THREE-DIMENSIONAL CONSERVATIVE STAR FLOWS ARE ANOSOV

A divergence-free vector field satisfies the star property if any divergence-free vector field in some $C^1$-neighborhood has all the singularities and all closed orbits hyperbolic. In this article we prove that any divergence-free star vector field defined in a closed three-dimensional manifold is Anosov. Moreover, we prove that a $C^1$-structurally stable three-dimensional conservative flow is Anosov.


(Communicated by Lan Wen)
Abstract. A divergence-free vector field satisfies the star property if any divergence-free vector field in some C 1 -neighborhood has all the singularities and all closed orbits hyperbolic. In this article we prove that any divergencefree star vector field defined in a closed three-dimensional manifold is Anosov. Moreover, we prove that a C 1 -structurally stable three-dimensional conservative flow is Anosov.
1. Introduction, basic definitions and statement of the results. Let M be a three-dimensional closed and connected C ∞ Riemannian manifold endowed with a volume-form and let µ denote the Lebesgue measure associated to it. We say that a vector field X : M → T M is divergence-free if its divergence is equal to zero or equivalently if the measure µ is invariant for the associated flow, X t , t ∈ R. In this case we say that the flow is conservative or volume-preserving. We denote by X r µ (M ) (r ≥ 1) the space of C r divergence-free vector fields on M and we endow this set with the usual C 1 Whitney topology. Let also denote by X r (M ) ⊃ X r µ (M ) (r ≥ 1) the space of C r (dissipative) vector fields on M .
Given X ∈ X 1 (M ) let Sing(X) denote the set of singularities of X and R := M \ Sing(X) the set of regular points.
Given x ∈ R we consider its normal bundle N x = X(x) ⊥ ⊂ T x M and define the linear Poincaré flow by P t X (x) : is the projection along the direction of X(X t (x)). Let Λ ⊂ R be an X t -invariant set and N = N 1 ⊕ N 2 be a P t X -invariant splitting over Λ; as X is conservative these bundles are one-dimensional. We say that this splitting is an ℓ-dominated splitting for the linear Poincaré flow if there exists an ℓ ∈ N such that for all x ∈ Λ we have: 2000 Mathematics Subject Classification. Primary: 37D30; Secondary: 37C75. Key words and phrases. Divergence-free vector fields, Anosov flows, Dominated splitting. The first author is partially supported by Fundação para a Ciência e a Tecnologia, SFRH/BPD/20890/2004.

MÁRIO BESSA AND JORGE ROCHA
This definition is weaker than hyperbolicity where it is required that When Λ is compact this definition is equivalent to the usual definition of hyperbolic flow ([9, Proposition 1.1]).
The simplest examples of hyperbolic sets are singularities and closed orbits and it is well-know that these sets are stable by C 1 -perturbations, that is, any other sufficiently C 1 -close system has equivalent behavior or, in other words, it is possible to find a change of coordinates conjugating locally the two dynamics (for more details see [12]). Other classical examples are the Anosov ones where M is hyperbolic, and they form an open set of X 1 µ (M ) (see e.g. [14]). We say that a vector field is Axiom A if the closure of the union of the closed orbits and the singularities is the non-wandering set, denoted by Ω(X), and this set is hyperbolic. Since, by Poincaré recurrence theorem, for conservative vector fields the non-wandering set is equal to M , a conservative vector field that is Axiom A is actually an Anosov system. In the dissipative case, in order to obtain stability we must check if there exists no cycles. Recall that, by the spectral decomposition of an Axiom A flow, we have that Ω(X) = ∪ k i=1 Λ i where each Λ i is a basic piece. We define an order relation by We say that X has a cycle if there exists a cycle with respect to ≺ (see [14] for details).
We say that Theorem 1.1. If X ∈ G 1 (M d ) and Sing(X) = ∅ then X is Axiom A without cycles.
