Robustly shadowable chain transtive sets and hyperbolicity
aa r X i v : . [ m a t h . D S ] M a r ROBUSTLY SHADOWABLE CHAIN TRANSITIVE SETSAND HYPERBOLICITY
MOHAMMAD REZA BAGHERZAD AND KEONHEE LEE
Abstract.
We say that a compact invariant set Λ of a C -vector field X on a compact bound-aryless Riemannian manifold M is robustly shadowable if it is locally maximal with respect toa neighborhood U of Λ, and there exists a C -neigborhood U of X such that for any Y ∈ U , thecontinuation Λ Y of Λ for Y and U is shadowable for Y t . In this paper, we prove that any chaintransitive set of a C -vector field on M is hyperbolic if and only if it is robustly shadowable. Introduction
The main goal of the study of differentiable dynamical systems is to understand the structure ofthe orbits of vector fields (or diffeomorphisms) on a compact boundaryless Riemannian manifold.To descirbe the dynamics on the underlying manifold, it is usual to use the dynamic propertieson the tangent bundle such as hyperbolicity and dominated splitting. A fundamental problem inrecent years is to study the influence of a robust dynamic property (i.e., property that holds fora given system and all C -nearby systems) on the behavior of the tangent map on the tangentbundle (e.g., see [4, 6–8, 10]).Recently, several results dealing with the influence of a robust dynamics property of a C -vectorfield were appeared. For instance, Lee and Sakai [6] proved that a nonsingular vector field X isrobustly shadowable (i.e., X and its C -nearby systems are shadowable) if and only if it satisfiesboth Axiom A and the strong transversality condition (i.e., it is structurally stable). Afterwards,Pilyugin and Tikhomirov [10] gave a description of robustly shadowable oriented vector fieldswhich are structurally stable. In particular, it is proved in [7] that any robustly shadowable chaincomponent C X ( γ ) of X containing a hyperbolic periodic orbit γ does not contain a hyperbolicsingularity, and it is hyperbolic if C X ( γ ) has no non-hyperbolic singularity. Here we say that thechain component C X ( γ ) is robustly shadowable if there is a C -neighbohood U of X such that forany Y ∈ U , the continuation C Y ( γ Y ) of Y containing γ Y is shadowable for Y t , where γ Y is thecontinuation of γ with respect to Y . Very recently, Gan et al. [4] showed that the set of all robustlyshadowable oriented vector fields is contained in the set of vector fields with Ω-stability. In thisdirection, the following question is still open: if the chain component C X ( γ ) of a C -vector field X on a compact boundaryless Riemannian manifold M containing a hyperbolic periodic orbit γ isrobustly shadowable, then is it hyperbolic? Date : July 22, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Chain transitive sets, dominated splitting, hyperbolicity, robustly shadowing.
In this paper, we study the dynamics of robustly shadowable chain transitive sets. More precisely,we prove that any chain transitive set of a vector field X is hyperbolic if and only if it is robustlyshadowable. For this, we first show that if a compact invariant set Λ ⊂ M is robustly shadowablethen every singularity and periodic orbit in Λ Y are hyperbolic for Y t , where Λ Y is the continuationof Λ with repect to a C -nearby vector field Y . Moreover, we see that any robustly shadowablechain transitive set Λ does not contain a singularity. Finally we show that Λ admits a dominatedsplitting, and it is indeed a hyperbolic splitting.Now we round out the introduction with some notations, definitions and main theorem whichwe will use throughout the paper. Let M be a compact boundaryless Riemannian manifold withdimension n . Denote by X ( M ) the set of all C vector fields of M endowed with the C topology.Then every X ∈ X ( M ) generates a C flow X t : M × R → M , that is, a family of diffeomorphismson M such that X s ◦ X t = X t + s for