aa r X i v : . [ m a t h . D S ] J un A RESCALED EXPANSIVENESS FOR FLOWS
XIAO WEN AND LAN WEN
Abstract.
We introduce a new version of expansiveness for flows. Let M be acompact Riemannian manifold without boundary and X be a C vector field on M that generates a flow ϕ t on M . We call X rescaling expansive on a compactinvariant set Λ of X if for any ǫ > δ > x, y ∈ Λand any time reparametrization θ : R → R , if d ( ϕ t ( x ) , ϕ θ ( t ) ( y ) ≤ δ k X ( ϕ t ( x )) k for all t ∈ R , then ϕ θ ( t ) ( y ) ∈ ϕ [ − ǫ,ǫ ] ( ϕ t ( x )) for all t ∈ R . We prove that everymultisingular hyperbolic set (singular hyperbolic set in particular) is rescalingexpansive and a converse holds generically. Introduction
Expansiveness is a strong symbol of chaotic dynamics that has been studiedextensively. Recall for systems of discrete time, say for a homeomorphism f of acompact metric space M , expansiveness states that there is δ > x and y in M , d ( f n ( x ) , f n ( y )) < δ for all n ∈ Z implies x = y . In other words,expansiveness requests that any two different points x and y must get separatedin a uniform distance δ at certain moment n . For systems of continuous time, thesituation is quite different. As Bowen-Walters [BW] point out, for a flow ϕ t on M ,any orbit itself is a priori “non-expansive” because, for any δ >
0, there is η > y = ϕ s ( x ) with s ∈ ( − η, η ) then d ( ϕ t ( x ) , ϕ t ( y )) = d ( ϕ t ( x ) , ϕ s ( ϕ t ( x ))) < δ for all t ∈ R . Thus the best possible expansive property one could expect for aflow seems to be that for any ǫ > δ > y δ -shadows x then y = ϕ s ( x ) with s ∈ ( − ǫ, ǫ ). To make the definition a conjugacy invariant and torule out some pathological behavior one needs to allow time-reparametrizations.This leads to the following definition introduced by Komuro [Kom1]:A flow ϕ t is expansive on a compact invariant set Λ of ϕ t if for any ǫ > δ > x and y in Λ and any surjective increasingcontinuous functions θ : R → R , if d ( ϕ t ( x ) , ϕ θ ( t ) ( y ) ≤ δ for all t ∈ R , then ϕ θ ( t ) ( y ) ∈ ϕ [ − ǫ,ǫ ] ( ϕ t ( x )) for some t ∈ R . Komuro [Kom1] proved that thegeometrical Lorenz attractor [Lor][Gu] is expansive. Araujo-Pacifico-Pujals-Viana[APPV] proved that every singular hyperbolic (see definition below) attractor in a3-dimensional manifold is expansive.In this paper we introduce another version of expansiveness for flows, in whichthe shadowing condition d ( ϕ t ( x ) , ϕ θ ( t ) ( y ) ≤ δ is rescaled by the flow speed: Definition 1.1.
A flow ϕ t generated by a C vector field X is rescaling expansive on a compact invariant set Λ if for any ǫ > δ > Mathematics Subject Classification.
Primary 37C10,37D30.
Key words and phrases.
Rescaling expansive, Singular hyperbolic, Multi-singular hyperbolic,Linear Poincar´e flow, Sectional Poincar´e map. x, y ∈ Λ and any increasing continuous functions θ : R → R , if d ( ϕ t ( x ) , ϕ θ ( t ) ( y ) ≤ δ k X ( ϕ t ( x )) k for all t ∈ R , then ϕ θ ( t ) ( y ) ∈ ϕ [ − ǫ,ǫ ] ( ϕ t ( x )) for all t ∈ R .Here θ is not assumed to be surjective. But we will see that, for small δ , theshadowing condition d ( ϕ t ( x ) , ϕ θ ( t ) ( y ) ≤ δ k X ( ϕ t ( x )) k forces θ to be surjective. Alsonote that in the definition if x is a singularity then y = x , and if x is regular then y is regular if δ is small. Similarly, in the definition of expansiveness of Komuro, ifone of the two points x and y is a hyperbolic singularity and if δ is small, then theother point must be the same singularity by the Hartman-Grobman theorem. Thusthe expansive property for flows has nontrivial behavior only when both x and y are regular points.The idea of rescaling the size of neighborhoods of a regular point by the flowspeed comes from the classical work of Liao on standard systems of differentialequations [L1, L2]. The recent paper of Gan-Yang [GY] extracts geometrically theideas of Liao to form an important tool in their work. See also [HW], [SGW] and[Y] for some relevant applications. We remark that for nonsingular flows the twodefinitions, expansiveness and rescaled expansiveness, are equivalent. (There is adiscussion on the equivalence in the appendix at the end of this paper.) Neverthelessfor flows with singularities we do not know if the two definitions imply one another.In this paper we prove that every multisingular hyperbolic set, singular hyperbolicset in particular, is rescaling expansive and a converse holds generically. We firststate the definition of singular hyperbolic set, which is introduced by Morales,Pacifico and Pujals [MPP], partly to characterize the celebrated geometrical Lorenzattractor [Lor][Gu].Let M be a d -dimensional compact Riemannian manifold without boundary and X be a C vector field on M . Denote ϕ t = ϕ Xt the flow generated by X , andΦ t = dϕ t : T M → T M the tangent flow of X . We call x ∈ M a singularity of X if X ( x ) = 0. Denote Sing( X ) the set of singularities of X . We call x ∈ M a regularpoint if x ∈ M \ Sing( x ).Let Λ be an invariant set of X . Let C > , λ > t -invariant splitting T Λ M = E ⊕ F is a ( C, λ ) -dominated splitting with respect toΦ t if k Φ t | E x k · k Φ − t | F ϕt ( x ) k < Ce − λt for any x ∈ Λ and t > Definition 1.2.
Let Λ be a compact invariant set of X . Let C > , λ > positively ( C, λ ) -singular hyperbolic for X if there is a( C, λ )-dominated splitting T Λ M = E ⊕ F such that the following three conditionsare satisfied:(1) the subbundle E is ( C, λ )-contracting with respect to Φ t , that is, k Φ t | E x k < Ce − λt , for any x ∈ Λ and t > F is ( C, λ )-area-expanding with respect to Φ t , that is, | det(Φ − t | L ) | < Ce − λt , for any x ∈ Λ and any two dimensional subspace L ⊂ F x and any t > singular hyperbolic for X if Λ is positively singular hyperbolicfor X or − X . RESCALED EXPANSIVENESS FOR FLOWS 3
We will prove that every singular hyperbolic set is rescaling expansive. Thiswill be a corollary of Theorem A stating that every multisingular hyperbolic set isrescaling expansive. The notion of multisingular hyperbolic set, introduced recentlyby Bonatti-da Luz [BL], is more general than the notion of singular hyperbolicset. (Proposition 4.3 explains that every singular hyperbolic set is multisingularhyperbolic.) First we recall the linear Poincar´e flow defined on the normal bundleof X over regular points of X . For x ∈ M \ Sing( X ), denote the normal space of X ( x ) to be N x = N x ( X ) = { v ∈ T x M : v ⊥ X ( x ) } . Denote the normal bundle of X to be N = N ( X ) = [ x ∈ M \ Sing( X ) N x . The linear Poincar´e flow ψ t : N → N of X is then defined to be the orthogonalprojection of Φ t | N to N , i.e., ψ t ( v ) = Φ t ( v ) − h Φ t ( v ) , X ( ϕ t ( x )) ik X ( ϕ t ( x )) k X ( ϕ t ( x ))for any v ∈ N x , where h· , ·i denotes the Riemannian metric.The notion of multisingular hyperbolicity is formulated using the extended linearPoincar´e flow [LGW], a “compactification” of the usual linear Poincar´e flow. Denote SM = { e ∈ T M : k e k = 1 } the unit sphere bundle of M and j : SM → M thebundle projection defined by j ( e ) = x if e ∈ SM ∩ T x M . Note that SM is compact.The tangent flow Φ t induces a flowΦ t : SM → SM Φ t ( e ) = Φ t ( e ) / k Φ t ( e ) k . For any e ∈ SM , let N e = { v ∈ T j ( e ) M : v ⊥ e } be the normal space of e . Denote N = N SM = [ e ∈ SM N e . Then N is a d − SM , irrelevant to vectorfields. This bundle and the normal bundle N = N ( X ) of a vector field X over M \ Sing( X ) are both abbreviated as N , which should not cause a confusion fromthe context. Define the extended linear Poincar´e flow to be˜ ψ t : N → N ˜ ψ t ( v ) = Φ t ( v ) − h Φ t ( v ) , Φ t ( e ) i · Φ t ( e ) , if v ∈ N e . Thus ˜ ψ t covers the flow Φ t of SM , that is, ι ◦ ˜ ψ t = Φ t ◦ ι, where ι : N SM → SM is the bundle projection.If e = X ( x ) / k X ( x ) k where x ∈ M \ Sing( X ), then N e = N x ( X ) andΦ t ( e ) = X ( ϕ t ( x )) / k X ( ϕ t ( x )) k . Hence ˜ ψ t ( v ) = Φ t ( v ) − h Φ t ( v ) , X ( ϕ t ( x )) k X ( ϕ t ( x )) k i · X ( ϕ t ( x )) k X ( ϕ t ( x )) k = ψ t ( v ) . XIAO WEN AND LAN WEN
In other words, the extended linear Poincar´e flow ˜ ψ t over the subset { X ( x ) / k X ( x ) k : x ∈ M \ Sing( X ) } of SM can be identified with the usual linear Poincar´e flow ψ t over M \ Sing( X ).Let Λ ⊂ M be a compact invariant set of X . Denote˜Λ = { X ( x ) / k X ( x ) k : x ∈ Λ \ Sing( X ) } , where the closure is taken in SM . The set ˜Λ is compact and Φ t -invariant. Due tothe parallel feature of vector fields near regular points, at every x ∈ Λ \ Sing( X ), ˜Λgives a single unit vector X ( x ) / k X ( x ) k . Thus in a sense ˜Λ is a “compactification”of Λ \ Sing( X ). At a singularity x ∈ Λ ∩ Sing( X ) however, ˜Λ usually gives a bunchof unit vectors.Now we give the definition of multisingular hyperbolic set of Bonatti-da Luz[BL1, BL2]. Let Λ be a compact invariant set of X . A continuous function h : ˜Λ × R → (0 , + ∞ ) is called a cocycle of X on ˜Λ if for any e ∈ ˜Λ and any s, t ∈ R , h ( e, s + t ) = h ( e, s ) · h (Φ s ( e ) , t ) . We often write h ( e, t ) as h t ( e ) . Two most importantexamples of cocycles are h t ( e ) = k Φ t ( e ) k and h t ( e ) ≡ . A cocycle is called pragmatical with respect to a singularity σ if there is an isolating neighborhood U of σ in M such that if e and Φ t ( e ) are both in j − U then h t ( e ) = k Φ t ( e ) k , andif e and Φ t ( e ) are both outside j − U then h t ( e ) = 1 (see [BL1] for a good figureillustration). A reparametrizing cocycle is a (finite) product of pragmatical cocycleswith disjoint isolating neighborhoods. Definition 1.3.