In this paper we deal with these issues in the setting of three-dimensional divergence-free vector fields and our approach is of a completely different nature. We consider flows that are star flows restricted to the conservative setting, which we denote by G 1 µ (M ). That is, X ∈ G 1 µ (M ) if there exists a neighborhood V of X in X 1 µ (M ) such that any Y ∈ V, has all the closed orbits and all the singularities hyperbolic. Our main result states that such a flow has no singularities and is hyperbolic (Anosov). We note that Gan and Wen must consider non-singular flows due, in particular, to the fact that the Lorenz strange attractor is in G 1 (M ). However, Arbieto and Matheus ([2, Corollary 4.1]) proved that, in the conservative setting, there are no geometrical Lorenz sets, which could indicate that it should be possible to remove the hypothesis of the non-existence of singularities.
Let us now state our main result.
then Sing(X) = ∅ and actually X is Anosov. We point out that the proof of this result is a consequence of several recent results on conservative three-dimensional flows. We believe that the previous result is also true in any dimension and its proof should be obtained by generalizing these recent results to any dimension 1 and eventually following the strategy of the cited work of Gan and Wen, namely by using the fact that vector fields in G 1 µ (M ) cannot have heterodimensional cycles.
µ denote the open set of divergence-free Anosov vector fields on a threedimensional manifold M .
It is clear that As a consequence of Theorem 1.2 we also obtain the following result. [12]).
Combining Theorem 1.2 with previous results of the first author with P. Duarte ( [5]) and with V. Araújo ([1]) we are able to prove the stability conjecture for C 1 conservative 3-flows.
A one-parameter linear family {A t } t∈R associated to Γ(p, τ ) and V is defined as follows: , in particular det(A t ) = 1, for all t ∈ R, and • the family A t is C ∞ on the parameter t. In this paper we will consider n = 3 and so V = N p and dim(V ) = 2. The following result is a kind of Franks' Lemma for volume-preserving flows and was proved in [6].
Theorem 2.1. Given ǫ > 0 and a vector field X ∈ X 4 µ (M ) there exists ξ 0 = ξ 0 (ǫ, X) such that ∀τ ∈ [1,2], for any periodic point p of period greater than 2, for any sufficient small flowbox T of Γ(p, τ ) and for any one-parameter linear family Let us state three crucial results that will help us to prove Theorem 1.2. They can be obtained, between other arguments (see [5]), by using Theorem 2.1, the fact that elliptic closed orbits are stable and also using a smoothness perturbation result due to Zuppa ([16]).
The first two lemmas give us different contexts where one can create a nearby elliptic closed orbit via a small perturbation, thus far from G 1 µ (M ).

Lemma 2.2. ([5, Proposition 3.8])
Let X ∈ X 1 µ (M ) and ǫ > 0 be given. There exists θ = θ(ǫ, X) > 0 such that if a hyperbolic closed orbit O for X has angle between its stable and unstable directions smaller than θ, then we can find an ǫ-C 1 -close divergence-free vector field Y such that O is an elliptic closed orbit for Y t . Assume that all divergence-free vector fields Y which are ǫ-C 1 -close to X do not admit elliptic closed orbits. Then, for every such Y , all closed orbits with period larger than T are hyperbolic, m-dominated and with angle between its stable and unstable directions bounded from below by θ.
We also recall the C 1 -Closing Lemma adapted to the setting of volume-preserving flows by Pugh and Robinson [13,Section 8(c)] and also the contribution of Arnaud ([3]) for a more accurate version, which in particular assures a) and b) bellow.
The X t -orbit of a recurrent point x can be approximated for a very long time T > 0 by a closed orbit of a flow Y which is C 1 -close to X. In fact, given r, T > 0 we can find a ǫ-C 1 -neighborhood U ⊂ X 1 µ (M ) of X, a closed orbit p of Y ∈ U with period π,T > T and a map g : [0, T ] → [0, π] close to the identity such that Another ingredient of the proofs of our theorems is a generalization of Bochi's dichotomy (see [8, Theorem A]) for the continuous-time class. This result was obtained recently by combining a Theorem of [4, Theorem 1], corresponding to the case when X has no singularities, and a Theorem of [1, Theorem A], that corresponds to case when X can have singularities. More precisely the following result was obtained.
Theorem 2.5. There exists a C 1 -residual set R ⊂ X 1 µ (M ) such that if X ∈ R then X is Anosov or else almost every point in M has zero Lyapunov exponents.
