Attractors and orbit-flip homoclinic orbits for star flows

We study star flows on closed 3-manifolds and prove that they either have a finite number of attractors or can be $C^1$ approximated by vector fields with orbit-flip homoclinic orbits.


Introduction
The notion of attractor deserves a fundamental place in the modern theory of dynamical systems. This assertion, supported by the nowadays classical theory of turbulence [27], is enlightened by the recent Palis conjecture [24] about the abundance of dynamical systems with finitely many attractors absorbing most positive trajectories. If confirmed such a conjecture means the understanding of a great part of dynamical systems in the sense of their long-term behaviour.
Here we attack a problem which goes straight to the Palis conjecture: The finitude of the number of attractors for a given dynamical system. Such a problem have been solved positively under certain circunstances. For instance we have the work by Lopes [16] who, based upon early works by Mañé [18] and extending previous ones by Liao [15] and Pliss [25], studied the structure of the C 1 structural stable diffeomorphisms and proved the finitude of attractors for such diffeomorphisms. His work was largely extended by Mañé himself in the celebrated solution of the C 1 stability conjecture [17]. On the other hand, the japanesse researchers S. Hayashi [11] and N. Aoki [2] studied the star diffeomorphisms, i.e., diffeomorphisms which cannot be C 1 approximated by ones with nonhyperbolic periodic points, and proved that they are Axiom A and so with only a finite number of attractors. Their investigation triggered the study of the star flows, i.e., vector fields which cannot be C 1 approximated by ones with nonhyperbolic closed orbits. Indeed, although it was known from the very beginning that these flows are not necessarily Axiom A [1], [7], [8], the aforementioned works by Liao and Pliss proved that they display finitely many attracting closed orbits.
A progress toward understanding star flows was tackled in 2003 by the author in collaboration with Pacifico [20]. Indeed, these authors proved on closed 3-manifolds that, except in a meager set, all such flows are singular-Axiom A and so with only a finite number of attractors. Soon later the chinesse authors Gan and Wen [5] extended the Aoki-Hayashi's conclusion to nonsingular star flows on closed manifolds impliying that these flows has a finite number of attractors too. In light of these works it seems quite promissing to prove the finiteness of the number of attractors for star flows in any closed manifold.
In this paper we shall provide a result which though partial provides an insight for a positive solution of this problem. Basically, we present the so-called orbitflip homoclinic orbits as obstruction for the finiteness of attractors of star flows on closed 3-manifolds. More precisely, we show that a star flow on a closed 3-manifold either has a finite number of attractors or can be C 1 approximated by vector fields exhibiting orbit-flip homoclinic orbits. Orbit-flip homoclinic orbits are very rich dynamical structures which have been studied during decates [3], [13], [14], [22], [23], [28], [29]. Let us state this result in a precise way.
Hereafter M will denote a compact connected boundaryless Riemannian manifold of dimension n ≥ 2 (a closed n-manifold for short). We shall consider a C 1 vector field X in M together with its induced one-parameter group X t , t ∈ R, the so-called flow of X. The space of C 1 vector fields in M comes equipped with the C 1 -topology which, roughly speaking, measures the distance between vector fields and their corresponding derivatives.
The long-time behavior of a point x ∈ M is often analyzed through its omegalimit set ω(x) = y ∈ M : y = lim n→∞ X tn (y) for some sequence t n → ∞ .
A compact invariant set is transitive if it coincides with the omega-limit set of one of its points, whereas, in this work, an attractor will be a transitive set of the form for some neighborhood U of it. The most representative example of attractors are the sinks, that is, hyperbolic closed orbits of maximal Morse index. Sometimes we use the term source referring to a sink for the time reversal vector field −X.
We call it orbit-flip as soon as the eigenvalues λ 1 , λ 2 , λ 3 of σ are real, satisfy the eigenvalue inequalities λ 2 < 0 < λ 3 < λ 1 and Γ ⊂ W uu (σ) where W uu (σ), the strong unstable manifold [12], is the unique invariant manifold of X which is tangent at σ to the eigenspace associated to the eigenvalue λ 1 (c.f. Figure 1).  With such definitions and notations we can state our result.
Main Theorem. A star flow on a closed 3-manifold either has a finite number of attractors or can be C 1 approximated by vector fields exhibiting orbit-flip homoclinic orbits.
The proof relies on recent results in the theory of star flows [5] together with some techniques resembling those in [19].
