On the essential hyperbolicity of sectional-Anosov flows
aa r X i v : . [ m a t h . D S ] J un ON THE ESSENTIAL HYPERBOLICITY OFSECTIONAL-ANOSOV FLOWS
S. BAUTISTA, C. A. MORALES
Abstract.
We prove that every sectional-Anosov flow of a compact 3-manifold M exhibits a finite collection of hyperbolic attractors and singularities whosebasins form a dense subset of M . Applications to the dynamics of sectional-Anosov flows on compact 3-manifolds include a characterization of essentialhyperbolicity, sensitivity to the initial conditions (improving [3]) and a rela-tionship between the topology of the ambient manifold and the denseness ofthe basin of the singularities. Introduction
A smooth vector field on a differentiable manifold is called essentially hyperbolic ifit exhibits a finite collection of hyperbolic attractors whose basins form an openand dense subset of the manifold [2], [12]. Basic examples are the Axiom A ones(by the spectral decomposition theorem [16], [27]), including the
Anosov flows , butnot the geometric Lorenz attractor [1], [14]. On the other hand, there is a classof systems, the sectional-Anosov flows [21], whose representative examples are theAnosov flows, the geometric Lorenz attractors, the saddle-type hyperbolic attract-ing sets, the multidimensional Lorenz attractors [10] and the examples in [22], [23].They motivate search of necessary and sufficient conditions for a sectional-Anosovflows to be essentially hyperbolic, or, if there is a sort of essential hyperbolicityfor them. At first glance it is tempting to say that for every sectional-Anosov flowof a compact manifold there is a finite collection of sectional-hyperbolic attractorswhose basins form an open and dense subset. However, this is false even in dimen-sion three as shown [5], [9]. Nevertheless, as proved in [11], every vector field closeto a transitive sectional-Anosov flow with singularities of a compact 3-manifoldsatisfies that generic points have a singularity in their omega-limit set. This wasimproved later in [3] by proving that all such vector fields satisfy that the basin ofthe singularities is dense in the manifold . In general, we can combine [3] and [25] toobtain that, for every compact 3-manifold M , there is a C open and dense subsetof sectional-Anosov vector fields all of whose elements exhibit a finite collection ofhyperbolic attractors and singularities whose basins form a dense subset of M . In Mathematics Subject Classification.
Primary 37D30; Secondary 37D45.
Key words and phrases.
Anosov Flow, Sectional-Anosov Flow, Sensitive, Essentially hyper-bolic, 3-manifold.Partially supported by CNPq, FAPERJ and PRONEX/DS from Brazil.SB was partially supported by the Universidad Nacional de Colombia, Bogot´a, Colombia.CAM was partially supported by CNPq, FAPERJ and PRONEX/Dynam. Sys. from Brazil andthe Universidad Nacional de Colombia from Colombia. CAM would like to thank the UniversidadNacional de Colombia, Bogot´a, Colombia, for its kindly hospitality during the preparation of thispaper. this paper we strengthen this last assertion by proving that every sectional-Anosovflow of every compact 3-manifold M exhibits a finite collection of hyperbolic at-tractors and singularities whose basins form a dense subset of M . This fact hassome consequences in the study of the dynamics of the sectional-Anosov flows X on compact 3-manifolds. The first one is that X is essentially hyperbolic if andonly if the basin of its set of singularities is nowhere dense. Another application isrelated to a result in [3] asserting that every vector field of a compact 3-manifoldthat is C close to a nonwandering sectional-Anosov flow is sensitive to the initialconditions. Indeed, we