An entropy dichotomy for singular star flows
AAN ENTROPY DICHOTOMY FOR SINGULAR STAR FLOWS
MARIA JOS´E PACIFICO, FAN YANG AND JIAGANG YANG
Abstract.
We show that non-trivial chain recurrent classes for generic C star flows satisfy a dichotomy: either they have zero topological entropy, orthey must be isolated. Moreover, chain recurrent classes for generic star flowswith zero entropy must be sectional hyperbolic, and cannot be detected by anynon-trivial ergodic invariant probability. As a result, we show that C genericstar flows have only finitely many Lyapunov stable chain recurrent classes. Introduction
A milestone towards the final proof of the stability conjecture for diffeomor-phisms is the independent discovery of the star systems by Liao [19] and Ma˜n´e [22].Recall that a diffeomorphism with star condition is a C diffeomorphism for whichall the periodic orbits of all nearby diffeomorphisms are hyperbolic (orbit-wise,not uniformly); in other words, there is no periodic orbit bifurcation. It was firstproven in [10, 19] that Ω-stability implies the star condition, and later proven byHayashi [16] with the help of the connecting lemma that star condition is equiv-alent to Axiom A with the no-cycle condition. For smooth vector fields withoutsingularities, the same result was obtained by Gan and Wen [13] using a differentapproach.For singular flows, that is, flows exhibiting singularities, the situation is muchmore complicated. Here, let us introduce the precise definition of the star conditionfor singular flows, taking into account the presence of the singularities. Definition . A C vector field X is a star vector field if there exists a neighborhood X ∈ U ⊂ X ( M ) such that for all vector fields Y ∈ U , all the critical elements of Y i.e., singularities and periodic orbits, are hyperbolic. The collection of C starvector fields is denoted by X ∗ ( M ).It is well known that singular star flows can exhibit rich dynamics and bifur-cation phenomenon comparing to their non-singular counterpart. Certain singularstar flows, such as the famous Lorenz flows, do not have hyperbolic nonwanderingsets [14] and are not structural stable [15]. There are also examples where theset of the periodic orbits is not dense in the nonwandering set [9], and where thenonwandering set is Axiom A but no-cycle condition fails [17].In order to properly describe the hyperbolicity of an invariant set that con-tains singularity, Morales, Pacifico and Pujals [24] proposed the notion of singularhyperbolicity for three-dimensional flows. They require the flow to have a one-dimensional uniformly contracting direction, and a two-dimensional subbundle con-taining the flow direction of regular points, on which the flow is volume expanding.This is later generalized to higher dimensions as sectional hyperbolicity [18, 23]. SeeDefinition 3 in the next section for more details. Date : January 26, 2021.M.J.P. and J.Y. are partially supported by CNPq, FAPERJ and PROEX-CAPES. J.Y. ispartially supported by NSFC 11871487 of China. F.Y. would like to thank the hospitality ofSouthern University of Science and Technology of China (SUSTC), where part of this work isdone. a r X i v : . [ m a t h . D S ] J a n MARIA JOS´E PACIFICO, FAN YANG AND JIAGANG YANG
Next, let us introduce the notion of chain recurrent classes, which plays animportant role in the study of the stability conjecture. Roughly speaking, chainrecurrent classes are the largest non-trivial invariant set of a topological dynamicalsystem, outside which the system behaves like a gradient system.
Definition . For ε > , T >
0, a finite sequence { x i } ni =0 is called an ( ε, T ) -chain ifthere exists { t i } n − i =0 such that t i > T and d ( φ t i ( x i ) , x i +1 ) < ε for all i = 0 , . . . , n − y is chain attainable from x , if there exists T > ε > ε, T )-chain { x i } ni =0 with x = x and x n = y . It is straightforwardto check that chain attainability is an equivalent relation on the closure of the set { x : ∀ t > , there is an ( ε, T )-chain with x = x n = x and (cid:80) i t i > t } . Each equiv-alent class under this relation is then called a chain recurrent class . A chain recur-rent class C is non-trivial if it is not a singularity nor a periodic orbit.As the first step towards the complete understanding of singular star flows, thefollowing conjecture was raised in [29]. Conjecture . (Generic) singular star flows have only finitely many chain recurrentclasses, all of which are sectional hyperbolic for X or − X .A partial affirmative answer to this conjecture was obtained in [28], where it isproven that for generic star flows, every non-trivial Lyapunov stable chain recurrentclass is sectional hyperbolic. In fact, they obtained a complete characterization onthe stable indices of singularities in the same chain recurrent classes. This result iscrucial to our current paper, and is explained in more detail in Section 2.4.A breakthrough was later obtained by da Luz and Bonatti [3, 8] which gives anegative answer to the second half of this conjecture. They construct an examplewhich has two singularities with different indices robustly contained in the samechain recurrent class. As a result, such classes cannot be sectional hyperbolic. In [3]they propose the notion of multi-singular hyperbolicity and prove that, generically,star condition is equivalent to multi-singular hyperbolicity [3, Theorem 3]. To avoidtechnical difficulty, we will not provide the definition of multi-singular hyperbolicityhere, and invite the interested readers to [7] for a simpler yet useful definition.However, the first half of Conjecture 1.1 is still left unanswered: must a (generic)singular star flow have only finitely many chain recurrent classes?
The goal of thecurrent paper is to provide a partial answer to this question, using the recentprogress in the entropy theory [26]:
Theorem A.
There is a residual set R of C star flows, such that for every X ∈ R and every non-trivial chain recurrent class C of X , we have(1) if h top ( X | C ) > , then C contains some periodic point p and is isolated;(2) if h top ( X | C ) = 0 , then C is sectional hyperbolic for X or − X , and containsno periodic orbits. In this case, every ergodic invariant measure µ with supp( µ ) ⊂ C must satisfy µ = δ σ for some σ ∈ Sing( X ) ∩ C . In other words, a chain recurrent class with zero topological entropy cannot bedetected by any non-trivial invariant ergodic probability measure, in the sense thatit can only support point masses of singularities. We call such chain recurrentclasses singular aperiodic classes .Note that in the second case, C cannot be Lyapunov stable due to [26, CorollaryD]. Also note that the example constructed by Bonatti and da Luz belongs to thefirst case and is isolated.An immediate corollary of Theorem A is: Corollary B. C generic star flows have only finitely many Lyapunov stable chainrecurrent classes. INGULAR STAR FLOWS 3
Now the first half of Conjecture 1.1, namely the finiteness of chain recurrentclasses for singular star flows, is reduced to the following conjecture:
Conjecture . For (generic) singular star flows, singular aperiodic classes do notexist. Consequently, (generic) singular star flows have only finitely many chainrecurrent classes, all of which are homoclinic classes of some periodic orbits.
Organization of the paper.
In Section 2 we provide the readers with preliminarieson singular flows: dominated splitting, (extended and scaled) linear Poincar´e flowsand Liao’s shadowing lemma. We also include in Section 2.4 a classification on thesingularities in the same chain recurrent classes, which is taken from [28]. Section 3contains a detailed analysis on the dynamics near Lorenz-like singularities and, moreimportantly, on the transverse intersection between invariant manifolds of nearbyperiodic orbits with that of points in the class. Finally, we provide the proof ofTheorem A in Section 4 by showing that every chain recurrent class with positivetopological entropy must contain a periodic orbit and, consequently, contains allperiodic orbits that are sufficiently close to the class.2.
Preliminaries
Throughout this paper, X will be a C vector field on a d -dimensional compactmanifold M . Denote by Sing( X ) (sometimes we also write Sing( φ t )) the set ofsingularities of X , φ t the flow generated by X , and f = φ the time-one map of φ t .We will write Φ t for the tangent flow, i.e., Φ t = Dφ t : T M → T M .This section includes the necessary background for the proof of Theorem A.Most notably, we will introduce the scaled and extended linear Poincar´e flow byLiao [20, 21], and the shadowing lemma which was first introduced by Liao [20, 21],and further developed by Gan [11]. We also collect some previously establishedresults on generic singular star flows in Section 2.4, most of which can be foundin [28] and [18].2.1.
Dominated splitting and invariant cones.
A dominated splitting for aflow φ t is defined similarly to the case of diffeomorphisms. The invariant set Λadmits a dominated splitting T Λ M = E ⊕ F if this splitting is invariant under Φ t ,and if there exist C > λ < x ∈ Λ, and every pair ofunit vectors u ∈ E x and v ∈ F x , one has (cid:107) (Φ t ) x ( u ) (cid:107) ≤ Cλ t (cid:107) (Φ t ) x ( v ) (cid:107) for t > . We invite the readers to [4, Appendix B] and [1] for more properties on the dom-inated splitting. The next lemma states the relation between dominated splittingfor the flow and its time-one map.
