aa r X i v : . [ m a t h . D S ] M a r STABILITY PROPERTIES OF DIVERGENCE-FREEVECTOR FIELDS
C´ELIA FERREIRA
Abstract.
A divergence-free vector field satisfies the star prop-erty if any divergence-free vector field in some C -neighborhoodhas all singularities and all closed orbits hyperbolic.In this paper we prove that any divergence-free vector field de-fined on a Riemannian manifold and satisfying the star property isAnosov. It is also shown that a C -structurally stable divergence-free vector field can be approximated by an Anosov divergence-freevector field. Moreover, we prove that any divergence-free vectorfield can be C -approximated by an Anosov divergence-free vectorfield, or else by a divergence-free vector field exhibiting a heterodi-mensional cycle. MSC 2000: primary 37D20, 37D30, 37C27; secondary 37C10.
Keywords:
Divergence-free vector field; Anosov vector field; Domi-nated splitting; Structurally stable vector field; Heterodimensional cy-cle. 1.
Introduction and statement of the results
Let M , sometimes denoted by M n , be an n -dimensional, n ≥ µ , called the Lebesguemeasure. Let X r ( M ) be the set of vector fields and let X rµ ( M ) be the setof divergence-free vector fields, both defined on M and endowed withthe C r Whitney topology, r ≥
1. We emphasize that in this paper weare restricted to the C topology, since our proofs use several technicalresults which are just proved for this topology (see results in Section 2).Take X ∈ X r ( M ). We denote by P er ( X ) the union of the closedorbits of X and by Sing ( X ) the union of the singularities of X . Sin-gularities and closed orbits are called critical elements , denoted by Crit ( X ). If p / ∈ Sing ( X ) then p is called a regular point and M is saidto be regular if Sing ( X ) = ∅ .Take x ∈ M a regular point for X ∈ X ( M ) and let N x := X ( x ) ⊥ ⊂ T x M denote the (dim( M ) − X at x . Since, in general, N x is not DX tx -invariant, we define the linearPoincar´e flow P tX ( x ) := Π X t ( x ) ◦ DX tx , where Π X t ( x ) : T X t ( x ) M → N X t ( x ) is the canonical orthogonal projec-tion. Recently, Li, Gan and Wen generalized the notion of the linearPoincar´e flow, in order to include singularities in their study (see [22]).Now, we state some basic definitions. Definition 1.1.
Let X ∈ X ( M ) . An X t -invariant, compact and reg-ular set Λ ⊂ M is called hyperbolic if N Λ has a P tX -invariant splitting N s Λ ⊕ N u Λ such that there is ℓ > satisfying k P ℓX ( x ) | N sx k ≤ and k P − ℓX ( X ℓ ( x )) | N uXℓ ( x ) k ≤ , for any x ∈ Λ . A vector field X is said to be Anosov if the manifold M is hyperbolic.Let A µ ( M ) denote the C -open set of divergence-free Anosov vectorfields defined on M . In the divergence-free context, a hyperbolic criticalpoint p must be of saddle type, having both N sp and N up with dimensionbetween 1 and n −
2. A vector field X is called isolated in the boundaryof A µ ( M ) if X is not Anosov and, given a small neighborhood U of X ,any vector field Y ∈ U \ X is Anosov.Now, we state the definition of dominated splitting , a more relaxednotion of splitting. Definition 1.2.
Take X ∈ X ( M ) and Λ ⊂ M a compact, X t -invariantand regular set. A P tX -invariant splitting N Λ = N ⊕ N is said a dom-inated splitting if there is ℓ > such that k P ℓX ( x ) | N x k · k P − ℓX ( X ℓ ( x )) | N Xℓ ( x ) k ≤ , ∀ x ∈ Λ . One of the most important conjectures in the field of dynamical sys-tems, posed by Palis and Smale in 1970, is to know if a C r -structuralstable system satisfies the Axiom A and the strong transversality con-ditions. This is the so called structural stability conjecture . Noticethat, by the Poincar´e recurrence theorem, in the conservative settingwe conclude that the non-wandering set coincides with the whole man-ifold. So, a conservative diffeomorphism (or a divergence-free vectorfield) which is Axiom A is actually Anosov.The study of the previous conjecture motivated Ma˜n´e, in the early1980’s, to define the set F ( M ) of dissipative diffeomorphisms havinga C -neighborhood U such that every diffeomorphism inside U has allclosed orbits of hyperbolic type. We call f ∈ F ( M ) a star diffeomor-phism . TABILITY PROPERTIES OF DIVERGENCE-FREE VECTOR FIELDS 3
It is known that Ω-stable diffeomorphisms belong to F ( M ) (see[13]) and that if f ∈ F ( M ) then Ω( f ) = P er ( f ) (see [25]). Thus, thestructural stability conjecture is contained in the following one Conjecture 1.
Does a star system have its non-wandering set hyper-bolic?
On [24], Ma˜n´e proved the previous conjecture for surfaces: everydissipative diffeomorphism of F ( M ) satisfies the Axiom A and theno-cycle condition. Later, Hayashi extended this result for higher di-mensions (see [19]).In 1988, Ma˜n´e presented a proof of the stability conjecture for C -diffeomorphisms (see [23]).For the continuous-time case, a star vector field is defined as follows. Definition 1.3.
A vector field X ∈ X ( M ) is a star vector field ifthere exists a C -neighborhood U of X in X ( M ) such that if Y ∈ U then every point in Crit ( Y ) is hyperbolic. Moreover, a vector field X ∈ X µ ( M ) is a divergence-free star vector field if there existsa C -neighborhood U of X in X µ ( M ) such that if Y ∈ U then everypoint in Crit ( Y ) is hyperbolic. The set of star vector fields is denotedby G ( M ) and the set of divergence-free star vector fields is denoted by G µ ( M ) . Note that G ( M ) and G µ ( M ) are C -open in X ( M ) and X µ ( M ),respectively.Once that the previous definition concerns critical elements only andthe hyperbolicity put on critical elements is merely orbit-wise, the star-condition looks, a priori, quite weak. However, as we will see in Theo-rem 1.2 and Theorem 1, for the divergence-free setting it is not.A star vector field may fail to have hyperbolic non-wandering set,as the famous Lorenz attractor shows (see [17]), or may fail to havethe critical elements dense in the non-wandering set (see [12]) or, evenwith Axiom A satisfied, still may fail to satisfy the no-cycle condition(see [21]). However, for star vector fields, all these counterexamplesexhibit singularities. So, recently, Gan and Wen (see [15]) proved thefollowing remarkable result about dissipative star vector fields definedon an n -dimensional manifold, where n ≥ Theorem 1.1. If X ∈ G ( M n ) and Sing ( X ) = ∅ then X is Axiom Awithout cycles. On [15], it is also shown that a dissipative star vector field exhibitno heterodimensional cycles.
