Dominated chain recurrent class with singularities
aa r X i v : . [ m a t h . D S ] J un Dominated chain recurrent class with singularities
Christian Bonatti Shaobo Gan Dawei Yang ∗ Abstract
We prove that for generic three-dimensional vector fields, domination impliessingular hyperbolicity.
One goal of differential dynamical systems is to understand the global dynamics of mostdynamical systems. The dynamics of hyperbolic vector fields are well understood. Thereis no good description of robustly non-hyperbolic dynamics. A sequence of conjectures ofPalis [30, 31, 32] gives us a beautiful perspective. There are already many results in thisdirection, mainly for diffeomorphisms. If one considers smooth vector fields, [32] has thefollowing conjectures:
Conjecture 1.
Every robustly non-hyperbolic vector field can be C r approximated by avector field with a homoclinic tangency or a heterodimensional cycle or a singular cycle. Conjecture 2.
Every robustly non-hyperbolic three-dimensional vector field can be C r approximated by a vector field with a singular cycle or a Lorenz-like attractor or a Lorenz-like repeller. The difficulty for vector field is the fact that hyperbolic singularities can be accumu-lated by recurrent regular points. Usually in this case, one can get a homoclinic orbit ofthe singularity by perturbation (using C connecting lemmas). Then one can see a dif-ferent phenomena compared with homoclinic orbits of periodic orbits: a homoclinic orbitof a singularity cannot be transverse, thus it cannot be robust under perturbation. Thefamous Lorenz attractor [23] gives us a very strange phenomenon: it’s a robustly non-hyperbolic chaotic attractor. To understand the hyperbolic properties of Lorenz-like at-tractor, [28, 29] gave the definition of singular hyperbolic . Moreover, for three-dimensionalflows, [28, 27, 29] proved: • Every robustly transitive set is a singular hyperbolic attractor or repeller. • If X cannot be accumulated by infinitely many sinks or sources, then X is singularAxiom A . ∗ D. Yang thanks the support of NSFC 11001101 and Ministry of Education of P. R. China20100061120098. S. Gan is supported by 973 program 2011CB808002 and NSFC 11025101.
1n the spirit of results for diffeomorphisms (see [33, 7, 13, 14, 15]) and vector fields,one can have several conjectures. To be more precise, we give some definitions. Let M be a compact C ∞ Riemannian manifold without boundary. Let X r ( M ) be the Banachspace of the C r vector fields on M with the usual C r norm. For X ∈ X ( M ), φ Xt is theflow generated by X . Φ Xt = dφ Xt is the tangent flow of X . If there is no confusion, wewill use φ t and Φ t for simplicity. For a compact invariant set Λ of φ t , one says that Λhas a dominated splitting with respect to Φ t if there is a continuous invariant splitting T Λ M = E ⊕ F , and two constants C > λ > x ∈ Λ and t > k Φ t | E ( x ) kk Φ − t | F ( φ t ( x )) k ≤ Ce − λt . If dim E is a constant, then dim E is called the index of the dominated splitting. Aninvariant bundle E ⊂ T Λ M is called contracting if there are two constants C > λ > x ∈ Λ and t >
0, one has k Φ t | E ( x ) k ≤ Ce − λt . An invariant bundle F iscalled expanding if it’s contracting for − X . If T Λ M = E ⊕ F is a dominated splitting andeither E is contracting or F is expanding, then one says that Λ is partially hyperbolic . Wealso need the notions of singular hyperbolicity. A continuous invariant bundle E ⊂ T Λ M is called sectional contracting if there are two constants C > λ > x ∈ Λ, t > L ⊂ E ( x ), one has | DetΦ t | L | ≤ Ce − λt . Acontinuous invariant bundle F is called sectional expanding if it is sectional contracting for − X . A compact invariant set Λ is called singular hyperbolic if Λ has a partially hyperbolicsplitting T Λ M = E ⊕ F , and either E is sectional contracting and F is sectional expanding,or E is contracting and F is sectional expanding. Singular hyperbolicity is a generalizationof hyperbolicity