Lemma 2.1. [26, Lemma 2.6]
Let Λ be an invariant set. A splitting T Λ M = E ⊕ F is a dominated splitting for the flow φ t | Λ if and only if it is a dominated splittingfor the time-one map f | Λ . Moreover, if φ t | Λ is transitive, then we have either X | Λ \ Sing( X ) ⊂ E or X | Λ \ Sing( X ) ⊂ F. Definition . A compact invariant set Λ of a flow X is called sectional hyperbolic ,if it admits a dominated splitting E s ⊕ F cu , such that E s is uniformly contracting,and F cu is sectional-expanding: there are constants C, λ > x ∈ Λ and any subspace V x ⊂ F cux with dim V x ≥
2, we have | det Dφ t ( x ) | V x | ≥ Ce λt for all t > . We call λ the sectional volume expanding rate on F cu . MARIA JOS´E PACIFICO, FAN YANG AND JIAGANG YANG
Remark . If the dominated splitting E ⊕ F is sectional hyperbolic, then Φ t on E is uniformly contracting by definition. Since the flow speed (cid:107) X ( x ) (cid:107) is bounded andthus cannot be backward exponentially expanding, we must have X | Λ \ Sing( X ) ⊂ F .For more detail, see [26, Lemma 3.10].Let E ⊕ F be a dominated splitting for the flow φ t . For a > x ∈ M , a( a, F ) -cone on the tangent space T x M is defined as C a ( F x ) = { v : v = v E + v F where v E ∈ E, v F ∈ F and (cid:107) v E (cid:107) < a (cid:107) v F (cid:107)} ∪ { } . When a is sufficiently small, the cone field C a ( F x ), x ∈ M , is forward invariantby Φ , i.e., there is λ < x ∈ M , Φ ( C a ( F x )) ⊂ C λa ( F f ( x ) ).Similarly, we can define the ( a, E )-cone C a ( E x ), which is backward invariant byΦ . When no confusing is caused, we call the two families of cones by F cones and E cones.For a C disk with dimension at most dim F , we say that D is tangent to the ( a, F ) -cone if for any x ∈ D , T x D ⊂ C a ( F x ). The same can be said for the ( a, E )-cone if T x D ⊂ C a ( E x ) for every x ∈ D .2.2. Scaled linear Poincar´e flows and a shadowing lemma by Liao.
In thissection, µ will be a non-trivial ergodic measure with a dominated splitting E ⊕ F for the time-one map f = φ on supp µ .The linear Poincar´e flow ψ t is defined as following: denote the normal bundle of φ t over Λ by N Λ = (cid:91) x ∈ Λ \ Sing( X ) N x , where N x is the orthogonal complement of the flow direction X ( x ), i.e., N x = { v ∈ T x M : v ⊥ X ( x ) } . Denote the orthogonal projection of T x M to N x by π x and the projection of T Λ M to N Λ by π . Given v ∈ N x for a regular point x ∈ M \ Sing( X ) and recalling thatΦ t is the tangent flow, we can define ψ t ( v ) as the orthogonal projection of Φ t ( v )onto N φ t ( x ) , i.e., ψ t ( v ) = π φ t ( x ) (Φ t ( v )) = Φ t ( v ) − < Φ t ( v ) , X ( φ t ( x )) > (cid:107) X ( φ t ( x )) (cid:107) X ( φ t ( x )) , where < ., . > is the inner product on T x M given by the Riemannian metric. Thefollowing is the flow version of the Oseledets theorem: Proposition 2.3.
For µ almost every x , there exists k = k ( x ) ∈ N and real numbers ˆ λ ( x ) > · · · > ˆ λ k ( x ) and a ψ t invariant measurable splitting on the normal bundle: N x = ˆ E x ⊕ · · · ⊕ ˆ E kx , such that lim t →±∞ t log (cid:107) ψ t ( v i ) (cid:107) = ˆ λ i ( x ) for every non-zero vector v i ∈ ˆ E ix . Now we state the relation between Lyapunov exponents and the Oseledets split-ting for ψ t and for f = φ : Theorem 2.4.
For µ almost every x , denote by λ ( x ) > · · · > λ k ( x ) the Lyapunovexponents and T x M = E x ⊕ · · · ⊕ E kx the Oseledets splitting of µ for f . Then N x = π x ( E x ) ⊕ · · · ⊕ π x ( E kx ) INGULAR STAR FLOWS 5 is the Oseledets splitting of µ for the linear Poincar´e flow ψ t . Moreover, the Lya-punov exponents of µ (counting multiplicity) for ψ t is the subset of the exponentsfor f obtained by removing one of the zero exponent which comes from the flowdirection.Definition . A non-trivial measure µ is called a hyperbolic measure for the flow φ t if it is an ergodic measure of φ t and all the Lyapunov exponents for the linearPoincar´e flow ψ t are non-vanishing. In other words, if we view µ as an invariantmeasure for the time-one map f , then µ has exactly one exponent which is zero,given by the flow direction. We call the number of the negative exponents of µ ,counting multiplicity, its index .The scaled linear Poincar´e flow , which we denote by ψ ∗ t , is the normalization of ψ t using the flow speed:(1) ψ ∗ t ( v ) = (cid:107) X ( x ) (cid:107)(cid:107) X ( φ t ( x )) (cid:107) ψ t ( v ) = ψ t ( v ) (cid:107) Φ t |
The following statements hold:(1) Assume that { φ t ( x ) } [0 ,T ] is ( λ, T ) -forward contracting for the bundle E ,and { φ t ( φ T ( x )) } [0 ,T ] is ( λ, T ) -forward contracting for the bundle ψ T ( E ) .Then { φ t ( x ) } [0 ,T + T ] is ( λ, T ) -forward contracting for the bundle E .(2) Assume that the scaled linear Poincar´e flow has a dominated splitting E ⊕ F . Let { φ t ( x ) } [0 ,T ] be ( λ, T ) -forward contracting for the bundle E , andassume that φ T ( x ) is a ( λ, T ) -forward hyperbolic time for the bundle E .Then x is a ( λ, T ) -forward hyperbolic time for the bundle E . The proof is standard and thus omitted.By the classic work of Liao [20], there exists δ = δ ( λ, T ) > x is a( λ, T )-backward hyperbolic time, then x has unstable manifold with size δ (cid:107) X ( x ) (cid:107) .Similarly, if x is a ( λ, T )-forward hyperbolic time then it has stable manifold withsize δ (cid:107) X ( x ) (cid:107) . In both cases, we say that x has unstable/stable manifold up to theflow speed .The next lemma can be seen as a C version of the Pesin theory for flows. Theproof can be found in [26, Section 4] Lemma 2.7.
Let µ be a hyperbolic measure for the flow φ t . For almost everyergodic component ˜ µ of µ with respect to f = φ , there are L (cid:48) , η, T > and acompact set Λ ⊂ supp µ \ Sing( X ) with positive ˜ µ measure, such that for every x satisfying f n ( x ) ∈ Λ for n > L (cid:48) , the orbit segment { φ t ( x ) } [0 ,n ] is ( η, T ) ∗ quasi-hyperbolic with respect to the splitting N x = π x ( E x ) ⊕ π x ( F x ) and the scaled linearPoincar´e flow ψ ∗ t . Next. we introduce a shadowing lemma by Liao [20] for the scaled linear Poincar´eflow. See [11] and [26] for the current version.
Lemma 2.8.
Given a compact set Λ with Λ ∩ Sing( X ) = ∅ and η ∈ (0 , , T > ,for any ε > there exists δ > , L > and δ > , such that for any ( η, T ) ∗ quasi-hyperbolic orbit segment { φ t ( x ) } [0 ,T ] with respect to a dominated splitting N x = E x ⊕ F x and the scaled linear Poincar´e flow ψ ∗ t , if x, φ T ( x ) ∈ Λ with d ( x, φ T ( x )) <δ , then there exists a point p and a C strictly increasing function θ : [0 , T ] → R ,such that(a) θ (0) = 0 and | θ (cid:48) ( t ) − | < ε ;(b) p is a periodic point with φ θ ( T ) ( p ) = p ;(c) d ( φ t ( x ) , φ θ ( t ) ( p )) ≤ ε (cid:107) X ( φ t ( x )) (cid:107) , for all t ∈ [0 , T ] ;(d) d ( φ t ( x ) , φ θ ( t ) ( p )) ≤ Ld ( x, φ T ( x )) ;(e) p has stable and unstable manifold with size at least δ .(f ) if Λ ⊂ Λ for a chain recurrent class Λ , then p ∈ Λ . Extended linear Poincar´e flows.