all t, s ∈ R , X = Id and dX t dt | t =0 = X ( p ) for any p ∈ M .Throughout the paper, for X, Y, . . . ∈ X ( M ), we always denote the generated flows by X t , Y t , . . . ,respectively. For x ∈ M , let us denote the orbit { X t ( x ) , t ∈ R } of the flow X t (or X ) through x by orb ( x, X t ), or O ( x ) if no confusion is likely. We say that a point x ∈ M is a singularity of X if X ( x ) = 0; and an orbit O ( x ) is closed (or periodic ) if it is diffeomorphic to a circle S . Let d be thedistance induced from the Riemannian structure on M . A sequence { ( x i , t i ) : x i ∈ M ; t i ≥ a
0, all points x, y ∈ M can be connected by a δ -pseudo orbit.Let Rep be the set of all increasing homeomorphisms (called reparametrizations ) h : R → R suchthat h (0) = 0 . We say that a compact invariant set Λ of X t is shadowable if for any ε >
0, thereis δ > δ -pseudo orbit { ( x i , t i ) : −∞ ≤ i ≤ ∞} in Λ,there exist a point y ∈ M and h ∈ Rep such that for all t ∈ R we have d ( X h ( t ) ( y ) , x ∗ t ) < ε, where x ∗ t = X t − S i ( x i ) for any t ∈ [ S i , S i +1 ], and S i is given by S i = P i − j =0 t j for i > , i = 0 , − P − j = i t j for i < . Note that the above concept of pseudo orbit is slightly different from that of pseudo orbit in [6, 10].However we point out here that a compact invariant set Λ is shadowable for X t under the abovedefinition if and only if it is shadowable for X t under the definition in [6, 10]. A point x ∈ M iscalled chain recurrent if for any δ >
0, there exists a δ -pseudo orbit { ( x i , t i ) : 0 ≤ i < n } with n > x = x and d ( X t n − ( x n − ) , x ) < δ . The set of all chain recurrent points of X t is called the chain recurrent set of X t , and denote it by CR ( X t ). For any x, y ∈ M , we say that OBUSTLY SHADOWABLE CHAIN TRANSTIVE SETS AND HYPERBOLICITY 3 x ∼ y , if for any δ >
0, there are a δ -pseudo orbit { ( x i , t i ) : 0 ≤ i < n } with n > x = x and d ( X t n − ( x n − ) , y ) < δ and a δ -pseudo orbit { ( x ′ i , t ′ i ) : 0 ≤ i < m } with m > x ′ = y and d ( X t ′ n − ( x ′ n − ) , x ) < δ . It is easy to see that ∼ gives an equivalence relationon the set CR ( X t ). An equivalence class of ∼ is called a chain component of X t (or X ). Wesay that a compact invariant set Λ of X t is chain transitive if for any x, y ∈ Λ and any δ > δ -pseudo orbit { ( x i , t i ) ∈ Λ × R | t i ≥ , ≤ i < n } with n > x = x and d ( X t n − ( x n − ) , y ) < δ .A compact invariant set Λ of X t is called hyperbolic if there are constants C > λ > DX t : T Λ M → T Λ M leaves a continuous invariant splitting T Λ M = E s ⊕ h X i ⊕ E u satisfying (cid:13)(cid:13) DX t | E s ( x ) (cid:13)(cid:13) ≤ Ce − λt and (cid:13)(cid:13) DX − t | E u ( x ) (cid:13)(cid:13) ≤ Ce − λt for any x ∈ Λ and t >
0, where h X i denotes the subspace generated by the vector field X . For anyhyperbolic closed orbit γ , the sets W s ( γ ) = { x ∈ M : X t ( x ) → γ as t → ∞} and W u ( γ ) = { x ∈ M : X t ( x ) → γ as t → −∞} are said to be the stable manifold and unstable manifold of γ , respectively. We say that thedimension of the stable manifold W s ( γ ) of γ is the index of γ , and denoted by ind ( γ ).The homoclinic class of X t associated to γ , denoted by H X ( γ ), is defined as the closure of thetransversal intersection of the stable and unstable manifolds of γ , that is; H X ( γ ) = W s ( γ ) ⋔ W u ( γ ) . By definition, we easily see that the set is closed and X t -invariant. Let C X ( γ ) be the chaincomponent of X t containing a hyperbolic periodic orbit γ . Then we have H X ( γ ) ⊂ C X ( γ ), but theconverse is not true in general. For two hyperbolic closed orbits γ and γ of X t , we say γ and γ are homoclinically related , denoted by γ ∼ γ , if W s ( γ ) ⋔ W u ( γ ) = ∅ and W s ( γ ) ⋔ W u ( γ ) = ∅ . By Birkhoff-Smale’s theorem (see [1]), we know that H X ( γ ) = { γ ′ : γ ′ ∼ γ } . A point x ∈ M is called nonwandering