Let Λ be a compact invariant set of X . Let C > , λ > C, λ ) -multisingular hyperbolic set of X if there is a ˜ ψ t -invariantsplitting N ˜Λ = ∆ s ⊕ ∆ u such that(1) ∆ s ⊕ ∆ u is ( C, λ )-dominated with respect to ˜ ψ t , that is, k ˜ ψ t | ∆ s ( e ) k·k ˜ ψ − t | ∆ u (Φ t ( e )) k
Let Λ be a multisingular hyperbolic set of a C vector field X on M .Then Λ is rescaling expansive. In fact, there is ǫ > such that for any < ǫ ≤ ǫ ,any x ∈ Λ and y ∈ M , and any increasing continuous functions θ : R → R , if RESCALED EXPANSIVENESS FOR FLOWS 5 d ( ϕ θ ( t ) ( y ) , ϕ t ( x )) ≤ ( ǫ/ k X ( ϕ t ( x )) k for all t ∈ R , then ϕ θ ( t ) ( y ) ∈ ϕ [ − ǫ,ǫ ] ( ϕ t ( x )) for all t ∈ R . Thus, for a multisingular hyperbolic set, the number δ in Definition 1.1 can bespecified to be ǫ/ δ >
0, we say that a sequence { ( x i , t i ) : t i ∈ M, t i ≥ } a
1. Given x, y ∈ M , we say that y is chain attainable from x if for any δ >
0, there is a ( δ, { ( x i , t i ) } ≤ i ≤ n , n >
1, such that x = x, x n = y . A compact invariant set Λ is called chain transitive if every pair of points x, y ∈ Λ are chain attainable from each other through points of
Λ, that is, for any δ >
0, there are a ( δ, { ( x i , t i ) } ≤ i ≤ n with x = x, x n = y and a ( δ, { ( y i , t i ) } ≤ i ≤ n with y = y, y n = x , where all x i , y i are in Λ.A compact invariant set Λ of X is called isolated if there is a neighborhood U ⊂ M of Λ such that Λ = \ t ∈ R ϕ t ( U ) . An isolated invariant set Λ is called locally star for X if there are a neighborhood U of X in X ( M ) and a neighborhood U of Λ in M such that, for every Y ∈ U ,every singularity and every periodic orbit of Y that is contained (entirely) in U ishyperbolic.Let X ( M ) be the space of C vector fields endowed with the C topology. Asubset R ⊂ X ( M ) is called residual if it is an intersection of countably open anddense subset of X ( M ). Theorem B.
There is a residual set
R ⊂ X ( M ) such that for any X ∈ R andany isolated chain transitive set Λ , the following three conditions are equivalent: (1) Λ is rescaling expansive for X . (2) Λ is locally star for X . (3) Λ is multisingular hyperbolic for X . Time-reparametrizations
A basic tool to what follows will be a “uniform relative” version of the classicalflowbox theorem. We first work with a Euclidean space.Let ¯ X be a C vector field on R n with k D ¯ X ( x ) || ≤ L for all x ∈ R n , where L > ϕ t be the flow generated by ¯ X . Here we use the notationsof ¯ X and ¯ ϕ t with a bar just to distinguish from the notations of the vector filed X and the flow ϕ t on the manifold M below.For every regular point x ∈ R n \ Sing( ¯ X ) and every r >
0, denote by¯ U x ( r k ¯ X ( x ) k ) = { v + t ¯ X ( x ) : v ∈ N x , k v k ≤ r k ¯ X ( x ) k , | t | ≤ r } the tangent box of relative size r at x , where N x denotes the normal space to thespan of ¯ X ( x ).Note that the size of the tangent box ¯ U x ( r k ¯ X ( x ) k ) is r k ¯ X ( x ) k but not r , andthis is why we have called r the relative size of the box, that is, the size relative tothe flow speed k ¯ X ( x ) k . XIAO WEN AND LAN WEN
Define the flowbox map F x of ¯ X at x to be F x : ¯ U x ( r k ¯ X ( x ) k ) → R n F x ( v + t ¯ X ( x )) = ¯ ϕ t ( x + v ) . Thus, for every v ∈ N x with k v k ≤ r k ¯ X ( x ) k , F x maps the line interval v +[ − r k ¯ X ( x ) k , r k ¯ X ( x ) k ] onto the orbital arc { ¯ ϕ t ( x + v ) : | t | ≤ r } . Note that F x (0) = x. Recall m ( A ) denotes the mininorm of a linear operator A , i.e., m ( A ) = inf {k A ( v ) k : v ∈ R n , k v k = 1 } . Proposition 2.1.
Let ¯ X be a C vector filed on R n such that k D ¯ X ( x ) || ≤ L for all x ∈ R n . There is r > such that, for any regular point x of ¯ X , F x :¯ U x ( r k ¯ X ( x ) k ) → R n is an embedding whose image contains no singularities of ¯ X ,and m ( D p F x ) ≥ / and k D p F x k ≤ for every p ∈ ¯ U x ( r k ¯ X ( x ) k ) . We call the image F x ( ¯ U x ( r k ¯ X ( x ) k )) a flowbox of ¯ X of relative size r at x .Proposition 2.1 says that, although the set of regular points of ¯ X is non-compact,the relative size r of flowboxes for all regular points can be chosen uniform.The idea and the term of “flowbox” are classical. See for instance Pugh-Robinson[PR] for the definition of flowbox. The “uniform relative” version like Proposition2.1 is probably new. Proof.
Since sup {k D ¯ X k} ≤ L , the vector field ¯ X on R n is Lipschitz with a Lipschitzconstant L . Hence if k y − x k ≤ L k ¯ X ( x ) k then k ¯ X ( y ) − ¯ X ( x ) k ≤ L k x − y k ≤ k ¯ X ( x ) k . ( ∗ )In particular, if y − x ∈ ¯ U x ( 14 L k ¯ X ( x ) k ) ∩ N x then ¯ X ( y ) = 0. This means that, for every regular point x ∈ R n , the flowbox F x ( ¯ U x ( r k ¯ X ( x ) k )) contains no singularities if r ≤ L . Claim 1. If k y − x k ≤ L k ¯ X ( x ) k and | t | ≤ L , then k ¯ ϕ t ( y ) − x k ≤ L k ¯ X ( x ) k . Proof.
Suppose for the contrary there are y and t with k y − x k ≤ L k ¯ X ( x ) k , | t | ≤ L but k ¯ ϕ t ( y ) − x k > L k ¯ X ( x ) k . Without loss of generality we assume t >
0. Then there is t with 0 < t < t such that k ¯ ϕ t ( y ) − x k = 14 L k ¯ X ( x ) k but k ¯ ϕ t ( y ) − x k < L k ¯ X ( x ) k RESCALED EXPANSIVENESS FOR FLOWS 7 for every 0 < t < t . By ( ∗ ), k ¯ X ( ¯ ϕ t ( y )) k ≤ k ¯ X ( x ) k for all 0 ≤ t ≤ t . Then k ¯ ϕ t ( y ) − y k = k Z t ¯ X ( ¯ ϕ t ( y )) dt k ≤ Z t k ¯ X ( ¯ ϕ t ( y )) k dt ≤ Z t k ¯ X ( x ) k dt = 54 k ¯ X ( x ) k t < k ¯ X ( x ) k L = 18 L k ¯ X ( x ) k . Here the strict inequality is guaranteed by t < t , t ≤ L .
Hence k ¯ ϕ t ( y ) − x k ≤ k ¯ ϕ t ( y ) − y k + k y − x k < L k ¯ X ( x ) k , contradicting k ¯ ϕ t ( y ) − x k = 14 L k ¯ X ( x ) k . This proves Claim 1.Now let r = 110 L .
We verify that r satisfies the requirement of the proposition. Claim 2.
For any p = v + t ¯ X ( x ) ∈ ¯ U x ( r k ¯ X ( x ) k ), k D p F x − id k ≤ / . Proof.