C´ELIA FERREIRA
However, in the divergence-free setting it is possible to prove thata star vector field does not exhibit singularities. So, generalizing Ganand Wen’s result, Bessa and Rocha (see [10]) recently proved an anal-ogous result on volume preserving vector fields, using techniques onconservative dynamics.
Theorem 1.2. If X ∈ G µ ( M ) then Sing ( X ) = ∅ and X is Anosov. From this result, note that we have G µ ( M ) = A µ ( M ). However,this result cannot be trivially extended to higher dimensions because itsproof, in dimension 3, assumes that the normal bundle is splitted in twoone-dimensional subbundles. So, using volume-preserving arguments,the authors were able to prove the existence of a dominated splittingfor the linear Poincar´e flow and then the hyperbolicity.On [6], the authors prove Conjecture 1 for Hamiltonian vector fields,defined on a three-dimensional, compact and with no singularities con-nected component of an energy level of a four-dimensional symplecticmanifold.In higher dimensions, the subbundles of the normal bundle may havedimension strictly larger than one, meaning that a vector field with adominated splitting structure is not necessarily hyperbolic. In thispaper, we prove the higher-dimensional version of Theorem 1.2. Theorem 1. If X ∈ G µ ( M n ) then Sing ( X ) = ∅ and X is Anosov, n ≥ . Notice that the converse of Theorem 1 is trivially true due to theopenness of Anosov vector fields set. So, Theorem 1 implies that G ( M n ) ∩ X µ ( M n ) = G µ ( M n ) = A µ ( M n ) , n ≥ . The next result is a consequence of Theorem 1.
Corollary 1.
The boundary of the set A µ ( M n ) has no isolated points, n ≥ . A vector field X ∈ X ( M ) is said to be Kupka-Smale if every el-ement of
Crit ( X ) is hyperbolic and its invariant manifolds intersecttransversely. Considering M a manifold with dimension greater than3, we have that the set of Kupka-Smale vector fields KS ( M ) is a C -residual subset of X ( M ) (see [37]). In [35], it is shown that thisproperty is also true for a residual subset of all divergence-free vec-tor fields, meaning that the set of Kupka-Smale divergence-free vectorfields KS µ ( M ) is a C -residual subset of X µ ( M ). Remark 1.
From Theorem 1.2 and Theorem 1, it is straightforwardto conclude that if X ∈ X µ ( M n ) is in the interior of KS µ ( M n ) then TABILITY PROPERTIES OF DIVERGENCE-FREE VECTOR FIELDS 5 X ∈ A µ ( M n ) , n ≥ . This is an immediate proof for divergence-freevector fields of the result shown by Toyoshiba in [39] for vector fields. A vector field X is C - structurally stable if there exists a C -neigh-borhood U of X in X ( M ) such that every Y ∈ U is topologically con-jugated to X (see for instance [29]). The notion of structural stabilitywas first introduced in the mid 1930’s by Andronov and Pontrjagin (see[1]).We point out that, after the proof of the C -structural stability con-jecture for diffeomorphisms, Gan proved this conjecture for dissipa-tive C -flows (see [14]) and Bessa and Rocha presented a proof on [10]for the C -divergence-free context, but considering a three-dimensionalmanifold. In this paper, we generalize this last result to higher dimen-sions. Theorem 2. If X ∈ X µ ( M n ) is C -structurally stable then it can be C -approximated by Y ∈ A µ ( M n ) , n ≥ . Nevertheless, the C r -structural stability conjecture remains wideopen for higher topologies ( r ≥ C -perturbation arguments, as the closing lemma,the connecting lemma and the Franks lemma, are either unknown orthey are false in higher topologies (see further details in [18, 31, 34]).At the second half of the 1960’s, it was already clear that uniformhyperbolicity could not be presented for every system of a dense subsetin the universe of all dynamics. So, it triggered the start of the searchof a answer to the question: Is it possible to look for a general scenariofor dynamics? This search draw the attention to homoclinic orbits,that is, orbits that in the past and in the future converge to the sameperiodic orbit, which has been first considered by Poincar´e, almost acentury before. The creation or destruction of such orbits is, roughlyspeaking, what its meant by homoclinic bifurcations (see [30]). Basedon these and other subsequent developments, Palis formulated, in the1990’s, the following conjecture (see [30, 28]): Conjecture 2.
The diffeomorphisms exhibiting a homoclinic bifurca-tion are C r -dense in the complement of the closure of the hyperbolicones, r ≥ . Pujals and Sambarino (see [33]) provided a proof of this conjecturein the case of diffeomorphisms defined on a compact surface in the C topology. Recently, Bessa and Rocha proved this conjecture forvolume-preserving diffeomorphisms on [11]. The authors show thata volume-preserving diffeomorphism can be C -approximated by an C´ELIA FERREIRA
Anosov volume-preserving map, or else by a volume-preserving diffeo-morphism displaying a heterodimensional cycle. The authors also showa similar result for symplectomorphisms.On [4], Arroyo and Hertz proved an analogous statement of the pre-vious conjecture in the context of C -vector fields defined on a three-dimensional, compact manifold. In this context, besides homoclinictangencies , the singular cycles are another homoclinic phenomenonthat must be considered: Theorem 1.3.
Any vector field X ∈ X ( M ) can be approximated byanother one Y ∈ X ( M ) showing one of the following phenomena: (1) Uniform hyperbolicity with the no-cycles condition; (2)
A homoclinic tangency; (3)
A singular cycle.
On the conservative setting, Bessa and Rocha, considering a three-dimensional manifold M , proved the next result on [8]: Theorem 1.4.
Any vector field X ∈ X µ ( M ) can be C -approximatedby another one Y ∈ X µ ( M ) which is Anosov or else has a homoclinictangency. On that paper, the authors left open the following question: canany X ∈ X µ ( M n ) be C -approximated by a divergence-free vector fieldexhibiting some form of hyperbolicity in M n , or by one exhibiting ho-moclinic tangencies or else by one having a heterodimensional cycle(see Definition 2.1 bellow), for n ≥
4? In the present paper we alsoanswer to this question.
