n .
The function f is non-zero on a set of positive measure, and since χ(k) is a non-trivial pth root of the unit, we get that (f • T n ) n≥0 is exactly p-periodic. Hence, the system (M × Z p , ν ⊗ Leb, T ) does not exhibit Keplerian shear.