A straightforward computation of the directional derivative of F x at p alongthe direction ¯ X ( x ) gives D p F x · ¯ X ( x ) = ¯ X ( F x ( p )) . Now | t | ≤ L and k v k < L k ¯ X ( x ) k hence, by Claim 1, k F x ( p ) − x k = k ¯ ϕ t ( x + v ) − x k ≤ L k ¯ X ( x ) k . Since ¯ X has Lipschitz constant L , we have k ¯ X ( F x ( p )) − ¯ X ( x ) k ≤ k ¯ X ( x ) k . That is, k D p F x · ¯ X ( x ) − ¯ X ( x ) k ≤ k ¯ X ( x ) k . Or k D p F x | < ¯ X ( x ) > − id k ≤ / . Likewise, for any u ∈ N x , a straightforward computation of the directionalderivative of F x at p along the direction u gives D p F x · u = D x + v ϕ t · u. Then k D p F x | N x − id k ≤ | e Lt − | ≤ e / − < / . XIAO WEN AND LAN WEN
Thus, for every p ∈ ¯ U x ( r k ¯ X ( x ) k ), k D p F x − id k ≤ k D p F x | < ¯ X ( x ) > − id k + k D p F x | N x − id k < / . This proves Claim 2.In particular, for every p ∈ ¯ U x ( r k ¯ X ( x ) k ), D p F x is a linear isomorphism. Bythe inverse function theorem, F x is a local diffeomorphism. To prove that F x isan embedding it suffices to prove F x is injective, i.e., to prove that for any z ∈ R n there is at most one y ∈ ¯ U x ( r k ¯ X ( x ) k ) such that F x ( y ) = z. By the generalizedmean value theorem, Claim 2 givesLip( F x − id ) ≤ / . Then we can write F x = id + φ, where φ : ¯ U x ( r k ¯ X ( x ) k ) → R n is Lipschitz with Lip( φ ) ≤ / . We need to prove that, for any z ∈ R n , F x ( y ) = y + φ ( y ) = z has at most one solution for y or, equivalently, y = z − φ ( y ) has atmost one solution for y . Define T = T z : ¯ U x ( r k ¯ X ( x ) k ) → R n to be T ( y ) = z − φ ( y ) . It suffices to prove that T has at most one fixed point. It is sufficient to verify that T is a contraction mapping. This is straightforward because k T ( y ) − T ( y ′ ) k ≤ Lip( φ ) k y − y ′ k ≤ k y − y ′ k . This proves that F x is an embedding. Clearly, m ( D p F x ) ≥ / , k D p F x k ≤ p ∈ ¯ U x ( r k ¯ X ( x ) k ). This ends the proof of Proposition 2.1. (cid:3) Now we come back to our manifold. As usual, denote T x M ( r ) = { v ∈ T x M : k v k ≤ r } ,B r ( x ) = exp x ( T x M ( r )) ,N x ( r ) = { v ∈ N x : k v k ≤ r } . By the compactness of M and the C smoothness of X , there are constants L > a > x ∈ M the vector fields¯ X = (exp − x ) ∗ ( X | B a ( x ) )in T x M ( a ) are locally Lipschitz vector fields with Lipschitz constant L . We call L a local Lipschitz constant of X . We may assume m ( D p exp x ) > / , k D p exp x k < / p ∈ T x M ( a ). For every x ∈ M \ Sing( x ), denote U x ( r k X ( x ) k ) = { v + tX ( x ) ∈ T x M : v ∈ N x , k v k ≤ r k X ( x ) k , | t | ≤ r } the tangent box of relative size r at x . Define a C map F x : U x ( r k X ( x ) k ) → M RESCALED EXPANSIVENESS FOR FLOWS 9 to be F x ( v + tX ( x )) = ϕ t (exp x ( v )) . Then Proposition 2.2 gives directly the following proposition whose proof is omitted.
Proposition 2.2.
For any C vector field X on M , there is r > such that forany regular point x of X , F x : U x ( r k X ( x ) k ) → M is an embedding whose imagecontains no singularities of X , and m ( D p F x ) ≥ / and k D p F x k ≤ for every p ∈ U x ( r k X ( x ) k ) . Here in the statement the constant 2 is changed to 3 because of the involvementof the exponential maps exp x in the proof.Now we analyze the time-reparametrizations θ in the definition of the rescaledexpansiveness. We will see that if δ is sufficiently small, the rescaled shadowingcondition d ( ϕ θ ( t ) ( y ) , ϕ t ( x )) ≤ δ k X ( ϕ t ( x )) k for all t ∈ R of Definition 1.1 will force θ to be nearly a translation. The key to the proof is tocontrol the time-difference | t | by the distance d ( x, ϕ t ( x )). We know by continuitythat if | t | is small then d ( x, ϕ t ( x )) is small. The converse is not true if, for instance, x is a singularity or x is periodic and t is the period. Nevertheless in some situationsa converse could be true. The next lemma states such a converse: in some situations d ( x, ϕ t ( x )) ≤ δ k X ( x ) k implies | t | ≤ δ . This will play a crucial role in the proof ofLemma 2.4.In what follows r always denotes the constant given in Proposition 2.2. Ideasof Komuro [Kom2] are helpful to the rest part of this section. Lemma 2.3.
Let x ∈ M \ Sing( X ) be given. (1) For any < δ ≤ r / and t ∈ [ − r , r ] , d ( x, ϕ t ( x )) ≤ δ k X ( x ) k implies | t | ≤ δ . (2) For any < δ ≤ r / , ϕ [0 ,t ] ( x ) ⊂ B ( x, δ k X ( x ) k ) implies | t | ≤ δ .Proof. (1) Since F x (0 x ) = x and m ( D p F x ) ≥ / k D p F x k ≤ p ∈ U x ( r k X ( x ) k ), we have F x ( U x ( r k X ( x ) k )) ⊃ B ( x, ( r / k X ( x ) k ) . Assume 0 < δ ≤ r / t ∈ [ − r , r ], and d ( x, ϕ t ( x )) ≤ δ k X ( x ) k . Take a geodesic γ connecting x and ϕ t ( x ). Then γ ⊂ B ( x, ( r / k X ( x ) k ). Since t ∈ [ − r , r ], F − x ( ϕ t ( x )) = tX ( x ). Then F − x ( γ ) is a curve in U x ( r k X ( x ) k ) connecting 0 and tX ( x ). Hence k tX ( x ) k ≤ l ( F − x ( γ )) ≤ l ( γ ) = 3 d ( x, ϕ t ( x )) ≤ δ k X ( x ) k . Thus | t | ≤ δ .(2) Assume 0 < δ ≤ r / ϕ [0 ,t ] ( x ) ⊂ B ( x, δ k X ( x ) k ). To prove | t | ≤ δ ,by (1), it suffices to verify t ∈ [ − r , r ]. Suppose t / ∈ [ − r , r ]. Without loss ofgenerality suppose t > r . Take s ∈ ( r , t ) slightly larger than r . Then ϕ s ( x ) / ∈ B ( x, ( r / k X ( x ) k ), contradicting ϕ [0 ,t ] ( x ) ⊂ B ( x, δ k X ( x ) k ). This proves Lemma2.3. (cid:3) Remark . In the proof of Lemma 2.3, without the condition t ∈ [ − r , r ], F − x ( ϕ t ( x ))may not be equal to tX ( x ). For instance this is the case when x is periodic and t is the period of x . Lemma 2.4.
For any ǫ > there is δ > such that, for any x ∈ M \ Sing( X ) ,any y ∈ M and any T ∈ [ r / , r ] , if there is an increasing continuous function θ : [0 , T ] → R such that d ( ϕ t ( x ) , ϕ θ ( t ) ( y )) ≤ δ k X ( ϕ t ( x )) k for all t ∈ [0 , T ] , then | θ ( T ) − θ (0) − T | ≤ ǫT .Proof. Let L be a local Lipschitz constant given in the paragraph right beforeProposition 2.2. First we recall two formulas from ODE about the continuousdependence of solutions with respect to initial conditions:(1) For any x ∈ M \ Sing( X ) and t ∈ R , k X ( ϕ t ( x )) kk X ( x ) k ∈ [ e − L | t | , e L | t | ] . (2) d ( ϕ t ( x ) , ϕ t ( y )) ≤ e L | t | d ( x, y ) . We also fix a fact that can be proved like the inequality ( ∗ ) in the proof ofProposition 2.1: Fact.
There is c > such that for any z, z ′ ∈ M \ Sing( X ) , if d ( z, z ′ ) < c k X ( z ) k then (1 / k X ( z ) k < k X ( z ′ ) k < k X ( z ) k . Now let ǫ > δ = min { r e Lr , c e Lr , ǫr e Lr ) } . Here the three expressions are just some rough estimates that will work.Assume we are given x ∈ M \ Sing( X ), y ∈ M , T ∈ [ r / , r ] and an increasingcontinuous function θ : [0 , T ] → R such that d ( ϕ t ( x ) , ϕ θ ( t ) ( y )) ≤ δ k X ( ϕ t ( x )) k for all t ∈ [0 , T ]. We prove | θ ( T ) − θ (0) − T | ≤ ǫT. Replacing θ by η with η ( t ) = θ ( t ) − θ (0) if necessary, we assume θ (0) = 0. Thus we prove | θ ( T ) − T | ≤ ǫT. Note that θ (0) = 0 implies d ( x, y ) ≤ δ k X ( x ) k . Most of the proofs will be to estimate the distance d ( ϕ θ ( T ) ( y ) , ϕ T ( y )). At last wewill convert it to the time-difference | θ ( T ) − T | , using Lemma 2.3.First assume θ ( T ) ≤ T . Then d ( ϕ θ ( T ) ( y ) , ϕ T ( y )) ≤ d ( ϕ θ ( T ) ( y ) , ϕ T ( x )) + d ( ϕ T ( x ) , ϕ T ( y )) ≤ δ k X ( ϕ T ( x )) k + e LT δ k X ( x ) k≤ δ k X ( ϕ T ( x )) k + e LT δ k X ( ϕ T ( x )) k = (1 + e LT ) δ k X ( ϕ T ( x )) k . Since d ( ϕ T ( x ) , ϕ T ( y )) ≤ e LT δ k X ( ϕ T ( x )) k , and since e LT δ ≤ c (by the choice of δ ), by the above Fact, k X ( ϕ T ( y )) k > / k X ( ϕ T ( x )) k . RESCALED EXPANSIVENESS FOR FLOWS 11
Then d ( ϕ θ ( T ) ( y ) , ϕ T ( y )) ≤ e LT ) δ k X ( ϕ T ( y )) k . Since T ∈ [ r / , r ], and since θ is increasing, θ (0) = 0, and θ ( T ) ≤ T , we have | θ ( T ) − T | ∈ [ − r , r ]. By the choice of δ ,2(1 + e LT ) δ ≤ ǫr . Then by the first part of Lemma 2.3, | θ ( T ) − T | ≤ ǫr ≤ ǫT. Now assume θ ( T ) > T . There is 0 ≤ S ≤ T such that θ ( S ) = T . Then d ( ϕ T ( x ) , ϕ S ( x )) ≤ d ( ϕ T ( x ) , ϕ T ( y )) + d ( ϕ T ( y ) , ϕ S ( x )) ≤ e LT δ k X ( ϕ T ( x )) k + δ k X ( ϕ S ( x )) k≤ e LT δ k X ( ϕ T ( x )) k + e LT δ k X ( ϕ T ( x )) k = ( e LT + e LT ) δ k X ( ϕ T ( x )) k . Here we have used the fact k X ( ϕ S ( x )) k ≤ e L ( T − S ) k X ( ϕ T ( x )) k ≤ e LT k X ( ϕ T ( x )) k . By the choice of δ , ( e LT + e LT ) δ ≤ r / . Then by the first part of Lemma 2.3,0 ≤ T − S ≤ e LT + e LT ) δ ≤ e LT δ. Here we replace 3( e LT + e LT ) by 6 e LT just to shorten the expression. Note thatby the choice of δ , 6 e LT δ ≤ r . Consider the flowbox U ϕ T ( x ) ( r k X ( ϕ T ( x )) k ) around ϕ T ( x ). By the definition ofthe flowbox map and Proposition 2.2, for every s ∈ [ S, T ], d ( ϕ s ( x ) , ϕ T ( x )) ≤ T − s ) k X ( ϕ T ( x )) k≤ T − S ) k X ( ϕ T ( x )) k ≤ e LT δ k X ( ϕ T ( x )) k . Now for every t ∈ [ T, θ ( T )], take s ∈ [ S, T ] such that θ ( s ) = t . Then d ( ϕ θ ( T ) ( y ) , ϕ t ( y )) ≤ d ( ϕ θ ( T ) ( y ) , ϕ T ( x )) + d ( ϕ T ( x ) , ϕ s ( x )) + d ( ϕ s ( x ) , ϕ t ( y )) ≤ δ k X ( ϕ T ( x )) k + 18 e LT δ k X ( ϕ T ( x )) k + δ k X ( ϕ s ( x )) k . By the choice of δ , 18 e LT δ ≤ c. Then k X ( ϕ s ( x )) k ≤ k X ( ϕ T ( x )) k . Therefore, for every t ∈ [ T, θ ( T )], d ( ϕ θ ( T ) ( y ) , ϕ t ( y )) ≤ (1 + 18 e LT ) δ k X ( ϕ T ( x )) k + 2 δ k X ( ϕ T ( x )) k = (3 + 18 e LT ) δ k X ( ϕ T ( x )) k≤ e LT ) δ k X ( ϕ θ ( T ) ( y )) k . Here we have used the fact d ( ϕ θ ( T ) ( y ) , ϕ T ( x )) ≤ δ k X ( ϕ T ( x )) k and hence the fact k X ( ϕ θ ( T ) ( y )) k ≥ (1 / k X ( ϕ T ( x )) k . Now by the choice of δ ,2(3 + 18 e LT ) δ ≤ ǫr . Then by the second part of Lemma 2.3, | θ ( T ) − T | ≤ ǫr ≤ ǫT. This ends the proof of Lemma 2.4. (cid:3)
Lemma 2.5.