Theorem 3. If X ∈ X µ ( M n ) then X can be C -approximated by anAnosov divergence-free vector field, or else by a divergence-free vectorfield exhibiting a heterodimensional cycle, n ≥ . This result rules out the C -approximation by a vector field exhibit-ing a homoclinic tangency in the higher-dimensional divergence-freesetting.This article is organized in five additional sections. In Section 2, wehave compiled some definitions and auxiliary results, that will be usedto prove the main theorems. Section 4 presents the proof of Theorem 1,which uses the results proved in Section 3, and also contains the proofof Corollary 1. The proof of Theorem 2 is provided in Section 5. Fi-nally, in Section 6 we prove some auxiliary results that, jointly withTheorem 1, allow us to easily conclude the proof of Theorem 3. TABILITY PROPERTIES OF DIVERGENCE-FREE VECTOR FIELDS 7 Definitions and auxiliary results
In this section, we state some definitions and present some resultsthat will be used in the proofs.Let O X ( p ) denote the orbit of p ∈ Crit ( X ) and π X ( p ) its period.By period, we mean the least period. If p is a singularity of X , we set π X ( p ) = 0 and O X ( p ) = p and if O X ( p ) is a hyperbolic set, its stable and unstable manifolds are defined as W sX ( O X ( p )) = { q ∈ M : dist ( X t ( q ) , O X ( p )) → , t → + ∞} and W uX ( O X ( p )) = { q ∈ M : dist ( X − t ( q ) , O X ( p )) → , t → + ∞} . We observe that both W sX ( O X ( p )) and W uX ( O X ( p )) do not dependon q ∈ O X ( p ). Therefore, we can write W sX ( O X ( p )) = W sX ( q ) and W uX ( O X ( p )) = W uX ( q ), for some q ∈ O X ( p ). These manifolds are re-spectively tangent to the subspaces E sq ⊕ R X ( q ) and R X ( q ) ⊕ E uq of T q M , q ∈ O X ( p ). Observe thatdim( W sX ( O X ( p ))) + dim( W uX ( O X ( p ))) = dim( M ) + 1 . Take p ∈ Crit ( X ) a hyperbolic saddle for a vector field X ∈ X ( M ).The index of p is defined as the dimension of the unstable bundle W uX ( p ) and will be denoted by ind ( p ). Now, we state the notion ofheterodimensional cycle. Definition 2.1.
Take X ∈ X ( M ) and let p, q be two distinct hyperboliccritical points of saddle type such that ind ( p ) < ind ( q ) . A vector field X exhibits a heterodimensional cycle associated to p and q if theinvariant manifolds of p and q intersect cyclically, that is W sX ( p ) ⊤∩ W uX ( q ) = ∅ and W uX ( p ) ∩ W sX ( q ) = ∅ , where ⊤∩ means that the intersection is transversal. This definition can be trivially extended to a finite number of hyper-bolic saddles.
Remark 2.1.
The condition ind ( p ) < ind ( q ) , stated in the previousdefinition, ensures that the connection W sX ( p ) ⊤∩ W uX ( q ) is C -persistentand that the connection W uX ( p ) ∩ W sX ( q ) is not C -persistent. Let HC ( M ) and HC µ ( M ) denote the subsets of X ( M ) and X µ ( M ),respectively, whose elements exhibit heterodimensional cycles. A het-erodimensional cycle is said to be periodic if it is composed just byclosed orbits, singular if it is composed just by singularities and mixed if it contains at least one singularity and one closed orbit.A vector field X ∈ X ( M ) is said to be far from heterodimensionalcycles , say X ∈ F C ( M ), if there exists a C -neighborhood U of X C´ELIA FERREIRA on X ( M ) such that every Y ∈ U does not exhibit heterodimensionalcycles. Moreover, if X and U are taken in X µ ( M ) and X satisfy theprevious property, we say that X ∈ F C µ ( M ). Remark 2.2.
We point out that: • Heterodimensional cycles do not exist if dim( M ) < because, inthis case, we cannot find hyperbolic critical points of saddle-typeand with different indices. • If dim( M ) = 3 , M does not support periodic heterodimensionalcycles since, in this case, the stable and the unstable manifoldsof any closed orbit are both two-dimensional. However, it ispossible to find singular heterodimensional cycles and also mixedheterodimensional cycles, where a link connecting two closed or-bits is not allowed. Mixed heterodimensional cycles just appearin the case that the singularities have index , since the indexof every closed orbit is . The first auxiliary result stated is due to Zuppa (see [43]) and allowsus to C -approximate any divergence-free vector field by a C ∞ onekeeping the divergence-free property. Theorem 2.1.
The set of C ∞ divergence-free vector fields is C -densein X µ ( M ) . The next result is a Pasting Lemma (see [2]) and it allows us torealize C -local perturbations in the divergence-free setting. Theorem 2.2.
Given ǫ > there exists δ > such that if X ∈ X µ ( M ) , K ⊂ M is a compact set and Y ∈ X ∞ µ ( M ) is δ - C -close to X in a smallneighborhood U ⊃ K , then there exist Z ∈ X ∞ µ ( M ) and open sets V and W , such that K ⊂ V ⊂ U ⊂ W , satisfying the properties: • Z | V = Y ; • Z | int ( W c ) = X ; • Z is ǫ - C -close to X . The following result is a version of Franks’ lemma for divergence-freevector fields (see [9] for more details). Under some conditions, it allowsus to realize a perturbation on the linear Poincar´e flow by a vector fieldwhich is C -close to the original one. Theorem 2.3.
Given ǫ > and a vector field X ∈ X µ ( M ) , there exists ξ = ξ ( ǫ, X ) such that for any τ ∈ [1 , , for any periodic point p ofperiod greater than , for any sufficient small flowbox T of X [0 ,τ ] ( p ) andfor any one-parameter linear family { A t } t ∈ [0 ,τ ] such that k A ′ t A − t k < ξ , TABILITY PROPERTIES OF DIVERGENCE-FREE VECTOR FIELDS 9 for all t ∈ [0 , τ ] , there exists Y ∈ X µ ( M ) satisfying the followingproperties: (1) Y is ǫ - C -close to X ; (2) Y t ( p ) = X t ( p ) , ∀ t ∈ R ; (3) P τY ( p ) = P τX ( p ) ◦ A τ ; (4) Y | T c = X | T c . In the sequence, we state a version of the C -Closing Lemma forvolume-preserving flows, firstly proved by Pugh and Robinson (see [32])and that, more recently, was improved by Arnaud, that presented asimpler proof (see [3]). It states that the orbit of a recurrent point canbe approximated by a long time closed orbit of a C -perturbation ofthe original vector field. Theorem 2.4.
Take X ∈ X µ ( M ) and x a X t -recurrent point. Given ǫ, r, T > , there is an ǫ - C -neighborhood U ⊂ X µ ( M ) of X , a closedorbit p of Y ∈ U with period π arbitrarily large, a map g : [0 , T ] → [0 , π ] close to the identity and ˜ T > T such that • dist (cid:0) X t ( x ) , Y g ( t ) ( p ) (cid:1) < ǫ , for every ≤ t ≤ ˜ T ; • Y = X on M \ B r (cid:0) X [0 , ˜ T ] ( x ) (cid:1) . A conservative version of Pugh and Robinson’s
General DensityTheorem (see [32]), also proved by Arnaud in [3], asserts that, C -generically, the closed orbits are dense in M . We denote by PR µ ( M )this residual set in X µ ( M ).The next result correspond to a dichotomy for conservative vectorfields. It requires the existence of a closed orbit with arbitrarily largeperiod and it is obtained following the ideas presented on [9, Proposi-tion 2.4]. Theorem 2.5.
