Robust transitivity of singular hyperbolic attractors
RRobust transitivity of singular hyperbolic attractors
Sylvain Crovisier ∗ Dawei Yang † January 22, 2020
Abstract
Singular hyperbolicity is a weakened form of hyperbolicity that has been introduced forvector fields in order to allow non-isolated singularities inside the non-wandering set. Atypical example of a singular hyperbolic set is the Lorenz attractor. However, in contrast touniform hyperbolicity, singular hyperbolicity does not immediately imply robust topologicalproperties, such as the transitivity.In this paper, we prove that open and densely inside the space of C vector fields of acompact manifold, any singular hyperbolic attractors is robustly transitive. Lorenz [L] in 1963 studied some polynomial ordinary differential equations in (cid:82) . Hefound some strange attractor with the help of computers. By trying to understand thechaotic dynamics in Lorenz’ systems, [ABS, G, GW] constructed some geometric abstractmodels which are called geometrical Lorenz attractors : these are robustly transitive non-hyperbolic chaotic attractors with singularities in three-dimensional manifolds.In order to study attractors containing singularities for general vector fields, Morales-Pacifico-Pujals [MPP] first gave the notion of singular hyperbolicity in dimension 3. Thisnotion can be adapted to the higher dimensional case, see [BdL, CdLYZ, MM, ZGW].In the absence of singularity, the singular hyperbolicity coincides with the usual notionof uniform hyperbolicity; in that case it has many nice dynamical consequences: spectraldecomposition, stability, probabilistic description,... But there also exist open classes ofvector fields exhibiting singular hyperbolic attractors with singularity: the geometricalLorenz attractors are such examples. In order to have a description of the dynamics ofgeneral flows, we thus need to develop a systematic study of the singular hyperbolicity inthe presence of singularity. This paper contributes to that goal.We do not expect to describe the dynamics of arbitrary vector fields. Instead, oneconsiders the Banach space X r p M q of all C r vector fields on a compact manifold M withoutboundary and focus on a subset G which is dense and as large as possible. A successfulapproach consists in considering subsets that are C -residual (i.e. containing a dense G δ subset with respect to the C -topology), but this does not handle immediately smoothersystems. For that reason it is useful to work with subsets G Ă X p M q that are C -openand C -dense and to address for each dynamical property the following question: Knowing that a given property holds on a C -residual subset of vector fields, is it satisfiedon a C -open and dense subset? ∗ S.C. was partially supported by the ERC project 692925
NUHGD . † D.Y. was partially supported by NSFC (11822109, 11671288, 11790274, 11826102). a r X i v : . [ m a t h . D S ] J a n recise setting. Given a vector field X P X p M q , the flow generated by X is denotedby p ϕ Xt q t P (cid:82) , and sometimes by p ϕ t q if there is no confusion. A point σ is a singularity of X if X p σ q “
0. A point p is periodic if it is not a singularity and there is T ą ϕ T p p q “ p . We denote by Sing p X q the set of singularities and by Per p X q the set of periodicorbits of X . The union Crit p X q : “ Sing p X q Y Per p X q is the set of critical elements of X .We will mainly discuss the recurrence properties of the dynamics. An invariant compactset Λ is transitive if it contains a point x whose positive orbit is dense in Λ. More generallyΛ is chain-transitive if for any ε ą x, y P Λ, there exists x “ x, x , . . . , x n “ y in Λand t , t , . . . , t n ´ ě d p x i ` , ϕ t i p x i qq ă ε for each i “ , . . . , n ´
1. A compactinvariant set Λ is said to be a chain-recurrence class if it is chain-transitive, and is not aproper subset of any other chain-transitive compact invariant set. The chain-recurrenceclasses are pairwise disjoint.Among invariant sets, important ones are those satisfying an attracting property. Aninvariant compact set Λ is an attracting set is there exists a neighborhood U such that X t ą ϕ t p U q “ Λ and an attractor if it is a transitive attracting set. More generally Λis
Lyapunov stable if for any neighborhood V there exists a neighborhood U such that ϕ t p U q Ă V for all t ą robustlytransitive if there exist neighborhoods U of Λ and U of X in X p M q such that for any Y P U , the maximal invariant set X t P (cid:82) ϕ Yt p U q is transitive, and coincides with Λ when Y “ X .As said before, singular hyperbolicity is a weak notion of hyperbolicity that has beenintroduced in order to characterize some robust dynamical properties. A compact invariantset Λ is singular hyperbolic if for the flow p ϕ t q generated by either X or ´ X , there are acontinuous Dϕ t -invariant splitting T Λ M “ E ss ‘ E cu and T ą x P Λ: • E ss is contracted: } Dϕ T | E ss p x q } ď { • E ss is dominated by E cu : } Dϕ T | E ss p x q }} Dϕ ´ T | E cu p ϕ T p x qq } ď { • E cu is area-expanded: | det Dϕ ´ T | P | ď { P Ă E cu p x q .Note that if Λ is a singular hyperbolic set, and Λ X Sing p X q “ H , then Λ is a hyperbolicset.Some robust properties or generic assumptions imply the singular hyperbolicity. Indimension 3, robustly transitive sets are singular hyperbolic [MPP] and any generic vectorfield X P X p M q far from homoclinic tangencies supports a global singular hyperbolicstructure [CY1, CY2]. In higher dimension, the transitive attractors of generic vector field X P X p M q satisfying the star property are singular hyperbolic [SGW]. Statement of the results.
It is well known that for uniformly hyperbolic sets: • chain-transitivity and local maximality (i.e. the set coincides with the maximal in-variant set in one of its neighborhoods) imply the robust transitivity; • Lyapunov stability implies that the set is an attracting set.We do not know whether these properties extend to general singular hyperbolic sets butthis has been proved in [PYY] for C -generic vector fields: generically Lyapunov stablechain-recurrence classes which are singular hyperbolic are transitive attractors. We showthat this holds robustly. Theorem A.