For any ǫ > there is δ > such that, for any x ∈ M \ Sing( X ) , any y ∈ M , and any T ≥ r , if there is an increasing continuous function θ : [0 , T ] → R such that d ( ϕ t ( x ) , ϕ θ ( t ) ( y )) ≤ δ k X ( ϕ t ( x )) k for all t ∈ [0 , T ] , then | θ ( T ) − θ (0) − T | ≤ ǫT .Proof. This is a corollary of Lemma 2.4. We prove the case T ≥ r . Divide [0 , T ]into several intervals as0 = T < T < · · · < T n − < T n = T with r / ≤ T i − T i − < r . Then we choose δ > θ : [0 , T ] → R be an increasing continuous function such that d ( ϕ t ( x ) , ϕ θ ( t ) ( y )) ≤ δ k X ( ϕ t ( x )) k for all t ∈ [0 , T ]. Without loss of generality we assume θ (0) = 0. Then for any1 ≤ i ≤ n , | θ ( T i ) − θ ( T i − ) − ( T i − T i − ) | ≤ ǫ ( T i − T i − ) . Hence we have | θ ( T ) − T | ≤ ǫT . This ends the proof of Lemma 2.5. (cid:3) Corollary 2.6.
There is δ > such that, for any x ∈ M \ Sing( X ) and any y ∈ M ,if there is an increasing continuous function θ : R → R such that d ( ϕ t ( x ) , ϕ θ ( t ) ( y )) ≤ δ k X ( ϕ t ( x )) k for all t ∈ R , then θ is surjective.Proof. Fix 0 < ǫ <
1. Let δ > x ∈ M \ Sing( X ), y ∈ M , and an increasingcontinuous function θ : R → R such that d ( ϕ t ( x ) , ϕ θ ( t ) ( y )) ≤ δ k X ( ϕ t ( x )) k for all t ∈ R . By Lemma 2.5, for any T ≥ r , | θ ( T ) − θ (0) − T | ≤ ǫT . Then θ ( T ) → ∞ as T → ∞ . Symmetrically, the same argument proves θ ( T ) → −∞ as T → −∞ .Thus θ : R → R is surjective. (cid:3) Sectional Poincar´e maps
In this section we discuss the sectional Poincar´e maps. Let X be a C vectorfield on M as before. By Proposition 2.2, the flowbox F x ( U x ( r k X ( x ) k )) containsa ball of radius r k X ( x ) k centered at x . For any y ∈ B r k X ( x ) k ( x ), if y = F x ( v, t ),then define P x ( y ) = v . In other words, define a map P x : B r k X ( x ) k ( x ) → N x to be P x = π x ◦ F − x , where π x denotes the orthogonal projection of T x M to N x . Since π x has norm ≤ k DP x | B r k X ( x ) k ( x ) k ≤ . For any t ∈ R , let r = r ( t ) = e − L | t | r , RESCALED EXPANSIVENESS FOR FLOWS 13 where L is chosen as in the previous paragraphs with the property sup {k DX k} < L .Since k X ( ϕ t ( x )) k ≤ e Lt k X ( x ) k , we have ϕ t ( B r k X ( x ) k ( x )) ⊂ B e L | t | r k X ( x ) k ( ϕ t ( x )) ⊂ B e L | t | r k X ( ϕ t ( x )) k ( ϕ t ( x )) = B r k X ( ϕ t ( x )) k ( ϕ t ( x )) . Hence for any t ∈ R , we can define a map P x,t : N x ( r k X ( x ) k ) → N ϕ t ( x ) to be P x,t = P ϕ t ( x ) ◦ ϕ t ◦ exp x , called the sectional Poincar´e map at x of time t ([GY]). Note that P x,t and P x aredifferent maps.The sectional Poincar´e map P x,t is defined in Gan-Yang [GY] using holonomymaps generated by orbit arcs. The definition here using flowboxes is equivalent butformally slightly different.The following proposition presents some uniform ( in a relative sense) propertyabout the family of the derivatives D v P x,T of the sectional Poincar´e maps P x,T at v ∈ N x ( r k X ( x ) k ). Note that, at the origin 0 x of N x , D x P x,T = ψ T | N x . Proposition 3.1. [GY]
The family of sectional Poincar´e maps { P x,T } has thefollowing properties: (1) k D v P x,T k is uniformly bounded in the following sense: for any T ∈ R ,there is K > such that k D v P x,T k ≤ K for any x ∈ M \ Sing( X ) and any v ∈ N x ( r k X ( x ) k ) . (2) D v P x,T is uniformly continuous in the following sense: Given T ∈ R , forany ǫ > there is δ > such that for any x ∈ M \ Sing( X ) and any v, v ′ ∈ N x ( r k X ( x ) k ) with k v − v ′ k < δ k X ( x ) k , k D v P x,T − D v ′ P x,T k < ǫ . Here x ranges over the non-compact set M \ Sing( X ) and v ranges over aneighborhood of 0 x in N x of uniform relative size r = r ( T ). This propositionextracts some old ideas from the work of Liao [L1, L2] and plays an important toolin the recent work of Gan-Yang [GY]. See also [HW], [SGW] and [Y] for somerelevant applications. Since our definition of sectional Poincar´e maps { P x,T } isformally slightly different from the one given originally in [GY], we give a sketch ofthe proof for Proposition 3.1. Proof.
The proof for (1) is immediate because k D v P x,T k = k D v ( P ϕ T ( x ) ◦ ϕ T ◦ exp x ) k≤ k D ϕ T (exp x ( v )) P ϕ T ( x ) k · k D exp x ( v ) ϕ T k · k D v exp x k≤ · e L | T | ·
32 = 92 e L | T | for any v ∈ N x ( r k X ( x ) k ).The key to the proof of (2) is that the derivatives of the flowbox map F x areuniformly continuous in the following relative sense. Modulo exponential mapsexp x , assume we work in a Euclidean space. Claim.