for compact invariant set with singularities, it’s special for vector fieldswith singularities. For instance, the classical Lorenz attractor is singular hyperbolic, butnot hyperbolic (see [1]). Here a compact invariant set Λ is called hyperbolic if thereis a continuous invariant splitting T Λ M = E s ⊕ < X > ⊕ E u , where E s is uniformlycontracting, E u is uniformly expanding, and < X > is the subspace generated by thevector field. If dim E s is constant, then dim E s is called the index (or stable index) of thehyperbolic set Λ.We will consider compact invariant set with recurrence. The most general setting isthe chain recurrence. For any point x, y ∈ M , if for any ε >
0, there is a sequence ofpoints { x i } ni =0 and a sequence of times { t i } ni =0 such that • x = x and x n = y , • t i ≥ i , • d ( φ t i ( x i ) , x i +1 ) < ε for any 0 ≤ i ≤ n − x is in the chain stable set of y . If x is in the chain stable set of y and y is also in the chain stable set of x , then one says that x and y are chain related . If x is chain related with itself, then x is called a chain recurrent point . Let CR( X ) be theset of chain recurrent points of X . It’s clear that chain related relation is an equivalentrelation. By using this equivalent relation, one can divide CR( X ) into equivalent classes.Each equivalent class is called a chain recurrent class . A chain recurrent class is called non-trivial if it’s not reduced to a periodic orbit or a singularity. X is called singular Axiom A without cycle if X has only finitely many chain recurrentclasses, and each chain recurrent class is singular hyperbolic. Singular Axiom A vector2elds is a generalization of hyperbolic vector fields. One says that X has a homoclinictangency if X has a hyperbolic periodic orbit γ , and W s ( γ ) and W u ( γ ) have some non-transversal intersection. One says that X has a heterodimensional cycle if X has twohyperbolic periodic orbits γ and γ with different indices such that W s ( γ ) ∩ W u ( γ ) = ∅ and W u ( γ ) ∩ W s ( γ ) = ∅ .In the spirit of Palis and the previous results, one has the following conjectures: Conjecture 3.
Every vector field can be C r approximated by a vector field with a horse-shoe or by a Morse-Smale vector field. Conjecture 4.
Every vector field can be C r approximated by one of the following threekinds of vector fields: • a vector field which is singular Axiom A without cycle, • a vector field with a homoclinic tangency, • a vector field with a heterodimensional cycle. These two conjectures could be viewed as the continuations of the conjectures of Palisfor flows. If dim M = 3, Conjecture 4 was given by [27]. One notices that X is called star if there is a neighborhood U of X such that every critical element of Y ∈ U ishyperbolic. [39] conjectured that every star vector field is singular Axiom A withoutcycle. If Conjecture 4 is true, one will have that every generic star vector field is singularAxiom A without cycle.The main result of this work deals with a special case of Conjecture 4: it concernswhen one can get singular hyperbolicity for three-dimensional vector fields. To avoid somepathological phenomena, one consider residual set of X r ( M ): it contains a countableintersection of dense open subset of X r ( M ). Since X r ( M ) is complete, one has everyresidual set is dense in X r ( M ). A residual set of X r ( M ) is also called a dense G δ set. Theorem A.
Assume that dim M = 3 . There is a dense G δ set G ⊂ X ( M ) such that if X ∈ G and C ( σ ) is a non-trivial chain recurrent class of a singularity σ with the followingproperties: • C ( σ ) contains a periodic point p , • C ( σ ) admits a dominated splitting T C ( σ ) M = E ⊕ F with respect to Φ t ,then C ( σ ) is singular hyperbolic. As a corollary, C ( σ ) is an attractor or a repeller de-pending on the index of σ . About the condition that the chain recurrent class contains a periodic point, [4] hasthe following conjecture:
Conjecture 5.