Note that the (scaled) linear Poincar´eflow is only defined at the regular points M \ Sing( X ). To solve this issue, weintroduce the extended linear Poincar´e flow , which is a useful tool developed byLiao [20, 21] and Li et al [18] to study hyperbolic singularities.Denote by G = { L : L is a 1-dimensional subspace of T x M, x ∈ M } INGULAR STAR FLOWS 7 the Grassmannian manifold of M . Given a C flow φ t , the tangent flow Φ t actsnaturally on G by mapping each L to Φ t ( L ).Write β : G → M and ξ : T M → M the bundle projection. The pullbackbundle of T M : β ∗ ( T M ) = { ( L, v ) ∈ G × T M : β ( L ) = ξ ( v ) } is a vector bundle over G with dimension dim M . The tangent flow Φ t lifts natu-rally to β ∗ ( T M ): Φ t ( L, v ) = (Φ t ( L ) , Φ t ( v )) . Recall that the linear Poincar´e flow ψ t projects the image of the tangent flow tothe normal bundle of the flow direction. The key observation is that this projectioncan be defined not only w.r.t the bundle perpendicular to the flow, but to theorthogonal complement of any section { L x : x ∈ M } ⊂ G .To be more precise, given L = { L x : x ∈ M } we write N L = { ( L x , v ) ∈ β ∗ ( T M ) : v ⊥ L x } . Then N , consisting of vectors perpendicular to L , is a sub-bundle of β ∗ ( T M ) over G with dimension dim M −
1. The extended linear Poincar´e flow is then definedas ψ t : N L → N L , ψ t ( L x , v ) = π (Φ t ( L x , v )) , where π is the orthogonal projection from fibres of β ∗ ( T M ) to the correspondingfibres of N along L .If we consider the the map ζ : Reg( X ) → G that maps every regular point x to the unique L x ∈ G with β ( L x ) = x such that L x is generated by the flow direction at x , then the extended linear Poincar´e on ζ (Reg( X )) can be naturally identified with the linear Poincar´e flow defined earlier.On the other hand, given any invariant set Λ of the flow φ t , consider the set:˜Λ = ζ (Λ ∩ Reg( X )) . If Λ contains no singularity, then ˜Λ can be seen as a natural copy of Λ in G equipped with the direction of the flow on Λ. If σ ∈ Λ is a singularity, then ˜Λcontains all the direction in β − ( σ ) that can be approximated by the flow directionat regular points in Λ. The extended Poincar´e flow restricted to ˜Λ can be seen asthe continuous extension of the linear Poincar´e flow on Λ. The same treatment canbe applied to the scaled linear Poincar´e flow ψ ∗ t .2.4. Classification of chain recurrent classes and singularities for genericstar flows.
In this subsection we recap the main result in [28] on C generic starflows. We begin with the following classification on the singularities. Definition . Let σ be a hyperbolic singularity contained in a non-trivial chainrecurrent class C ( σ ). Assume that the Lyapunov exponents of σ are: λ ≤ · · · ≤ λ s < < λ s +1 ≤ · · · ≤ λ dim M . Write Ind( σ ) = s for the stable index of σ . We say that(1) σ is Lorenz-like , if λ s + λ s +1 > λ s − < λ s (this implies that e λ s < t | T σ M ), and W ss ( σ ) ∩ { σ } = { σ } , where W ss ( σ )is the stable manifold of σ corresponding to λ , . . . , λ s − ; regular orbits in C ( σ ) can only approach σ along E cu ( σ ) cone, where E cu is the Φ t -invariantsubspace correspond to λ s , . . . , λ dim M ; MARIA JOS´E PACIFICO, FAN YANG AND JIAGANG YANG (2) σ is reverse Lorenz-like , if it is Lorenz-like for − X ; in this case, regularorbits in C ( σ ) can only approach σ along E cs ( σ ) cone. See Figure 1. Figure 1.
Lorenz-like and reverse Lorenz-like singularitiesThen it is shown in [18] and [28] that (all the theorems are labeled according to [28]): • for a star vector field X , if a chain recurrent class C is non-trivial, thenevery singularity in C is either Lorenz-like or reverse Lorenz-like (Theorem3.6); the original proof can be found in [18]; • there exists a residual set R ⊂ X ∗ ( M ) such that for every X ∈ R , if aperiodic orbit p is sufficiently close to a singularity σ , then: – when σ is Lorenz-like, the index of the p must be Ind( σ ) − – when σ is reverse Lorenz-like, the index of the p is Ind( σ ) (Lemma4.4);furthermore, the dominated splitting on σ induced by such periodic orbitscoincides with the hyperbolic splitting on σ (proof of Theorem 3.7); • For every chain recurrent class C there exists an integer Ind C >
0, suchthat every periodic orbit contained in a sufficiently small neighborhood of C has the same stable index which equals Ind C (Theorem 5.7); • combine the previous two results, we see that all the singularity in C hasindex either Ind C +1 (in which case it must be Lorenz-like) or Ind C (reverseLorenz-like); • if all the singularities in C are Lorenz-like, then C is sectional hyperbolic(Theorem 3.7); if all the singularities in C are reverse Lorenz-like, then C is sectional hyperbolic for − X (Theorem 3.7); • if C contains singularity with different indices (note that they can onlydiffer by one), then there is no sectional hyperbolic splitting on C ; one ofsuch examples was constructed by Bonatti and da Luz [3, 8].3. Flow orbits near singularities
This section contains some general results on hyperbolic singularities of C vec-tor fields (Section 3.1), and on Lorenz-like singularities for star flows (Section 3.2).The key results are Lemma 3.2 and 3.4, which state that the time near a singu-larity where the orbit is “make the turn” is bounded from above. For Lorenz-likesingularities of star flows, we prove in Lemma 3.12 that for a periodic orbit Orb( p )approaching a singularity σ while exhibiting backward hyperbolic times, the unsta-ble manifold of Orb( p ) must transversally intersect with W s ( σ ). Then Lemma 3.13 In [28] and [7] both cases are called Lorenz-like. Here we distinguish between the two sinceour main argument are different in each case. See the proof of Lemma 4.6 for more details.
INGULAR STAR FLOWS 9 deals with the case where a sequence of forward hyperbolic times approaches aLorenz-like singularity. These two results will allow us to show in the next sectionthat such periodic orbit must be in the same chain recurrent class as σ .3.1. Flow orbits near hyperbolic singularities.
In this section we will establishsome geometric properties for flow orbits in a small neighborhood of a hyperbolicsingularity. Our result applies to all C vector fields X which are not necessarilystar.For this purpose, let σ be a hyperbolic singularity with the hyperbolic splitting E sσ ⊕ E uσ . Without loss of generality, we can think of σ as the origin in R n , andassume that E sσ and E uσ are perpendicular (which is possible if one changes themetric). In particular, we will assume that E sσ = R s is the s -dimensional subspaceof R n with the last dim M − s coordinates being zero. Here s = dim E sσ is the stableindex of σ . Similarly, E uσ is the subspace of R n where the first s coordinates arezero. As before we will write f = φ for the time-one map of the flow.Since the vector filed X is C , we can take a neighborhood U = B r ( σ ) with r small enough, such that: • the flow in U can be written as(4) φ t ( x ) = e At x + C small perturbation , where A is a matrix no eigenvalue on the imaginary axis; • for x ∈ U , the tangent map Df x = Φ | T x M are small perturbations of thehyperbolic matrix e A , with eigenvalues bounded away from 1.For each x ∈ U , denote by x s its distance to E uσ and x u its distance to E sσ . Thenfor every α > α -cone on the manifold , denote by D iα ( σ ), i = s, u , as follows: D sα ( σ ) = { x ∈ U : x u < αx s } , D uα ( σ ) = { x ∈ U : x s < αx u } . Note that the hyperbolic splitting T σ M = E sσ ⊕ E uσ can be extended to U ina natural way: for each x ∈ U , put E s ( x ) as the s -dimensional hyperplane thatis parallel to E sσ ; the same can be done for E u ( x ). This allows us to considerthe α -cones C α ( E i ), i = s, u , on the tangent bundle as defined in Section 2.1.The next lemma easily follows from the smoothness of the vector field X and thehyperbolicity of σ : Lemma 3.1.
There exists L ≥ , such that for all α > small enough,(1) for every x ∈ Cl( D sα ( σ )) , we have X ( x ) ∈ C Lα ( E s ) ;(2) for every x ∈ U , if X ( x ) ∈ C α ( E s ) , we have x ∈ D sLα ( σ ) .Moreover, the same holds for D uα ( σ ) and C α ( E u ) . Let us fix some α > x ∈ U \ ( D sα ( σ ) ∪ D uα ( σ )), we lose control on the direction of X ( x ). One can think of the region U \ ( D sα ( σ ) ∪ D uα ( σ )) as the place where the flowis ‘making the turn’ from the E s cone to the E u cone. The next lemma states thatthe time that an orbit segment spend in this region is uniformly bounded. To thisend, we write, for each x ∈ U , t + ( x ) = sup { t > φ [0 ,t ] ( x ) ⊂ U } , t − ( x ) = sup { t > φ [ − t, ( x ) ⊂ U } . Then the orbit segment φ ( − t − ,t + ) ( x ) contains x and is contained in U . With slightabuse of notation, we will frequently drop the depends of t ± ( x ) on x . Lemma 3.2.