For any ǫ > , there is δ > such that for any x ∈ M \ Sing ( X ) and any p, q ∈ U x ( r k X ( x ) k ) , if d ( p, q ) < δ k X ( x ) k , then k D p F x − D q F x k < ǫ . To prove the claim, we estimate k D p F x − D q F x k in two directions, the onedimensional space < X ( x ) > spanned by X ( x ) and the normal space N x .Let p = v + t X ( x ) , q = v + t X ( x ) ∈ U x ( r k X ( x ) k ), where v , v ∈ N x . If d ( p, q ) < δ k X ( x ) k , then k ( D p F x − D q F x )( X ( x )) k = k X ( F x ( p )) − X ( F x ( q )) k≤ L k F x ( p ) − F x ( q ) k < Ld ( p, q ) < Lδ k X ( x ) k . Hence k ( D p F x − D q F x ) |
On the other hand, for any u ∈ N x , k ( D p F x − D q F x )( u ) k = k D x + v ϕ t ( u ) − D x + v ϕ t ( u ) k≤ k D x + v ϕ t ( u ) − D x + v ϕ t ( u ) k + k D x + v ϕ t ( u ) − D x + v ϕ t ( u ) k≤ k Id − D ϕ t ( x + v ) ϕ t − t k · k D x + v ϕ t ( u ) k + k D x + v ϕ t ( u ) − D x + v ϕ t ( u ) k≤ | e L | t − t | − | · e L | t | · k u k + k D x + v ϕ t ( u ) − D x + v ϕ t ( u ) k≤ e / L | t − t | · k u k + k D x + v ϕ t ( u ) − D x + v ϕ t ( u ) k < Lδ · k u k + k D x + v ϕ t − D x + v ϕ t k · k u k . Since M is compact, there is δ > y , y ∈ M with d ( y , y ) < δ and any t ∈ [ − r , r ], one has k D y ϕ t − D y ϕ t k < ǫ . Then for any ǫ > δ > d ( p, q ) < δ k X ( x ) k then k ( D p F x − D q F x ) |
0, there is δ > y , y ∈ F x ( U x ( r k X ( x ) k )), if d ( y , y ) < δ k X ( x ) k , then k D y F − x − D y F − x k < ǫ . Thus k D y P x − D y P x k < ǫ. Then for any v, v ′ ∈ N x ( r k X ( x ) k ), k D v P x,T − D v ′ P x,T k = k D v ( P ϕ T ( x ) ◦ ϕ T ◦ exp) − D v ′ ( P ϕ T ( x ) ◦ ϕ T ◦ exp) ≤ k D ϕ T (exp x ( v )) P ϕ T ( x ) − D ϕ T (exp x ( v ′ )) P ϕ T ( x ) k · k D exp x ( v ) ϕ T k · k D v exp x k + k D ϕ T (exp x ( v ′ )) P ϕ T ( x ) k · k D exp x ( v ) ϕ T − D exp x ( v ′ ) ϕ T k · k D v exp x k + k D ϕ T (exp x ( v ′ )) P ϕ T ( x ) k · k D exp x ( v ′ ) ϕ T k · k D v exp x − D v ′ exp x k . RESCALED EXPANSIVENESS FOR FLOWS 15
By the uniform bound of k DP x k and k Dϕ T k and k D exp x k and the continuity of D v P x discussed above, it is straightforward to verify item 2. We omit the details. (cid:3) Given the rescaled shadowing condition d ( ϕ t ( x ) , ϕ θ ( t ) ( y )) ≤ δ k X ( ϕ t ( x )) k , thereis a time sequence { θ ( T k ) } for y that corresponds to the time sequence { kT } for x as described in the next proposition. Proposition 3.2.
For any
T > r , there is δ = δ ( T ) > with the followingproperty: for any x ∈ M \ Sing ( X ) and y ∈ M and any increasing continuousfunction θ : R → R with d ( ϕ t ( x ) , ϕ θ ( t ) ( y )) ≤ δ k X ( ϕ t ( x )) k for any t ∈ R , there is asequence { T k } k ∈ Z such that (1) ϕ θ ( T k ) ( y ) ∈ exp( N ϕ kT ( x ) ) ; (2) k exp − ϕ kT ( x ) ( ϕ θ ( T k ) ( y )) k ≤ δ k X ( ϕ kT ( x )) k ;(3) | θ ( kT ) − θ ( T k ) | ≤ δ ; (4) P ϕ kT ( x ) ,T (exp − ϕ kT ( x ) ( ϕ θ ( T k ) ( y ))) = exp − ϕ ( k +1) T ( x ) ( ϕ θ ( T k +1 ) ( y )) for any k ∈ Z . Let us briefly explain the statement. If we think of exp x ( N x ( r )), r small, a localcross section at x in M , then the flow ϕ t is transverse to the local sections andinduces a holonomy map from exp x ( N x ( r )) to exp ϕ t ( x ) ( N ϕ t ( x ) ). Item (1) just saysthat θ ( T k ) is the time when the orbit of y cuts the local section exp ϕ kT ( x ) ( N ϕ kT ( x ) )under the hololomy. Item (2) says the cut point is near the “origin” ϕ kT ( x ). Item(4) just says these cuts are in the same orbit (of y ). Proof.
Let
T > r be given. Choose ǫ > ǫT < r / . By Lemma 2.5 there is δ < r / x ∈ M \ Sing( X ), any y ∈ M , and any increasing continuousfunction θ : [0 , T ] → R , if d ( ϕ t ( x ) , ϕ θ ( t ) ( y )) ≤ δ k X ( ϕ t ( x )) k for all t ∈ [0 , T ], then | θ ( T ) − θ (0) − T | ≤ ǫT. We also require δ < r / r ( T ) / . Assume we are given x ∈ M \ Sing( X ) and y ∈ M with a increasing continuousfunction θ : R → R such that d ( ϕ t ( x ) , ϕ θ ( t ) ( y )) ≤ δ k X ( ϕ t ( x )) k for all t ∈ R . We will look at the sequence of points ϕ kT ( x ) on Orb( x ). For each k ∈ Z , we will consider a flowbox of relative size r around ϕ kT ( x ). The “shadowingpoint” ϕ θ ( kT ) ( y ) is in the flowbox and is near the center ϕ kT ( x ), but generally notin the normal direction of X ( ϕ kT ( x )). Thus we need to consider a third point, to bedenoted ϕ θ ( T k ) ( y ), the projection of ϕ θ ( kT ) ( y ) along Orb( y ) to the normal directionof X ( ϕ kT ( x )). That is, on Orb( x ) we will consider the point of time kT , while onOrb( y ) we will consider the two points of time θ ( kT ) and θ ( T k ), respectively. Precisely, let F ϕ kT ( x ) be the flowbox map of relative size r at ϕ kT ( x ). Write F − ϕ kT ( x ) ( ϕ θ ( kT ) ( y )) = u k + t k · X ( ϕ kT ( x )) , where u k ∈ N ϕ kT ( x ) , t k ∈ R . Then by the definition of the flowbox map, ϕ t k (exp ϕ kT ( x ) ( u k )) = ϕ θ ( kT ) ( y ) . Since k DF − x k ≤ d ( ϕ kT ( x ) , ϕ θ ( kT ) ( y )) ≤ δ k X ( ϕ kT ( x )) k , we have k u k k ≤ δ k X ( ϕ kT ( x )) k and | t k | ≤ δ .By Corollary 2.6, we may assume that θ is surjective. Then there is T k such that θ ( T k ) = θ ( kT ) − t k . We prove the sequence { T k } k ∈ Z satisfies Proposition 3.2.Since ϕ θ ( T k ) ( y ) = ϕ θ ( kT ) − t k ( y ) = exp ϕ kT ( x ) ( u k ) ∈ exp( N ϕ kT ( x ) ) , item (1) holds. Since k exp − ϕ kT ( x ) ( ϕ θ ( kT ) − t k ( y )) = k u k k ≤ δ k X ( ϕ kT ( x )) k , item (2) holds. Since | θ ( kT ) − θ ( T k ) | = | t k | ≤ δ, item (3) holds.It remains to prove item (4), which is equivalent to P ϕ kT ( x ) ,T ( u k ) = u k +1 . Note that k u k k ≤ δ k X ( ϕ kT ( x )) k and δ < r /
3, hence the sectional Poincar´e mapis well defined at u k . By the definition of P ϕ kT ( x ) ,T , this is the same as P ϕ ( k +1) T ( x ) ( ϕ T (exp ϕ kT ( x ) ( u k ))) = u k +1 . Thus it suffices to find s ∈ [ − r , r ] such that ϕ T (exp ϕ kT ( x ) ( u k )) = F ϕ ( k +1) T ( x ) ( u k +1 + sX ( ϕ ( k +1) T ( x ))) , or ϕ T (exp ϕ kT ( x ) ( u k )) = ϕ s (exp ϕ ( k +1) T ( x ) ( u k +1 )) . This is the same as ϕ T + θ ( kT ) − t k ( y ) = ϕ s + θ (( k +1) T ) − t k +1 ( y ) . Let s = T + θ ( kT ) − t k − θ (( k + 1) T ) + t k +1 . Then | s | ≤ | θ ( kT ) + T − θ (( k + 1) T ) | + | t k | + | t k +1 |≤ ǫT + 3 δ + 3 δ ≤ r / δ ≤ r . This proves item (4) and hence Proposition 3.2. (cid:3)
RESCALED EXPANSIVENESS FOR FLOWS 17 Proof of Theorems A
We will reduce the problem of expansiveness to the following Proposition 4.1 ofthe problem of uniqueness of fixed points.For any i ∈ Z , let E i be a d -dimensional Euclidean space. Let Y = Π ∞ i = −∞ E i .For any v = { v i } ∈ Y , denote k v k = sup {k v i k} . Let Y = { v ∈ Y : k v k < + ∞} . Then Y is a Banach space with norm k · k . For any i ∈ Z , let G i : E i → E i +1 be a map. These maps define a map G : Y → Y by( Gv ) i +1 = G i ( v i ) . In other words, G is defined to be “fiber-preserving” with respect to the shift map i → i + 1. Below in Proposition 4.1 and Theorem A the map G will be defined thisway.For any i ∈ Z , assume E i has a direct sum decomposition E i = ∆ si ⊕ ∆ ui . Define the angle between ∆ si and ∆ ui by ∠ (∆ si , ∆ ui )= inf {k u − v k : ( u ∈ ∆ si , v ∈ ∆ ui , k u k = 1) or ( u ∈ ∆ si , v ∈ ∆ ui , k v k = 1) } . Proposition 4.1.
Let η ∈ (0 , and α > be given. There is ξ > such that if forevery i ∈ Z the splitting E i = ∆ si ⊕ ∆ ui has angle ∠ (∆ si , ∆ ui ) > α , and if G : Y → Y has the form G i = L i + φ i : E i → E i +1 , where L i is a linear isomorphism of theblock form L i = (cid:18) A i D i (cid:19) with respect to the splittings of E i and E i +1 such that k A i k ≤ η, k D − i k ≤ η , and Lip φ i < ξ and φ i (0) = 0 , then for any v ∈ Y , G ( v ) = v implies v = 0 . Here G is a Lipschitz perturbation of a “hyperbolic” operator L = { L i } . Since φ i (0) = 0, we know v = 0 is a fixed point of G already. Then Proposition 4.1 statesthat v = 0 is the only fixed point of G . This is a classical result, see for instancePilyugin [P] and Gan [G]. Since there is some slight difference here, for conveniencewe give the proof. Proof.