if for any neighborhood U of x , there is t ≥ X t ( U ) ∩ U = ∅ . The set of all nonwandering points of X t is called the nonwandering set of X t ,denoted by Ω( X t ). Let Sing ( X ) be the set of all singularities of X , and let P O ( X t ) be the set ofall periodic orbits (which are not singularities) of X t . Clearly we have Sing ( X ) ∪ P O ( X t ) ⊂ Ω( X t ) ⊂ CR ( X t ) . We say that X satisfies Axiom A if P O ( X t ) is dense in Ω( X t ) \ Sing ( X ), and Ω( X t ) is hyperbolicfor X t . A point y ∈ M is said to be an ω limit point of x if there exists a sequence t i → + ∞ such MOHAMMAD REZA BAGHERZAD AND KEONHEE LEE that X t i ( x ) → y . Denote the set of all omega limit points of x by ω ( x ). We say that a compactinvariant set Λ of X t is transitive if there is x ∈ Λ such that ω ( x ) = Λ.Let Λ be a compact invariant set of X t . For any C -close Y to X and a neighbourhood U of Λ,the set Λ Y = \ t ∈ R Y t ( U )is called the continuation of Λ for Y and U . If there exists a neighbourhood U of Λ satisfyingΛ = T t ∈ R X t ( U ) , then we say that Λ is locally maximal with respect to U , and U is called an isolating block of Λ. Let γ be a hyperbolic closed orbit of X t . Then we know that there are a C neighbourhood U of X and a neighbourhood U of γ such that for any Y ∈ U , there is a uniquehyperbolic closed orbit γ Y in U which is equal to the set T t ∈ R Y t ( U ). Note that every γ Y is locallymaximal with respect to U . The chain component of Y ∈ U containing the continuation γ Y willbe denoted by C Y ( γ Y ).Now we give the definition of robust shadowability for invariant sets of vector fields. Definition 1.1.
We say that a compact invariant set Λ of X t is robustly shadowable if it hasan isolating block U , and there exists a C -neighborhood U of X such that for any Y ∈ U , thecontinuation Λ Y for Y and U is shadowable for Y t . Here U is said to be an admissible neighborhoodof X with repsect to Λ . In this paper, we prove the following main theorem.
Main Theorem.
Let X ∈ X ( M ) , and let Λ be a compact, invariant and chain transitive set for X t . Then Λ is hyperbolic if and only if it is robustly shadowable. Linear Poincar´e flows and quasi hyperbolic orbit arcs
Hereafter we assume that the exponential mapexp p : T p M (1) → M is well defined for all p ∈ M , where T p M ( r ) denotes the r -ball { v ∈ T p M : k v k ≤ r } in T p M . Forany regular point x ∈ M (i.e., X ( x ) = ), we let N x = (span X ( x )) ⊥ ⊂ T x M, and N x ( r ) the r -ball in N x . Let ˆ N x,r = exp x ( N x ( r )). Given any regular point x ∈ M and t ∈ R , wecan take a constant r > C map τ : ˆ N x,r → R such that τ ( x ) = t and X τ ( y ) ( y ) ∈ ˆ N X t ( x ) , for any y ∈ ˆ N x,r . Now we define the Poincar´e map f x,t : ˆ N x,r → ˆ N X t ( x ) , , f x,t ( y ) = X τ ( y ) ( y )for y ∈ ˆ N x,r . Let M X = { x ∈ M : X ( x ) = } . Then it is easy to check that for any fixed t there exists a continuous map r : M X → (0 ,
1) such that for any x ∈ M X , the Poincar´emap f x,t : ˆ N x,r ( x ) → ˆ N X t ( x ) , is well defined and the respective time function τ ( y ) satisfies2 t/ < τ ( y ) < t/ y ∈ ˆ N x,r ( x ) . OBUSTLY SHADOWABLE CHAIN TRANSTIVE SETS AND HYPERBOLICITY 5
Let t be fixed. At each x ∈ M X , one can consider a flow box chart ( ˆ U x,t ,δ , F x,t ) at x such thatˆ U x,t ,δ = { tX ( x ) + y : 0 ≤ t ≤ t , y ∈ N x ( δ ) } ⊂ T x M, where F x,t : ˆ U x,t ,δ → M is defined by F x,t ( tX ( x ) + y ) = X t (exp x y ). Then it is well known thatif X t ( x ) = x for any t ∈ (0 , t ], then there is δ > F x,t : ˆ U x,t ,δ → M is an embedding.For ε > r >
0, let N ε ( ˆ N x,r ) be the set of all diffeomorphisms φ : ˆ N x,r → ˆ N x,r such that supp ( φ ) ⊂ ˆ N x,r/ and d C ( φ, id ) < ε. Here d C is the usual C metric, id denotes the identity map and the supp ( φ ) is the closure of theset of points where it differs from id . Proposition 2.1.