extend this result by proving that every sectional-Anosovflows on every compact 3-manifold is sensitive to the initial conditions. Finally, weprove that every sectional-Anosov flow with singularities (all Lorenz-like) but with-out null homotopic periodic orbits of a compact atoroidal 3-manifold M satisfiesthat the basin of the set of singularities is dense in M . Let us state our results ina precise way.Consider a compact manifold M with possibly nonempty boundary ∂M . Toindicate its dimension n we will call it n -manifold. Consider also a vector field X with induced flow X t on M , inwardly transverse to ∂M if ∂M = ∅ (all vector fieldsin this paper will be assumed to be C ). Define the maximal invariant set of XM ( X ) = \ t ≥ X t ( M ) . We say that Λ ⊂ M ( X ) is invariant if X t (Λ) = Λ for every t ∈ R . Given x ∈ M we define the omega-limit set , ω ( x ) = (cid:26) y ∈ M : y = lim k →∞ X t k ( x ) for some sequence t k → ∞ (cid:27) . Define the basin (of attraction) of any subset B ⊂ M as the set of points x ∈ M such that ω ( x ) ⊂ B . An invariant set Λ is transitive if Λ = ω ( x ) for some x ∈ Λ.An attractor of X is transitive set A for which there is a compact neighborhood U satisfying A = \ t ≥ X t ( U ) . The nonwandering set Ω( X ) of X is defined as the set of points x ∈ M such thatfor every neighborhood U of x and T > t > T satisfying X t ( U ) ∩ U = ∅ .Clearly ω ( x ) ⊂ Ω( X ) ⊂ M ( X ) for every x ∈ M . By a singularity of X we mean apoint σ ∈ M satisfying X ( σ ) = 0. Definition 1.1.
A compact invariant set Λ of X is hyperbolic if there are a de-composition T Λ M = E s Λ ⊕ E X Λ ⊕ E u Λ of the tangent bundle over Λ as well as positiveconstants K, λ and a Riemannian metric k · k on M satisfying (1) k DX t ( x ) /E sx | ≤ Ke − λt , for every x ∈ Λ and t ≥ . (2) E X Λ is the subbundle generated by X . (3) m ( DX t ( x ) /E ux ) ≥ K − e λt , for every x ∈ Λ and t ≥ where m ( · ) indicatesthe conorm operation.If E sx = 0 and E ux = 0 for all x ∈ Λ we will say that Λ is a saddle-type hyperbolicset . A hyperbolic attractor is an attractor which is simultaneously a hyperbolic set.A singularity σ of X is hyperbolic if it is hyperbolic as a compact invariant set, or,equivalently, if the linear map DX ( σ ) has no purely imaginary eigenvalues. N THE ESSENTIAL HYPERBOLICITY OF SECTIONAL-ANOSOV FLOWS 3
Definition 1.2 ([19]) . A compact invariant set Λ of X is sectional-hyperbolic ifthere are a decomposition T Λ M = E s Λ ⊕ E c Λ of the tangent bundle over Λ as well aspositive constants K, λ and a Riemannian metric k · k on M satisfying (1) k DX t ( x ) /E sx k ≤ Ke − λt for every x ∈ Λ and t ≥ . (2) k DX t ( x ) /E sx k m ( DX t ( x ) /E cx ) ≤ Ke − λt , for every x ∈ Λ and t ≥ . (3) | det( DX t ( x ) /L x ) | ≥ K − e λt for every x ∈ Λ , t ≥ and every two-dimensional subspace L x of E cx . Definition 1.3 ([21]) . A sectional-Anosov flow is a vector field whose maximalinvariant set is sectional-hyperbolic. The following definition describes a certain type of singularities for sectional-Anosov flows.
Definition 1.4.
We say that a singularity σ of a vector field X on a -manifold M is Lorenz-like if, up to some order, the eigenvalues { λ , λ , λ } of DX ( σ ) : T σ M → T σ M satisfy the eigenvalue condition λ < λ < < − λ < λ . A sectional-Anosov flow with singularities of a compact 3-manifold may haveLorenz-like singularities or not [5], [8], [23]. With these definitions we can state ourmain theorem.
Theorem 1.5.
For every sectional-Anosov flow of a compact -manifold M there isa finite collection of hyperbolic attractors and Lorenz-like singularities whose basinsform a dense subset of M . Applying this result we obtain easily the equivalence below.