Let X ∈ X µ ( M ) and let U be a small C -neighborhoodof X . Then, for any ǫ > , there exist l, τ > such that, for any Y ∈ U and any closed orbit x of Y t of period π ( x ) > τ , • either P tY admits an l -dominated splitting over the Y t -orbit of x , or else • for any neighborhood U of x , there exists an ǫ - C -perturbation ˜ Y of Y , coinciding with Y outside U and along the orbit of x ,such that P π ( x )˜ Y ( x ) = id . Take X ∈ X µ ( M ). By Oseledets’s theorem (see [27]), µ -almost everypoint x in M has a splitting of the tangent bundle, T x M = E x ⊕· · · ⊕ E k ( x ) x , called the Oseledets splitting , and real numbers λ ( x ) > · · · > λ k ( x ) ( x ), called the Lyapunov exponents , 1 ≤ k ( x ) ≤ n , such that DX tx ( E ix ) = E iX t ( x ) and λ i ( x ) = lim t →±∞ t log k DX tx ( v i ) k , for any v i ∈ E ix \ { ~ } and i ∈ { , ..., k ( x ) } . The full µ -measure set ofthe Oseledets points is denoted by O ( X ). Remark 2.3.
As a consequence of Oseledets’s theorem one has that k ( x ) X i =1 λ i ( x ) · dim( E ix ) = lim t →±∞ t log | det DX tx | . However, since the vector field X is divergence-free, we deduce that | det DX t ( x ) | = 1 , for every t ∈ R and every x ∈ M . So, we concludethat k ( x ) X i =1 λ i ( x ) · dim( E ix ) = 0 , ∀ x ∈ O ( X ) . Note that if we do not take into account the multiplicities of theeigenvalues associated to the eigenspaces E x , · · · , E k ( x ) x , we have exactly n Lyapunov exponents λ ( x ) ≥ · · · ≥ λ n ( x ) . If we assume the absence of a dominated splitting, it is possible tomake a C -perturbation of the vector field in order to get a new onewith Lyapunov exponents arbitrarily close to zero, as it is shown in [7,Theorem 1].Now, we state that a singularity p is linear if there exist smoothlocal coordinates around p such that X is linear and equal to DX ( p )in these coordinates (cf. [40, Definition 4.1]). The next lemma statesthat any singularity can be turned into a linear one, by performing asmall perturbation of the vector field. Lemma 2.6. If X ∈ X µ ( M ) has a singularity then, for any neighbour-hood V of X , there is an open and nonempty set U ⊂ V such that any Y ∈ U has a linear hyperbolic singularity.Proof. Let p be a singularity of X ∈ X µ ( M ) and ǫ >
0. By a small C -conservative perturbation of X (see [9]), we can find X , ǫ - C -close to X , with a hyperbolic singularity p . Denote by V a C -neighbourhood of X in X µ ( M ) where the analytic continuation of p is well-defined. Now,by Zuppa’s Theorem (see [43]), there is a smooth vector field X ∈ V with a hyperbolic singularity p . If the eigenvalues of DX ( p ) satisfythe nonresonance conditions of the Sternberg linearization theorem (see[38]) then there is a smooth diffeomorphism conjugating X and its TABILITY PROPERTIES OF DIVERGENCE-FREE VECTOR FIELDS 11 linear part around p . If the nonresonance conditions are not satisfiedthen we can perform a C -conservative perturbation of X , so that theeigenvalues satisfy the nonresonance conditions. So, since the set ofdivergence-free vector fields satisfying the nonresonance conditions isan open and dense set in X µ ( M ), there is a C -neighbourhood U of X in V such that any vector field X ∈ U is conjugated to its linear part,meaning that X has a linear hyperbolic singularity. (cid:3) The next result, will be used to prove that a star vector field can notexhibit singularities.
Theorem 2.7. [40, Proposition 4.1] If X ∈ X ( M ) admits a linearhyperbolic singularity of saddle-type then the P tX does not admit anydominated splitting over M \ Sing ( X ) . The final presented auxiliary result asserts that, C -generically, avector field is topologically mixing, and so transitive. Theorem 2.8. [5, Theorem 1.1]
There exists a C -residual subset R ⊂ X µ ( M ) such that, if X ∈ R then X is a topologically mixing vector field. Auxiliary lemmas
We start this section by showing that a divergence-free star vectorfield does not have singularities.
Lemma 3.1. If X ∈ G µ ( M ) then Sing ( X ) = ∅ .Proof. Fix X ∈ G µ ( M ) and U a C -neighborhood of X in G µ ( M ), smallenough such that Theorem 2.5 holds. Recall that PR µ ( M ) is a residualset in X µ ( M ) such that any X ∈ PR µ ( M ) has the closed orbits densein M (see [32, § R be the residual set given by Theorem 2.8To obtain a contradiction, take p ∈ Sing ( X ), which is hyperbolic andof saddle-type, by definition of G µ ( M ), and so it persists to C -smallperturbations of X . By Lemma 2.6, there is Y ∈ U ∩ R ∩ PR µ ( M ), C -close to ˜ X , such that p ∈ Sing ( Y ) is linear hyperbolic of saddle-type,and Y has a closed orbit x , with arbitrarily large period. Therefore,as Y ∈ G µ ( M ), by Theorem 2.5, there exist constants ℓ, τ > P tY admits an ℓ -dominated splitting over the Y t -orbit of x withperiod π ( x ) > τ . Also, Y has a dense orbit because it belongs to R .So, by the volume preserving Closing Lemma (Theorem 2.4), there isa sequence of vector fields Y n ∈ U ∩ R , C -converging to Y , and, forevery n ∈ N , Y n has a closed orbit Γ n = Γ n ( t ) of period π n such thatlim n →∞ Γ n (0) = x and lim n →∞ π n = + ∞ . Therefore, by Theorem 2.5, P tY n admits an ℓ -dominated splitting over the orbit Γ n , for large n . Taking a subsequence if necessary, say n ∈ I ⊆ N , we have a sequence of Y n with closed orbit Γ n such that P tY n has an ℓ -dominated splitting andsuch that the dimensions of the invariant bundles do not depend on n .Then, given that M = lim sup n Γ n = \ N ∈ N (cid:18) ∞ [ n ≥ N Γ n (cid:19) , we conclude that there exists an ℓ -dominated splitting for P tY over M \ Sing ( Y ).However, since p is a linear hyperbolic singularity of saddle-type of Y , by Theorem 2.7, we conclude that P tY does not admit a dominatedsplitting over M \ Sing ( Y ). This is a contradiction. So, X has nosingularities. (cid:3) The next lemma states that, given a divergence-free star vector field,we can define a continuous splitting N = N ⊕ N over M . Lemma 3.2. If X ∈ G µ ( M ) then there exists a continuous splitting N x = N x ⊕ N x , for every x ∈ M .Proof. Take X ∈ G µ ( M ) and recall that, by Lemma 3.1, Sing ( X ) = ∅ .So, we have that N p = N sp and N p = N up , for any p ∈ P er ( X ). Toextend these fibers to any x ∈ M , fix y / ∈ P er ( X ) and a sequence { y n } n ∈ P er ( X ) such that lim n → + ∞ y n = y and lim n → + ∞ N , X t ( y n ) = N u,sX t ( y ) .So, any x ∈ M has attached the subspaces N , x such thatdim N x + dim N x = dim M − P tX ( x )( N , x ) = N , X t ( x ) . Notice that, by [25, Lemma 3.1], since X ∈ G µ ( M ) then P er ( X ) = Ω( X ) = M . So, the domination over P er ( X ), that can be extended to P er ( X ) = M , leads to N x ∩ N x = { } ,for any x ∈ M . This with (1) implies that N x = N x ⊕ N x and that thefibers depend continuously on x , for any x ∈ M . (cid:3) The proof of the next lemma uses a generalization, for the higher-di-mensional context, of the adopted techniques in the proof of Lemma 3.1in [10]. However, at this point, we already know that a vector field in G µ ( M ) has not singularities. Lemma 3.3. If X ∈ G µ ( M ) then P tX admits a dominated splitting over M .Proof. Take X ∈ G µ ( M ) and U a C -neighborhood of X in G µ ( M ),small enough such that Theorem 2.5 holds. By Lemma 3.1, we have TABILITY PROPERTIES OF DIVERGENCE-FREE VECTOR FIELDS 13 that M is regular for X . So, P tX is well defined on M . It follows thatthere exists V ⊂ U , a C -neighborhood of X in G µ ( M ), whose elementsdo not have singularities. By Lemma 3.2, we have a continuous splitting N = N ⊕ N over M . By contradiction, assume that this splitting isnot dominated. So, we claim that Claim 3.1.