Let X ∈ X ( M ) , and let U ⊂ X ( M ) be a neighborhood of X . For any constant t > , there are a constant ε > and a C -neighborhood V of X such that for any Y ∈ V , thereexists a continuous map r : M Y → (0 , satisfying the following property : for any x ∈ M Y satisfying Y t ( x ) = x for < t ≤ t and any φ ∈ N ε ( ˆ N x,r ( x ) ) , there is Z ∈ U such that Y ( z ) = Z ( z ) for all z ∈ M \ F x ( ˆ U x ) and Z t ( y ) = Y t ( φ ( y )) for any y ∈ ˆ N x,r ( x ) and t / < t < t / , where F x ( ˆ U x ) isthe flow box of Y at x .Proof. See [11, p. 293–295]. (cid:3)
Remark 2.2.
In the above proposition, it is easy to see that if φ ( x ) = x , then f x,t ◦ φ is thePoincar´e map of Z , where f x,t : ˆ N x,r ( x ) → ˆ N X t ( x ) , is the Poincar´e map of Y . For the study of stability conjecture (see [5]) posed by Palis and Smale, Liao [9] introduced thenotion of linear Poincar´e flow for a C -vector field as follows. Let N = S x ∈ M X N x be the normalbundle based on M X . Then we can introduce a flow (which is called a linear Poincar´e flow for X )Ψ t : N → N , Ψ t | N x = π N x ◦ D x X t | N x , where π N x : T x M → N x is the natural projection along the direction of X ( x ), and D x X t is thederivative map of X t . Then we can see thatΨ t | N x = D x f x,t and f x,t ◦ exp x = exp X t ( x ) ◦ Ψ t . Using Proposition 2.1, we can prove the following lemma which has the same philosophy with theFranks’ Lemma for diffeomorphisms. One can find another proof for the lemma in [2].
Lemma 2.3.
Let U be a C neighborhood of X ∈ X ( M ) . For any T > , there exists a constant η > such that for any tubular neighborhood U of an orbit arc γ = X [0 ,T ] ( x ) of X t and for any η -perturbation F of the linear Poincar´e flow Ψ T | N x , there exists a vector field Y ∈ U such that thelinear Poincar´e flow ˜Ψ T | N x associated to Y coincides with F , and Y coincides with X outside U and along X [ − t ,t ] ( x ) , where t = min { t > , X − t ( x ) ∈ ∂U } and t = min { t > , X t ( x ) ∈ ∂U } . We introduce the notions of dominated splitting and hyperbolic splitting for linear Poincar´e flowsas follows.
MOHAMMAD REZA BAGHERZAD AND KEONHEE LEE
Definition 2.4.
Let Λ be an invariant set of X t which contains no singularity. We call a Ψ t -invariant splitting N Λ = ∆ s ⊕ ∆ u as an l -dominated splitting (or Λ admits an l -dominated splitting)if (cid:13)(cid:13) Ψ t | ∆ s ( x ) (cid:13)(cid:13) · (cid:13)(cid:13) Ψ − t | ∆ u ( X t ( x )) (cid:13)(cid:13) ≤ for any x ∈ Λ and any t ≥ l , where l > is a constant. Moreover, if dim(∆ sx ) is constant forall x ∈ Λ , then we say that the splitting is a homogeneous dominated splitting. Furthermore, a Ψ t -invariant splitting N Λ = ∆ s ⊕ ∆ u is said to be a hyperbolic splitting if there exist C > and λ ∈ (0 , such that (cid:13)(cid:13) Ψ t | ∆ s ( x ) (cid:13)(cid:13) ≤ Cλ t and (cid:13)(cid:13) Ψ − t | ∆ u ( x ) (cid:13)(cid:13) ≤ Cλ t for any x ∈ Λ and t > . The following proposition which is crucial to prove the hyperbolicity of invariant sets was provedby Doering and Liao [3, 9]. For a detailed proof, see Proposition 1.1 in [3].