The Main Theorem motivates the obvious question if star flows which can be C 1 approximated by vector fields with orbit-flip homoclinic orbits exist on any closed 3-manifold. Actually this is true but, as the reader can see by himself [15], [19], [22], [25], the set of such flows constitute a meager subset of star flows. We therefore conclude that every closed 3-manifold comes equipped with an open and dense subset of star flows, all of whose elements have a finite number of attractors. However, it is worth noting that we can obtain exactly the same conclusion by making use of [19] and [20].
Another question is if, in the statement of the Main Theorem, we can replace the finitely many attractor's option by the stronger property of being singular-Axiom A (in the sense of [20]). Unfortunately, such a question has negative answer as we can easily find star flows in the 3-sphere which neither are singular-Axiom A nor can be C 1 approximated by vector fields with orbit-flip homoclinic loops. Finally let us mention that, in the statement of the Main Theorem, we can replace the term attractor by that of Lyapunov stable omega-limit set (in the spirit of [21]).

Proof
We denote by · the norm induced by a Riemannian metric in M and by m(·) its corresponding minimum norm. Given a C 1 vector field X with flow X t in M we denote by Sing(X, U ) the set of singularities of X in U and we write Sing(X) = Sing(X, M ). Likewise, the union of the periodic orbits of X is denoted by P er(X). The elements of P er(X) will be called periodic points.
• E X Λ is the subbundle generated by X in T Λ M . Recall that a closed orbit O (i.e. a periodic orbit or singularity) is hyperbolic if it does as a compact invariant set. A periodic point is hyperbolic if its corresponding orbit is. We say that a compact invariant set is nontrivial if it does not reduce to a single closed orbit.
For any given index i in between 0 and n − 1 we denote by P er i (X) the union of the hyperbolic periodic orbits O of X with dim(Ê s x ) = i for some (and hence for all) x ∈ O. As pointed out earlier by Wen [30] we can extended this set to include periodic orbits for nearby vector fields. More precisely, we denote by P er * i (X) the i-preperiodic set of X consisting of those x ∈ M for which there are sequences X k and x k of vector fields and points in P er i (X k ) such that X k → X and x k → x. We shall also use the notion of fundamental i-limit which are limits (in the Hausdorff metric) of sequences of hyperbolic periodic orbits O n ⊂ P er i (X n ) of vector fields X n → X.
Now we state four technical lemmas the first of which is Lemma 3.4 in [5]: Lemma 2.1. If X is a star flow on a closed n-manifold and Λ is a fundamental i-limit of X with Sing(X, Λ) = ∅, then Λ is a sink or a source depending on whether i = n − 1 or i = 0.
Denote by N → M \ Sing(X) the vector bundle with fiber N x = {v ∈ T x M : v ⊥ X(x)}. We define the linear Poincaré flow P t : N → N by where π : T M → N stands for orthogonal projection.
Lemma 2.2. Let X a star flow on a closed n-manifold and Λ be a compact invariant set with Sing(X, Λ) = ∅ for which there is a dominated ρ-splitting N Λ = ∆ − Λ ⊕ ∆ + Λ for P t over Λ with 1 ≤ ρ ≤ n − 2. If ∆ − Λ is not contracting there is a fundamental r-limit contained in Λ with r < ρ. Likewise, if ∆ + Λ is not expanding there is a fundamental r-limit contained in Λ with r > ρ.
The proof of the following result can be obtained as in Theorem 3.8 of [31] (see also the proof of Lemma 2.8 in [6]).

Lemma 2.3. If X is a star flow, then for every index
for P t over P er * i (X) \ Sing(X) such that x is the corresponding hyperbolic splitting along the orbit of x.
The following lemma is Theorem B in [5]. Denote by Cl(·) the closure operation. Lemma 2.4. If X is a star flow on a closed manifold, then every compact invariant set Λ ⊂ Cl(P er(X)) with Sing(X, Λ) = ∅ is hyperbolic.
These lemmas will be used to analyze attractors for star flows on closed 3manifold. To start with we extend the conclusion of Lemma 2.4 to all such attractors.
Proposition 2.5. If X is a star flow on a closed 3-manifold, then every attractor A of X with Sing(X, Λ) = ∅ is hyperbolic.