Let L : Y → Y and φ : Y → Y denotes the maps defined by ( L ( v )) i +1 = L i ( v i ) and ( φ ( v )) i +1 = φ i ( v i ) respectively for any v = ( v i ). Let I be the identitymap on Y . Then we know that I − L is invertible and( I − L ) − = (cid:18) ( I − A ) −
00 ( I − D ) − (cid:19) where A = ( A i ) and D = ( D i ). By the fact that k A k ≤ η and k D − k ≤ η we knowthat k ( I − A ) − k ≤ − η , k ( I − D ) − k ≤ η − η . For any v = v s + v u with k v k = 1, where v s ∈ ∆ s and v u ∈ ∆ u , we have k v s k v s k + v u k v s k k ≥ α by the definition of angle. Then k v s k ≤ /α . Similarly, k v u k ≤ /α . Hence k ( I − L ) − ( v ) k = k ( I − A ) − v s + ( I − D ) − v u k≤ − η k v s k + η − η k v u k ≤ ηα (1 − η ) . Hence we have k ( I − L ) − k ≤ ηα (1 − η ) . It is easy to see that G ( v ) = v is equivalent to Lv + φ ( v ) = v and equivalent to v = ( I − L ) − φ ( v ). Now we consider a map T : Y → Y defined by T ( v ) = ( I − L ) − φ ( v ) . Then T and G have the same set of fixed points. For any u, u ′ ∈ Y , we have k T ( u ) − T ( u ′ ) k = k ( I − L ) − ( φ ( u ) − φ ( u ′ )) k≤ ηα (1 − η ) · Lip φ · k u − u ′ k . Now we choose ξ > ηα (1 − η ) ξ < , then T is a contraction mapping on Y under the assumption Lip φ < ξ . We knowthat T has a unique fixed point 0 ∈ Y and so does G . In other words, if there is v ∈ Y such that G ( v ) = v , then v = 0. This ends the proof of Proposition 4.1. (cid:3) Now we prove theorems A, that is, every multisingular hyperbolic set is rescalingexpansive. In the proof we will not assume the full strength of multisingularhyperbolicity but only a naive version of it. Anyway let us give it a name anda definition. Let Λ be a compact invariant set of X . We will call a function h : (Λ \ Sing( X )) × R → (0 , + ∞ ) a naive cocycle of X on Λ \ Sing( X ) if thefollowing two conditions are satisfied:(1) for any x ∈ Λ \ Sing( X ) and any s, t ∈ R , h ( x, s + t ) = h ( x, s ) · h ( ϕ s ( x ) , t ) , (2) for any x ∈ Λ \ Sing( X ), there is K = K ( x ) > h ( x, t ) ≤ K forall t ∈ R .Following Bonatti-da Luz [BL1] we will call a compact invariant set Λ of X a naive multisingular hyperbolic set of X if, for some C > λ >
0, there is a ψ t -invariant splitting N Λ \ Sing( X ) = ∆ s ⊕ ∆ u such that(1) ∆ s ⊕ ∆ u is a ( C, λ )-dominated splitting with respect to ψ t ;(2) there is a naive cocycle h st of X such that ∆ s is ( C, λ )-contracting for h st · ψ t ;(3) there is a naive cocycle h ut of X such that ∆ u is ( C, λ )-expanding for h ut · ψ t .Note that this definition does not care about singularities and uses the usuallinear Poincar´e flow defined on Λ \ Sing( X ). Let h ( e, t ) be a pragmatical cocycle on˜Λ with respect to a singularity σ with isolating neighborhood U . It gives a cocycle h ( x, t ) : Λ \ Sing( X ) × R → R by h ( j ( e ) , t ) = h ( e, t ) for e ∈ j − (Λ \ Sing( X )) . It isnot hard to see that h ( x, t ) ≤ K ( x ) := max { sup x ∈ M k X ( x ) kk X ( x ) k , sup x ∈ M k X ( x ) k inf x ∈ ∂U k X ( x ) k } for any x ∈ Λ \ Sing( X ) and t ∈ R . Thus h ( x, t ) is a naive cocycle. Hence areparametrizing cocyle gives automatically a naive cocycle. Then one can easilycheck that every multisingular hyperbolic set is naive multisingular hyperbolic. RESCALED EXPANSIVENESS FOR FLOWS 19
Proof of Theorem A . In fact we prove every naive multisingular hyperbolic setis rescaling expansive. Let Λ be a naive multisingular hyperbolic set of X with a( C, λ )-dominated splitting N Λ \ Sing ( X ) = ∆ s ⊕ ∆ u . Let
L > X . Choose T > η = Ce − λT < . Since N Λ \ Sing( X ) ∆ s ⊕ ∆ u is a dominated splitting, there is α > ∠ (∆ s ( x ) , ∆ u ( x )) > α for every x ∈ Λ \ Sing( X ). Note that this is guaranteed by the (uniform) dominanceof the splitting on Λ \ Sing( X ), even though Λ \ Sing( X ) is non-compact. See [HW]for a proof.Now we determine the number ǫ > X and the set Λ (and hence on L , T , η , and α ) butnot on x , y and others.Let ξ > η and α . Take ǫ > ǫ ≤ min { r / , δ ( T ) } , where r = r ( T ) is the number in the definition of the sectional Poincar´e map,and δ ( T ) is the number in the statement of Proposition 3.2. Also, by item 2 ofProposition 3.1, we can take ǫ so that for any regular point z of X , if z ′ ∈ exp z ( N z )and d ( z ′ , z ) < ǫ k X ( z ) k then k D exp − z ( z ′ ) P z,T − D P z,T k < ξ η − e LT . Here ξ/ (5 η − e LT ) and 3 ǫ play the role of “ ǫ ” and “ δ ” in the statement of Proposition3.1. Since D P z,T = ψ T | N z , this is the same as k D exp − z ( z ′ ) P z,T − ψ T | N z k < ξ η − e LT . This settles the choice of ǫ > h st and h ut be two naive cocycles such that h st · ψ t | ∆ s is ( C, λ )-contractingand h ut · ψ t | ∆ u is ( C, λ )-expanding. Then for any x ∈ Λ \ Sing( X ),(a) k ψ T | ∆ sx k · k ψ − T | ∆ u ( ϕ T ( x )) k ≤ η ;(b) h sT ( x ) · k ψ T | ∆ s ( x ) k ≤ η ;(c) h uT ( x ) · m ( ψ T | ∆ u ( x ) ) ≥ η − .The key to the proof of Theorem A is the following Claim.
For every x ∈ Λ \ Sing( X ) , there is a sequence { c i = c i ( x ) > i ∈ Z } such that the following three conditions hold: ( A The set { c i ( x ) : x ∈ Λ \ Sing( X ) , i ∈ Z } of numbers is bounded. ( A For every x ∈ Λ \ Sing( X ) , c i ·k ψ T | ∆ s ( ϕ iT ( x )) k ≤ η , and c i · m ( ψ T | ∆ u ( ϕ iT ( x ) ) ≥ η − . ( A Denote b i = b i ( x ) = c · c · · · c i − for i > and b i = b i ( x ) = c − i · c − i +1 · · · c − − for i < . Then for every x ∈ Λ \ Sing( X ) , the sequence { b i ( x ) } i ∈ Z is bounded. Briefly, condition (A2) says that, replacing h sT ( ϕ iT ( x )) and h uT ( ϕ iT ( x )) both by c i , items (b) and (c) hold simultaneously. Condition (A3) says a “bounded product”property. Proof of the Claim . Let x ∈ Λ \ Sing( X ). We define c i by two different formulasdepending on i ≥ i <
0. If i ≥
0, let c i = c i ( x ) = η − m ( ψ T | ∆ u ( ϕ iT ( x )) ) . Then(1) η − e − LT ≤ c i ≤ η − e LT ;(2) c i · m ( ψ T | ∆ u ( ϕ iT ( x )) ) = η − ;(3) c i · k ψ T | ∆ s ( ϕ iT ( x )) k ≤ η ;(4) c i ≤ h uT ( ϕ iT ( x )).Thus (1) verifies condition (A1) for i ≥
0, and (2) and (3) verify condition (A2)for i ≥ i <
0, let c i = c i ( x ) = η k ψ T | ∆ s ( ϕ iT ( x )) k . Then(1*) ηe − LT ≤ c i ≤ ηe LT ;(2*) c i · k ψ T | ∆ s ( ϕ iT ( x )) k = η ;(3*) c i · m ( ψ T | ∆ u ( ϕ iT ( x )) ) ≥ η − ;(4*) c i ≥ h sT ( ϕ iT ( x )).Thus (1*) verifies condition (A1) for i <
0, and (2*) and (3*) verify condition(A2) for i < b = 1. For every i >
0, by (4), b i = b i ( x ) = c · c · · · c i − ≤ h uT ( x ) · h uT ( ϕ T ( x )) · · · h uT ( ϕ ( i − T ( x )) = h uiT ( x ) . For every i <
0, by (4*), b i = b i ( x ) = c − i · c − i +1 · · · c − − ≤ [ h sT ( ϕ iT ( x )) · h sT ( ϕ ( i +1) T ( x )) · · · h sT ( ϕ − T ( x ))] − = [ h s − iT ( ϕ iT ( x ))] − = h siT ( x ) . By the definition of naive cocycle, for fixed x , the two sequences { h uiT ( x ) } i ∈ Z and { h siT ( x ) } i ∈ Z are bounded. Thus the sequence { b i ( x ) } i ∈ Z is bounded. This verifiescondition (A3), proving the Claim.Now let 0 < ǫ ≤ ǫ . Let x ∈ Λ and y ∈ M and an increasing continuous function θ : R → R be given such that d ( ϕ t ( x ) , ϕ θ ( t ) ( y )) ≤ ( ǫ/ k X ( ϕ t ( x )) k for all t ∈ R . From now on x and y will be fixed till the end of the proof of TheoremA. We prove ϕ θ ( t ) ( y ) ∈ ϕ [ − ǫ, ǫ ] ( ϕ t ( x ))for all t ∈ R . We assume x / ∈ Sing( X ) because otherwise the situation would betrivial.Denote E i = N ϕ iT ( x ) . Let β : [0 , + ∞ ) → [0 ,
1] be a bump function such that(a) β ( t ) = 1 for t ∈ [0 , / β ( t ) = 0 for t ∈ [2 / , + ∞ ); RESCALED EXPANSIVENESS FOR FLOWS 21 (c) β ′ ( t ) ∈ [ − ,
0] for any t ∈ [0 , + ∞ ).Define P i : E i → E i +1 to be P i ( v ) = β ( k v k ǫ k X ( ϕ iT ( x )) k ) · P ϕ iT ( x ) ,T ( v ) + (1 − β ( k v k ǫ k X ( ϕ iT ( x )) k )) · ψ T ( v ) . Roughly, we use the bump function β to extend the local map P ϕ iT ( x ) ,T defined nearthe origin of E i to the whole E i so that it agrees with ψ T away from the origin.Precisely, P i = P ϕ iT ( x ) ,T inside the ball N ϕ iT ( x ) ( ǫ k X ( ϕ iT ( x )) k ), and P i = ψ T outside the ball N ϕ iT ( x ) (3 ǫ k X ( ϕ iT ( x )) k ). Note that 3 ǫ ≤ r , hence P ϕ iT ( x ) ,T is welldefined in the ball N ϕ iT ( x ) (3 ǫ k X ( ϕ iT ( x )) k ), and hence P i is well defined on thewhole E i . A direct computation gives k D v P i − ψ T | E i k < ξη − e LT , ∀ v ∈ E i . Remark.