Let σ be a hyperbolic singularity for a C vector field X . Then forevery α > small enough, there exists T α > such that for every r > smallenough and every x ∈ U = B r ( x ) , the set T ( x ) := { t ∈ ( − t − , t + ) : φ t ( x ) / ∈ D sα ( σ ) ∪ D uα ( σ ) } has Lebesgue measure bounded by T α .Proof. Recall that the Lebesgue measure on the interval ( − t − , t + ) corresponds tothe length of open intervals. Below we will prove that the set in question is containedin a subinterval of ( − t − , t + ) whose length is bounded from above.We take a small neighborhood x ∈ V ⊂ U , such that for every y ∈ V it holds φ ( − t − ( y ) ,t + ( y )) ( y ) ∩ D ∗ α ( σ ) (cid:54) = ∅ , ∗ = s, u. Note that if a orbit segment φ ( − t − ,t + ) ( x ) does not intersect with V , then t − + t + must be bounded. Therefore we only need to prove the lemma for orbit segmentsthat intersect with V .Shrinking V if necessary, we may assume that φ ( − t − ,t + ) ( x ) ∩ V (cid:54) = ∅ = ⇒ φ − t − ( x ) ∈ D sα ( σ ) , φ t + ( x ) ∈ D uα ( σ ) . We will also assume, by changing to a different point on the same orbit segment ifnecessary, that x ∈ V \ ( D sα ( σ ) ∪ D uα ( σ )). For such an orbit segment φ ( − t − ,t + ) ( x ),define t s = t s ( x ) = sup { t > φ ( − t − , − t ) ( x ) ⊂ D sα ( σ ) } , and t u = t u ( x ) = sup { t > φ ( t,t + ) ( x ) ⊂ D uα ( σ ) } . Clearly we have T ( x ) ⊂ ( − t s , t u ). Below we will show that t s + t u is bounded fromabove.Writing x s = φ − t s ( x ), x u = φ t u ( x ), Lemma 3.1(2) shows that X ( x s ) / ∈ C α/L ( E s ) , X ( x u ) / ∈ C α/L ( E u ) . In particular ,(5) || X ( x s ) u |||| X ( x s ) s || > α/L, || X ( x u ) u |||| X ( x u ) s || < L/α. On the other hand, by the hyperbolicity of σ , there is λ > α such that for each x ∈ U ∩ φ − ( U ), and for every v ∈ T x M it holds that || Φ ( v ) u |||| Φ ( v ) s || > λ || v u |||| v s || . Combine this with (5) and the observation that Φ t s + t u ( X ( x s )) = X ( x u ), we obtain λ t s + t u < L α , which implies that t s + t u < L − α log λ := T α . Also note that T α can be made uniform for r > L and λ canbe chosen independent of r . This concludes the proof of the lemma. (cid:3) Now let us look at this lemma from the perspective of invariant measures. Settingfor i = s, u , L i ( σ ) = { L ∈ G : β ( L ) = σ, L is parallel to E i } , then L i are invariant under Φ t | β − ( σ ) (note that this is the tangent flow on G ).Furthermore, the hyperbolicity of σ implies that L s is a repelling set while L u isan attracting set.Next we take a sequence of points { x i } ⊂ U with x i → σ as i → ∞ . To simplynotation, we will write t ± i = t ± ( x i ). Note that t ± i ↑ + ∞ . For every ε > φ ( − t − i ,t + i ) ( x i ) spend in the region U \ B ε ( σ ) Following our notation earlier, X ( x ∗ ) = X ( x ∗ ) s + X ( x ∗ ) u ∈ E s ⊕ E u . INGULAR STAR FLOWS 11 is uniformly bounded in i . As a result, the empirical measures supported on theseorbit segments behaves trivially:(6) ν i = 1 t − i + t + i (cid:90) t + i − t − i δ φ s ( x i ) ds i →∞ −−−→ δ σ , where δ σ is the atomic measure on σ .On the other hand, the map ζ : Reg( X ) → G defined in Section 2.3 lifts anymeasure µ on M with µ (Sing) = 0 to a measure ζ ∗ ( µ ) on G . Now consider thelifted empirical measures:(7) ˜ ν i = ζ ∗ ( ν i ) . If we take any weak-* limit ˜ µ of { ˜ ν i } (the limit exists since G is compact), ˜ µ mustbe invariant under Φ t and is supported on L s ∪ L u ⊂ β − ( σ ) since L s ∪ L u containsthe non-wandering set of Φ t | β − ( σ ) . Write U ∗ α = ζ ( D ∗ α ( σ )) for ∗ = s, u . Observethat by Lemma 3.1, U ∗ α each contains a neighborhood of L ∗ ( σ ) in G , ∗ = s, u .Furthermore, we have U sα ∩ U uα = ∅ . Combine this with Lemma 3.2, we obtain thefollowing lemma: Lemma 3.3.
For all α > small, we have ˜ ν i ( U sα ∪ U uα ) → as i → N . The next lemma states that the time that orbit segments φ ( − t − i ,t + i ) ( x i ) spend in D sα ( σ ) and D uα ( σ ) are comparable: Lemma 3.4.
There is a ∈ (0 , ) independent of α , such that for every sequence { x i } ⊂ U with x i → σ and every weak*-limit ˜ µ of the empirical measure ˜ ν i definedusing (6) and (7) , we have ˜ µ ( U sα ) > a and ˜ µ ( U uα ) > a. Proof.
We will show that for every orbit segment φ ( − t − i ,t + i ) ( x i ), the time it spendsin D ∗ α , ∗ = s, u are comparable, with a ratio that is uniform in i and α .First, note that since the vector field is C , the flow speed is a Lipschitz functionof d ( x, σ ): there is 0 < C < C such that (cid:107) X ( x ) (cid:107) d ( x, σ ) ∈ ( C , C ) . To simplify notation, we write x e,i = φ − t − i ( x i ), and x l,i = φ t + i ( x i )for the end points of φ ( − t − i ,t + i ) ( x i ) that enter and leave the neighborhood U . Byour construction, x e,i , x l,i ∈ ∂U = ∂B r ( x ). As a result, for every i it holds (cid:107) X ( x e,i ) (cid:107)(cid:107) X ( x l,i ) (cid:107) ∈ (cid:18) C C , C C (cid:19) . Denote by t i ∈ ( − t − i , t + i ) the time such that the point x i = φ t ( x i ) satisfies( x i ) s = ( x i ) u . One could think of x i as the point on the orbit segment φ ( − t − i ,t + i ) ( x i )where the flow speed is the lowest. We parse each orbit segment φ ( − t − i ,t + i ) ( x i ) intothree sub-segments (recall the definition of t s ( x i ) and t u ( x i ) in Lemma 3.2. Tosimplify notation we will write t ∗ i = t ∗ ( x i ), ∗ = s, u ): • write x si = φ t si ( x i ) for the point on φ ( − t − i ,t + i ) ( x i ) that is on the boundaryof D sα ( σ ); then the orbit from x e,i to x si is contained in D sα ( σ ); • write x ui = φ t ui ( x i ) for the point on φ ( − t − i ,t + i ) ( x i ) that is on the boundaryof D uα ( σ ); then the orbit from x ui to x l,i is contained in D uα ( σ ); • the orbit segment from x si to x ui is outside D ∗ α ( σ ), ∗ = s, u ; by Lemma 3.2, t ui − t si ≤ T α . Note that x i is contained in the orbit segment from x si to x ui . Since the flow timefrom x i to x ± i is bounded by T α and the flow is C , we obtain (cid:107) X ( x ui ) (cid:107)(cid:107) X ( x si ) (cid:107) = (cid:107) X ( x ui ) (cid:107)(cid:107) X ( x i ) (cid:107) (cid:107) X ( x i ) (cid:107)(cid:107) X ( x si ) (cid:107) ∈ (cid:0) || Φ T α || , || Φ T α || − (cid:1) . For the orbit segment from x e,i to x si , Lemma 3.1(1) shows that X ( x ) ∈ C Lα ( E s )for each x in this orbit segment. Since the flow speed is uniformly exponentiallycontracting in C Lα ( E s ) provided that α and r are small enough, we see that thetime length of this orbit segment satisfies t − i − t si = O (log (cid:107) X ( x e,i ) (cid:107)(cid:107) X ( x − i ) (cid:107) ) . Similarly, t + i − t ui = O (log (cid:107) X ( x l,i ) (cid:107)(cid:107) X ( x ui ) (cid:107) ) . Then the ratio is t − i − t si t + i − t ui = O (cid:18) log (cid:107) X ( x e,i ) (cid:107) − log (cid:107) X ( x − i ) (cid:107) log (cid:107) X ( x l,i ) (cid:107) − log (cid:107) X ( x ui ) (cid:107) (cid:19) = O (cid:18) log (cid:107) X ( x − i ) (cid:107) log (cid:107) X ( x ui ) (cid:107) (cid:19) = O (1) , where in the last equality we use the elementary fact that if a i → , b i → a i /b i is bounded from above and away from zero, then log a i / log b i → . Finally, note that even though the ratio (cid:107) X ( x ui ) (cid:107)(cid:107) X ( x si ) (cid:107) depends on α , t − i − t si t + i − t ui only de-pends on the exponential contracting/expanding rate in C Lα ( E ∗ ), which can bemade uniform for α small enough. This finishes the proof of the lemma. (cid:3) We conclude this subsection with the following lemma, which will be used laterto create transverse intersection between the unstable manifold of a periodic orbitand the stable manifold of the singularity σ . As before, s is the stable index of σ . Lemma 3.5.