Corollary 1.6.
A sectional-Anosov flow X of a compact -manifold M is essen-tially hyperbolic if and only if the basin of the set of singularities of X is nowheredense in M . Examples of sectional-Anosov flows for which the properties of the above corol-lary fail are the geometric Lorenz attractors. Further examples are the Anosovflows, the hyperbolic attracting sets (both without singularities) and the ones in[23]. We also observe that there are sectional-Anosov flows on certain compact 3-manifolds which are essentially hyperbolic but not Axiom A: They can be obtainedby modifying the singular horseshoe in [17].For the next corollary we shall use the following classical definition.
Definition 1.7.
We say that a vector field X of a manifold M is sensitive to theinitial conditions if there is δ > such that for every x ∈ M and every neighborhood U of x there are y ∈ U and t ≥ such that d ( X t ( x ) , X t ( y )) > δ . The number δ willbe referred to as a sensitivity constant of X This is a basic property of chaotic systems widely studied in the literature [4],[13],[18], [27], [28], [29], [30]. The following corollary asserts that this propertyholds for all sectional-Anosov flows on compact 3-manifolds. More precisely, wehave the following result.
Corollary 1.8.
Every sectional-Anosov flow of a compact -manifold is sensitiveto the initial conditions. S. BAUTISTA, C. A. MORALES
To finish we state a topological consequence of Theorem 1.8. Recall that acompact 3-manifold M is atoroidal if every two-sided embedded torus T on M , forwhich the homeomorphism of fundamental groups π ( T ) → π ( M ) induced by theinclusion is injective, is isotopic to a boundary component of M .By Corollary 2.6 in [21] we have that a sectional-Anosov flow with singularities,all Lorenz-like, but without null homotopic periodic orbits in an atoroidal compact3-manifold has no hyperbolic attractors. This together with Theorem 1.8 impliesthe following corollary yielding a relationship between topology and the densenessof the basin of the singularities. Corollary 1.9.
Let X be a sectional-Anosov flow of a compact atoroidal -manifold M . If X has singularities (all Lorenz-like) but not null homotopic periodic orbits,then the basin of the set of singularities of X is dense in M . An example where the hypotheses of the above corollary are fulfilled is the geo-metric Lorenz attractor.The proof of Theorem 1.5 relies on the techniques in [3], [21] but with some im-portant differences. For instance, the proof in [3] is based on the
Property (P) thatthe unstable manifold of every periodic point of X intersects the stable manifoldof a singularity of X . This property not only holds for every vector field close to anonwandering sectional-Anosov flow of a compact 3-manifold, but also implies thatthe basin of the singularities of X is dense in M .In our case we do not have this property since the vector fields under consider-ation are not close to a nonwandering sectional-Anosov flow in general. To bypassthis problem we will prove that every sectional-Anosov flow comes equipped witha positive constant δ such that every point whose omega-limit set passes δ -closeto some singularity is accumulated by the stable manifolds of the singularities. Toprove this assertion we combine some arguments from [3], [7] and [20]. This as-sertion is the key ingredient for the proof of Theorem 1.5. Corollary 1.8 will beobtained easily from this theorem and Lemma 2.8. Both results will be proved inthe last section. 2. Preliminars
In this section we prove some lemmas which will be used to prove our results.We start with some basic definitions. Let X be a C vector field on M inwardlytransverse to ∂M (if ∂M = ∅ ). For every x ∈ M ( X ) we define the sets W ss ( x ) = { y ∈ M : d ( X t ( x ) , X t ( y )) → t → ∞} ,W uu ( x ) = { y ∈ M : d ( X t ( x ) , X t ( y )) → t → −∞} W s ( x ) = [ t ∈ R W ss ( X t ( x )) and W u ( x ) = [ t ∈ R W uu ( X t ( x ))We denote by Sing ( X ) the set of singularities of X and denote by W s ( Sing ( X )) = [ σ ∈ Sing ( X ) W s ( σ )the basin of Sing ( X ). We say that a point p is periodic for X if there is a minimal t > X t ( p ) = p . Denote by P er ( X ) the set of periodic points of X . Weshall use the following auxiliary definition. N THE ESSENTIAL HYPERBOLICITY OF SECTIONAL-ANOSOV FLOWS 5
Definition 2.1. An intersection number for a vector field X is a positive num-ber δ such that if p ∈ P er ( X ) and W u ( p ) ∩ B δ ( Sing ( X )) = ∅ , then W u ( p ) ∩ W s ( Sing ( X )) = ∅ . Applying the connecting lemma [6] as in Lemma 1 of [20] we obtain the followingkey fact.