For all ℓ ∈ N , there exists an X t -invariant and measurableset Γ ℓ ∈ M such that µ (Γ ℓ ) > and Γ ℓ does not have an ℓ -dominatedsplitting for P tX . In fact, if the claim was not true, there would exist ℓ ∈ N suchthat M has an ℓ -dominated splitting for P tX , which contradicts ourassumption. The existence of these sets Γ ℓ without an ℓ -dominatedsplitting, for any ℓ ∈ N , allow us to use the techniques involved in theproof of [7, Theorem 1] in order to conclude that, for any ǫ >
0, thereexists ℓ ∈ N , large enough, such that, for any η > µ -almost every point x ∈ Γ ℓ , we can find t > X ∈ U , ǫ - C -close to X , satisfying exp ( − ηt ) < k P tX ( x ) k < exp ( ηt ) , ∀ t > t . Now, let R ⊂ Γ ℓ be the full µ -measure set of recurrent points, givenby the Poincar´e recurrence theorem with respect to X , and let Z η ⊂ Γ ℓ be the set of points with Lyapunov exponent, associated to X , lessthan η .So, fixing δ ∈ (cid:0) , log 2( n − ℓ (cid:1) and η < δ , given x ∈ Z η ∩ R , there exists t x ∈ R such that exp ( − δt ) < k P tX ( x ) k < exp ( δt ) , ∀ t > t x , where we can assume that t x ≥ T .Now, once x ∈ Z η ∩ R , by the volume preserving Closing Lemma(Theorem 2.4), the X t -orbit of x can be approximated by a closedorbit p with period π of a C -close vector field X ∈ U . So, letting r > π > max { τ, T } arbitrarily large, where τ > − δπ ) < k P πX ( p ) k < exp( δπ ) . (2)Note that X ∈ U , a C -neighborhood of X in G µ ( M ), and that p is a X -closed orbit with period π > τ , obviously hyperbolic. So, byTheorem 2.5, there is ℓ > P tX admits an ℓ -dominatedsplitting N q = N q ⊕ · · · ⊕ N kq , 2 ≤ k ≤ n −
1, such that k P ℓ X ( q ) | N iq k · k P − ℓ X ( q ) | N jq k ≤ , for every 0 ≤ i < j ≤ k and every q ∈ O X ( p ). Now, given that p is a hyperbolic saddle with period π for X , let usassume that P πX ( p ) admits the following Lyapunov spectrum: λ ( p ) ≥ ... ≥ λ r ( p ) > > λ r +1 ( p ) ≥ ... ≥ λ k ( p ) . So, let N up = N p ⊕ · · · ⊕ N rp and N sp = N r +1 p ⊕ · · · ⊕ N kp .Let [ a ] denote the integer part of a and observe that k P πX ( p ) | N sp k · k P − πX ( p ) | N up k = k P π − ℓ [ π/ℓ ]+ ℓ [ π/ℓ ] X ( p ) | N sp k · k P − π − ℓ [ π/ℓ ]+ ℓ [ π/ℓ ] X ( p ) | N up k≤ k P π − ℓ [ π/ℓ ] X ( p ) | N sp k · k P ℓ [ π/ℓ ] X ( X ℓ [ π/ℓ ]2 ( p )) | N sXℓ π/ℓ p ) k·· k P − π + ℓ [ π/ℓ ] X ( p ) | N up k · k P − ℓ [ π/ℓ ] X ( X − ℓ [ π/ℓ ]2 ( p )) | N uX − ℓ π/ℓ p ) k≤ C ( p, X ) [ π/ℓ ] Y i =1 k P ℓ X ( X ℓ ( p )) | N sXiℓ
02 ( p ) k · k P − ℓ X ( X − ℓ ( p )) | N uX − iℓ
02 ( p ) k≤ C ( p, X ) (cid:18) (cid:19) [ π/ℓ ] , where C ( p, X ) = sup ≤ t ≤ ℓ (cid:16) k P tX ( p ) | N sp k · k P − tX ( p ) | N up k (cid:17) . Once C ( p, X )depends continuously on X , in the C -topology, there exists a uniformbound for C ( p, · ), for every vector field which is C -close to X .As it was mentioned in Remark 2.3, we have that k X i =1 λ i ( p ) = 0. So,observing that k P πX ( p ) k = k P πX ( p ) | N p k , one has that1 π log k P πX ( p ) | N p k = λ r +1 ( p ) = − k X i =1 i = r +1 λ i ( p ) ≥ − ( k − λ ( p ) = − ( k − π log k P πX ( p ) | N up k = − ( k − π log k P πX ( p ) k . TABILITY PROPERTIES OF DIVERGENCE-FREE VECTOR FIELDS 15
So, from k P πX ( p ) | N sp kk P − πX ( p ) | N up k ≤ C ( p, X ) (cid:18) (cid:19) [ π/ℓ ] and knowingthat k P πX ( p ) | N up k − ≤ k P − πX ( p ) | N up k , we have that k P πX ( p ) | N sp kk P πX ( p ) | N up k − ≤ C ( p, X ) (cid:18) (cid:19) [ π/ℓ ] ⇔ log k P πX ( p ) | N sp k − log k P πX ( p ) | N up k ≤ log C ( p, X ) − [ π/ℓ ] log 2 ⇔ π log k P πX ( p ) k ≥ − log C ( p, X ) π + [ π/ℓ ] log 2 π + 1 π log k P πX ( p ) | N sp k⇔ π log k P πX ( p ) k ≥ − log C ( p, X ) π + [ π/ℓ ] log 2 π − ( k − π log k P πX ( p ) k . Now, taking π arbitrarily large,1 π log k P πX ( p ) k ≥ log 2 kℓ ≥ log 2( n − ℓ > δ. But this contradicts (2). Then P tX admits a dominated splitting N = N ⊕ N over M . (cid:3) Remark 3.1.