For convenience we sketch the computation. Abbreviate r = 3 ǫ k X ( ϕ iT ( x )) k .Then P i ( v ) − ψ T ( v ) = β ( k v k r ) · ( P ϕ iT ( x ) ,T ( v ) − ψ T ( v )) . We may assume k v k ≤ r because otherwise the value is 0. Hence D v P i − ψ T = β ′ ( k v k r ) · r · v k v k · ( P ϕ iT ( x ) ,T ( v ) − ψ T ( v )) + β ( k v k r ) · ( D v P ϕ iT ( x ) ,T − ψ T ) . By the generalized mean value theorem, k P ϕ iT ( x ) ,T ( v ) − ψ T ( v ) k ≤ ξ η − e LT k v k . Thus k D v P i − ψ T k ≤ | β ′ ( k v k r ) | · ξ η − e LT · k v k r + | β ( k v k r ) | · ξ η − e LT ≤ ξη − e LT . The last step uses the facts that | β ′ | ≤ k v k ≤ r , and | β | ≤
1. This ends theremark.Since ǫ/ ≤ ǫ / ≤ δ ( T ), by Proposition 3.2, there is a sequence { T i : i ∈ Z } such that ϕ θ ( T i ) ( y ) ∈ exp( N ϕ iT ( x ) ) . Let u i = exp − ϕ iT ( x ) ( ϕ θ ( T i ) ( y )) ∈ E i . By items 2 and 4 of Proposition 3.2, we have k u i k ≤ ǫ k X ( ϕ iT ( x )) k , P ϕ iT ( x ) ,T ( u i ) = u i +1 . That is, P i ( u i ) = u i +1 . Let u = ( u i ) i ∈ Z . Since M is compact, {k X ( z ) k} z ∈ M is bounded. Hence k u k = sup {k u i k : i ∈ Z } < + ∞ , i.e., u ∈ Y. Here Y consists of all bounded elements of Y , where Y = Π ∞ i = −∞ E i (see the beginning of section 4 for notations). Now define G i : E i → E i +1 to be G i ( v ) = b i +1 P i ( b − i v ) . Here b i = b i ( x ) is given by the Claim, where x is the point that has been fixed suchthat 0 x is the origin of E . Since b i +1 · b − i = c i ∈ [ ηe − LT , η − e LT ](condition (A1)), and k D v P i − ψ T | E i k < ξη − e LT , ∀ v ∈ E i , we have k D v G i − c i ψ T | E i k < ξ, ∀ v ∈ E i . Define G : Y → Y to be G | E i = G i . Since G (0) = 0 , the derivative condition k D v G i − c i ψ T | E i k < ξ guarantees that G maps a bounded element of Y to a bounded element of Y . That is, G maps Y into Y and hence G : Y → Y is well defined. Write G i = c i ψ T | E i + φ i . Then Lip( φ i ) < ξ. By condition (A2), c i ψ T | E i can serve as the operator L i of Proposition 4.1.Let w i = b i u i . Then G i ( w i ) = b i +1 P i ( b − i b i u i ) = b i +1 u i +1 = w i +1 . That is, G ( w ) = w, where w = ( w i ) i ∈ Z . By condition (A3), the sequence { b i } i ∈ Z is bounded. Then k w k = sup {k w i k : i ∈ Z } < + ∞ , i.e., w ∈ Y. Therefore, by Proposition 4.1, w = 0. From w i = 0 we get v i = 0 . Thatis, ϕ θ ( T i ) ( y ) = ϕ iT ( x ) . By Proposition 3.2, | θ ( T i ) − θ ( iT ) | ≤ · ( ǫ/
3) = ǫ. Then ϕ θ ( iT ) ( y ) = ϕ θ ( iT ) − θ ( T i ) ( ϕ θ ( T i ) ( y ))= ϕ θ ( iT ) − θ ( T i ) ( ϕ iT ( x )) ∈ ϕ [ − ǫ, ǫ ] ( ϕ iT ( x )) . Now for any τ ∈ R , set z = ϕ τ ( x ) , y = ϕ θ ( τ ) ( y ) , θ ( t ) = θ ( t + τ ) − θ ( τ ) . Then d ( ϕ θ ( t ) ( y ) , ϕ t ( z )) = d ( ϕ θ ( t + τ ) ( y ) , ϕ t + τ ( x )) ≤ ( ǫ/ k X ( ϕ t + τ ( x )) k = ( ǫ/ k X ( ϕ t ( z )) k . RESCALED EXPANSIVENESS FOR FLOWS 23
Hence y = ϕ θ (0) ( y ) ∈ ϕ [ − ǫ, ǫ ] ( z ). Thus ϕ θ ( τ ) ( y ) ∈ ϕ [ − ǫ, ǫ ] ( ϕ τ ( x )) . This ends the proof of Theorem A.
Corollary 4.2.
Let Λ be a singular hyperbolic set of a C vector field X on M .Then Λ is rescaling expansive. In fact, there is ǫ > such that for any < ǫ ≤ ǫ ,any x ∈ Λ and y ∈ M , and any increasing continuous functions θ : R → R , if d ( ϕ θ ( t ) ( y ) , ϕ t ( x )) ≤ ( ǫ/ k X ( ϕ t ( x )) k for all t ∈ R , then ϕ θ ( t ) ( y ) ∈ ϕ [ − ǫ,ǫ ] ( ϕ t ( x )) for all t ∈ R . The next proposition explains why this is a corollary of Theorem A.
Proposition 4.3.
Every singular hyperbolic set is multisingular hyperbolic.Proof.
Let Λ be a (
C, λ )-singular hyperbolic set of X with dominated splitting E ⊕ F of Φ t . Without loss of generality we assume Λ is positive singular hyperbolicfor X . Thus E is ( C, λ )-contracting and F is ( C, λ )-area-expanding with respectto Φ t . First we work on the usual linear Poincar´e flow on Λ \ Sing( X ). For x ∈ Λ \ Sing( X ), let h st ( x ) ≡ h ut ( x ) = k Φ t | h X ( x ) i k . Note that since k Φ t | h X ( x ) i k = k X ( ϕ t ( x )) k / k X ( x ) k , h ut satisfies the cocycle condition. We prove the followingthree items:(1) there is a dominated splitting N Λ \ Sing( X ) = ∆ s ⊕ ∆ u with respect to ψ t ;(2) ∆ s is contracting for ψ t ;(3) ∆ u is expanding for k Φ t |
0, proving (3).We verify that ∆ s ⊕ ∆ u is a dominated splitting with respect to ψ t . There is atricky point here as we have to look at the negative direction of the flow: For anyunit vectors u ∈ ∆ sx and v ∈ ∆ ux and any t > k ψ − t ( v ) kk ψ − t ( u ) k ≤ k Φ − t ( v ) kk π ϕ − t ( x ) Φ − t ( u ′ ) k ≤ k Φ − t ( v ) k K − k Φ − t ( u ′ ) k = K k Φ − t ( v ) kk u ′ kk Φ − t ( u ′ / k u ′ k ) k ≤ KCe − λt , where u ′ = ( π x | E x ) − ( u ). The last inequality uses the fact that u ′ ∈ E x and v ∈ F x .This proves (1).Now we extend everything to ˜Λ. DenoteΛ = { X ( x ) / k X ( x ) k : x ∈ Λ \ Sing( X ) } . Then Λ = ˜Λ. The cocycle h : (Λ \ Sing( X )) × R → (0 , ∞ ) h ( x, t ) = k Φ t | h X ( x ) i k = k Φ t ( X ( x ) k X ( x ) k ) k gives a cycle h : Λ × R → (0 , ∞ ) h ( e, t ) = k Φ t ( e ) k , which is uniformly continuous and hence extends to a (reparametrizing) cocycle˜ h : ˜Λ × R → (0 , ∞ )˜ h ( e, t ) = k Φ t ( e ) k . As usual, the dominated splitting ∆ s ⊕ ∆ u of ψ t extends to a dominated splitting(still denoted) ∆ s ⊕ ∆ u of ˜ ψ t such that items (1) through (3) still hold. This provesProposition 4.3. (cid:3) Proof of Theorem B
For preciseness we use sometimes the notation ϕ Xt to denote the flow generatedby the vector field X . Lemma 5.1.
Let X ∈ X ( M ) and let Q be a non-hyperbolic periodic orbit of X .For any neighborhood U of X , any neighborhood U of γ and any δ > , there is Y ∈ U such that: (1) X = Y outside U ; (2) there exist two distinct hyperbolic periodic orbits Q and Q of Y containedin U with a time reparametrization θ : R → R such that for some x ∈ Q and x ∈ Q , d ( ϕ Yt ( x ) , ϕ Yθ ( t ) ( y )) < δ k Y ( ϕ t ( x )) k for all t ∈ R . RESCALED EXPANSIVENESS FOR FLOWS 25
Proof.