Lemma 2.2.
Every sectional-Anosov flow of a compact -manifold has an inter-section number. Let X be a vector field on a compact manifold M inwardly transverse to ∂M (if ∂M = ∅ ). Given δ > H δ = \ t ∈ R X t ( M \ B δ ( Sing ( X ))) . If X is sectional-Anosov, then H δ is a saddle-type hyperbolic set [5], [26].As a first application of Lemma 2.2 we obtain the following corollary (which isalso true in higher dimensions). Corollary 2.3.
The number of attractors of a sectional-Anosov flow on a compact -manifold is finite.Proof. Suppose by contradiction that there is a sectional-Anosov flow of a compact3-manifold exhibiting an infinite sequence of attractors A k , k ∈ N . Since thefamily of attractors of X is pairwise disjoint, and Sing ( X ) is finite, we can assumethat none of such attractors have a singularity. By Lemma 2.2 we can fix anintersection number δ of X . If one of the attractors A k intersects B δ ( Sing ( X )),then we can select a periodic point p k ∈ A k ∩ B δ ( Sing ( X )). Since δ is an intersectionnumber we would have that W u ( p k ) ∩ W s ( Sing ( X )) = ∅ . But W u ( p k ) ⊂ A k (for A k is an attractor) so A k contains a singularity, a contradiction. Therefore, B δ ( Sing ( X )) ∩ ( T k A k ) = ∅ and so S k A k ⊂ H δ where H δ is given in (1). Since X is sectional-Anosov we have that H δ is a hyperbolic set and, since the numbers ofattractors on a hyperbolic set is finite, we obtain that the sequence A k is finite, acontradiction. This ends the proof. (cid:3) Next we recall the terminology of singular partitions [7].Consider a vector field X on a compact manifold M . By a cross section of X we mean a codimension one submanifold Σ which is transverse to X . The interiorand the boundary of Σ (as a submanifold) will be denoted by Int (Σ) and ∂ Σrespectively. Given a family of cross sections R we still denote by R the union ofits elements. We also denote ∂ R = [ Σ ∈R ∂ Σ and
Int ( R ) = [ Σ ∈R Int (Σ) . Definition 2.4. A singular partition of a compact invariant set Λ of X is a finitedisjoint collection of cross sections R satisfying Λ ∩ ∂ R = ∅ and Sing ( X ) ∩ Λ = { x ∈ Λ : X t ( x )
6∈ R , ∀ t ∈ R } . A cross section Σ of X is a rectangle if it is diffeomorphic to [0 , × [0 , ∂ Σ is formed by two vertical curves, with union ∂ v Σ, andtwo horizontal curves. If z ∈ Int (Σ) we say that the rectangle Σ is around z . Onthe other hand, it is well known from the invariant manifold theory [15] that the S. BAUTISTA, C. A. MORALES subbundle E s of a sectional-Anosov flow X can be integrated yielding a strongstable foliation W ss on M . As usual we denote by W ss ( x ) the leaf of this foliationpassing through x ∈ M . In the case when x ∈ Σ we denote by F s ( x, Σ) (or simply F s ( x )) the projection of W ss ( x ) onto Σ along the orbits of X .For any set A we denote by Cl ( A ) its closure and by B δ ( A ) (for δ >
0) we denotethe δ -ball centered at A .The following lemma uses intersection numbers to find singular partitions forcertain omega-limit sets. Its proof follows closely that of Theorem 3 in [3]. Lemma 2.5.