Notice that the previous lemma is also true if we supposethat X is an isolated point in the boundary of A µ ( M n ) , for fixed n ≥ .In fact, in Lemma 3.3 we need X ∈ G µ ( M n ) in order to ensure adominated splitting over a closed orbit x , with large enough period π ,for a vector field Y , C -close to X , given by Theorem 2.5. However,if we start the proof by assuming that X is an isolated point in theboundary of A µ ( M n ) , we must obtain the same conclusion, becauseany C -perturbation ˜ Y of Y must be Anosov, and so cannot satisfy P π ˜ Y ( x ) = id . Next lemma is an adaption of the ideas of Ma˜n´e ([24]) to our setting.
Lemma 3.4.
Take X ∈ X µ ( M ) and assume that M is regular andthat any x ∈ M admits a dominated splitting N x = N x ⊕ N x . If lim inf t →∞ k P tX ( x ) | N x k = 0 and lim inf t →∞ k P − tX ( x ) | N x k = 0 , for all x ∈ M ,then M is hyperbolic.Proof. By hypothesis, for any x ∈ M we can find t x such that k P t x X ( x ) | N x k < / x ∈ M has a neighborhood B ( x )such that every y ∈ B ( x ) satisfies k P t x X ( y ) | N y k < / Since M is compact, there are x , ..., x n ∈ M , such that M ⊂ n [ i =1 B ( x i ). So, given any y ∈ M , there is 1 ≤ i ≤ n such that y ∈ B ( x i ).Let K = sup {k P tX ( y ) | N y k , y ∈ B ( x i ) , ≤ t ≤ t x i , ≤ i ≤ n } , j besuch that K / j < / T > j sup { t x i , ≤ i ≤ n } . Let us seethat T is the uniform hyperbolicity constant.Take t i , ..., t i k +1 and l j = t i + ... + t i j , 1 ≤ j ≤ k + 1, such that(1) X l j ( y ) ∈ B ( x i j +1 ) , ≤ j ≤ k ,(2) l k ≤ T ≤ l k +1 .From the previous, we observe that k ≥ j and 0 ≤ T − l k ≤ t i k +1 .So, for any y ∈ M , k P T X ( y ) | N y k = k P T − l k + l k X ( y ) | N y k≤ k P T − l k X ( X l k ( y )) | N Xlk ( y ) k · k P t i X ( y ) | N y k·· k P t i X ( X l ( y )) | N Xl y ) k · · · k P t ik X ( X l k − ( y )) | N Xlk − y ) k≤ K k ≤ K j < . Changing P tX by P − tX , the second case can be derived from this one. (cid:3) In the following lemma, we show that a divergence-free star vectorfield has uniform hyperbolicity in the period, which is a crucial step toderive hyperbolicity from Lemma 3.3.
Lemma 3.5. (Uniform hyperbolicity in the period) Fix X ∈ G µ ( M ) .There exist U , a C -neighborhood of X on G µ ( M ) , and θ ∈ (0 , suchthat, for any Y ∈ U , if p ∈ P er ( Y ) has period π Y ( p ) and has thehyperbolic splitting N p = N sp ⊕ N up then: (a) k P π Y ( p ) Y ( p ) | N sp k < θ π Y ( p ) and (b) k P − π Y ( p ) Y ( p ) | N up k < θ π Y ( p ) .Proof. Take X ∈ G µ ( M ) and U a C -neighborhood of X in G µ ( M ).So, for every p ∈ P er ( Y ) with period π Y ( p ), where Y ∈ U , we havethat p is a hyperbolic saddle, meaning that N p = N sp ⊕ N up and thatthere is a constant θ p ∈ (0 ,
1) such that k P π Y ( p ) Y ( p ) | N sp k < θ π Y ( p ) p and k P − π Y ( p ) Y ( p ) | N up k < θ π Y ( p ) p . However, we want θ p to be uniform.Let us prove (a). Suppose that, by contradiction, for any θ ∈ (0 , Y ∈ U , C -arbitrarily close of X , and p ∈ P er ( Y ) with TABILITY PROPERTIES OF DIVERGENCE-FREE VECTOR FIELDS 17 period π Y ( p ), hyperbolic by hypothesis, such that θ π Y ( p ) ≤ k P π Y ( p ) Y ( p ) | N sp k . In order to apply Theorem 2.3, we must have a C -vector field. So,using Zuppa’s theorem (Theorem 2.1), we start by C -approximate Y by a vector field ˜ Y ∈ U ∩ X µ ( M ), such that ˜ p ∈ P er ( ˜ Y ) is the hyperboliccontinuation of p , so with period π ˜ Y (˜ p ) close to π Y ( p ), and θ π ˜ Y (˜ p ) ≤ k P π ˜ Y (˜ p )˜ Y (˜ p ) | N s ˜ p k . (3)For simplicity, assume that π ˜ Y (˜ p ) is an integer. By (3), θ ≤ k P Y (˜ q ) | N s ˜ q k ,for some ˜ q ∈ O ˜ Y (˜ p ).Let A t be a one-parameter family of linear maps, for t ∈ [0 , π ˜ p ],such that k A ′ t A − t k is arbitrarily small, for any t ∈ [0 ,
1] and sup-pose k P Y (˜ q ) | N s ˜ q k = 1 − γ, where, by expression (3), γ is such that0 < γ < − θ and θ is chosen arbitrarily close to 1. Now, take A t = id , for t ≤
0, and A t a homothetic transformation of ratio of order11 − γ , for t ∈ [0 , π ˜ Y (˜ p )], and with entry a ,n − = δα ( t ), where α ( t ) isa smooth function such that α ( t ) = 1, for t ≥ α ( t ) = 0, for t ≤ < α ′ ( t ) <
1, and δ > k A ′ t A − t k < δ − γ and that this norm can be taken arbitrarilysmall, by choosing δ > ǫ > π ˜ Y (˜ p ) = π ˜ Y (˜ q ) in π ˜ Y (˜ q )-one-time intervals. ByTheorem 2.3, there exist vector fields Z i ∈ G µ ( M ), ǫπ ˜ Y (˜ q ) - C -close to˜ Y , such that P Z i (˜ q ) = P Y (˜ q ) ◦ A , for i ∈ { , ..., π ˜ Y (˜ q ) } . So, by Theorem2.2, there exists Z ∈ G µ ( M ), ǫ - C -close to ˜ Y , such that P π ˜ Y (˜ q ) Z (˜ q ) hasa eigenvalue equal to 1 or −
1. This is a contradiction because, since Z ∈ G µ ( M ), ˜ q is a hyperbolic closed orbit of saddle-type and so itsspectrum must be disjoint from S . So, (a) must hold.Using a similar argument, (b) is proved. This finishes the proof. (cid:3) Proof of Theorem 1