There is an open and dense set U P X p M q such that for any X P U , anysingular hyperbolic Lyapunov stable chain-recurrence class Λ of X is a robustly transitiveattractor. et us mention that a more general notion of hyperbolicity, called multi-singular hy-perbolicity has been recently introduced in order to characterize star vector fields, in [BdL](see also [CdLYZ]). In contrast to Theorem A above, a multi-singular chain-recurrenceclass of a C -generic vector field may be isolated and not robustly transitive [dL].Under the setting of Theorem A, we have a more accurate description of the singularhyperbolic attractors in Theorem A. Two hyperbolic periodic orbits γ and γ are homo-clinically related if W s p γ q intersects W u p γ q transversely and W s p γ q intersects W u p γ q transversely. The homoclinic class H p γ q of a hyperbolic periodic orbit γ is the closure ofthe union of the periodic orbits that are homoclinically related to γ . This is a transitiveinvariant compact set. In dimension 3, any singular hyperbolic transitive attractor is ahomoclinic class [APu]; we show that this also holds in higher dimension for vector fieldsin a dense open set. Theorem B.
There is an open and dense set U P X p M q such that for any X P U ,any singular hyperbolic Lyapunov stable chain-recurrence class Λ of X (not reduced to asingularity) is a homoclinic class (in particular the set of periodic points is dense in Λ ).Moreover, any two periodic orbits contained in Λ are hyperbolic and homoclinicallyrelated. In dimension 3, we know more properties of chain-recurrence classes with singularitiesfor generic systems. This gives the following consequence:
Corollary C.
When dim M “ , there is an open dense set U P X p M q such that forany X P U , any singular hyperbolic chain-recurrence class is robustly transitive. Unless theclass is an isolated singularity, it is a homoclinic class. Theorem A in fact solves Conjecture 7.5 in [BM]: a C generic three-dimensional flowhas either infinitely many sinks or finitely many robustly transitive attractors (hyperbolicor singular hyperbolic ones) whose basins form a full Lebesgue measure set of M . With [M],only the robustness was unknown before this paper.
Further discussions.
It is natural to expect that the previous results hold for arbitrarysingular hyperbolic chain-recurrence class, also in higher dimension:
Question 1.
Does there exists (when dim p M q ě ) a dense and open subset U Ă X p M q such that for any X P U , any singular hyperbolic chain-recurrence class is robustly transi-tive? is a homoclinic class? This would imply in particular that if X P U is singular hyperbolic (i.e. each of its chain-recurrence class is singular hyperbolic), it admits finitely many chain-recurrence classesonly.One can also study stronger forms of recurrence. It is known that for C r -generic vectorfield, each homoclinic class is topologically mixing, see [AAB] and one may wonder if thisholds robustly. Question 2.
Does there exists a dense and open subset U Ă X p M q such that for any X P U , any singular hyperbolic transitive attractor is robustly topologically mixing. The answer is positive in the case of non-singular transitive attractors [FMT]. Also[AM] proves that X p M q contains a C -dense and C -open subset of vector fields suchthat any singular hyperbolic robustly transitive attractor with dim p E cu q “ Preliminaries
Let us consider a C vector field X and a compact subset K of M . We will use thefollowing notations. The orbit of a point x under the flow of X is denoted by Orb p x q . Twoinjectively immersed sub-manifolds W and W are said to intersect transversely at a point x P W X W if T x W ` T x W “ T x M . The set of the transverse intersections is denotedby W & W . Hyperbolicity.
A compact invariant set Λ of a vector field X is hyperbolic if there is acontinuous invariant splitting T Λ M “ E ss ‘ (cid:82) .X ‘ E uu , and constants C, λ ą x P Λ and any t ě } Dϕ t | E ss p x q } ď C e ´ λt and } Dϕ ´ t | E uu p x q } ď C e ´ λt . Note that dim (cid:82) .X p x q “ x is a singularity.To each point x in a hyperbolic set Λ is associated a strong stable manifold W ss p x q anda strong unstable manifold W uu p x q that are tangent to E ss and E uu , respectively. For ahyperbolic critical element γ we introduce the stable and unstable sets W s p γ q and W u p γ q .Note that: • for a hyperbolic singularity σ , we have W s p σ q “ W ss p σ q and W u p σ q “ W uu p σ q ; • for a hyperbolic periodic orbit γ , the sets W s p γ q and W u p γ q are injective immersedsub-manifolds and W s p γ q “ ď x P γ W ss p γ q , W u p γ q “ ď x P γ W uu p γ q . Homoclinic classes.
The homoclinic class of a hyperbolic periodic orbit γ is H p γ q “ W s p γ q & W u p γ q . The set H p γ q can also be defined in another way. Two hyperbolic periodic orbits γ and γ are said to be homoclinically related if W s p γ q & W u p γ q and W s p γ q & W u p γ q arenon-empty. The Birkhoff-Smale theorem implies that this is an equivalence relation andthat H p γ q is the union of the hyperbolic periodic orbits that are homoclinically relatedwith γ . Moreover H p γ q is a transitive set. See [N] and [APa, Section 2.5.5]. Chain-recurrence classes. If σ is a critical element of X , we denote by C p σ q thechain-recurrent class of X that contains σ . Singular hyperbolicity.
As for the uniform hyperbolicity, the singular hyperbolicityis robust: there exist a C -neighborhood U of X and a neighborhood U of Λ such that forany Y P U , any ϕ Y -invariant compact set Λ contained in U is singular hyperbolic.In this case, any point x P Λ admits a strong stable manifold W ss p x q tangent to E ss p x q .Moreover one can choose a neighborhood W ssloc p x q of x in W ss p x q which is an embeddeddisc which varies continuously for the C -topology with respect to x and to the vector field. .2 Genericity We recall several generic results. Some notations and definitions are given first. • For a hyperbolic critical element γ of X , its hyperbolic continuation will be denotedby γ Y for Y that is C -close to X . • A compact invariant set Λ is locally maximal if there is a neighborhood U of Λ suchthat Λ “ X t P (cid:82) ϕ t p U q . Proposition 2.1.