For each β > small and δ > , there is α > such that for all α < α and for every point x ∈ D sα ( σ ) , let W ( x ) be a (dim M − s ) -dimensionalsubmanifold that contains x and is tangent to C β ( E u ) . If diam W ( x ) > δ (cid:107) X ( x ) (cid:107) ,then W ( x ) (cid:116) W s ( σ ) (cid:54) = ∅ .Proof. Since the flow speed at x is a Lipschitz function of d ( x, σ ), we see thatdiam W ( x ) > C δd ( x, σ ) > C C (cid:48) δx s for some C (cid:48) >
0. Here C is the same constant as in the previous lemma.On the other hand, since W ( x ) is tangent to the β -cone of E u , there is C (cid:48)(cid:48) > W ( x ) > C (cid:48)(cid:48) x u = C (cid:48)(cid:48) d ( x, W s ( σ )), we must have W ( x ) (cid:116) W s ( σ ) (cid:54) = ∅ . Since x u < αx s in the cone D sα , the choice of α < α := C C (cid:48) δ/C (cid:48)(cid:48) guaranteesthat diam W ( x ) > CC (cid:48) δx s > C (cid:48)(cid:48) x u , therefore W ( x ) (cid:116) W s ( σ ) (cid:54) = ∅ . This concludesthe proof of the lemma. (cid:3) Near Lorenz-like singularities.
Now we turn our attention to Lorenz-likesingularities for star flows. Assume that σ is a Lorenz-like singularity containedin a non-trivial chain recurrent class C = C ( σ ). The discussion below applies toreverse Lorenz-like singularities if one considers the flow − X .Let E csσ ⊕ E uσ be the hyperbolic splitting on T σ M . We will write E ssσ the subspaceof T σ M corresponding to the exponents λ , . . . , λ s − and E cσ the subspace of T σ M corresponding to the exponent λ s . Then E ssσ ⊕ E cσ = E csσ .The discussion in the previous sub-section applies to σ without any modification(note that this time, we change the notation of E s to E cs and L s to L cs ). Fur-thermore, we can think of σ as the origin in R n with three bundles E ssσ , E cσ and E uσ INGULAR STAR FLOWS 13 perpendicular to one another (which is possible if one changes the metric). Thesebundles can be naturally extended to U = B r ( x ) as before. As a result, the conefield C α ( E ∗ ) can be defined for ∗ = ss, c, u, cu and cs . The same can be said aboutthe cones on the manifold, D ∗ α ( σ ).It is proven in [25], [18] and [28] that W ss ( σ ) ∩ C( σ ) = { σ } . Furthermore, it isshown that if the orbit segment is taken inside C ( σ ) (or if the orbit segment belongsto a periodic orbit that is sufficiently close to C ( σ )), then it can only approach thesingularity σ along the one-dimensional subspace E c in the following sense: write L c = { L ∈ G : β ( L ) = σ, L is parallel to E c } , then L c ⊂ L cs consists of a single point in G . If we take x i ∈ C ( σ ) with x i → σ and define the empirical measure ν i and its lift ˜ ν i according to (6) and (7), thenany weak*-limit ˜ µ of { ˜ ν i } must satisfysupp ˜ µ = L c ∪ L u . This observation leads to the following lemma, which is an improved version ofLemma 3.3 and 3.4.
Lemma 3.6.
Let { x i } ⊂ C ( σ ) , then the conclusion of Lemma 3.3 and Lemma 3.4remain true with U s ( α ) replaced by a neighborhood U c ( α ) of L c in G . The samecan be said if we take { p i } to be periodic points with p i → σ but not necessarily in C ( σ ) . The proof remains unchanged and is thus omitted.Now let us describe the hyperbolicity of the periodic orbits close to σ . Recallthat for a regular point x , N x ⊂ T x M is the orthogonal complement of the flowdirection X ( x ). Since X is a star vector field, every periodic orbit is hyperbolic. Itis proven in [28, Theorem 3.7] if p n is a sequence of periodic points near C ( σ ) andapproaches σ , then the hyperbolic splitting E s ⊕ E cu (with E cu = < X > ⊕ E u ) onOrb( p n ) extends to a dominated splitting on σ , which coincides with the splitting E ssσ ⊕ E cuσ where E cuσ = E cσ ⊕ E uσ . Since we assume that E ss is perpendicular to E cu ,and the flow direction is tangent to the cone C Lα ( E c ) as the orbit approaches thesingularity σ , it follows that for n large enough, the local stable manifold W s ( p (cid:48) n )(which has dimension s −
1) for p (cid:48) n ∈ Orb( p n ) ∩ D csα ( σ ) is tangent to a E ss cone. Remark . It is tempting to argue that for n large enough, W u ( p (cid:48) n ) must intersecttransversally with W cs ( σ ). However, this is not necessarily the case: as p n getscloser to the singularity σ , the size of the invariant manifolds of p will shrink withrates proportional to the flow speed, as observed by Liao [20]. As a result, even ifwe have a sequence of periodic points p n → p ∈ W cs ( σ ), there is still no guaranteethat the unstable manifold of p n will intersect with W cs ( σ ). To solve this issue, weconsider the hyperbolic times of p n as defined in Definition 5. But before that, letus first estimate the hyperbolicity of the orbit segment inside D csα and D uα . Lemma 3.8.
Let σ be a Lorenz-like singularity, then for r small enough, for everyorbit segment φ [0 ,T ] ( x ) ⊂ B r ( σ ) ∩ C ( σ ) , the scaled linear Poincar´e flow ψ ∗ t | E ssx isuniformly contracting, where E ssx ⊂ N x is the stable subspace for the scaled linearPoincar´e flow. The same holds true for every periodic orbit segment φ [0 ,T ] ( p ) ⊂ B r ( σ ) but not necessarily in C ( σ ) .Proof. We only need to estimate ψ ∗ t ( v ) = (cid:107) X ( x ) (cid:107)(cid:107) X ( φ t ( x )) (cid:107) ψ t | E ssx ( v ) , for v ∈ E ssx . We take ε > λ s − + 2 ε < λ s − ε < λ s < σ ). If we take r > B r ( σ ) we have (cid:107) ψ t | E ssx ( v ) (cid:107) ≤ e ( λ s − + ε ) t (cid:107) v (cid:107) . On the other hand, for (cid:107) X ( x ) (cid:107)(cid:107) X ( φ t ( x )) (cid:107) we have (in the worst case scenario, where theflow direction is tangent to the E c cone): (cid:107) X ( φ t ( x )) (cid:107) ≥ e λ s − ε (cid:107) X ( x ) (cid:107) . Indeed the flow speed is expanding while the direction is tangent to the E u cone.While the orbit is neither in D csα nor in the D uα , we lose all the estimate. However,the length of such orbit segment is uniformly bounded due to Lemma 3.2, and canbe safely ignored.As a result, we get (cid:107) ψ ∗ t ( v ) (cid:107) ≤ e ( λ s − + ε − λ s + ε ) t (cid:107) v (cid:107) ≤ e − εt (cid:107) v (cid:107) , and conclude the proof of the lemma. (cid:3) On the unstable subspace E ux ⊂ N x , the situation is different: when the orbitof x moves away from σ , it can only do so along D uα ( σ ). As a result, one loses thehyperbolicity along the E ux direction. On the other hand, when the orbit approaches σ , the orbit segment will be ‘good’ (in the sense that it is quasi-hyperbolic accordingto Definition 6) as long as the flow direction is tangent to the E c cone. This issummarized in the next lemma: Lemma 3.9.