In this section, we conclude the proof of Theorem 1, by adapting toour setting a technique due to Ma˜n´e in [24].Take X ∈ G µ ( M ). By Lemma 3.1, Lemma 3.2 and Lemma 3.3, wehave that M is regular for X and that P tX admits a dominated splitting N = N ⊕ N over M . We want to prove that P tX | N is uniformlycontracting on M and that P tX | N is uniformly expanding on M . Let us prove the first condition. By Lemma 3.4, it suffices to prove thatlim inf t →∞ k P tX ( x ) | N x k = 0 , ∀ x ∈ M. By contradiction, suppose that there is x ∈ M such thatlim inf t →∞ k P tX ( x ) | N x k > . Then, we can choose a subsequence { t n } n ∈ N such that t n → ∞ as n → ∞ and lim n →∞ t n log k P t n X ( x ) | N x k ≥ . (4)Let C ( M ) be the set of continuous functions on M and define ϕ : C ( M ) → R by ϕ ( p ) = ∂ h (log k P hX ( p ) | N p k ) h =0 . By the Riez Theo-rem, there is a X t -invariant Borel probability measure µ such that Z M ϕ dµ = lim t n → + ∞ t n Z t n ϕ ( X s ( x )) ds = lim t n → + ∞ t n Z t n ∂ h (log k P hX ( X s ( x )) | N Xs ( x ) k ) h =0 ds = lim t n → + ∞ t n log k P t n X ( x ) | N x k ≥ . Also, by the Birkhoff Ergodic Theorem, Z M ϕ dµ = Z M lim t → + ∞ t Z t ϕ ( X s ( x )) dsdµ ( x ) ≥ . Now, let Σ( X ) be the set of points x ∈ M such that, for any C -neighbourhood U of X in X µ ( M ) and δ >
0, there exist Y ∈ U anda Y -closed orbit y ∈ M of period π such that X = Y except on the δ -neighborhood of the Y -orbit of y , and that dist ( Y t ( y ) , X t ( x )) < δ ,for 0 ≤ t ≤ π . A conservative version of the Ergodic Closing Lemma,proved by Arnaud in [3], says that, given a X t -invariant Borel proba-bility measure µ , µ (Σ( X )) = 1. So, there is x ∈ Σ( X ) such thatlim t → + ∞ t Z t ϕ ( X s ( x )) ds = lim t → + ∞ t log k P tX ( x ) | N x k ≥ . (5)Let log θ < δ < θ ∈ (0 ,
1) is fixed andgiven by Lemma 3.5. Thus, there is t δ such that, for t ≥ t δ ,1 t log k P tX ( x ) | N x k ≥ δ. Since x ∈ Σ( X ), there are X n ∈ U , C -converging to X , and p n ∈ P er ( X n ) with period π n . Notice that π n → + ∞ as n → ∞ , otherwise, TABILITY PROPERTIES OF DIVERGENCE-FREE VECTOR FIELDS 19 x ∈ P er ( X ) with period π such that P πX ( x ) | N x expands, by (5). This isa contradiction since X ∈ G µ ( M ). So, assuming that π n > t δ for every n , by the continuity of the dominated splitting we have that, for n bigenough, k P π n X n ( p n ) | N pn k ≥ exp( δπ n ) > θ π n . But this contradicts (a) in Lemma 3.5, because X n ∈ U . So, P tX | N isuniformly contracting on M .Analogously, we prove that P tX | N is uniformly expanding on M ,using ( b ) of Lemma 3.5. Thus, M is Anosov. (cid:3) We end this section with the proof of Corollary 1.
Proof of Corollary 1.
By contradiction, assume that there exists an iso-lated vector field X on the boundary of A µ ( M n ), for fixed n ≥
4. Inthis case, we claim that
Sing ( X ) = ∅ . Let us suppose that this claimis not true. If p ∈ Sing ( X ) is hyperbolic, and so persistent to small C -perturbations of X , we can find a divergence-free vector field Y ,arbitrarily close to X , such that Sing ( Y ) = ∅ . But this is a contradic-tion because, since X is isolated on the boundary of A µ ( M n ), Y hasto be Anosov. If p is not hyperbolic, by Lemma 2.6, we can transform p in a hyperbolic singularity of a vector field Z , that is C -close to X .So, as before, we reach a contradiction.Now, by the previous claim and by Remark 3.1, we deduce that M is regular for X and that P tX admits a dominated splitting over M . So,we just have to follow the proof of Theorem 1, presented in Section 4,in order to conclude that X ∈ A µ ( M n ), which is a contradiction. So,the boundary of A µ ( M n ) cannot have isolated points. (cid:3) Proof of Theorem 2
Let X ∈ X µ ( M ) be a C -structurally stable vector field, where M isa manifold with dimension n ≥
4, and choose a C -neighborhood V of X , such that every Y ∈ V is topologically conjugated to X .We start the proof with the following claim. Claim 5.1. If X ∈ X µ ( M ) is C -structurally stable then Sing ( X ) = ∅ .Proof of Claim 5.1. Let X ∈ X µ ( M ) be a C -structurally stable vec-tor field and suppose that there exists p ∈ Sing ( X ). If p is a linearhyperbolic saddle then, perturbing X in V and proceeding as in theproof of Lemma 3.1, we get a contradiction. This happens because theexistence of a C -perturbation Y of X in V , such that P πY ( x ) = id ,is forbidden, for x ∈ P er ( Y ) with arbitrarily large period π , because the existence of a parabolic closed orbit prevents the structural stabil-ity (see [36]). Now, if p ∈ Sing ( X ) is not hyperbolic then, as statedin Lemma 2.6, it can be turned into a linear hyperbolic singularityby a small C -perturbation of X in V . Then, we just have to ap-ply the previous argument once again. So, a C -structurally stabledivergence-free vector field has no singularities, which concludes theproof of Claim 5.1. (cid:3) Now, we are in conditions to obtain the conclusion of the proof ofTheorem 2. Recall that PR µ ( M ) is a residual set in X µ ( M ), such thatany X ∈ PR µ ( M ) has its closed orbits dense in M . Let W be a small C -neighbourhood of X such that Theorem 2.5 holds. We start C -perturbing X in V ∩ W and taking a vector field Y ∈ V ∩ W ∩ PR µ ( M ).So, Y is a structurally stable divergence-free vector field and it has aclosed orbit y with arbitrarily large period π y . So, by Theorem 2.5,there is ℓ > P tY admits an ℓ -dominated splitting over the Y t -orbit of y because, since Y is structurally stable, the existence of aparabolic orbit associated to ˜ Y ∈ V is not allowed. Then, reproducingthe technique used in the proof of Lemma 3.1, we conclude that P tY admits a dominated splitting over M since, by Claim 5.1, Sing ( Y ) = ∅ .Finally, following the proof of Theorem 1, we show that Y ∈ A µ ( M n ). (cid:3) Remark 5.1.