For generic vector field X , if C ( σ ) is a non-trivial chain recurrent classcontaining a hyperbolic singularity σ , then C ( σ ) contains a hyperbolic periodic orbit. If we don’t assume that C ( σ ) contains a periodic point, we will get a partially hyper-bolic splitting. 3 heorem B. Assume that dim M = 3 . There is a dense G δ set G ⊂ X ( M ) such that if X ∈ G and C ( σ ) is a non-trivial chain recurrent class of a singularity σ with a dominatedsplitting T C ( σ ) M = E ⊕ F with respect to Φ t , then C ( σ ) is partially hyperbolic. Remark. [5] proved that for C generic vector field X , X has two kinds of chain recurrentclasses: • Either, a chain recurrent class contains a hyperbolic periodic orbit, then the chainrecurrent class is the homoclinic class of the hyperbolic periodic orbit, i.e., the closureof all transverse homoclinic orbits of the hyperbolic periodic orbit. • Or, a chain recurrent class contains no periodic orbits, then it’s called an aperiodicclass .For diffeomorphisms, it’s difficult to define the continuations of aperiodic classes. Butfor vector fields, if an aperiodic class contains singularities, it’s easy to define their contin-uations. One of the differences between periodic orbits and singularities is: singularitiescan’t have transverse homoclinic orbits.
We have some further remarks on the conjectures: • [2] proved Conjecture 1 for three-dimensional vector fields for C topology. • [18] proved Conjecture 3 for three-dimensional vector fields for C topology.In the spirit of conjectures of [4], we have the following conjectures: Conjecture 6.
Every vector field can be C approximated by a vector field which issingular Axiom A without cycle, or by a vector field with a robustly heterodimensionalcycle. Conjecture 7.
For C generic X ∈ X ( M ) which cannot be approximated by vector fieldswith a homoclinic tangency, X has only finitely many chain recurrent classes. If dim M = 3, Conjecture 6 can be restated as: C generic vector field is singular AxiomA without cycle. Compared with two-dimensional diffeomorphisms, this conjecture maybe called singular Smale conjecture . [16] proves that for C generic three-dimensionalvector field X , the singularities in a same chain recurrent class of X have the same index.There are also many results for higher dimensional vector fields. We give a partial listof them. • [10] gave an example of singular hyperbolic attractors for which the unstable man-ifolds of singularities have arbitrarily large dimension. • [20, 39, 26] proved that under the star condition , every robustly transitive set issingular hyperbolic. • [35] constructed an example on robustly wild strange (quasi)-attractor with singu-larities for four-dimensional vector fields for higher regularities. • [3] showed that there exist robustly chain transitive non-singular-hyperbolic attrac-tors for five-dimensional vector fields. • [9] showed that there exist robustly chain transitive non-singular-hyperbolic attrac-tors with different indices of singularities for four-dimensional vector fields.4 Preliminaries
As in the introduction, every vector field X ∈ X ( M ) generates a flow φ Xt . Weidentity the vector field and its flow as the same object. From the flow φ Xt , one can defineits tangent flow Φ Xt = dφ Xt : T M → T M . For every regular point x ∈ M \ Sing( X ), onecan define its normal space N x = { v ∈ T x M : < v, X ( x ) > = 0 } . Define the normal bundle on regular points as: N = G x ∈ M \ Sing( X ) N x . On the normal bundle N , one can define the linear Poincar´e flow ψ Xt : for each v ∈ N x ,one can define ψ t ( v ) = Φ t ( v ) − < Φ t ( v ) , X ( φ t ( x )) > | X ( φ t ( x )) | X ( φ t ( x )) . For an invariant (may not compact) set Λ ⊂ M \ Sing( X ), one says that Λ admits adominated splitting with respect to the linear Poincar´e flow if there are constants C > λ < N Λ = ∆ s ⊕ ∆ u such that for any x ∈ Λ, one has k ψ t | ∆ s ( x ) kk ψ − t | ∆ u ( φ t ( x )) k < Ce λt . dim ∆ s is called the index of the dominated splitting.If Λ is a compact invariant set without singularities, the existence of dominated split-ting for the linear Poincar´e flow is a robust property. Lemma 2.1.
For X ∈ X ( M ) , if Λ is a compact invariant set which is disjoint fromsingularities, and admits a dominated splitting with respect to the linear Poincar´e flowof index i , then there is ε > such that for each Y which is ε - C -close to X , for anycompact invariant set Λ Y contained in the ε neighborhood of Λ , Λ Y admits a dominatedsplitting with respect to the linear Poincar´e flow of index i . For dominated splittings of tangent flows, one will always have the robust propertyfor compact invariant sets with singularity or not.
Lemma 2.2.