Let σ be a Lorenz-like singularity, then there exists λ ∈ (0 , , T > , α > and r > , such that if α < α and x is a periodic orbit such that the orbitsegment φ [0 ,T ] ( x ) is contained in B r ( σ ) ∩ Cl( D csα ( σ )) , then the orbit segment is ( λ, T ) -backward contracting. If the orbit segment is in Cl( D uα ( σ )) , then it does nothave any sub-segment that is backward contracting.Proof. Case 1. φ [0 ,T ] ( x ) ⊂ Cl( D csα ( σ )) . To simplify notation we write y = φ T ( x ) for the endpoint of the orbit segment(note that it is the starting point for the same orbit segment under − X ). Take any t ∈ [0 , T ], we will estimate ψ ∗− t ( v ) = (cid:107) X ( y ) (cid:107)(cid:107) X ( φ − t ( y )) (cid:107) ψ − t ( v ) , for v ∈ E uy .Recall that λ s +1 is the smallest positive exponent of σ . Like in the previouslemma, we take ε > λ s + ε < λ s +1 − ε >
0, therefore − λ s +1 + λ s + 2 ε < r > α > (cid:107) ψ − t ( v ) (cid:107) ≤ e − ( λ s +1 − ε ) t (cid:107) v (cid:107) , and (cid:107) X ( φ − t ( y )) (cid:107) ≥ e − ( λ s + ε ) t (cid:107) X ( y ) (cid:107) , since the orbit segment is in Cl( D csα ( σ )). This gives (cid:107) ψ ∗− t ( v ) (cid:107) ≤ e ( − λ s +1 + ε + λ s + ε ) t (cid:107) v (cid:107) = e ( − λ s +1 + λ s +2 ε ) t (cid:107) v (cid:107) , which shows that the orbit segment is backward contracting. Case 2. φ [0 ,T ] ( x ) ⊂ Cl( D uα ( σ )) . We take any orbit segment φ [0 ,T ] ( x ) in Cl( D uα ( σ )) and take any y ∈ φ [0 ,T ] ( x ). ByLemma 3.1 the flow direction is almost parallel to E u if we take α small enough. INGULAR STAR FLOWS 15
As a result, we can take v ∈ E uy such that v is almost parallel to E c ( σ ). For such v we have (cid:107) ψ − t ( v ) (cid:107) ≥ e − ( λ s + ε ) t (cid:107) v (cid:107) , and the flow direction satisfies (cid:107) X ( φ − t ( y )) (cid:107) ≤ e ( − λ s +1 + ε ) t (cid:107) X ( y ) (cid:107) . This shows that (cid:107) ψ ∗− t ( v ) (cid:107) ≥ e ( − λ s − ε + λ s +1 − ε ) t (cid:107) v (cid:107) = e ( − λ s + λ s +1 − ε ) t (cid:107) v (cid:107) . For ε small enough, − λ s + λ s +1 − ε >
0. As a result, ψ ∗− t will never be contractingas long as the orbit segment is contained in Cl( D uα ( σ )). Therefore φ [0 ,T ] ( x ) doesnot contain any sub-segment that is backward contracting. (cid:3) Remark . The previous two lemmas have similar formulations for reverse Lorenz-like singularities.Recall the definition of t ± ( x ) from the previous section. Next, we introduce themain lemmas in this section, which will enable us to solve the issue mentioned inRemark 3.7 and create transverse intersection between the invariant manifolds of p with points in C . Lemma 3.11.
Let σ be a Lorenz-like singularity. Then for r > small enough,for every λ ∈ (0 , , T > there exists α > such that for all α < α , if y ∈ B r ( σ ) ∩ Cl( D csα ( σ )) is a ( λ, T ) -backward hyperbolic time on its orbit, then W u ( y ) (cid:116) W cs ( σ ) (cid:54) = ∅ .Proof. Since y is a ( λ, T )-backward hyperbolic time, according to the classic workof Liao [20], there is δ > W u ( y ) has size δ (cid:107) X ( y ) (cid:107) , tangent to C β ( E u )for some β > C β ( E uuy ) where E uuy is the unstable subspace in N y ; however, we may assume that E uuy and E u are almost parallel as long as theflow orbit remains in the cone Cl( D csα ( σ ))). Let α be given by Lemma 3.5 for such β and δ , we see that W u ( y ) (cid:116) W cs ( σ ) (cid:54) = ∅ . (cid:3) Lemma 3.12 (Backward hyperbolic times near σ ) . Let σ be a Lorenz-like singu-larity, and { p n } be a sequence of periodic points with p n → σ . For λ ∈ (0 , , T > and for α ∈ (0 , min { α , α } ) , assume that the set H n = { t ∈ ( − t − n , t + n ) : φ t ( p n ) is a ( λ, T ) -backward hyperbolic time. } has positive density: there exists a > such that for every n , ν n ( { φ t ( p n ) : t ∈ H n } ) > a > , where t ± n and ν n are taken according to (6) . Then there exists N > such that W u (Orb( p n )) (cid:116) W cs ( σ ) (cid:54) = ∅ , for all n > N. Proof.
Let α < min { α , α } where α is given by Lemma 3.9. We claim that { φ t ( x ) : t ∈ H n } ∩ D csα ( σ ) (cid:54) = ∅ .To this end, we parse the orbit segment φ ( − t − i ,t + i ) ( p n ) into three consecutiveparts like in the proof of Lemma 3.4:( − t − n , t + n ) = ( − t − n , − t sn ) ∪ ( − t sn , t un ) ∪ ( t un , t + n ) , such that: • − t sn is the first time in ( − t − n , t + n ) such that φ − t sn ( p n ) / ∈ D csα ( σ ); • t un is the first time in ( − t − n , t + n ) such that φ t un ( p n ) ∈ D uα ( σ ). Figure 2.
Transverse intersection between W u (Orb( p n )) and W cs ( σ )In other words, the orbit segment in ( t sn , t un ) is ‘making the turn’. Then accordingto Lemma 3.2, t un + t sn is uniformly bounded by T α . Therefore, we can take n largeenough such that ν n ( { φ t ( p n ) : t ∈ ( t sn , t un ) } ) < a . On the other hand, Lemma 3.9 states that for every t ∈ ( t un , t + n ) , φ t ( p n ) cannot bea backward hyperbolic time, since any sub-segment contained in φ ( t un ,t ) ( p n ) cannotbe backward contracting. As a result, we have H n ∩ ( t un , t + n ) = ∅ . It then followsthat ν n ( { φ t ( p n ) : t ∈ H n ∩ ( − t − n , − t sn ) } ) > a . In other words, there is a backward hyperbolic time y n = φ t ( p n ) contained in D csα ( σ ).It follows from Lemma 3.11 that W u ( y n ) (cid:116) W cs ( σ ) (cid:54) = ∅ . See Figure 2. Weconclude the proof of the lemma. (cid:3) Lemma 3.13 (Forward hyperbolic times near σ ) . Let σ be a Lorenz-like singularity.For β > small enough, λ ∈ (0 , , T > , there exists α > with the followingproperty:For every α < α , let D be a (dim E u + 1) -dimensional disk that contains σ and istangent to C β ( E cu ) , and let z be a ( λ, T ) -forward hyperbolic time that is containedin D cα ( σ ) ∩ B r ( σ ) for some r > small enough. Then we have W s ( z ) (cid:116) D (cid:54) = ∅ , for all n large enough . Proof.
Similar to Lemma 3.11, there is δ > W s ( z ) has size δ (cid:107) X ( z ) (cid:107) ,tangent to C β (cid:48) ( E ss ) for some β (cid:48) > W s ( z ) > C δd ( z, σ ) > CC (cid:48) δx c . On the other hand, since D is tangent to C β ( E cu ) and satisfiesdim D + dim W s ( z ) = dim E u + 1 + dim E ss = dim M, INGULAR STAR FLOWS 17 there exists C (cid:48)(cid:48)(cid:48) > W s ( z ) > C (cid:48)(cid:48) x ss we must have W s ( z ) (cid:116) D (cid:54) = ∅ . See Figure 3. Since x ss < αx c inside the center cone D cα ( σ ),the choice of α = C C (cid:48) δ/C (cid:48)(cid:48)(cid:48) satisfies the requirement of the lemma. (cid:3) Figure 3.
Transverse intersection between W s ( z ) and D Henceforth, we assume that r > λ ∈ (0 , , T > α < min { α , α , α } .4. Proof of Theorem A
This section contains the proof of Theorem A. The proof is three-fold:(1) every chain recurrent class C with positive entropy must contain a periodicorbit; note that the converse is also true: for C generic flows, a non-trivial chain recurrent class containing a periodic point p must coincidewith the homoclinic class of p ([26, Proposition 4.8]; see also [2]), thereforehas positive topological entropy;(2) there exists a neighborhood U of C , such that every periodic orbit in U must be contained in C ;(3) C is isolated.Recall that R is the residual set in X ∗ ( M ) described in Section 2.4. Denote by R ⊂ X ( M ) the residual set of Kupka-Smale vector fields. The next lemma takescare of Step (3) above. Lemma 4.1.
For a C vector field X ∈ R ∩ R , let C be a chain recurrent classwith the following property: there exists a neighborhood U of C such that everyperiodic orbit in U is contained in C . Then C is isolated.Proof. The proof is standard. Let C n be a sequence of distinct chain recurrentclasses approaching C in the Hausdorff topology. Without loss of generality wemay assume that C n ⊂ U , and C n (cid:54) = C for all n . Since X has only finitely manysingularities, we may assume that U is small enough such that all the singularities Recall that for C generic diffeomorphisms, every non-trivial chain recurrent class is approx-imated by periodic orbits. See [6]. in U are indeed contained in C . It then follows that C n ∩ Sing( X ) = ∅ for all n large enough.Since X is a star vector field, the main result of [13] shows that C n is uniformlyhyperbolic. In particular, there exists periodic orbit Orb( p n ) ⊂ C n ⊂ U . Byassumption we must have Orb( p n ) ⊂ C , so C n = C which is a contradiction. (cid:3) Next, we turn our attention to Step (1) and (2).4.1. C contains a periodic orbit. Step (1) of the proof is carried out in thefollowing two propositions, which are of independent interest.
Proposition 4.2.