3. Proofs of the results. Let us recall that a singularity σ of a vector field X is linear hyperbolic if σ is a hyperbolic singularity and there exists a smooth local change of coordinates around σ that conjugates X and DX σ (cf. [15,Definition 4.1]).
The proof of Theorem 1.2 is made in two steps. First we prove that if X ∈ G 1 µ (M ) then X has no singularities and P t X admits a dominated splitting over M (Lemma 3.1) and then we prove that if X ∈ G 1 µ (M ) is such that P t X admits a dominated splitting over M then X is Anosov (Lemma 3.2). Lemma 3.1. If X ∈ G 1 µ (M ) then X has no singularities and P t X admits a dominated splitting over M .
Proof. Let us first observe that G 1 µ (M ) is C 1 open in X 1 µ (M ). To prove the lemma let us fix X ∈ G 1 µ (M ) and a C 1 -neighborhood V of X in G 1 µ (M ). Let us choose Y ∈ V such that all the singularities of Y are linear hyperbolic. If M \ Sing(Y ) admits a dominated splitting for the linear Poincaré flow of Y then [15,Proposition 4.1] implies that Sing(Y ) = ∅. We observe that when Y is robustly transitive then the previous conclusion follows directly from [11,Supplement].
It follows that there exists U ⊂ V, Y ∈ U, whose elements do not have singularities and admit a dominated splitting for the associated linear Poincaré flow. So let us now assume that M \ Sing(Y ) does not admit a dominated splitting for the linear Poincaré flow of Y .
We claim that for all m ∈ N, there exists a Y t -invariant set Γ m ⊂ M \ Sing(Y ) such that µ(Γ m ) > 0 and Γ m do not have dominated splitting for P t Y . In fact, if this claim were false then there would exist m such that M \ Sing(Y ) has an m-dominated splitting which contradicts our assumption.
Since Y ∈ G 1 µ (M ), all divergence-free vector fields which are ǫ-C 1 -close to Y do not admit elliptic closed orbits. Then, from Lemma 2.4, for every such a vector field Y there are constants θ = θ(ǫ, Y ), m = m(ǫ, θ) and T = T (m) such that, for each closed orbit with period greater then T , one has: i) m-dominated splitting and ii) angle between its stable and unstable directions bounded from below by θ.
Observe that, since Y ∈ G 1 µ (M ), these closed orbits are hyperbolic. We will get a contradiction with the fact that there exists a positive measure set without domination.
Using the techniques involved in the proof of Theorem 2.5 (see [4]) it is straightforward to conclude that for m sufficiently large and positive η arbitrarily close to 0, there exist T 0 > 0 and Y 1 ∈ V, C 1 -close to Y , such that for a.e. x ∈ Γ m one has e −tη < P t Y1 (x) < e tη , for every t > T 0 . Actually, letÛ ⊂ Γ m be a measurable set with positive measure. Let R ⊂Û be the set given by the Poincaré recurrence theorem with respect to Y 1 . Then, since the set of periodic points is a zero measure set, it follows that almost every x ∈ R is not a periodic point and it returns toÛ infinitely many times under the flow Y t 1 . Let Z η denote the subset of points of Γ m having Lyapunov exponents, for Y 1 , less than η.
Let us fix δ ∈ 0, log(2) 2m and η < δ. Given x ∈ Z η ∩ R there exists T x ∈ R such that e −δt < P t Y1 (x) < e δt for every t ≥ T x . Note that we can assume that T x ≥ T .
By the ergodic C 1 -Closing Lemma ( [3]) there exists a point x ∈ Z η ∩ R such that the Y t 1 -orbit of x can be approximated, by a closed orbit of a C 1 -close flow Y 2 : given r, T > 0 and a small C 1 -neighborhood U of Y 1 in X 1 µ (M ), there exist a vector field Y 2 ∈ U,T > T , a periodic orbit p of Y 2 with period π and a map g : [0,T ] → [0, π] close to the identity such that Letting r > 0 be small enough we obtain also that where π > T . Now, by construction, it follows that Y 2 is C 1 -close to Y , so that the orbit of p under Y 2 satisfies the conclusion of Lemma 2.4. In particular we have that Since the subbundles N s and N u are one-dimensional we write p i := Y 2 im (p) for i = 0, . . . , [π/m] with [t] := max{k ∈ Z : k ≤ t} and where C(p, the C 1 topology. Then, there exists a uniform bound for C(p, ·) for all vector fields which are C 1 -close to Y . Notice that we can take π > T arbitrarily large by letting r > 0 be small enough in the arguments described above. Therefore, the inequality in (2), ensures that P π Y2 (p) = P π Y2 |N u p and also Moreover, since P π Y2 is area-preserving we have that the sum of the Lyapunov exponents is zero, that is (recalling that π is the period of p) The constants in inequality (2) do not depend on π so taking the period very large we can deduce that This contradicts (1). Therefore Sing(Y ) = ∅ and P t Y admits a dominated splitting over M .