For X ∈ X ( M ) , if Λ is a compact invariant set with a dominated splittingwith respect to the tangent flow of index i , then there is ε > such that for each Y whichis ε - C -close to X , for any compact invariant set Λ Y contained in the ε neighborhood of Λ , Λ Y admits a dominated splitting with respect to the tangent flow of index i . By the definition of linear Poincar´e flow, one has the following lemma:
Lemma 2.3.
For X ∈ X ( M ) , if Λ is a compact invariant set with a dominated splitting T Λ M = E ⊕ F with respect to the tangent flow and X ( x ) ∈ F ( x ) for any x ∈ Λ , then N Λ \ Sing( X ) admits a dominated splitting of index dim E with respect to the linear Poincar´eflow. .2 Minimally non-hyperbolic set and C arguments Sometimes we need to discuss non-hyperbolic set. Its non-hyperbolicity will concen-trate on some smaller parts, which are called minimally non-hyperbolic set from Liao [21]and Ma˜n´e [25]. A compact invariant set Λ is called minimally non-hyperbolic if Λ is nothyperbolic and every compact invariant proper subset of Λ is hyperbolic. From [2, 33],one has the following two lemmas.
Lemma 2.4.
Assume that dim M = 3 , if Λ is a minimally non-hyperbolic set of a vectorfield X ∈ X ( M ) such that • Λ ∩ Sing( X ) = ∅ , • N Λ admits a dominated splitting with respect to the linear Poincar´e flow,then Λ is transitive. X is called weak-Kupka-Smale if every periodic orbit or singularity is hyperbolic. Lemma 2.5.
Assume that dim M = 3 and X is a C weak-Kupka-Smale , if Λ is a transitive minimally non-hyperbolic set of a vector field X such that Λ ∩ Sing( X ) = ∅ ,then Λ is a normally hyperbolic torus and the dynamics on Λ is equivalent to an irrationalflow. A compact invariant set Λ is called chain transitive if for any ε >
0, for any x, y ∈ Λ,there are { x i } ni =0 ⊂ Λ and { t i } n − i =0 ⊂ [1 , ∞ ) such that x = x , x n = y and d ( φ t i ( x i ) , x i +1 ) <ε for each 0 ≤ i ≤ n −
1. For chain transitive sets and hyperbolic periodic orbits orsingularities, by using λ -lemma, one has Lemma 2.6. If Λ is a non-trivial chain transitive set and Λ contains γ , where γ isa hyperbolic periodic orbit or a hyperbolic singularity, then Λ ∩ W s ( γ ) \ { γ } 6 = ∅ and Λ ∩ W u ( γ ) \ { γ } 6 = ∅ . As a corollary of the above folklore lemma, one has:
Lemma 2.7. If Λ is a non-trivial chain transitive such that • Λ admits a dominated splitting T Λ M = E ⊕ F with respect to the tangent flow and X ( x ) ∈ F ( x ) for any x ∈ Λ , • Λ contains a hyperbolic singularity σ ,Then ind( σ ) > dim E .Proof. We will prove this lemma by absurd. If the lemma is not true, one has ind( σ ) ≤ dim E for some hyperbolic singularity σ ∈ Λ. Since Λ has the dominated splitting, onehas E s ( σ ) ⊂ E ( σ ). By Lemma 2.6, one has there is x ∈ W s ( σ ) ∩ Λ \ { σ } . Thus, X ( φ t ( x )) ⊂ T φ t ( x ) W s ( σ ) for any t >
0. By the assumption one has X ( φ t ( x )) ⊂ F ( φ t ( x )).On the other hand, one has lim t →∞ < X ( φ t ( x )) > ⊂ E s ( σ ) ⊂ E ( σ ). This fact contradictsto the continuity of dominated splittings. 6or each compact set K ( K may be not invariant), one can define the chain recurrentset in K : CR( X, K ). We says that x ∈ CR(
X, K ) if there is a chain transitive set Λ ⊂ K such that x ∈ Λ. CR(
X, K ) has some upper-semi continuity property.
Lemma 2.8.
For given X and K , if there is a sequence of vector fields { X n } and asequence of compact sets K n such that • X n → X as n → ∞ in the C topology, • K n → K as n → ∞ in the Hausdorff topology,then lim sup n →∞ CR( X n , K n ) ⊂ CR(
X, K ) . By the upper-semi continuity property, one has
Lemma 2.9.