Let δ be an intersection number of a sectional-Anosov flow on acompact -manifold M . If x / ∈ Cl ( W s ( Sing ( X ))) satisfies ω ( x ) ∩ B δ ( Sing ( X )) = ∅ ,then ω ( x ) has a singular partition.Proof. By Proposition 2 in [3] it suffices to prove that for every z ∈ ω ( x ) \ Sing ( X )there is a cross section Σ z through z such that ω ( x ) ∩ ∂ Σ z = ∅ . So fix z ∈ ω ( x ) \ Sing ( X ).We claim that ω ( x ) ∩ W ss ( z ) has empty interior in W ss ( z ). If ω ( x ) has asingularity this follows from the Main Theorem of [24] (indeed, the proof in [24]was done for transitive sets but works for omega-limit sets also). Then, we canassume that ω ( x ) has no singularities, and so, it is a hyperbolic set [5], [26]. If ω ( x ) ∩ W ss ( z ) has nonempty interior in W ss ( z ) then ω ( x ) contains a local strongstable manifold W ssǫ ( y ) for some y ∈ ω ( x ). From this and the hyperbolicity of ω ( x )we obtain x ∈ ω ( x ) and so x ∈ Ω( X ). As x / ∈ Cl ( W s ( Sing ( X ))) the closinglemma in [20] implies that there is a sequence p n ∈ P er ( X ) converging to x .As ω ( x ) ∩ B δ ( Sing ( X )) = ∅ the above convergence implies that the orbit of p n intersects B δ ( Sing ( X )) for n large. As δ is an intersection number we obtain W u ( p n ) ∩ W s ( Sing ( X )) = ∅ for all n large. From this and the Inclination Lemma[16] we obtain that x ∈ Cl ( W s ( Sing ( X ))) which is absurd. The claim follows.Using the claim we obtain a rectangle R z around z such that ω ( x ) ∩ ∂ v R z = ∅ . Ifthe positive orbit of x intersects only one component of R z \ F s ( z ) we select a point x ′ in that component and a point z ′ in the other component. In such a case wedefine Σ z as the subrectangle of R z bounded by F s ( x ′ ) and F s ( z ′ ). We certainlyhave that ω ( x ) ∩ F s ( z ′ ) = ∅ . On the other hand, if ω ( x ) intersects F s ( x ′ ) in apoint h (say) then h ∈ Ω( X ) and so, by the closing lemma [20], h is approximatedby periodic points or by points whose omega-limit set is a singularity. The latteroption must be excluded (for it would imply x ∈ Cl ( W s ( Sing ( X )))) so h is thelimit of a sequence p n ∈ P er ( X ). But h ∈ F s ( x ′ ) so ω ( h ) = ω ( x ′ ). As x ′ and x belongs to the same orbit we also have ω ( x ′ ) = ω ( x ) yielding ω ( h ) = ω ( x ).As ω ( x ) ∩ B δ ( Sing ( X )) = ∅ we obtain ω ( h ) ∩ B δ ( Sing ( X )) = ∅ too. Since p n approaches h we conclude that the orbit of p n intersects B δ ( Sing ( X )) for all n large. As δ is an intersection number we obtain that W u ( p n ) ∩ W s ( Sing ( X )) = ∅ and so h ∈ Cl ( W s ( Sing ( X ))) by the Inclination Lemma as before. From this andthe uniform size of the stable manifolds we obtain x ∈ Cl ( W s ( Sing ( x ))) which isa contradiction. Therefore, ω ( x ) ∩ F s ( x ′ ) = ∅ . As ∂ Σ z consists of ∂ v Σ z togetherwith F s ( x ′ ) ∪ F s ( z ′ ) we obtain ω ( x ) ∩ ∂ Σ z = ∅ . The construction of Σ z is similarin the case when ω ( x ) intersects both components of R z \ F s ( z ). This finishes theproof. (cid:3) A second auxiliary definition is as follows.