There is a dense G δ set G in X p M q such that for any X P G , we have:1. X is Kupka-Smale: each critical element of X is hyperbolic; the intersections of thestable manifold of one critical element with the unstable manifold of another criticalcritical element are all transverse.2. For any σ P Crit p X q , the map Y ÞÑ C p σ Y q is continuous at X for the Hausdorfftopology.3. If a chain-recurrence class contains a hyperbolic periodic orbit γ , then it coincideswith the homoclinic class H p γ q of γ .4. If a chain-recurrence class of a critical element σ is locally maximal, then it is robustlylocally maximal: there are a neighborhood U of X in X p M q and a neighborhood U of C p σ q such that C p σ Y q is the maximal invariant set in U for any Y P U .5. For any critical elements γ , γ in a same chain-recurrence class C p γ q and satisfying dim W u p γ q ě dim W u p γ q , then any neighborhood U x of a point x P C p γ q X W sloc p γ q contains a point y P W u p γ q & W sloc p γ q .6. Any hyperbolic periodic orbits in a same chain-recurrence class and with the samestable dimension are homoclinically related.7. For γ P Crit p X q , if C p γ q contains the unstable manifold W u p γ q , then it is Lyapunovstable and W u p γ q “ C p γ q . Moreover W u p γ Y q Ă C p γ Y q still holds for any Y that is C -close to X .8. If σ P Sing p X q has unstable dimension equal to one, then either C p σ q “ t σ u or C p σ q is Lyapunov stable. The properties in Proposition 2.1 are well known. We give some comments:– Item 1 is the classical Kupka-Smale theorem [K, S].– Item 2 follows from the upper semi-continuity of the chain-recurrence class C p σ Y q with respect to the vector field: the continuity holds at generic points.– Item 3 and Item 4 have been proved for diffeomorphisms in [BC, Remarque 1.10 andCorollaire 1.13]. The proofs for flows are similar.– Item 5 is an application of the connecting lemma in [BC]. For any point x as in thestatement, by using the connecting lemma, there is a point y close to x such that y P W u p γ q & W sloc p γ q for Y close to X . Then one can apply a Baire argument toconclude.– Item 6 is a consequence of Item 5.– Item 7 and 8 are applications of the connecting lemma for pseudo-orbits in [BC], seealso [GY, Lemmas 3.13, 3.14 and 3.19].We know the following theorem from [PYY, Corollary C] and [ALM, Theorem 1.1] (andpreviously [MP, Theorem D] in the three-dimensional case). heorem 2.2. There is a dense G δ set G Ă X p M q such that for any vector field X P G ,if C p σ q is a singular hyperbolic Lyapunov stable chain-recurrence class (and not reduced to σ ), then C p σ q contains periodic orbits and is an attractor. The following proposition gives some open and dense properties for chain-recurrenceclasses.
Proposition 2.3.
There is an open and dense set U Ă X p M q such that any X P U has aneighborhood U X with the following property. For each σ P Sing p X q and Y P U X ,1. W u p σ X q Ă C p σ X q ô W u p σ Y q Ă C p σ Y q ;2. C p σ X q is non-trivial (i.e. not reduced to t σ X u ) if and only if C p σ Y q is non-trivial.Proof. We take the dense G δ set G provided by Proposition 2.1 and we consider the subset G n Ă G of vector fields with exactly n singularities. Consider X P G n and denote the set ofits singularities by t σ , ¨ ¨ ¨ , σ n u .Note that Closure p W u p γ X qq varies lower semi-continuously with respect to the vectorfield X and C p γ X q varies upper semi-continuously with respect to the vector field X .So, for any (hyperbolic) critical element γ of X , if W u p γ X qz C p γ q ‰ H , then there is aneighborhood U X such that W u p γ Y qz C p γ Y q ‰ H . Similarly, if C p σ X q “ t σ X u , there existsa neighborhood U X such that C p σ Y q is contained in a small neighborhood of C p σ X q . Since σ X is hyperbolic, C p σ Y q “ t σ Y u .Together with the Items 2 and 7 of Proposition 2.1, for each σ i , there is an open set U i,X such that1. either for any Y P U i,X we have W u p σ i,Y qz C p σ i,Y q ‰ H , or for any Y P U i,X we have W u p σ i,Y q Ă C p σ i,Y q ;2. C p σ i q is non-trivial if and only if C p σ i,Y q is non-trivial for any Y P U i,X .By reducing U i,X if necessary for each i , any singularity of Y P X ni “ U i,X is the contin-uation of a singularity of X . Now we take U “ ď n P (cid:78) U n , where U n “ ď X P G n pX ni “ U i,X q . It is clear that G Ă U . Thus U is dense. Now we check that the open set U has therequired properties. For any vector field Y P U , there is n P (cid:78) and a vector field X P G n such that Y P X ni “ U i,X . Thus, any singularity σ Y of Y is a continuation of a singularity σ i,X of X . By the choice of U i,X , we have1. W u p σ Y q Ă C p σ Y q if and only if W u p σ Z q Ă C p σ Z q for any Z P X ni “ U i,X .2. C p σ Y q is non-trivial if and only if C p σ i,Z q is non-trivial for any Z P X ni “ U i,X .This implies the proposition. Proposition 2.4.
Assume that C p σ q is a transitive singular hyperbolic attractor of a vectorfield X containing a hyperbolic singularity σ (and not reduced to σ ). Then there is aneighborhood U X of X such that the continuation C p σ Y q of any Y P U X satisfies:1. C p σ Y q admits a singular hyperbolic splitting T C p σ Y q M “ E ssY ‘ E cuY for Y such that E ssY is Dϕ YT -contracted and E cu is Dϕ YT -area-expanded for some T ą .2. For each x P C p σ Y q , we have that Y p x q Ă E cuY p x q . . The stable space E sY p ρ q of any singularity ρ P C p σ Y q has a dominated splitting E sY p ρ q “ E ssY p ρ q ‘ E cY p ρ q such that dim E cY “ . Moreover W ssY p ρ q X C p σ Y q “ t ρ u .Proof. We first prove Item 1 for the vector field X . Let us recall a result from [BGY,Lemma 3.4]: for a transitive set Λ with a dominated splitting T Λ M “ E ‘ F we have • either, X p x q P E p x q for any point x P Λ, • or, X p x q P F p x q for any point x P Λ.Now we consider a transitive singular hyperbolic attractor C p σ q . Assume by contra-diction that it has a singular hyperbolic splitting T C p σ q M “ E cs ‘ E uu such that E uu isexpanded. Since it is an attractor, the strong unstable manifold of σ is contained in C p σ q .For a point z P W uu p σ q , we have X p z q P E uu p z q . Thus, for any point x P C p σ q , we have X p x q P E uu p x q . But (cid:82) .X cannot be uniformly expanded everywhere and one gets a contra-diction. Thus the singular hyperbolic splitting on C p σ q has the form T C p σ q M “ E ss ‘ E cu and Item 1 is proved for X . Since the singular hyperbolic splitting is robust, and the chain-recurrence class varies upper semi-continuously, C p σ Y q admits the same kind of splittingfor any Y close to X . This proves Item 1.Now we prove Items 2 and 3 for the vector field X . By using the result in [BGY] again,if for some regular point x , we have X p x q P E ss p x q , then this property holds for every point.But (cid:82) .X cannot be uniformly contracted and one gets a contradiction again. Thus Item 2for X is proved. Since for any point x P C p σ q , we have X p x q P E cu p x q , the strong stablemanifold W ss p σ q cannot contain a regular orbit. Thus W ss p σ q X C p σ q “ t σ u . On the otherhand, the stable manifold W s p σ q contains regular orbits of C p σ q since C p σ q is transitive.Thus dim E s p σ q ą dim E ss p σ q . If dim E s p σ q ą dim E ss p σ q `