For given X and K , if CR(
X, K ) is hyperbolic, then there is a C neigh-borhood U of X and a neighborhood U of K such that CR(
Y, U ) is hyperbolic. Lemma 2.10.
For given X and K , if CR(
X, K ) = ∅ , then there is a C neighborhood U of X and a neighborhood U of K such that CR(
Y, U ) = ∅ . Ma˜n´e’s ergodic closing lemma [24] was established for flows by Wen [36]. x ∈ M \ Sing( X ) is called strongly closable if for any C neighborhood U of X , for any δ > Y ∈ U and p ∈ M , π ( p ) > • φ Yπ ( p ) ( p ) = p , • X ( x ) = Y ( x ) for any x ∈ M \ ∪ t ∈ [0 ,π ( p )] B ( φ t ( x ) , δ ), • d ( φ Xt ( x ) , φ Yt ( p )) < δ for each t ∈ [0 , π ( p )].Let Σ( X ) be the set of strongly closable points of X . Lemma 2.11. [Ergodic closing lemma for flows [36]] µ (Σ( X ) ∪ Sing( X )) = 1 for every T > and every φ XT -invariant probability Borel measure µ . We list all known generic results we need in this paper.
Lemma 2.12.
There is a dense G δ set G ⊂ X ( M ) such that for each X ∈ G , one has1. Every periodic orbit or every singularity of X is hyperbolic.2. For any non-trivial chain recurrent class C ( σ ) , where σ is a hyperbolic singularity ofindex dim M − , then every separatrix of W u ( σ ) is dense in C ( σ ) . As a corollary, C ( σ ) is transitive.3. Given i ∈ [0 , dim M − . If there is a sequence of vector fields { X n } such that • lim n →∞ X n = X , each X n has a hyperbolic periodic orbits γ X n of index i such that lim n →∞ γ X n =Λ ,then there is a sequence of hyperbolic periodic orbits γ n of index i of X such that lim n →∞ γ n = Λ . Remark.
Item 1 is the classical Kupka-Smale theorem [19, 34]. Item 2 is a corollary ofthe connecting lemma for pseudo-orbits [5]. There is no explicit version like this. [27,Section 4] gave some ideas about the proof of Item 2 without using of the terminology ofchain recurrence. Item 3 is fundamental, one can see [37] for instance.
Assume that dim M = d . For a hyperbolic singularity σ of X ∈ X r ( M ), one can listall eigenvalues of DX ( σ ) as { λ , λ , · · · , λ i , λ i +1 , · · · , λ d } such thatRe( λ ) ≤ Re( λ ) ≤ · · · ≤ Re( λ i ) < < Re( λ i +1 ) ≤ · · · ≤ Re( λ d ) . Then one says that I ( σ ) = Re( λ i ) + Re( λ i +1 ) is the saddle value of σ .By using the C connecting lemma for pseudo-orbits [5] and an estimation of Liao[22], [18] proved that Lemma 2.13.
Assume that dim M = 3 . There is a residual set G ⊂ X ( M ) such thatfor any X ∈ G , if σ is a hyperbolic singularity of index and I ( σ ) < and the normsof eigenvalues of DX ( σ ) are mutually different, then σ is isolated in CR( X ) : there is aneighborhood U of σ such that U ∩ CR( X ) = { σ } . Remark.
We give some rough idea of the proof of Lemma 2.13. Let σ be a singularityas in Lemma 2.13. If it’s not isolated form other chain recurrent points (i.e., C ( σ ) isnon-trivial), by using the C connecting lemma, one can get a homoclinic loop associateto the singularity. By an extra perturbation, one can assume that the homoclinic loop isnormally hyperbolic. By another small perturbation, one can put the unstable manifold ofthe singularity in the stable manifold of a sink: and this is a robust property! Thus, theunstable manifold of the singularity is in the stable manifold of a sink generically, whichgives a contradiction. One notices that [29] proved that singularities with the properties in Lemma 2.13 isdisjoint from robustly transitive sets for three-dimensional flows.
Lemma 3.1.