N THE ESSENTIAL HYPERBOLICITY OF SECTIONAL-ANOSOV FLOWS 7
Definition 2.6.
Let X be a vector field of a compact manifold M . Given δ ≥ wesay that X satisfies (C) δ if { x ∈ M : ω ( x ) ∩ B δ ( Sing ( X )) = ∅} ⊂ Cl ( W s ( Sing ( X ))) . We shall use the previous lemma to prove the following result. Its proof followsclosely that of Theorem 4 in [3].
Lemma 2.7. If δ is an intersection number of a sectional-Anosov flow X of acompact -manifold M , then X satisfies (C ) δ .Proof. Suppose by contradiction that (C) δ fails. Then, there is x ∈ M such that ω ( x ) ∩ B δ ( Sing ( X )) = ∅ but x / ∈ Cl ( W s ( Sing ( X ))).By Lemma 2.5 we have that ω ( x ) has a singular partition R . Using that x / ∈ Cl ( W s ( Sing ( X ))) we can choose an interval I around x , tangent to E cx , which doesnot intersect W s ( Sing ( X )). On the other hand, we have clearly that ω ( x ) is not asingularity. Then, by Theorem 11 in [7], there are S ∈ R , sequence x n ∈ S (in thepositive orbit of x ), a sequence I n of intervals around x n (in the positive orbit of I )such that both components of I n \ { x n } have length bounded away from zero. Wecan assume that x n → w for some w ∈ S and further I n converges to an interval J around w tangent to E cw . But w ∈ ω ( x ) so w ∈ Ω( X ) and, then, by the closinglemma [20], we have that w is accumulated by periodic points or by points whoseomega-limit set is a singularity. In the latter case we have from the uniform size ofthe stable manifolds that J ∩ W s ( Sing ( X )) = ∅ and so J n ∩ W s ( Sing ( X )) = ∅ forall n large. As J n belongs to the orbit of I we obtain I ∩ W s ( Sing ( X )) = ∅ whichis a contradiction. Therefore, there is a sequence p n ∈ P er ( X ) converging to w . If W u ( p n ) ∩ B δ ( Sing ( X )) = ∅ for infinitely many n ’s we obtain from the fact that δ is an intersection number that W u ( p n ) ∩ W s ( Sing ( X )) = ∅ for such integers n .Applying the Inclination Lemma as before we obtain that x ∈ Cl ( W s ( Sing ( X ))),a contradiction. From this we conclude that Cl ( W u ( p n )) ∩ B δ ( Sing ( X )) = ∅ forall n large. In particular, every p n belongs to the set H δ defined in (1) whichis hyperbolic, and so, the unstable manifold W u ( p n ) has uniformly large size forlarge n . As both p n and x n converges to w we obtain that there is a point in thepositive orbit of x whose omega-limit set is contained in Cl ( W u ( p n )) for some n . Itfollows that ω ( x ) ⊂ Cl ( W u ( p n )) and, then, Cl ( W u ( p n )) ∩ B δ ( Sing ( X )) = ∅ whichis absurd. This contradiction concludes the proof. (cid:3) To prove Corollary 1.8 we need the following generalization of Proposition 1 in[3].
Lemma 2.8.