1, then the dimension of theinvariant space E c p σ q : “ E s p σ q X E cu p σ q is at least 2. But this space is not area-expanded,this contradicts the definition of the singular hyperbolicity. Thus, we have dim E c p σ q “ X is complete.Now we give the proof of Item 3 for any Y close to X . Since C p σ q is an attractorof X , there is a neighborhood U of C p σ q such that U X Sing p X q Ă C p σ q . Consequently,for Y close to X , any singularity ρ Y P C p σ Y q is a continuation of some singularity ρ X in C p σ q . Since Item 3 holds for ρ X , for Y close to X the singularity ρ Y still has a dominatedsplitting E sY p ρ q “ E ssY p ρ q ‘ E cY p ρ q with dim p E cY “ ρ Y P C p σ Y q .Let us assume by contradiction that there is a sequence of vector fields p X n q Ñ X anda singularity ρ X P C p σ q such that W ssX n p ρ X n q X C p σ X n qzt ρ X n u ‰ H . Thus, there is ε that is independent of n such that the local manifold W ssX n ,ε p ρ X n q ofsize ε intersects C p σ X n q for each n P (cid:78) . Since the homoclinic class C p σ Y q is upper semi-continuous with respect to Y , we have that the boundary of W ssε p ρ q intersects C p σ q . Thiscontradicts Item 3 for X . Thus the proof of Item 3 of this proposition is complete.Let us assume by contradiction that Item 2 fails: this means that there is a sequenceof vector fields X n Ñ X and a sequence of points x n P C p σ X n q such that X n p x n q is notcontained in E cuX n p x n q .From the dominated splitting E ss ‘ E cu , by replacing x n by a backward iterate, onecan assume furthermore that the angle between (cid:82) .X n p x n q and E ssX n p x n q is less than 1 { n .Since for regular points y close to a singularity ρ , the angle between (cid:82) .X n p y q and E ssX n p y q s uniformly bounded away from zero, the points x n are far from the singularities. One canthus assume that p x n q converges to a regular point x P C p σ q . By taking the limit, we have X p x q P E ss p x q . This contradicts the Item 2 for X . Thus Item 2 holds for any Y close to X . This section is devoted to the following result which will be used to prove the densityof the unstable manifold.
Theorem 3.1.
There is a dense G δ set G in X p M q such that for any X P G , for any sin-gular hyperbolic Lyapunov stable chain-recurrence class C p σ q of X , and for any hyperbolicperiodic orbit γ in C p σ q , there exists a neighborhood U X of X with the following property.For any Y P U X and for any x P C p σ Y q , either x belongs to the unstable manifold of asingularity or W ssloc,Y p x q intersects W uY p γ Y q transversely. Let G be a dense G δ subset in X p M q given by Proposition 2.1. Let X P G . We firstprove some preliminary lemmas. Lemma 3.2.
For any singularity ρ P C p σ q , and for any x P p W s p ρ q X C p σ qqzt ρ u , thesubmanifolds W ssloc p x q and W u p γ q have a transverse intersection point.Proof. Consider x P p W s p ρ q X C p σ qqzt ρ u . Without loss of generality, we may assume that x belongs to W sloc p ρ q .By the Item 5 of Proposition 2.1, there exists a transverse intersection point y between W sloc p ρ q and W u p γ q near x . The flow preserves W u p γ q , W s p ρ q and the strong stable foliationof W s p ρ q ; moreover since the strong stable foliation is one-codimensional inside W s p ρ q andsince x, y are two points close in W s p ρ qz W ss p ρ q , there exists t P (cid:82) such that ϕ t p W ss p y qq “ W ss p x q . In particular, ϕ t p y q is a transverse intersection point between W ss p x q and W u p γ q .That point can be chosen arbitrarily close to x in W ss p x q , hence belongs to W ssloc p x q .Since the unstable manifold W u p γ Y q and the leaves of the strong stable foliation W ssloc,Y p x q vary continuously for the C topology with respect to x and Y , a compactness argumentgives: Corollary 3.3.
For any singularity ρ P C p σ q , let K ρ be a compact subset of W s p ρ q X C p σ qqzt ρ u . Then there exists a neighborhood V ρ of K ρ and a C -neighborhood U ρ of X such that for any Y P U ρ and any y P V ρ X C p σ Y q , the submanifolds W ssloc,Y p y q and W u p γ Y q have a transverse intersection point. Lemma 3.4.
For any invariant compact set K reg Ă C p σ qz Sing p X q , there exists a neigh-borhood V reg of K reg and a C -neighborhood U reg of X such that for any Y P U reg and any y P V reg X C p σ Y q , the submanifolds W ssloc,Y p y q and W u p γ Y q have a transverse intersectionpoint.Proof. Since C p σ q is singular hyperbolic, the invariant compact set K reg Ă C p σ qz Sing p X q is hyperbolic. By the shadowing lemma, and a compactness argument, there exist finitelymany periodic orbits γ , . . . , γ (cid:96) in an arbitrarily small neighborhood of K reg such that forany y P K reg , the submanifolds W ssloc p y q intersects transversally some W u p γ i q . Moreovereach γ i is in the chain-recurrence class of a point of K reg , hence is contained in C p σ q . Bythe Item 6 of Proposition 2.1, γ i is homoclinically related to γ . The inclination lemma thenimplies that for any y P K reg , the submanifolds W ssloc p y q intersects transversally W u p γ q . Asbefore this property is C -robust. roof of Theorem 3.1. Let ρ , . . . , ρ (cid:96) be the singularities contained in C p σ q . For each ofthem, one chooses a compact set K i Ă W s p ρ i q X C p σ qqzt ρ i u which meets each orbit of W s p ρ i q X C p σ qqzt ρ i u . The Corollary 3.3 associates open sets V i and U i . There exists aneighborhood O i of ρ i such that any point z P O i z W uloc p ρ i q has a backward iterate in acompact subset of V i .Let K reg be the maximal invariant set of C p σ qz Y i p V i Y O i q . The Lemma 3.4 associatesopen sets U reg and V reg . By construction, any point z P C p σ q either belongs to some W u p ρ i q , has a backward iterate in a compact subset of some V i , or belongs to K reg . Since C p σ Y q varies upper semi-continuously with Y , this property is still satisfied for Y in a C -neighborhood U return of X .We set U X “ U X ¨ ¨ ¨ X U (cid:96) X U reg X U return . For any Y P U X and any x P C p σ q whichdoes not belong to the unstable manifold of a singularity ρ i,Y , there exists a backwarditerate ϕ Y ´ t p x q which belongs either to some V i or to V reg . In both cases, W ssloc,Y p ϕ Y ´ t p x qq intersects W uY p γ Y q transversely. This concludes. This section is devoted to the following result which will be used to prove the densityof the stable manifold.