Proposition 2.5.
Let Λ ⊂ M be a compact invariant set of X t such that Λ ∩ Sing ( X ) = ∅ . Then Λ is hyperbolic for X t if and only if the linear Poincar´e flow Ψ t restricted on Λ has a hyperbolicsplitting N Λ = ∆ s ⊕ ∆ u . Proposition 2.6.
Let Λ be a locally maximal set of X t with an isolating block U . Suppose that X has a C -neighbourhood U such that for any Y ∈ U , every periodic orbit and singularity of Y in U are hyperbolic. Then X has a neighbourhood ˜ U , together with two uniform constants ˜ η > and ˜ T > such that for any Y ∈ ˜ U ,(i) whenever x is a point on a periodic orbit of Y t in U and ˜ T ≤ t < ∞ , then t [ log m (Ψ Yt | E ux ) − log k Ψ Yt | E sx k ] ≥ η ; (ii) whenever P is a periodic orbit of Y t in U with period T, x ∈ P , and whenever an integer m ≥ and a partition t < t < · · · < t l = mT of [0 , mT ] are given that satisfy t k − t k − ≥ ˜ T , k = 1 , , ..., l, then mT l − X k =0 log k Ψ Yt k +1 − t k | E sXtk − x ) k ≤ − ˜ η, and mT l − X k =0 log m (Ψ Yt k +1 − t k | E uXtk − x ) ) ≥ ˜ η. Proof.
See Theorem 2.6 in [8]. (cid:3)
Let Λ ⊂ M X be a closed invariant set of X t that has a continuous Ψ t -invariant splitting N Λ =∆ s ⊕ ∆ u with dim ∆ s = p , 1 ≤ p ≤ dimM −
2. For two real numbers
T > η >
0, an orbitarc ( x, t ) = X [0 ,t ] ( x ) will be called ( η, T, p )- quasi hyperbolic orbit arc of X t with respect to the OBUSTLY SHADOWABLE CHAIN TRANSTIVE SETS AND HYPERBOLICITY 7 splitting ∆ s ⊕ ∆ u if [0 , t ] has a partition0 = T < T < ... < T l = t such that T ≤ T i − T i − < T , i = 1 , , ..., l, and the following three conditions are satisfied:1 T k k X j =1 log k Ψ T j − T j − | ∆ s ( X Tj − )( x ) k≤ − η, T l − T k − l X j = k log m (Ψ T j − T j − | ∆ u ( X Tj − )( x ) ) ≥ η, log k Ψ T k − T k − | ∆ s ( X Tk − )( x ) k − log m (Ψ T k − T k − | ∆ u ( X Tk − )( x ) ) ≤ − η, for k = 1 , , ..., l .Liao [9] proved the following shadowing result which says that any quasi hyperbolic orbit arcwith close enough end points can be shadowed by a hyperbolic periodic orbit. Proposition 2.7.
Let Λ be a compact invariant set of X t without singularities. Assume that thereexists a continuous invariant splitting N Λ = ∆ s ⊕ ∆ u with dim ∆ s = p , ≤ p ≤ dim M − .Then for any η > , T > , and ε > , there exists δ > such that if ( x, τ ) is an ( η, T, p ) -quasihyperbolic orbit arc of X t with respect to the splitting ∆ s ⊕ ∆ u and d ( X τ ( x ) , x ) < δ then there existsa hyperbolic periodic point y ∈ M and an orientation preserving homeomorphism g : [0 , τ ] → R with g (0) = 0 such that d ( X g ( t ) ( y ) , X t ( x )) < ε for any t ∈ [0 , τ ] and X g ( τ )( y ) = y. From robust shadowing to dominated splitting
In this section, we prove that if a nontrivial chain transitive subset Λ of X t is robustly shadowable,then it admits a dominated splitting. For this, we first show that any continuition Λ Y of Λ doesnot contain both a non-hyperbolic sigularity and a non-hyperbolic periodic orbit. Next we showthat Λ does not contain a singularity. Finally we prove that Λ admits a dominated splitting, Lemma 3.1.