Proof. First we show that A ⊂ P er * 1 (X) unless A is a sink or a source. Indeed, if A ⊂ P er * 1 (X) we can select y ∈ A \ P er * i (X). As A has no singularities and y ∈ A we have that y is a regular point (i.e. X(y) = 0). Then, it follows from the Pugh's closing lemma [26] that y ∈ P er * 0 (X) ∪ P er * 2 (X) and so there exist a fundamental i-limit with i = 0, 2 intersecting A. As A is an attractor we conclude that such a fundamental i-limit is contained in A. Therefore, by Lemma 2.1, it would exist a sink or a source contained in A. In such a case A is a sink or a source. Then, we can assume that A ⊂ P er * 1 (X). So, Lemma 2.3 implies that there is a dominated 1-splitting were not contracting it would exist a fundamental 0-limit in A in virtue of Lemma 2.2. Therefore A is a source and so hyperbolic. Hence we can assume that ∆ − A is contracting and analogously that ∆ + A is expanding. Then, A is hyperbolic by the Doering's criterium.
is the complement of the region of Σ in between the graphs of γ 1 and γ 2 of u 1 and u 2 respectively (i.e. the complement of the shadowed region in Figure  2-(b)).
If (iv) holds, then β (which is contained in A) intersects γ (which is contained in W s (σ)) whence σ ∈ A which is absurd since A is a nontrivial hyperbolic attractor. Therefore, (v) holds and so β ∩ R = ∅ yielding P er * 1 (X) ∩ R = ∅ again an absurdity. These contradictions prove the result. Now we prove the following key result.
Proposition 2.8. Let X be a star flow with singularities on a closed 3-manifold which cannot be C 1 approximated by vector fields with orbit-flip homoclinic orbits. Then, there is a neighborhood U of Sing(X) such that if A is an attractor of X then A ∩ U = ∅ if and only if Sing(X, A) = ∅.
Proof. Otherwise there is a sequence of attractors A n with Sing(X, A n ) = ∅ and σ ∈ Λ ∩ Sing(X), where Since star flows have finitely many sinks ( [15], [25]) we can assume that each A n is nontrivial and they are all hyperbolic by Proposition 2.5. It follows that every A n is a nontrivial hyperbolic attractor and so Λ ⊂ P er * 1 (X). We clearly have that σ is neither a sink nor a source (otherwise it could not be accumulated by periodic orbits which is the case for σ). So, we can order its eigenvalues λ 1 , λ 2 , λ 3 in a way that either λ 2 or λ 1 is real and, in each case, with Re(·) denoting real part.
In the first case σ clearly satisfies hypothesis (iv) of Proposition 2.7. In the second we must have that both λ 3 and λ 1 are real (otherwise the dominated 1splitting claimed to exist in Lemma 2.3 would not exist) and, since X cannot be approximated by vector fields with orbit-flip homoclinic loops, we still have λ 3 < λ 1 . In other words, in such a case σ satisfies hypothesis (v) of Proposition 2.7. On the other hand, in both cases it is certainly possible to find x ∈ (W s (σ) \ {σ}) ∩ Λ. In particular, x ∈ P er * 1 (X) \ Sing(X) and so, by Proposition 2.7, there is δ > 0 such that d(A n , x) ≥ δ for all n. But this is clearly impossible due to the definition of Λ so the result is true.
Proof of the Main Theorem. Let X be a star flow on a closed 3-manifold which cannot be C 1 approximated by vector fields exhibiting orbit-flip homoclinic orbits. We can assume without any loss of generality that X has singularities (if not we apply [5]). Suppose by contradiction that it has infinitely many distinct attractors A n , n ∈ N. Since X has finitely many singularities and sinks, and, since the attractors are pairwise disjoint, we can assume that each A n is not a sink and satisfies Sing(X, A n ) = ∅. In particular, each A n is a nontrivial hyperbolic attractor by Proposition 2.5. Moreover, by Proposition 2.8, there is a neighborhood U of Sing(X) such that A n ∩ U = ∅ for all n. It follows that the closure Cl ( n A n ) has no singularities. Since each A n is a nontrivial hyperbolic attractor we have A n ⊂ Cl(P er(X)) and so Cl ( n A n ) is also a compact invariant set in Cl(P er(X)). Applying Lemma 2.4 we conclude that Cl ( n A n ) is a hyperbolic set. However, as is well known, hyperbolic sets contains only a finite number of attractors which is certainly not the case for Cl ( n A n ). We obtain so a contradiction which proves the result.