We do not show that the vector field X in the previousresult is itself Anosov, which would prove the stability conjecture forhigher dimensional divergence-free vector fields (see example in [16] ). Proof of Theorem 3
Let us start by proving the following lemmas. Recall that
F C µ ( M )is the set of divergence-free vector fields that are far from heterodimen-sional cycles and that KS µ ( M ) denotes the Kupka-Smale C -residualset in X µ ( M ). Lemma 6.1.
There exists a residual set
S ⊂ F C µ ( M ) such that, forevery X ∈ S , all the critical elements of X are hyperbolic and theirindex is constant.Proof. Take S = F C µ ( M ) ∩ KS µ ( M ), a residual set in F C µ ( M ), andassume that X ∈ S admits two critical elements ∆ X and Γ X withdifferent indices, say ind (∆ X ) < ind (Γ X ). Notice that ∆ X and Γ X can be closed orbits or singularities. Let U be an arbitrarily small C -neighborhood of X , in X µ ( M ), such that the hyperbolic continuationsof ∆ X and of Γ X are well defined. TABILITY PROPERTIES OF DIVERGENCE-FREE VECTOR FIELDS 21
By Theorem 2.8, there exists Y ∈ U ∩ S which is topologically mix-ing, so transitive. Let ∆ ′ Y ∈ O Y (∆ Y ) and Γ ′ Y ∈ O Y (Γ Y ). So, if we take p ∈ W sY (∆ ′ Y ) and q ∈ W uY (Γ ′ Y ), there exists an orbit of Y which passesarbitrarily close to p and q . Now, applying the conservative version ofthe Connecting Lemma for flows (see [42]), we get Z ∈ U such that W sZ (∆ ′ Z ) and W uZ (Γ ′ Z ) intersects transversely. Finally, we can repeatthe previous argument, in order to get ˜ Z , C -arbitrarily close to Z ,such that W u ˜ Z (∆ ′ ˜ Z ) ∩ W s ˜ Z (Γ ′ ˜ Z ) = ∅ , but also W s ˜ Z (∆ ′ ˜ Z ) ∩ W u ˜ Z (Γ ′ ˜ Z ) = ∅ ,because the first connection is robust and so it persists to small per-turbations. Thus, ˜ Z exhibits a heterodimensional cycle in S , that canbe a periodic, a singular or a mixed heterodimensional cycle. So, wereach a contradiction. Then every critical element of any X in S ishyperbolic and has constant index. (cid:3) Lemma 6.2. If X ∈ X µ ( M ) \G µ ( M ) then X can be C -approximatedby a divergence-free vector field Y that exhibits a heterodimensionalcycle. Notice that, in particular, F C µ ( M ) ⊂ G µ ( M ) .Proof. Take X ∈ X µ ( M ) \G µ ( M ) and Y a vector field, C -sufficientlyclose to X , such that Y ∈ (cid:0) X µ ( M ) \G µ ( M ) (cid:1) ∩ KS µ ( M ) ∩ PR µ ( M ),where KS µ ( M ) denotes the Kupka-Smale C -residual subset of X µ ( M )(see [37]) and PR µ ( M ) denotes a C -residual set in X µ ( M ) such thatany X ∈ PR µ ( M ) has its closed orbits dense in M . So, we can take p Y a hyperbolic closed orbit of Y , with period π Y and index u . Let W bea small C -neighborhood of Y such that the hyperbolic continuationof p Y is well defined.As Y ∈ X µ ( M ) \G µ ( M ), and once that this set is open, for ev-ery C -neighborhood V of Y in X µ ( M ) \G µ ( M ), we can find a vec-tor field Z ⊂ W ∩ V , C -arbitrarily close to Y , such that Z has ahyperbolic closed orbit p Z , corresponding to the hyperbolic continua-tion of p Y , with index u and period π Z close to π Y . However, since Z ∈ X µ ( M ) \G µ ( M ), it has to have a non-hyperbolic critical point q Z ,that can be a singularity or a closed orbit.If q Z is a non-hyperbolic singularity of Z , by a C -small perturbationon the vector field, it can be turned on a hyperbolic singularity withindex v = u . Observe that it can appear another non-hyperbolic criticalpoints but, since we already have two hyperbolic critical points withdifferent indices, we can build a heterodimensional cycle, as shown inLemma 6.1.Now, assume that q Z is a non-hyperbolic closed orbit of Z , so un-stable. In this case, we start by applying Theorem 2.1 to increase thedifferentiability of the vector field Z from C to C , in order to be able to use Theorem 2.3, that ensures the existence of a vector field W ∈ X µ ( M ), C -close to Y , such that p W (the hyperbolic continuationof p Z ) and q W are now hyperbolic closed orbits for W with differentindices. Again, by Lemma 6.1, we can C -proximate W by a vectorfield ˜ W exhibiting a heterodimensional cycle. (cid:3) Remark 6.1.
For the dissipative setting, in [15] the authors show that G ( M ) is a subset of F C ( M ) . Now, we are ready to prove Theorem 3. Suppose that X ∈ X µ ( M ),where dim( M ) ≥
4, cannot be C -approximated by a divergence-freevector field exhibiting a heterodimensional cycle, meaning that X ∈F C µ ( M ). Then, by Lemma 6.2, and due to the openness of F C µ ( M )in X µ ( M ), we see that X can be C -approximated by a divergence-free vector field Y , satisfying that Y ∈ F C µ ( M ) ∩ G µ ( M ). Finally,Theorem 1 ensures that Y is Anosov, which concludes the proof. (cid:3) Acknowledgements
I would like to thank my supervisors, M´ario Bessa and Jorge Rocha,whose encouragement and guidance enabled me to develop this work.The author was supported by Funda¸c˜ao para a Ciˆencia e a Tecnolo-gia, SFRH/BD/ 33100/2007.
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