Theorem 4.1.
There is a dense G δ set G in X p M q such that for any X P G , for anysingular hyperbolic transitive attractor C p σ q of X , and for any hyperbolic periodic orbit γ in C p σ q , there exist a neighborhood U of C p σ q and a neighborhood U X of X with thefollowing property.For any Y P U X , the stable manifold W sY p γ Y q is dense in U . Moreover, for any periodicorbit γ Ă C p σ Y q , the set of transverse intersections W uY p γ q & W sY p γ Y q is dense in W uY p γ q . The set G is the dense G δ subset of vector fields satisfying the Items 1 and 3 of Proposi-tion 2.1. In the whole section, we fix X P G and C p σ q , γ as in the statement of Theorem 4.1. We consider the singular hyperbolic splitting T C p σ q M “ E ss ‘ E cu on C p σ q (see theItem 1 of Proposition 2.4). These bundles may be extended as a continuous (but maybenot invariant) splitting T U M “ r E ss ‘ r E cu on a neighborhood U of C p σ q . For x P U and α ą center-unstable cone C α p x q “ t v P T x M : v “ v s ` v c , v s P r E ss p x q , v c P r E cu p x q , } v c } ď α } v s }u . Since the splitting is uniformly dominated, we have the following lemma.
Lemma 4.2.
For any α ą β ą , there are T ą , a C -neighborhood U of X and neigh-borhood U Ă U of C p σ q such that for any Y P U and for any orbit segment ϕ Y r ,t s p x q Ă U with t ě T , Dϕ Yt p C α p x qq Ă C β p ϕ Yt p x qq . Proof.
The proof for the vector field X above the class C p σ q is standard, hence is omitted.The conclusion extends to neighborhoods of X and C p σ q by continuity of the cone fieldsand of Dϕ Yt p x q , t P r T, T s , with respect to p x, t, Y q . n the following we fix α ą C “ C a . Note that (by increasing T , reducing U , U and using the singular hyperbolicity) the following property holds: for any Y P U ,any orbit segment ϕ Y r ,t s p x q Ă U with t ě T and any 2-dimensional subspace P Ă C α p x q , | det p Dϕ Yt q| P | ě . (1)The inner radius r of a submanifold Γ is the supremum of R ą B p , R q Ă T x Γ Ñ Γ with respect to the metric induced on Γ by the Riemannianmetric of M is well defined and injective for some x P Γ. Note that Γ always contain asubmanifold Γ Ă γ with same inner radius r and with diameter smaller than or equal to2 r : indeed one considers a sequence of balls B p x k , R k q Ă Γ with R k Ñ R that are theimages of exponential maps and consider a limit point x for p x k q ; the submanifold Γ is theball B p x, R q Ă Γ.Theorem 4.1 is a consequence of the next proposition.
Proposition 4.3.
Under the setting of Section 4.1, there exists ε ą , a C -neighborhood U Ă U of X and a neighborhood U of C p σ q with the following property.For any Y P U and any submanifold Γ Ă U of dimension dim p E cu q , tangent to thecenter-unstable cone field C and satisfying Y p x q P T x Γ for each x P Γ , there is t ą such that ϕ Y r ,t s p Γ q : “ Y s Pr ,t s ϕ Ys p Γ q contains a submanifold tangent to C , with dimension dim p E cu q and inner radius larger than ε . The proof of this proposition is postponed to the end of the Section 4.
Proof of Theorem 4.1.
Since C p σ q is a transitive attractor, it is the chain-recurrence classof γ . By Item 3 of Proposition 2.1, C p σ q coincides with the homoclinic class H p γ q of γ . Hence, the stable manifold of γ is dense in C p σ q , hence intersects transversally anysubmanifold of dimension dim p E cu q tangent to C of inner radius ε and which meets a smallneighborhood U Ă U of C p σ q . By continuation of the stable manifold, this property is stillsatisfied for vector fields Y in a small C -neighborhood U X Ă U of X . Since C p σ q is anattractor, one may reduce U X , choose a smaller neighborhood U Ă U of C p σ X q and assumethat ϕ Yt p U q Ă U for any t ą Y P U X . Let O be a small isolating neighborhood ofSing p X q . By the Item 2 of Proposition 2.4, X is tangent to C on C p σ X qz O . Up to reducethe neighborhoods U and U , one can thus require that the vector fields Y P U are tangentto C on U z O .In order to check that W sY p γ Y q is dense in U for any Y P U X , we choose an arbi-trary non-empty open subset V Ă U z O and a submanifold Γ Ă V as in the statement ofProposition 4.3: such a submanifold can be built by choosing a small disc Σ of dimensiondim p E cu q ´ C and transverse to X ; then we set Γ “ Y | t |ă δ ϕ Yt p Σ q for δ close to0; since Y is tangent to C on U z O , the submanifold Γ is tangent to C as required. From thechoice of U and the Proposition, there exists t ą ϕ Yt p Γ q has inner radius largerthan ε , intersects U , is still tangent to C , hence meets W s p γ Y q transversally. This showsthat W s p γ Y q intersects V as required. Let us now consider an open set V Ă O . Since O is isolating for Y , there exists a forward iterate ϕ Yt p V q which meets U z O . In particular W sY p γ Y q meets ϕ Yt p V q , hence V . We have proved that W sY p γ Y q is dense in U .For the last part of the Theorem, one considers a small open subset of W uY p γ q . It is asubmanifold Γ as considered above, hence it intersects transversally W sY p γ Y q . W s p ρ q Figure 1: Construction of the cross-sections close to a singularity.