Let us now prove that X has no singularities and that P t X admits a dominated splitting over M . In fact if X has a singularity then there exists Y 0 ∈ V such that Y 0 has at least one linear hyperbolic singularity. Now we proceed as before to Y ∈ V, arbitrarily close to Y 0 , having all the singularities linear hyperbolic and with Sing(Y ) = ∅. Repeating the arguments above we get that Sing(Y ) = ∅, which is a contradiction. Therefore, in those arguments we can take Y = X and then conclude that X has a dominated splitting for the linear Poincaré flow.
is such that P t X admits a dominated splitting over M then X is Anosov.
Proof. Since P t X admits a dominated splitting over M one gets that there exists ℓ ∈ N such that where N = N 1 ⊕ N 2 , and these subbundles are P t X -invariant and are one dimensional.
For any i ∈ N we have ∆(x, iℓ) ≤ 1/2 i . For every t ∈ R we may write t = iℓ+r and since P r X is bounded, say by L, take C = 2 r ℓ L 2 and σ = 2 − 1 ℓ to get ∆(x, t) ≤ Cσ t , for every x ∈ M and t ∈ R. Denote by α t the angle ∠(N 1 X t (x) , N 2 X t (x) ). We already know, by domination, that this angle is bounded bellow from zero, say by β. Since we do not have singularities there exists K > 1 such that for all x ∈ M , K −1 ≤ X(x) ≤ K. Since the flow is conservative and the subbundles are both one dimensional we have that So, Analogously we get These two inequalities show that M is hyperbolic for the linear Poincaré flow. Then by [9, Proposition 1.1] we obtain that M is a hyperbolic set, thus X is Anosov. This ends the proof of the lemma.
Proof. (of Corollary 1) We claim that an isolated point X of the boundary of A 3 µ do not have singularities. In fact if Sing(X) = ∅ then, since Anosov vector fields do not have singularities, the singularities of X must be all nonhyperbolic. A nonhyperbolic singularity can be made hyperbolic by a small perturbation, thus there are vector fields arbitrarily close to X having (stably) hyperbolic singularities which is a contradiction because X is an isolated point of the boundary of A 3 µ . Now we just have to follow the proof of Theorem 1.2, taking Y = X (where we don't need to assume anymore that X ∈ G 1 µ (M )), concluding that the linear Poincaré flow of X admits a dominated splitting over M . Now, as in the proof of the previous corollary, it follows that X is Anosov.
Proof. (of Theorem 1.3) Let us fix a C 1 -structurally stable vector field in X 1 µ (M ) and choose a neighborhood V of X whose elements are topologically equivalent to X. If X / ∈ A 3 µ = G 1 µ (M ) then it follows that V ∩ A 3 µ = ∅. Using [5,1] one gets that there exists a residual subset R ⊂ V such that for every Y ∈ R the set of elliptic closed orbits is dense in M . Let us fix Y ∈ R and choose a small neighborhood of Y , W ⊂ V.
Let x be an elliptic point of large period, say π. Using Zuppa's theorem ( [16]) and the stability of elliptic points, we can approximated Y , in the C 1 topology, by a C 4 -vector field Z ∈ W such that the analytic continuation of x is also an elliptic point with period close to π. Now, if π is large enough, we apply Theorem 2.1 several times, by concatenating small rotations (the maps A t ), in order to obtain a new vector field W ∈ W exhibiting a parabolic closed orbit. Since the existence of a parabolic point prevents structural stability and W ∈ W we get a contradiction. Therefore X ∈ A 3 µ , which ends the proof.