Every vector field X of a compact manifold M exhibiting a finitecollection of saddle-type hyperbolic attractors and singularities whose basins form adense subset of M is sensitive to the initial conditions.Proof. If X has no saddle-type hyperbolic attractors, then the result follows fromProposition 1 in [3]. So, we can assume that X has at least one saddle-type hyper-bolic attractor.Let { A , · · · , A r } and { σ , · · · , σ l } be the collection of saddle-type hyperbolicattractors and singularities of X whose basins form a dense subset of M . As is well-known (p.9 in [27]) for every i = 1 , · · · , r there is β i > X restricted to B β i ( A i ) is sensitive to the initial conditions. Let δ i be the corresponding sensitivityconstant for i = 1 , · · · , r . S. BAUTISTA, C. A. MORALES
To conclude the proof we shall prove that any positive number δ less thanmin (cid:26) β , · · · , β r , δ , · · · ,δ r , min { d ( B, C ) :
B, C ∈ { A , · · · , A r , σ , · · · , σ l } , B = C }} is a sensitivity constant of X .Indeed, take x ∈ M and suppose by contradiction that there is a neighborhood U of x such that d ( X t ( x ) , X t ( y )) ≤ δ for every t ≥
0. Suppose for a while thatthere is y ∈ U such that ω ( y ) ⊂ A i for some i = 1 , · · · , r . Then, d ( X t ( y ) , A i ) < β i for some t ≥ d ( X t ( x ) , A i ) ≤ d ( X t ( x ) , X t ( y )) + d ( X t ( y ) , A i ) ≤ δ + β i < β i β i β i thus X t ( x ) ∈ B β i ( A i ). From this we can find T ≥ t and also z ∈ U such that d ( X T ( x ) , X T ( z )) ≥ δ i ≥ δ . Therefore, we can assume that there is no y ∈ U withinthe union of the basins of the attractors { A , · · · , A r } , and so, W s ( { σ , · · · , σ l } ) ∩ U is dense in U by the hypothesis. Now we can proceed as in the proof of Proposition1 in [3] to obtain the desired contradiction.More precisely, we have two possibilities, namely, either x ∈ W s ( σ i ) for some i = 1 , · · · , l or not. In the first case we can select y ∈ U outside W s ( σ i ) since W s ( σ i )has no interior (recall σ i is saddle-type). Since the positive orbit of x converges to σ i , and that of y does not, we eventually find t > d ( X t ( x ) , X t ( y )) ≥ δ which is absurd. In the second case can use the hypothesis to select y ∈ W s ( σ i ) ∩ U for some i = 1 , · · · , l since W s ( { σ , · · · , σ l } ) ∩ U is dense in U . Again we argue thatsince the positive orbit of y converges to σ i , and that of x does not, we eventuallyfind t > d ( X t ( x ) , X t ( y )) ≥ δ which is absurd too. These contradictionsprove that δ as above is a sensitivity constant of X and the result follows. (cid:3) Proof of Theorem 1.5 and Corollary 1.8
Proof of Theorem 1.5.
Let X be a sectional-Anosov flow of a compact 3-manifold.By Lemma 2.2 we have that X has an intersection number δ and by Lemma 2.7 wehave that X satisfies (C) δ . For such a δ we let H δ be as in (1).Now take x Cl ( W s ( Sing ( X ))). Since Cl ( W s ( Sing ( X ))) is closed there isa neighborhood U of x such that U ∩ Cl ( W s ( Sing ( X ))) = ∅ . By (C) δ we have ω ( y ) ∩ B δ ( Sing ( X )) = ∅ and then ω ( y ) ⊂ H δ for every y ∈ U . But H δ is hyperbolic,so, there is an open and dense subset of U all of whose points belong to the basin ofa hyperbolic attractor of X in H δ . This proves that the union of the basins of thehyperbolic attractors form together with W s ( Sing ( X )) a dense subset of M . As theunion of the stable manifolds of the non-Lorenz-like singularities is nowhere dense(c.f. [5], [8]) we obtain that the union of the basins of the hyperbolic attractors andthe Lorenz-like singularities is dense in M . As X has only a finite number of bothhyperbolic attractors (by Corollary 2.3) and Lorenz-like singularities (for they arehyperbolic) we are done. (cid:3) Proof of Corollary 1.8.
By Theorem 1.5 we have that every sectional-Anosov flowof a compact 3-manifold satisfies the hypotheses of Lemma 2.8. So, the resultfollows from this lemma. (cid:3)
N THE ESSENTIAL HYPERBOLICITY OF SECTIONAL-ANOSOV FLOWS 9
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