Let Y be a vector field whose singularities ρ are hyperbolic. One associates to each ofthem their local stable and unstable manifold W sY,loc p ρ q , W sY,loc p ρ q . Definition 4.4. A cross section of Y is a co-dimension sub-manifold D Ă M z Sing p Y q such that there exists α ą and a compact subset ∆ Ă D satisfying:– D X Sing p X q “ H and for each x P D , the angle between Y p x q and T x D is larger than α ,– the interior of ∆ in D intersects any forward orbit in M z Y ρ W sY,loc p ρ q and anybackward orbit in M z Y ρ W uY,loc p ρ q ,– For each singularity ρ , D intersects W sY,loc p ρ q along a fundamental domain containedin the interior of ∆ : the intersection is a one-codimensional sphere inside W sY,loc p ρ q which meets exactly once each orbit in W sY,loc p ρ q ; similarly D intersects W uY,loc p ρ q along a fundamental domain contained in the interior of ∆ .Such a compact set ∆ Ă D is called a core of the cross section D . Lemma 4.5.
There exists a cross section D for Y .Proof. We cover the set of regular orbits by finitely small transverse sections Σ such thateach local stable or unstable manifold of a singularity ρ P C p σ q meets only one of thesesections, along a one-codimensional sphere. The union of these small sections satisfies thedefinition, but is maybe not a submanifold. One can always perturb them so that eachintersection Σ X Σ is transverse. We modify them so that they become pairwise disjoint:each time two small sections Σ , Σ intersect along a submanifold N , one removes somesmall neighborhoods U , U of N in each of them and one adds two new small disjointsections which are the images of U , U by small times of the flow.Let ∆ be the union of the obtained transverse sections. The section D is built byenlarging slightly ∆.We now fix the cross section D for Y . Note that it still satisfies the definition of crosssection (with the same core ∆) for vector fields Z that are C -close to Y . We also consideran open set U Ă M . efinition 4.6. A holonomy for p Y, D, U q is a C -diffeomorphism π : V Ñ ˜ V between opensets V, ˜ V Ă D X U such that there exists a continuous function t : V Ñ p , `8q satisfyingfor each x P V : • π p x q “ ϕ t p x q p x q P ˜ V , • ϕ s p x q P U for any s P r , t s .The time t is called transition time of x for π . (We do not require t to be the first returntime to ˜ V . ) Construction and continuation of holonomies.
One can build a holonomy by consid-ering two open subsets V , ˜ V Ă D , an open set W Ă U and an open interval I in p , `8q satisfying: (H) For any piece of orbit t ϕ Ys p x q , s P r , t su in W with t P I , at most one point ϕ Ys p x q with s P r , t s X I belongs to ˜ V .One then considers all the connected pieces of orbit contained in W which meet both V and ˜ V at points x P V and ϕ Yt p x q P ˜ V for some t P I : the set V (resp. ˜ V ) is the setof the points x (resp. ϕ Yt p x q ) and we set π p x q “ ϕ t p x q . By property (H), the map π iswell defined; since the flow is transverse to D , the sets V and ˜ V are open in D . If theproperty (H) still holds for vector fields Z that are C -close to Y , this construction definesalso holonomies π Y : V Z Ñ ˜ V Z for Z , called continuation of π .The dynamics can be covered by a finite collection of holonomies admitting continua-tions. Since we may want to localize the dynamics, one considers an attracting invariantcompact set Λ with an attracting neighborhood U : there exists t ą p ϕ s p x qq s ą t of any point x P U is contained in U and accumulates on a subset of Λ. Proposition 4.7.
Let us consider for Y a cross section D with a core ∆ , an attractingset Λ with an attracting neighborhood U and T ą . Then, there exist a finite collectionof holonomies π Yi : V Yi Ñ ˜ V Yi , i “ , . . . , (cid:96) for p Y, D, U q , a C -neighborhood U of Y , aneighborhood O of Λ X ∆ in D , and ε ą such that for each Z P U ,(i) the holonomies admit continuations π Zi : V Yi Ñ ˜ V Zi ,(ii) O z Y ρ W sloc p ρ Z q Ă Y i V Zi and Y i ˜ V Zi Ă ∆ ,(iii) the transition time of each honolomy π Zi is bounded from below by T ,(iv) if a set A Ă D z Y ρ W sloc p ρ Z q is contained in the ε -neighborhood of Λ X ∆ and hasdiameter smaller than ε , then A is contained in a V Zi .Proof. The holonomies are built in two different ways depending if the points are close to W sloc p Sing p Y qq or not. Claim.
For any ρ P Λ X Sing p Y q , there exist a neighborhood O ρ of W sloc p ρ q X Λ X ∆ in D and a holonomy π admitting continuations for any Z close to Y , such that the domain V Z of π Z contains O ρ z W sloc p ρ Z q , the image ˜ V Z is contained in ∆ and the transition time isbounded from below by T .Proof. By definition of the cross section, the sets ∆ s “ W sloc p ρ q X ∆ and ∆ u “ W uloc p ρ q X ∆are compact fundamental domains in the local stable and unstable manifolds. Let V and˜ V be neighborhoods of W sloc p ρ q X Λ X ∆ and W uloc p ρ q X Λ X ∆ in D and let W Ă U be asmall neighborhood of Λ X p W sloc p ρ q Y W uloc p ρ qq . A piece of orbit in W intersects ˜ V at mostone, hence the property (H) is satisfied for the time interval I “ p , `8q and any Z close o Y . As a consequence, this defines a holonomy π : V Ñ ˜ V which admits a continuationfor vector fields Z that are C -close to Y .For any Z close to Y , the intersection ∆ X W sloc p ρ Z q is contained in a small neighborhood O ρ of ∆ s in D and V Z contains O ρ z W sloc p ρ Z q . Moreover by definition of the cross sections, W uloc p ρ q X Λ X ∆ is contained in the interior of ∆ in D , hence ˜ V Ă ∆. We may have chosen W small enough so that the transition times of the holonomies are larger than T . Claim.