Let Λ be a chain transitive set of X t . If Λ is robustly shadowable, then it is transitive.Proof. The proof is straightforward. (cid:3)
Using the perturbation technique developed by Pugh and Robinson [11], Pilyugin and Tikhomirov[10] showed that if M is robustly shadowable for X t then there is a C -neighbourhood U of X suchthat for any Y ∈ U , every critical element of Y t is hyperbolic. Here we prove that any continuitionΛ Y of a robustly shadowable chain transitive set Λ does not contain both a non-hyperbolic sigularityand a non-hyperbolic periodic orbit Proposition 3.2.
Let Λ be a robustly shdaowable set of X t . Then there exists a C -neighbourhood U of X such that for any Y ∈ U , every singularity and periodic orbit of Y t in Λ Y are hyperbolicfor Y t . MOHAMMAD REZA BAGHERZAD AND KEONHEE LEE
Proof.
Suppose Λ is a robustly shadowable set of X t . Then there exist a C -neighborhood U of X and a neighborhood U of Λ such that for any Y ∈ U , the continuation Λ Y = ∩ t ∈ R Y t ( U ) isshadowable for Y t . Case
1: Suppose there is Y ∈ U such that Λ Y contains a non-hyperbolic singularity σ . By usingthe Taylor’s theorem, we may assume that in a neighbourhood of σ the dynamical system inducedby Y is expressed by the following differential equation:˙ x = Ax + K ( x ) , where A ∈ M n × n ( R ) and K : R n → R n is a continuous map satisfyinglim x → K ( x ) k x k = 0 . Since σ is not hyperbolic, there is an eigenvalue λ of A with zero real part. First we assume that λ = 0. By changing coordinate, if necessary, we may assume that there is a n × n -matrix D closeenough to A such that(1) D = " B , where B is a ( n − × ( n − x in a neighbourhood of σ by x = ( y, z ) with respect to D . Let ε >
0, and choose a real valued C ∞ bump function β : R → R that satisfies the following conditions: β ( x ) ⊂ [0 ,
1] for x ∈ R ,β ( x ) = 0 for | x |≥ ε,β ( x ) = 1 for | x |≤ ε , ≤ β ′ ( x ) < ε for x ∈ R . Define ρ : R n → R by ρ ( x ) = β ( k x k ). By taking ε small enough, one can see that the vector field Z obtained from the following differential equation˙ x = Dx + (1 − ρ ( x )) K ( x )is C -close to Y . Moreover, we have B ε ( σ ) ⊂ U . Consequently we see that Z ∈ U , σ ∈ Sing ( Z ) ∩ Λ Z and Λ Z is shadowable for Z . Since ρ ( x ) = 1 for k x k < ε , in the ε neighbourhood of σ , thedifferential equation associated to Z is given by ( ˙ y = 0˙ z = Bz .
By considering coordinates represented in (1), for any x = ( y, z ) ∈ B ε ( σ ) , we have Z t ( x ) = Z t ( y, z ) = ( y, exp ( Bt ) z ) . This implies that if | y |≤ ε then ( y, ∈ Sing ( Z ) ∩ U , and so { ( y,
0) : | y | < ε } ⊂ Λ Z . Let δ > Y for ε . Choose α = 0 < α < OBUSTLY SHADOWABLE CHAIN TRANSTIVE SETS AND HYPERBOLICITY 9 ... < α n = ε such that | α i − α i − | < δ for i = 1 , ...n . Let x i = ( y i , z i ) and t i = 1 for i = 1 , ..., n. Clearly { ( x i , t i ) | i = 0 , . . . , n } is a finite δ -pseudo orbit of Z t in Λ Z . Since x and x n aresingularities we can put x i = x , t i = 1 for i ≤
0; and x i = x n , t i = 1 for i > n. Then { ( x i , t i ) | i ∈ Z } is a δ -pseudo orbit of Z t in Λ Z . Since Λ Z is shadowable, there are ( y, z ) ∈ M and a reparametrization h such that d ( X h ( t ) ( y, z ) , x ∗ t ) < ε t ∈ R . This implies O ( y ) ⊂ B ε (0). Since the intersections of planes formulated by { ( y, z ) | y = c } with B ε (0) are invariant ( c is a constant), there is c ∈ ( − ε , − ε ) such that O ( y ) ⊂ { ( y, z ) | y = c } . Without loss of generality, we may assume c = 0. Then we get a contradiction since d ( X t ( y, z ) , x ) ≥ ε for all t ∈ R .Suppose that λ = ib for some nonzero b ∈ R . By the same techniques as above, we can constructa vector field Z which is C -close to Y and in a neighbourhood of σ , the differential equationassociated to Z is given by(2) ˙ x = Ax = " C B yz , where C = " cos ( b ) sin ( b )-sin ( b ) cos ( b ) . By considering the coordinates obtained from (2) in the ε neighbourhood of σ , we can see that every point x = ( y , y ,