Assume that dim M = 3 . For C generic X ∈ X ( M ) , if Λ is a chaintransitive set with the following properties: • Sing( X ) ∩ Λ = ∅ , • N Λ = ∆ s ⊕ ∆ u is a dominated splitting with respect to ψ t ,then Λ is hyperbolic. roof. We take a countable basis { U n } of M . Let O = { O n } n ∈ N such that each O n is theunion of finite elements in { U n } . For each n , one can define • H n ⊂ X ( M ) is a subset with the following property: X ∈ H n if and only ifCR( X, O n ) is hyperbolic or CR( X, O n ) = ∅ . By Lemma 2.9 and Lemma 2.10, H n is an open set. • N n ⊂ X ( M ) is a subset with the following property: X ∈ N n if and only if there isa C neighborhood U ⊂ X ( M ) of X such that for any Y ∈ U , CR( Y, O n ) is neitherhyperbolic nor empty.By definitions, one has H n ∪ N n is open and dense in X ( M ). Now one takes G = \ n ∈ N ( H n ∪ N n ) . It’s clear that G is a dense G δ set. For each X ∈ G , we assume that Λ is a non-singularchain transitive set with a dominated splitting N Λ = ∆ s ⊕ ∆ u on the normal bundle N Λ with respect to the linear Poincar´e flow ψ t . We will prove that Λ = CR( X, Λ) (since Λis chain transitive) is hyperbolic. If not, Λ contains a minimally non-hyperbolic set. ByLemma 2.4, Λ contains a minimally non-hyperbolic set Γ, which is transitive. Take n ∈ N such that • Γ is contained in O n . • There is a C neighborhood U of X such that the maximal invariant set in O n of Y ∈ U has a dominated splitting on the normal bundle with respect to ψ Yt byLemma 2.1.Since Γ is not hyperbolic, one has X ∈ N n . Take a C weak-Kupka-Smale vector field Y ∈ N n ∩ U . Since Y in N n , CR( Y, O n ) is not hyperbolic. But since Y ∈ U , the maximalinvariant set has a dominated splitting on the normal bundle with respect to the linearPoincar´e flow. By Lemma 2.5, one has the splitting CR( Y, O n ) = Λ ∩ Λ such that • Λ is the union of chain transitive sets, and each chain transitive set is hyperbolic.In other words, Λ is hyperbolic. • Λ = ∪ ≤ i ≤ m T i , each T i is a normally hyperbolic torus, and the dynamics on T i is equivalent to an irrational flow. In other words, T i is isolated from other chainrecurrent points.By an arbitrarily small perturbation, there is Z C -close to Y such that • CR(
Z, O n ) = Λ ∪ Λ . • Λ is still a hyperbolic set of Z . • The dynamics on T i of Z is Morse-Smale.As a corollary, CR( Z, O n ) is hyperbolic. This fact contradicts to Z ∈ N n .9 emma 3.2. Assume that dim M = 3 . For C generic X ∈ X ( M ) , if Λ is a compactinvariant set with a dominated splitting T Λ M = E ⊕ F of index 1 with respect to thetangent flow Φ t such that • There is
T > such that for every singularity σ ∈ Λ , one has | Det(Φ T | F ( σ ) ) | > . • For every x ∈ Λ \ Sing( X ) , one has < X ( x ) > ⊂ F ( x ) . • F is not sectional expanding,then there is a sequence of sinks { P n } such that lim n →∞ P n = Γ ⊂ Λ .Proof. One define ϕ ( x ) = log | Det(Φ T | F ( x ) ) | for each x ∈ Λ. We will prove this lemma byabsurd. If for any x ∈ Λ, there is n ( x ) ∈ N such that ϕ ( φ n ( x ) T ( x )) >