There exists a residual set ˜ R ⊂ X ∗ ( M ) such that for every X ∈ ˜ R , let C be a chain recurrent class of X with singularities of different indices,then C contains a periodic point and has positive topological entropy.Proof. Let X ∈ R ∩ R as before. Following the discussion in Section 2.4, let σ + ∈ C be a Lorenz-like singularity, and σ − ∈ C be reverse Lorenz-like. We haveInd( σ + ) − σ − ) = Ind C , where Ind C is also the stable index of the periodic orbits sufficiently close to C . Wewill construct a periodic point p such that Lemma 3.11 can be applied at σ + for X , and at σ − for − X . This shows that p ∈ C , which also implies that C = H ( p )where H ( p ) is the homoclinic class of p , therefore has positive topological entropy,by [2].Given a periodic orbit γ , we denote by Π( γ ) its primary period. By [28, Lemma2.1] which is originally due to Liao, there is ˜ λ ∈ (0 , T > γ of X with periodic Π( γ ) longer than T , we have [Π( γ ) /T ] − (cid:89) i =0 (cid:107) ψ T | N s ( φ iT ( x )) (cid:107) ≤ ˜ λ Π( γ ) , and a similar estimate holds on N u . Here N s ⊕ N u is the hyperbolic splitting on N x for the linear Poincar´e flow ψ t . By the Pliss Lemma [27], for every λ ∈ (˜ λ, λ, T )-backward hyperbolic times for N u along the orbit of γ . Moreover,such points have positive density along the orbit of γ .Fix such λ and r > n >
0, consider the followingproperty:(P(n)): there is a hyperbolic periodic orbit p n , such that (see Figure 4): • the time that Orb( p n ) spends inside B r ( σ + ) is more than (1 / − /n )Π( p n ); • the time that Orb( p n ) spends inside B r ( σ − ) is more than (1 / − /n )Π( p n ); • the time that Orb( p n ) spends outside B r ( σ − ) ∪ B r ( σ − ) is less than n Π( p n ).Clearly this is an open property in X ( M ). On the other hand, note that W cs ( σ + )and W cu ( σ − ) must have transverse intersection due to the Kupka-Smale theorem.Since σ ± are in the same chain recurrent class, using the connecting lemma we cancreate an intersection between W u ( σ + ) and W s ( σ − ), which in turn creates a loopbetween σ + and σ − . Then standard perturbation technique (see [28] for instance)will allow one to create periodic orbits that satisfy the conditions above. Thereforethe following property is generic:(P’): there exists periodic orbits { p n } n that approach both σ ± , such that PropertyP(n) holds for every n .As a result, passing to the generic subset R where property (P’) holds, we mayassume that X itself has periodic orbits { p n } satisfying Property (P’).Below we will show that there exists ( λ, T )-backward hyperbolic times con-tained in Orb( p n ) ∩ B r ( σ + ) ∩ D csα ( σ + ), for n large enough. This allows us to apply INGULAR STAR FLOWS 19
Figure 4.
The periodic points { p n } satisfying Property (P’)Lemma 3.11 to show that W u (Orb( p n )) has transverse intersection with W cs ( σ + ).The same argument applied to the flow − X shows the transverse intersection be-tween W s (Orb( p n )) and W cu ( σ − ), thus p n is contained in C for n large enough.For convenience, we assume that p n → σ + such that p n lies on the boundary of D csα ( σ + ), where α ∈ (0 , min { α , α } ) as specified at the end of the previous section.Denote by p σ + n = φ t n ( p n )where t n < p σ + n ∈ ∂B r ( σ + ), and p σ − n = φ t n ( p n )where t n = sup { t < t n : φ t ( p n ) ∈ B r ( σ − ) } . See the Figure above. Since theorbit segment φ [ t n ,t n ] ( p n ) ⊂ M \ ( B r ( σ + ) ∪ B r ( σ − )), Property (P’) dictates that t n − t n < n Π( p n ).On the other hand, the Pliss Lemma [27] shows that the set of ( λ, T )-backwardhyperbolic times on the orbit of p n have density a >
0. Thus for n large enough,there must be hyperbolic times on the subsegment Orb( p n ) ∩ B r ( σ ± ). We considerthe following cases:(1) If there exists a backward hyperbolic time p (cid:48) n contained in the orbit segmentOrb( p n ) ∩ B r ( σ + ), then by Lemma 3.9 p (cid:48) n / ∈ D uα ( σ + ); on the other hand,the transition time from D csα ( σ + ) to D uα ( σ + ) is bounded (Lemma 3.2); thisshows that p (cid:48) n ∈ D csα ( σ + );(2) if there exists a backward hyperbolic times p (cid:48) n ∈ Orb( p n ) ∩ B r ( σ − ), thenLemma 3.8 (applied to − X ) shows that the orbit segment from p (cid:48) n to p σ − n is backward contracted by the scaled linear Poincar´e flow. By Lemma 2.6, p σ − n itself must be a backward hyperbolic time.Next, by Lemma 3.9, the orbit segment from p σ + n to p n is backward con-tracting, and the orbit segment from p σ − n to p σ + n has very small lengthcomparing to the former. As a result, p n is a backward hyperbolic time.It then follows from both cases, that there exists a backward hyperbolic time in-side Cl( D csα ( σ + )). As a result of Lemma 3.11, W u (Orb( p n )) must intersect transver-sally with W cs ( σ + ). The same argument applied to the flow − X shows that the stable manifold ofOrb( p n ) intersects transversally with the unstable manifold of σ − . We concludethat Orb( p n ) is contained in the chain recurrent class of σ ± , finishing the proof ofthis proposition. (cid:3) Proposition 4.3.
There exists a residual set ˜ R ⊂ X ∗ ( M ) such that for every X ∈ ˜ R , if C is a chain recurrent class of X with singularity and positive topologicalentropy then C contains a periodic point p .Proof. In view of the discussion in Section 2.4, we consider the following two casesfor X ∈ R ∩ R : Case 1.
All singularities in C have the same index. Then they must be of the sametype: either they are all Lorenz-like or all reverse Lorenz-like. By [28, Theorem3.7], C is sectional hyperbolic for X or − X . We may assume that C is sectionalhyperbolic for X . When h top ( φ t | C ) >
0, [26, Theorem B] guarantees that there areperiodic orbits p contained in C . Case 2. C contains singularities with different indices. Then the proposition followsfrom Proposition 4.2. (cid:3) This finishes the proof of Step (1).4.2.
All periodic orbits close to C must be contained in C . Now we turn ourattention to Step (2). The main result of this subsection is Lemma 4.6 which showsthat whenever a periodic orbit Orb( p ) gets sufficiently close to a chain recurrentclass containing a periodic orbit, we must have W u ( p ) (cid:116) W s ( x ) for some x ∈ C .The proof of this lemma requires a careful analysis on how the backward hyper-bolic times along the orbit of p approach points in C . There are essentially threecases:(1) backward hyperbolic times near a regular point; this is the easy case;(2) backward hyperbolic times near a Lorenz-like singularity σ ; this has beentaken careful of by Lemma 3.12;(3) backward hyperbolic times near a reverse Lorenz-like singularity σ ; in thiscase we consider − X , under which we have a sequence of forward hyperbolictimes near a Lorenz-like singularity σ ; this is taken care of by the followinglemma, as well as Lemma 3.13. Lemma 4.4.
There exists a residual set ˜ R ⊂ X ∗ ( M ) such that for every X ∈ ˜ R ,let C be a chain recurrent class of X that contains some Lorenz-like singularity σ and a hyperbolic periodic point q . Let { p n } be a sequence of periodic pointswith lim H Orb( p n ) ⊂ C . Furthermore, assume that there exists x n ∈ Orb( p n ) thatare ( λ, T ) -forward hyperbolic times for λ ∈ (0 , , T > independent of n , with x n → σ . Then for n large enough, W s (Orb( p n )) (cid:116) W u (Orb( q )) (cid:54) = ∅ . Proof.
Recall that E ssσ is the subspace spanned by the (generalized) eigenspacescorresponding to the eigenvalues λ , . . . , λ s − ; furthermore we have E ssσ ⊕ E cσ = E csσ .Consider the strong stable manifold of σ , W ss ( σ ) that is tangent to E ssσ at σ . Itexists because of the dominated splitting T σ M = E ssσ ⊕ E cuσ with E cuσ = E cσ ⊕ E uσ .Moreover, W ss ( σ ) locally divides the stable manifold W s ( σ ) into two branches,which we denote by W s, ± ( σ ).We may assume that p n themselves are forward hyperbolic times with p n → σ .Fix some r > t n < φ t n ( p n ) ∈ ∂B r ( σ ). In particular, z n := φ t n ( p n ) → z ∈ W s, + ( σ ) ∩ C ∩ ∂B r ( σ ). Sincethe orbit of p n can only approach σ along the center direction E cσ , we must have z n ∈ INGULAR STAR FLOWS 21
Cl( D cβ ( σ )), for some β = β ( r ) >
0. By Lemma 3.8, the orbit segment φ [ t n , ( p n ) isforward contracting. It follows from Lemma 2.6 that z n itself is a ( λ, T )-forwardhyperbolic time, and possesses stable manifold with size proportional to the flowspeed at z n .On the other hand, we have W s, + ( σ ) ∩ W u (Orb( q )) (cid:54) = ∅ , a courtesy of theconnecting lemma [2] (see also [26, Proposition 4.9]). Let a be a point of intersectionbetween W s, + ( σ ) and W u (Orb( q )), and D a disk in W u (Orb( q )) with a ∈ D ⊂ W u (Orb( q )). By the inclination lemma, φ t ( D ) approximates W u ( σ ) as t → ∞ .Note that ˜ D := { φ t ( D ) : t > } is tangent to the cone C β ( E cu ) and has di-mension dim E uσ + 1. See figure 3. Shrinking r such that β ( r ) < α and applyingLemma 3.13, we obtain W s ( z n ) (cid:116) { φ t ( D ) : t > } (cid:54) = ∅ for all n large enough, withwhich we conclude the proof. (cid:3) Remark . The proof of the lemma is reminiscent of [26, Corollary E] with twomajor differences. Firstly, in [26], C is Lyapunov stable. This guarantees not onlythe existence of a periodic orbit q ∈ C , but the disk D is contained in the class C ,which immediately allows one to conclude that p n ∈ C for n large enough. WithoutLyapunov stability, we only obtain the chain attainability from C to p n . Secondly, C is sectional hyperbolic in [26]. Therefore nearby periodic orbits have dominatedsplitting E s ⊕ F cu and, consequently, have stable manifolds with uniform size.Now we are ready to state the main lemma of this subsection, which allows us toestablish the chain attainability from nearby periodic orbits to the class C . Notethat the lemma does not impose any condition on the type of singularities containedin C , therefore can be applied to − X . Lemma 4.6.