0) is periodic. Since the intersectionsof cylinders formulated by { ( y , y , z ) | y + y = c, c ∈ R } and B ε ( σ ) are invariant, we can derivea contradiction by using the same techniques as above. Case
2: Suppose there is Y ∈ U such that Λ Y contains a non-hyperbolic periodic orbit γ . Let p ∈ γ , and denote the period of γ by π ( p ). Then the linear Poincar´e map Ψ π ( p ) : N p → N p has aneigenvalue of modulus 1. Hence we can find a linear map P : N p → N p arbitrarily close to Ψ π ( p ) that has an eigenvalue λ of modulus 1, the multiplicity of λ is 1, and λ is a root of unity (i.e., λ n = 1 for some n ∈ N ). Using Lemma 2.3, we may assume that Ψ π ( p ) = P . By changing thecoordinates in N P , if necessary, we may assume that(3) Ψ π ( p ) = " C B and C w = λw for some ( w, ∈ N p , where C is a 1 × × r > N r ⊂ U and the Poincar´e map f p,π ( p ) : ˆ N x,r → ˆ N p, is well defined. Since f p,π ( p ) is a C map,using the same techniques as in Case 1, we can find a map g p,π ( p ) : ˆ N x,r → ˆ N p,
10 MOHAMMAD REZA BAGHERZAD AND KEONHEE LEE which is arbitrarily C -close to f p,π ( p ) and exp − p ◦ g ◦ exp p | N x, r = Ψ π ( p ) | N x, r . By Proposition2.1, we may assume that f p,π ( p ) = g .By the tubular flow theorem for closed orbits in Section 2.5.2 in [1], we can find constants s, δ , l > x ∈ ˆ N p ∩ B s ( p ), y ∈ M and ε ∈ (0 , δ ) then d ( x, y ) < ε implies y = Y t ′ ( y ′ ) , for some y ′ ∈ ˆ N p and | t ′ | , d ( y ′ , x ) < lε. Let δ > ε < min { δ , s l } obtained from the shadowing property of Λ Y . Let v be a scalar multiplication of w which obtainedin equation (3) satisfying k v k = s . To make a δ -pseudo orbit, fix N > x i = p i ≤ ,exp p ( iN C i v,
0) 0 ≤ i ≤ N − ,exp p ( C N v, i ≥ N, and t i = τ ( x i ) , where τ is the first return map. Then we get d ( X t i ( x i ) , x i +1 ) = k iN C i +1 v − i + 1 N C i +1 v k = k λ i +1 N v k = k N v k < δ, for sufficient large N . Since C n v = λ n v = v , we see that each { x i } is periodic and O ( x i ) ⊂ U forall i ∈ Z . Consequently, we get x i ∈ Λ Y for all i ∈ Z . Since Λ Y satisfies the shadowing property,there are x ∈ M and h ∈ Rep such that O ( x ) ⊂ B s ( γ ) and d ( X h ( t ) ( x ) , x ∗ t ) < ε for all t ∈ R . Hence there are t , t ∈ R such that d ( Y t ( x ) , p ) < ε and d ( Y t ( x ) , x N ) < ε. By the above fact, we can choose t ′ and t ′ in R such that(4) d ( Y t ′ ( x ) , p ) < lε < s , d ( Y t ′ ( x ) , x N ) < lε < s , and Y t ′ ( x ) , Y t ′ ( x ) ∈ ˆ N p . Suppose that Y t ′ ( x ) = exp p ( v , w ) and Y t ′ ( x ) = exp p ( v , w ) . Then (4) implies that(5) k ( v , w ) k < s k ( v , w ) − ( C N v, k < s . Moreover, we see that ( v , w ) and ( v , w ) belongs to the same orbit of Ψ π ( p ) . Hence, withoutloss of generality, we may assume that there is j ∈ N such that v = C j v . Consequently, we get k v k = k C j v k = k v k . But (5) implies that k v k < s k v − C N v k < s . On the other hand, we have k C n v k = k v k = s , and so the contradiction completes the proof ofour proposition. (cid:3) Recently, Gan et al. [4] showed that if M is robustly shadowable for X ∈ X ( M ), then thereis no singularity σ ∈ Sing ( X ) exhibiting homoclinic connection. Here the homoclinic connection OBUSTLY SHADOWABLE CHAIN TRANSTIVE SETS AND HYPERBOLICITY 11 is the closure of a orbit of a regular point which is contained in both the stable and the unstablemanifolds of σ . Proposition 3.3.