0, then by a compactargument one can get that F is sectional expanding. Thus, there is an ergodic invariantmeasure µ with supp( µ ) ⊂ Λ such that Z ϕdµ ≤ . By Lemma 2.11, for the set of strongly closable set Σ( X ), one has µ (Σ( X ) ∪ Sing( X )) = 1.Since for each singularity σ ∈ Λ one has Det(Φ T | F ( σ ) ) > µ (Σ( X ) \ Sing( X )) = 1. Since ϕ is a continuous function, by Birkhoff’s ergodic theorem,one has for almost every point x ∈ supp( µ ) ∩ Σ( X ) with respect to µ such thatlim n →∞ n n − X i =0 ϕ ( φ iT ( x )) = Z ϕdµ. Without loss of generality, one has that x is not periodic. Otherwise, since X is C generic, one has that x is a hyperbolic periodic point. This will imply that the orbit of x is a periodic sink in Λ. Thus one can get the conclusion.Since x is a strong closable point, for any ε > Y which is ε - C -close to X and p ε ∈ M , π ( p ε ) > • φ Yπ ( p ε ) ( p ε ) = p ε , • d ( φ Xt ( x ) , φ Yt ( p ε )) < δ for each t ∈ [0 , π ( p ε )].Since x is non-periodic, one has π ( p ε ) → ∞ as ε →
0. By the continuity property ofdominated splittings, one has the orbit of p ε with respect to Y also a dominated splitting E ε ⊕ F ε and F ε → F , E ε → E as ε → ε → π ( p ε ) /T ] n − X i =0 log | Det( D Φ YT | F ε ( φ iT ( p ε )) ) | ≤ . Since one has the dominated splitting on the orbit of each periodic orbit, one has foreach p ε the largest Lyapunov exponent along the orbit of p ε tends to zero as ε → { Orb( p ε ) } to be smaller by an arbitrarily small perturbation. As a corollary, there is asequence of vector fields { X n } such that 10 lim n →∞ X n = X in the C topology. • Each X n has a sink γ n such that lim n →∞ γ n = Γ.Since X is C generic, by Lemma 2.12, one can gets the conclusion.Now we will manage to prove Theorem B. Assume that we are under the assump-tions of Theorem B. First we have
Lemma 3.3.
For every regular point x ∈ C ( σ ) , one has • either, X ( x ) ∈ E ( x ) , • or, X ( x ) ∈ F ( x ) .Proof. By Lemma 2.12, C ( σ ) is transitive. Thus, the set˜ C = { x ∈ C ( σ ) : ω ( x ) = α ( x ) = C ( σ ) } is dense in C ( σ ).By the invariance property, if for some y ∈ ˜ C , one has either X ( y ) ∈ E ( y ) or X ( y ) ∈ F ( y ), then one can get the conclusion. Thus, one can assume that it’s not true. Thus,for any y ∈ ˜ C , one has X ( y ) / ∈ E ( y ) ∪ F ( y ). Take y ∈ ˜ C . There are a sequence of times { t n } such that • lim n →∞ t n = ∞ . • lim n →∞ φ t n ( y ) = y .By the dominated property, one will have lim n →∞ Φ t n ( X ( y )) ∈ F ( y ). Since Φ t n ( X ( y )) = X ( φ t n ( y )) and the vector field X is continuous with respect to the space variable x ∈ M ,one has X ( y ) = lim n →∞ X ( φ t n ( y )) = lim n →∞ Φ t n ( X ( y )). This will imply that X ( y ) ∈ F ( y ), which gives a contradiction. Corollary 3.3.1. If ind( σ ) = 2 , then for any regular point y ∈ C ( σ ) , X ( y ) ∈ F ( y ) . Asa corollary, singularities in C ( σ ) will have the same index.Proof. The fact the ind( σ ) = 2 implies that E ( σ ) ⊂ E s ( σ ). We will prove this corollaryby absurd. If it’s not true, by Lemma 3.3, one has X ( x ) ∈ E ( x ) for any regular point x ∈ C ( σ ). This implies that regular points will approximate σ only in the stable subspace E s ( σ ). By λ -lemma, one knows that regular points will accumulate both W s ( σ ) and W u ( σ ). This fact gives a contradiction.If singularities in C ( σ ) have different indices, then there are hyperbolic singularities σ , σ ∈ C ( σ ) such that ind( σ ) = 1 and ind( σ ) = 2. Thus by previous arguments, onehas for every regular point x , one has X ( x ) ∈ E ( x ) and X ( x ) ∈ F ( x ). This contradictionends the proof. Lemma 3.4. If ind( σ ) = 2 , then dim E = 1 and E is contracting. roof. First by Corollary 3.3.1, one has for every regular point x , X ( x ) ∈ F ( x ). Ifdim E = 1 is not true, one has dim E = 2. This means that regular points approximate σ only in the unstable subspace of σ , which contradicts to the fact that C ( σ ) ∩ W s ( σ ) \{ σ } 6 = ∅ . We will prove that E is contracting. One notices that by Corollary 3.3.1, every sin-gularity σ ′ in C ( σ ) has index 2 and E ( σ ′ ) ⊂ E s ( σ ′ ). For any point x ∈ Σ, there are twocases:1. ω ( x ) ⊂ Sing( X ),2. ω ( x ) \ Sing( X ) = ∅ .In the first case, one has there is t x > k Φ t x | E ( x ) k <