For each x P p Λ X ∆ qz Y ρ W sloc p ρ q , there exist a neighborhood O x of x in D anda holonomy π admitting continuations for any Z close to Y , such that the domain V Z of π Z contains O x , the image ˜ V Z is contained in ∆ and the transition time is bounded frombelow by T .Proof. By definition of the cross sections, there exists t ą T such that ϕ t p x q belongs tothe interior of ∆. We choose small open neighborhoods V Ă D and ˜ V Ă ∆ in D , a smallneighborhood W Ă U of the piece of orbit t ϕ s p x q , s P r , t su Ă Λ and the time interval p t ´ δ, t ` δ q for δ ą Z closeto Y with transition time bounded from below by T .By compactness of Λ X ∆, we can select from the holonomies provided by the previousclaims a finite number of them such that the union O of the open sets O ρ and O x containΛ X ∆. These holonomies admit continuations for vector fields Z in a C -neighborhood U of Y and satisfy the items (i), (ii) and (iii). There exists ε such that if A Ă D has diametersmaller than ε and is contained in the ε -neighborhood of Λ X ∆, then A is included insome open set O ρ or O x ; the item (iv) follows.As in Section 4.1, we consider a center-unstable cone field on an open neighborhood U of a singular hyperbolic chain-recurrence class C p σ q . Definition 4.8.
Let Z be a vector field C -close to Y . A cu-section of Z is a submanifold N Ă D with dimension dim p E cu q ´ such that T x N ‘ Z p x q Ă C p x q for each x P N . The forward invariance of the cone field (see Lemma 4.2) implies:
Lemma 4.9.
Let us consider open sets V , ˜ V and W Ă U satisfying (H) and the associatedholonomy π . Then the image by π of a cu-section N is still a cu-section. The minimal norm of a linear map A between two euclidean spaces is defined by m p A q : “ inf t} Av } : unit vector v u . Definition 4.10.
A holonomy π : V Ñ ˜ V is if there exists χ ą suchthat for any cu-section N Ă V , the derivative Dπ | N has minimal norm larger than χ withrespect to the induced metrics on N and π p N q . The definition of the singular hyperbolicity (in particular condition (1)) and the uniformtransversality between D and the vector field imply: Lemma 4.11.
Let us consider a cross section D and an open neighborhood U of a singularhyperbolic chain-recurrence class C p σ q as in Section 4.1. There exists T ą such that anyholonomy π : V Ñ r V for p Y, D, U q , whose transition times are bounded below by T , is -expanding. .3 Proof of the Proposition 4.3 Let us consider the neighborhoods U of C p σ X q , U of X and the cone field C satisfyingcondition (1). Let us consider a cross section D for X with a core ∆ Ă D (as given byLemma 4.5). One can always replace C and U by forward iterates under ϕ X , so that C isarbitrarily close to the bundle E cu on C p σ X q . Claim. (Up to replace C and U by forward iterates), for each singularity ρ one can assumethat C is transverse to D X W sloc p ρ q at points of C p σ q .Proof. For each singularity ρ , the intersection D X W sloc p ρ q is a one-codimensional spherein W sloc p ρ q transverse to X . Since E cu X T W sloc p ρ q “ (cid:82) X at regular points of C p σ q , onededuces that at points of D X W sloc p ρ q X C p σ q , the submanifold D X W sloc p ρ q is transverseto E cu , hence to C .Since C p σ X q is an attractor, it admits a neighborhood U Ă U which is attracting suchthat Λ : “ C p σ X q is the maximal invariant set in U . By the claim above, up to reduce theneighborhoods U and U , one can require that for any Y P U the cone field C is transverseto D X W sloc p ρ Y q X U for each singularity ρ Y P U of Y . In particular, there exists ε ą N of Y P U with diameter smaller than ε can intersect W sloc p ρ Y q in at most one point.Let T ą π Yi : V Yi Ñ ˜ V Yi , i “ , . . . , (cid:96) , defined for any vector field Y in a neighborhood U Ă U of X and whose transition times are bounded from below by T . The union Y i V Yi covers a uniform neighborhood O of C p σ X q X ∆. Since C p σ X q is an attractor for ϕ X andby our choice of the cross section D and of ∆, one can reduce the neighborhood U of X and choose a neighborhood U Ă U of C p σ X q such that for any Y P U and any x P U :– x belongs to the stable manifold of a singularity ρ Y or has a forward iterate by ϕ Y which belongs to the interior of ∆ in D ,– the forward orbit of x is contained in U .Moreover there exists ε satisfying the item (iv) of Proposition 4.7. We can always reduce ε to be smaller than ε .The angle between the cross section D and the vector field is bounded away from zeroby α . Hence if ε is reduced enough and ε ą N Ă ∆ with inner radius ε and diameter less than 3 ε , the set Yt ϕ Ys p N q : | s | ď ε u is asubmanifold with inner diameter larger than ε .Let us consider Y P U and Γ be a submanifold of dimension dim p E cu q as in thestatement of Proposition 4.3. Note that C p σ Y q is not a sink (it contains γ Y ), hence σ Y has a non-trivial unstable space. From the Item 3 of Proposition 2.4, this implies that thebundle E cu has dimension at least 2. From the area expansion (1), the stable manifoldsof the singularities are meager in Γ. By definition of the cross section and of its core ∆,one can thus find x P Γ having a forward iterate ϕ Yt p x q in ∆ z Y ρ W s p ρ Y q . By constructionand forward invariance the set Γ : “ Y | t ´ t |ă δ ϕ Yt p Γ q is still a submanifold tangent to C if δ ą Y p z q P T z Γ , the intersection N : “ Γ X D is a cu-section. SeeFigure 2We inductively build a sequence of cu-sections N n Ă D z Y W sloc p ρ Y q with diametersmaller than ε and contained in the orbit Y t ą ϕ Yt p N q of N . We denote by r n their innerradius. As explained in Section 4.1, we can reduce N n without reducing the inner radiusin a such a way that the diameter of N n is smaller than 3 r n .Since by definition a cu-section is transverse to W s p ρ Y q , one can choose N Ă N z W sloc p ρ Y q .The cu-section N n is then built inductively from N n ´ by: W sloc p ρ q Γ DND X W sloc p ρ q Figure 2: Construction of the cu-section N . a- Choosing a domain V Yi intersecting N n ´ : if N n ´ is disjoint from W sloc p ρ Y q for anysingularity ρ Y , then by Item (iv) of Proposition 4.7, there exists a domain V Yi whichcontains N n ´ ; otherwise, there exists a singularity ρ Y such that N n ´ z W sloc p ρ Y q iscontained in a domain V Yi . Claim.