There exists a residual set ˜ R ⊂ X ∗ ( M ) such that for every X ∈ ˜ R ,let C be a chain recurrent class of X with singularities and a hyperbolic periodicpoint. Then there exists a neighborhood U of C such that every periodic orbit Orb( p ) ⊂ U satisfies W u ( p ) (cid:116) W s ( x ) (cid:54) = ∅ for some x ∈ C .Proof. We prove by contradiction. Assume that there is a sequence of periodicorbits Orb( p n ) ⊂ U n with ∩ n U n = C , such that W u ( p ) (cid:116) W s ( x ) = ∅ for every x ∈ C . It is easy to see that the period of p n must tend to infinity; otherwise, wecould take a subsequence of { p n } that converges to a periodic orbit p ∈ C ; by thestar assumption, p is hyperbolic, and must be homoclinically related with p n for n large enough, a contradiction.By [28, Theorem 5.7], the index of p n coincides with Ind C for all n large enough.By [28, Lemma 2.1], there exists λ ∈ (0 , , T > p n ) contains( λ, T )-backward hyperbolic times x n . Moreover, the collection of such pointsΛ n = { x n ∈ Orb( p n ) : x n is a backward hyperbolic time } have positive density(independent of n ) in Orb( p n ) with respect to the empirical measure on Orb( p n ),thanks to the Pliss lemma [27].Taking subsequence if necessary, we write ˜ C ⊂ C for the Hausdorff limit ofOrb( p n ), and Λ = lim sup n →∞ Λ n ⊂ ˜ C (note that it is not invariant). We mayalso assume that the empirical measure µ n on Orb( p n ) converges to an invariantmeasure µ supported on ˜ C . The backward hyperbolic times having uniform positivedensity implies that µ (Λ) > ˜ D defined in this way dos not contain σ . However it can be extended to a disk D that contains σ in its interior. Note that D and ˜ D coincide within the upper half of the cone D cα ( σ ), whichcontains the point of the transverse intersection with W s ( z n ). We consider the following two cases:
Case 1.
There is an ergodic component of µ , denoted by µ , which satisfies µ (Λ) > µ (Sing) = 0.Then µ must be a non-trivial hyperbolic measure, thanks to [28, Theorem5.6]. The same argument used in Lemma 2.8 (f) shows that for n large enough, W u (Orb( p n )) has transverse intersection with the stable manifold of some point insupp µ . Roughly speaking, every regular point in Λ ∩ supp µ have stable manifold(with uniform size if we choose a subset of Λ away from all singularities; note thatsuch subset still has positive µ measure), which must intersect transversally withthe unstable manifold of the backward hyperbolic times x n ∈ Λ n (recall that suchpoints have unstable manifold up to the flow speed; if we take them uniformlyaway from all singularities, then their unstable manifolds also have uniform size),a contradiction. Case 2.
Every ergodic component µ of µ with µ (Λ) > { σ } .As per our discussion in Section 2.4, singularities in C must be either Lorenz-likeor reverse Lorenz-like. We further consider two subcases: Subcase 1.
One of those σ is reverse Lorenz-like .In this case we consider the vector field − X . Note that backward hyperbolictimes for X are forward hyperbolic times for − X . Therefore we have a sequence offorward hyperbolic times on the orbit of p n , approaching a Lorenz-like singularity σ . We are in a position to apply Lemma 4.4, which shows that W s, − X (Orb( p n )) (cid:116) W u, − X ( q ) (cid:54) = ∅ for a hyperbolic periodic point q ∈ C . Reverse back to X , we see that W u (Orb( p n )) (cid:116) W s ( x ) (cid:54) = ∅ , which contradictions with our assumption on C . Subcase 2.
One of those σ is Lorenz-like.In this case we have backward hyperbolic times in a neighborhood of a Lorenz-likesingularity with positive density. By Lemma 3.12, W u (Orb( p n )) intersect trans-versely with W cs ( σ + ), which is a contradiction.The proof is now complete. (cid:3) In the proof of Lemma 4.6, the assumption that C contains a periodic orbit isonly used in Case 2, Subcase 1. Since this case is impossible when C is sectionalhyperbolic, we obtain the following proposition which has intrinsic interest. Proposition 4.7.
Let C be a sectional hyperbolic chain recurrent class for C generic star flow X . Assume that C contains a singularity σ . Then there exists aneighborhood U of C , such that C is chain attainable from all periodic orbits in U . Proof of Theorem A.
Now we are ready to prove Theorem A. By Propo-sition 4.3, every singular chain recurrent class with positive entropy contains aperiodic point. We then apply Lemma 4.6 to C for X and − X . This shows thatthere exists a neighborhood U of C , such that all periodic orbits in U are homo-clinically related with C , therefore, are contained in C . Finally we use Lemma 4.1to conclude that C is isolated.The only remaining case is chain recurrent classes with zero entropy. By Propo-sition 4.2, singularities in such classes must have the same stable index. Then [28,Theorem 3.7] shows that C is sectional hyperbolic for X or − X . C cannot containany periodic orbit; otherwise it must coincide with the homoclinic class of saidperiodic orbit, resulting in positive topological entropy.To conclude the proof, we need to show that the only ergodic invariant measuressupported on C are point masses of singularities. Note that this case does not happen when C is sectional hyperbolic. INGULAR STAR FLOWS 23
For this purpose, let µ be an ergodic invariant measure with supp µ ⊂ C . Weprove by contradiction and assume that µ (cid:54) = δ σ for any σ ∈ Sing( X ). By [28], µ isa non-trivial hyperbolic measure whose support contains regular orbits of X . ByLemma 2.7 and 2.8 (f), there exists a periodic orbit in C , a contradiction.4.4. Proof of the corollary.
We conclude this paper with the proof of the Corol-lary.
Proof of Corollary B.
In [20] it is proven that C generic star flows have onlyfinitely many periodic sinks. As a result, if X has infinitely many distinct Lya-punov stable chain recurrent classes { C n } ∞ n =1 , then we may assume that C n ’s arenon-trivial and approach, under Hausdorff topology, to a chain recurrent class C .Note that C cannot be trivial since trivial chain recurrent classes, i.e., periodicorbits and singularities, are all hyperbolic and isolated due to the star assumption.Therefore C is non-trivial and sectional hyperbolic with zero topological entropydue to Theorem A.Denote by λ C > F cu , then by thecontinuity of the dominated splitting (see [4, Appendix B] and [1]), nearby classes C n must be sectional hyperbolic and have sectional volume expanding rate λ C n >λ C /
2. Since C n are Lyapunov stable, we are in a position to apply [26, TheoremC] on C n , and see that h top ( X | C n ) > λ C /
2. By the variational principle, we cantake µ n an ergodic invariant measure supported on C n , such that h µ n ( X ) ≥ λ C / X is robustly entropy expansivein s small neighborhood of C ; in particular, this together with [5] shows that themetric entropy must be upper semi-continuous in a small neighborhood of C . Let µ be any limit point of µ n in weak-* topology, then we see that supp µ ⊂ C and h µ ( X ) ≥ λ C /
2. It then follows from the variational principle that h top ( X | C ) ≥ λ C /
2, a contradiction. (cid:3)
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Instituto de Matem´atica, Universidade Federal do Rio de Janeiro, C. P. 68.530, CEP21.945-970, Rio de Janeiro, RJ, Brazil.
Email address : [email protected] Department of Mathematics, Michigan State University, East Lansing, MI, USA.
Email address : [email protected] Department of Mathematics, Southern University of Science and Technology ofChina, Guangdong, China; and Departamento de Geometria, Instituto de Matem´atica eEstat´ıstica, Universidade Federal Fluminense, Niter´oi, Brazil.
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