Let Q be a non-hyperbolic periodic orbit of X . Take q ∈ Q and denote N q ( r )the r -ball of the center the origin 0 q in the normal space N q . There is r > f Xq : N p ( r ) → N p of X is well defined such that D p f Xq has an eigenvalue λ on the unit circle. With an arbitrarily small perturbation ifnecessary, we can assume λ is a simple eigenvalue and there are no other eigenvaluesof D p f Xq on the unit circle except λ and ¯ λ . Let V be the eigenspace associated to λ . If λ is complex, with an arbitrarily small perturbation near Q if necessary, wecan assume that D p f Xq | V is a rational rotation. In any case we can assume that( D p f Xq ) k | V = id for some positive integer k . Then by a standard perturbationargument (see Lemma 1.3 of [MSS] for a precise proof), there is Y arbitrarily closeto X that keeps the orbit Q unchanged such that f Y p = exp p ◦ D q f Xq ◦ exp − q in asmall neighborhood of 0 p where f Y p denotes the first return map of Y on N q . Notethat all perturbations here can be supported on an arbitrarily small neighborhoodof Q . That is, we can assume Y = X on M \ U for any given neighborhood U of Q . Thus we may assume that U contains no singularities of X , and hence there is a > k X ( x ) k > a for every x ∈ U . Since Y can be chosen arbitrarilyclose to X , we can assume k Y ( x ) k > a for all x ∈ U . Since ( f Y p ) k = id in asmall disc or arc in exp p ( V ) centered at 0 p , for any δ >
0, we can choose distinct x, y ∈ exp p ( V ) arbitrarily close to q together with an increasing homeomorphism θ : R → R such that d ( ϕ Y t ( x ) , ϕ Y θ ( t ) ( y ) < aδ for all t ∈ R . With an arbitrarilysmall perturbation Y of Y that keeps the orbits of x, y unchanged, we may assume Q = Orb( x ) and Q = Orb( y ) are hyperbolic. This ends the proof of Lemma5.1. (cid:3) Proposition 5.2.
There is a residual set R ⊂ X ( M ) such that, for any X ∈ R ,if there are X n → X and non-hyperbolic periodic orbits Q n of X n that converge toa compact set Γ in the Hausdorff metric, then there are two sequences of hyperbolicperiodic points { p n } , { q n } of X with the following properties:(1) for any n , Orb( p n ) = Orb( q n ) , and there is an increasing homeomorphism θ n : R → R such that d ( ϕ t ( p n ) , ϕ θ n ( t ) ( q n )) < (1 /n ) k X ( ϕ t ( p n )) k for all t ∈ R .(2) Orb( p n ) and Orb( q n ) converge to Γ in the Hausdorff metric.Proof. Let K ( M ) be the space of nonempty compact subsets of M with the Hausdorffmetric, and {O n } ∞ n =1 be a countable basis of K ( M ). For each pair of positiveintegers n and k , denote by H n,k the subset of X ( M ) such that any Y ∈ H n,k hasa C neighborhood V in X ( M ) such that every Z ∈ V has two hyperbolic periodicpoints p and q such that (a) Orb( p ) and Orb( q ) are distinct and both in O k , (b)there is an increasing homeomorphism θ : R → R such that d ( ϕ Zt ( p ) , ϕ Zθ ( t ) ( q )) < n k Z ( ϕ Zt ( p )) k for all t ∈ R .Let N n,k be the complement of the C -closure of H n,k . Clearly, for every pair( n, k ), H n,k ∪ N n,k is C open and dense in X ( M ). Let KS denote the set ofKupka-Smale systems in X ( M ). Denote R = ( \ n,k ∈ N ( H n,k ∪ N n,k )) ∩ KS . Then R is C residual. Let X ∈ R . Assume X n → X . Also assume there are non-hyperbolic periodicorbits Q n of X n that converge to a compact set Γ in Hausdorff metric. Then forany neighborhood O of Γ in K ( M ), there is O k with Γ ∈ O k ⊂ O . By Lemma5.1, for any positive integer n and any neighborhood U of X , there are Z ∈ U andtwo hyperbolic periodic points p and q of Z such that (a) Orb( p ) and Orb( q ) aredistinct and both in O k , (b) there is an increasing homeomorphism θ : R → R suchthat d ( ϕ Zt ( p ) , ϕ Zθ ( t ) ( q )) < n k Z ( ϕ Zt ( p )) k for all t ∈ R . Since a hyperbolic periodic orbit is persistent under C perturbations, Z ∈ H n,k . Hence for any pair of positive integers ( n, k ), X is in the closureof H n,k and hence not in N n,k . This means X ∈ H n,k for all ( n, k ). Hencefor any neighborhood O of Γ and any positive integer n , X has two hyperbolicperiodic points p and q of distinct orbits that are in O together with an increasinghomeomorphism θ : R → R such that d ( ϕ t ( p ) , ϕ θ ( t ) ( q )) < (1 /n ) k X ( ϕ t ( p ) k for all t ∈ R . This ends the proof of Proposition 5.2. (cid:3) We need the recent result of Bonatti-da Luz:
Proposition 5.3. ([BL1, BL2])
There is a residual set R ⊂ X ( M ) such thatany X ∈ R is a star flow if and only if any chain class Λ of X is multisingularhyperbolic. Now we prove Theorem B.
Proof of Theorem B . Let R = R ∩ R . We prove R satisfies Theorem B. Thus let X ∈ R and let Λ be an isolated chaintransitive set of X . We prove the three items of Theorem B circularly. Since (2) ⇒ (3) is guaranteed by Proposition 5.3 (Proposition 5.3 is global, but it obviouslyapplies to our case of an isolated chain transitive set) and (3) ⇒ (1) is guaranteedby Theorem A, it remains to prove (1) ⇒ (2). Proof.
Assume Λ is rescaling expansive. We prove Λ is locally star. Suppose forthe contrary there are X n → X with non-hyperbolic periodic orbits Q n of X n thatconverge to a compact set Γ ⊂ Λ in the Hausdorff metric. By Proposition 5.2, thereare two sequences of hyperbolic periodic points { p n } , { q n } of X with the followingproperties:(1) for any n , Orb( p n ) = Orb( q n ), and there is an increasing homeomorphism θ n : R → R such that d ( ϕ t ( p n ) , ϕ θ n ( t ) ( q n )) < (1 /n ) k X ( ϕ t ( p n )) k for all t ∈ R .(2) Orb( p n ) and Orb( Q n ) converge to Γ in the Hausdorff metric.Since Λ is isolated for X , p n , q n ∈ Λ for large n . This contradicts the assumptionthat Λ is rescaling expansive. This proves (1) ⇒ (2) and hence Theorem B. Remark.
We may add item (4) as to be “Λ is naive multisingular hyperbolic for X ”. Then the four items are equivalent. This is because (3) ⇒ (4) is obvious and(4) ⇒ (1) is contained in (the proof of) Theorem A. In other words, generically,naive multisingular hyperbolicity is equivalent to multisingular hyperbolicity. RESCALED EXPANSIVENESS FOR FLOWS 27 Appendix
In this appendix we discuss the equivalence between the rescaled expansivenessand the Komuro expansiveness for non-singular flows. We also include a thirdcondition, the expansiveness of Bowen-Walters [BW] and Keynes-Sears [KS].
Proposition 6.1.
Let ϕ t be a continuous flow on a compact metric space M withoutsingularities. Then the following three conditions are equivalent: (1) For any ǫ > , there is δ > such that for any x, y ∈ M and any increasingcontinuous functions θ : R → R , if d ( ϕ t ( x ) , ϕ θ ( t ) ( y )) ≤ δ for all t ∈ R , then ϕ θ ( t ) ( y ) ∈ ϕ [ − ǫ,ǫ ] ( ϕ t ( x )) for all t ∈ R ; (2) For any ǫ > , there is δ > such that for any x, y ∈ M and any increasingcontinuous functions θ : R → R , if d ( ϕ t ( x ) , ϕ θ ( t ) ( y )) ≤ δ for all t ∈ R , then ϕ θ (0) ( y ) ∈ ϕ [ − ǫ,ǫ ] ( x ) ; (3) For any ǫ > , there is δ > such that for any x, y ∈ M and any surjectiveincreasing continuous functions θ : R → R , if d ( ϕ t ( x ) , ϕ θ ( t ) ( y )) ≤ δ for all t ∈ R ,then ϕ θ ( t ) ( y ) ∈ ϕ [ − ǫ,ǫ ] ( ϕ t ( x )) for some t ∈ R ; In case ϕ t is generated by a C vector field X , item (1) of Proposition 6.1 isjust the rescaled expansiveness because, in the non-singular case, k X ( x ) k has anupper bound and also a positive lower bound. Item (3) is just the expansiveness ofKomuro [Kom1]. Item (2) is the expansiveness of Bowen-Walter [BW] and Keynes-Sears [KS]. Thus Proposition 6.1 says that, for nonsingular flows, the three versionsof expansiveness are equivalent.Note that for flows with singularities item (2) and item (3) are not equivalent:Komuro [Kom1] has proved that the geometrical Lorenz attractor satisfies item (3)but not item (2). Proof.
That (2) ⇔ (3) is proved by Oka [O]. Since (1) ⇒ (2) is obvious, we onlyprove (2) ⇒ (1).Assume item (2) holds. Let δ be chosen in item (2) associated to ǫ . Let x, y ∈ M and any surjective increasing continuous functions θ : R → R be given such that d ( ϕ θ ( t ) ( y ) , ϕ t ( x )) ≤ δ for all t ∈ R . For any τ ∈ R , set z = ϕ τ ( x ) , y = ϕ θ ( τ ) ( y ) , θ ( t ) = θ ( t + τ ) − θ ( τ ) . Then d ( ϕ θ ( t ) ( y ) , ϕ t ( z )) = d ( ϕ θ ( t + τ ) ( y ) , ϕ t + τ ( x )) ≤ δ. Hence by item (2) we have y = ϕ θ (0) ( y ) ∈ ϕ [ − ǫ, ǫ ] ( z ). Thus ϕ θ ( τ ) ( y ) ∈ ϕ [ − ǫ, ǫ ] ( ϕ τ ( x )) . This means item (1) holds, proving Proposition 6.1.
Acknowledgement.
The first author is supported by National Natural ScienceFoundation of China (No. 11671025 and No. 11571188) and the FundamentalResearch Funds for the Central Universities. The second author is supported byNational Natural Science Foundation of China (No. 11231001). We thank ShaoboGan, Ming Li and Dawei Yang for many discussions and communications. We alsothank Christian Bonatti and Adriana da Luz for an early preprint of their recentpaper.
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