1. In the second case,one choose y ∈ ω ( x ) \ Sing( X ). Take a small neighborhood U y of y such that for any y , y ∈ U y , one has 12 ≤ | X ( y ) || X ( y ) | ≤ . Choose a sequence of times { t n } such that • lim n →∞ t n = ∞ . • φ t n ( x ) ∈ U y .Thus, | X ( x ) || X ( φ t n ( x )) | = | X ( x ) || X ( φ t ( x )) | | X ( φ t ( x )) || X ( φ t n ( x )) | ≤ | X ( x ) || X ( φ t ( x )) | . Since E ⊕ F is a dominated splitting and X ⊂ F , one has there are constants λ < C > k Φ t n | E ( x ) k ≤ Ce λt n | X ( x ) || X ( φ t n ( x )) | ≤ Ce λt n | X ( x ) || X ( φ t ( x )) | . When n is large enough, one has k Φ t n | E ( x ) k < x ∈ C ( σ ), there is t x > k Φ t x | E ( x ) k <
1. By a classical compact argument, one has E is uniformlycontracting.Since dim M = 3, every hyperbolic singularity in a non-trivial chain recurrent classhas either index 1 or index 2, Lemma 3.4 ends the proof of Theorem B.We will manage to prove Theorem A now. Proof of Theorem A.
Without loss of generality, one can assume that every singularityin C ( σ ) has index 2. Thus, C ( σ ) is Lyapunov stable. • By Lemma 3.4, C ( σ ) has a partially hyperbolic splitting T C ( σ ) M = E s ⊕ F withdim E s = 1. Moreover, every singularity in C ( σ ) has index 2. • By Lemma 2.13, I ( σ ′ ) > σ ′ ∈ C ( σ ).12e will prove Theorem A by absurd. If C ( σ ) is not singular hyperbolic, then F is notsectional expanding by Theorem B. By Lemma 3.2, one has that there is a sequence ofsinks { P n } such that lim n →∞ P n = Λ ⊂ C ( σ ). There are two cases:1. Λ contains a singularity σ ′ ∈ C ( σ ).2. Λ ∩ Sing( X ) = ∅ .In the second case, by Lemma 3.1, Λ is a hyperbolic set. Then, either Λ is a sink, orΛ cannot be accumulated by sinks. But if Λ is a sink, Λ cannot be contained in the chainrecurrent class C ( σ ), which shows that the second case is impossible.Now we are in the first case. Since Λ contains a singularity σ ′ ∈ C ( σ ), one hasΛ ∩ W u ( σ ′ ) = ∅ . Since X is C generic, one has that every separatrix of W u ( σ ′ ) is dense in C ( σ ) by Lemma 2.12. Thus, Λ = C ( σ ). As a corollary, Λ contains the hyperbolic periodicpoint p . Thus, there are p n ∈ P n such that lim n →∞ p n = p . Since C ( σ ) is Lyapunov stableby Lemma 2.12, one has W uloc (Orb( p )) ⊂ C ( σ ). It’s clear that W uloc (Orb( p )) is a two-dimensional manifold and transversal to the strong stable direction E s . In particular, onehas that W s ( W uloc (Orb( p ))) contains Orb( p ) as its interior. As a corollary, ω ( x ) ⊂ C ( σ )for any x ∈ W s ( W uloc (Orb( p ))) and W s ( W uloc (Orb( p ))) contains no sinks. This contradictsto the fact that p can be accumulated by sinks.Now notice that one of the main theorems in [27] asserts that every singular hyperbolicchain recurrent class with a singularity is an attractor or a repeller for three-dimensionalflows. References [1] V. Araujo and M. Pacifico, Three-dimensional flows, Springer-Verlag, 2010.[2] A. Arroyo and F. Rodriguez Hertz, Homoclinic bifurcations and uniform hyperbolic-ity for threedimensional flows,
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