The inner diameter of N n ´ X V Yi is larger than r n ´ { .Proof. If N n ´ Ă V Yi , there is nothing to prove. Otherwise by our choice of ε ą ε ,the intersection N n ´ X W sloc p ρ Y q contains only one point. Hence the inner radius of N n ´ z W sloc p ρ Y q is larger than one third of the inner radius of N n ´ . See Figure 3.b- Considering the image A n : “ π Yi p N n ´ X V Yi q and the inner radius r n of A n . N n ´ π ´ i p N n q N n ´ X W sloc p ρ q Figure 3: Construction of N n from N n ´ .15 - Choosing N n as a subset of A n having the same inner radius as A n and satisfyingDiam p V n q ă r n (as explained in Section 4.1). By Lemma 4.9, N n is a cu-section.By Lemma 4.11, the holonomies are 10-expanding, hence any ball in N n ´ X V Yi has radiussmaller than r n {
10. This gives r n ´ { ă r n {
10. This implies that the sequence of radiiincreases until r n ě ε (and then the construction stops).By our choice of ε, ε , the set Yt ϕ Ys p N n q : | s | ď ε u contains a submanifold tangent to C with inner diameter larger than ε ą
0. This ends the proof of the Proposition 4.3 and ofTheorem 4.1.
We will first prove Theorem A in a generic setting.
Theorem A’.
There is a dense G δ set G P X p M q such that for any X P G , any singularhyperbolic Lyapunov stable chain-recurrence class C p σ q of X is robustly transitive.More precisely, there are neighborhoods U X of X and U of C p σ q such that the maximalinvariant set of U for any Y P U X is an attractor which coincides with C p σ Y q . If it is notan isolated singularity, it is a homoclinic class.Proof. We consider a dense G δ set G P X p M q whose elements satisfy the conclusions ofProposition 2.1 and Theorems 2.2, 3.1, 4.1.By Theorem 2.2, we know that C p σ q is in fact an attractor for ϕ X , hence it is locallymaximal. By Item 4 of Proposition 2.1, C p σ q is robustly locally maximal, i.e. thereare a neighborhood U X of X and a neighborhood U of C p σ q such that C p σ Y q is themaximal invariant set in U for any Y P U X . Since C p σ q is an attractor, we can assumethat ϕ XT p U q Ă U for some T ą Y close to X . As aconsequence, up to reduce U X and U , the class C p σ Y q is also an attractor for Y P U X . Theremaining is to prove that C p σ Y q is a homoclinic class (if C p σ q is not trivial).By Theorem 2.2, and up to reduce U X , there exists a periodic orbit γ Ă C p σ q such that γ Y Ă C p σ Y q for any Y P U X . Furthermore, the conclusions of Theorems 3.1 and 4.1 hold.Consider a point x P C p σ Y q . We will show that x is accumulated by transverse homoclinicpoints of γ Y . Claim.
Any neighborhood V x of x intersects W uY p γ Y q .Proof of the Claim. If α p x q is not a single singularity, then by Theorem 3.1, the unstablemanifold W uY p γ Y q intersects W ssloc,Y p ϕ Y ´ t p x qq for any t ą
0. By choosing t ą W uY p γ Y q intersects W ssloc,Y p x q at a point arbitrarily close to x .Let us assume now that α p x q is a single singularity ρ Y . By the Item 5 of Proposition 2.1,the unstable manifold of γ Y intersect W sY p ρ Y q transversely. Hence by the inclination lemma,there is a point y P V x X W uY p γ Y q . The claim is verified in both cases.By Theorem 4.1, the transverse intersection points between W uY p γ Y q and W sY p γ Y q aredense in W uY p γ Y q . This concludes that C p σ Y q is a homoclinic class. Since a homoclinicclass is transitive, we have that C p σ Y q is transitive.We can now conclude the proof of the main results. Proof of Theorems A and B.
We first notice that these theorems hold for singular hyper-bolic attractors which do not contain any singularity and for isolated hyperbolic singularity: n these cases the attractors are uniformly hyperbolic and the proof is classical. In the fol-lowing we only consider non-trivial classes which contain at least singularity.Let G be a dense G δ subset of X p M q satisfying the conclusions of Theorem A’, Propo-sitions 2.3, 2.1 and Theorems 3.1, 4.1. For X P G , the singularities are hyperbolic and finiteand there exists a neighborhood U X where the singularities admit a continuation, satisfyProposition 2.3 and such that (as in Theorem A’, 3.1 and 4.1) the following property holds:if C p σ q is a (non-trivial) Lyapunov stable chain-recurrence class of a singularity σ for X ,then there exists a periodic orbit γ Ă C p σ q such that for any Y P U X :– the continuation C p σ Y q is a transitive attractor and coincides with the homoclinicclass H p γ Y q ,– for any x P C p σ Y q which does not belong to the unstable manifold of a singularity,there exists a transverse intersection between W ssY p x q and W uY p γ Y q .– for any periodic orbit γ Y P C p σ Y q , there exists a transverse intersection between W sY p γ Y q and W uY p γ Y q .We define the dense open set U “ ď X P G U X . Now we will verify that Theorem A holds in this open dense set U .Take Y P U , there is a vector field X P G such that Y P U X . Consider a singularhyperbolic Lyapunov stable chain-recurrence class C p σ Y q of Y : it has the property that W uY p σ Y q Ă C p σ Y q . Hence by the first property of Proposition 2.3, we have that W uX p σ X q Ă C p σ X q . Hence C p σ X q is Lyapunov stable by Item 7 of Proposition 2.1. By Theorem A’, C p σ Y q is a robustly transitive attractor since Y P U X . Hence Theorem A holds.Since Y P U X , we also conclude that C p σ Y q “ H p γ Y q is a homoclinic class and containa dense subset of periodic points. Moreover, for any periodic orbit γ Y Ă C p σ Y q , thereis a transverse intersection between W sY p γ Y q and W uY p γ Y q . Considering any x P γ Y , byTheorem 3.1, there also exists a transverse intersection between W ssY p x q and W uY p γ Y q be-cause that x is not contained in the unstable manifolds of singularities. The periodic orbits γ Y and γ Y are thus homoclinically related. Beeing homoclinically related is a transitiverelation on hyperbolic periodic orbits (by the inclination lemma), hence any two periodicorbits in C p σ Y q are homoclinically related. This concludes the proof of Theorem B. Proof of Corollary C.
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Laboratoire de Math´ematiques d’Orsay School of Mathematical SciencesCNRS - Universit´e Paris-Sud Soochow UniversityOrsay 91405, France Suzhou, 215006, P.R. China