Let Λ be a nontrivial chain transitive set of X t . If Λ is robustly shadowablethen it does not contain a singularity of X .Proof. Let U be an isolating block of Λ, and suppose U contains a singularity σ . By Proposition3.2, it must be hyperbolic.First we show that there is z ∈ W s ( σ ) ∩ W u ( σ ) such thatΓ := { σ } ∪ O ( z ) ⊂ Λ . Choose x ∈ Λ \ { σ } , and let η > W sη ( σ ) andthe local unstable manifold W uη ( σ ) of σ are embedded submanifolds of M . Take δ > [ y ∈ Λ B δ ( y ) ⊂ U . Let δ > ε = min { η , δ , d ( σ,x )2 } obtained fromthe shadowability of Λ. Since Λ is transitive, there are two finite δ -pseudo orbits in Λ { ( x ′ i , t ′ i ) | t ′ i ≥ , i = 1 , . . . , n } and { ( x ′′ i , t ′′ i ) | t ′′ i ≥ , i = 1 , . . . , m } such that x ′ = x ′′ m = σ , and x ′ n = x ′′ = x . Define an infinite δ -pseudo orbit in Λ as follows:( x i , t i ) = ( σ, i < , ( x ′ i , t ′ i ) 0 ≤ i < n, ( x ′′ i − n , t ′′ i − n ) n ≤ i < n + m, ( σ, i ≥ n + m. Then there are z ∈ M and h ∈ Rep such that d ( X h ( t ) ( z ) , x ∗ t ) < ε for all t ∈ R . This implies that there is T > d ( X t ( z ) , σ ) < η for all t > T and t < − T. By our construction, we see that z ∈ W s ( σ ) ∩ W u ( σ ) . Hence we havesup t ∈ R ,y ∈ Λ ( d ( X t ( z ) , y )) < ε < δ . This implies that O ( z ) ⊂ U and z ∈ Λ.Second we show that there is x ∈ W s ( σ ) ∩ W u ( σ ) such thatΓ ′ := { σ } ∪ O ( x ) ⊂ Λ and x / ∈ O ( z ) . Let ε > S x ∈ Γ B ε ( x ) ⊂ U, and let δ be a corresponding constant for ε obtained fromthe shadowing property of Λ. Since z ∈ W s ( σ ) ∩ W u ( σ ), there is m ∈ N such that d ( X n ( z ) , σ ) < δ d ( X − n ( z ) , σ ) < δ for all n ≥ m . Consider a δ -pseudo orbit in Λ( x i , t i ) = ( ( X i ( z ) ,
1) for i ≤ m, ( X i − m ( z ) ,
1) for i > m.
Then there are x ∈ M and h ∈ Rep such that d ( X h ( t ) ( x ) , x ∗ t ) < ε. We also easily check that x ∈ W s ( σ ) ∩ W u ( σ ) , O ( x ) ⊂ U and x O(z) . This implies that dimE s = dimW s ( σ ) = k ≥
2. By applying Lemma 3.5 in [4], we can assumethat there is a dominated splitting E s = E c ⊕ E ss such that dimE c = 1. We also perturb Γ andΓ ′ to make sure that (Γ ∪ Γ ′ ) ∩ W ss ( σ ) = { σ } , where W ss ( σ ) be the strong stable submanifold of M tangent to E ss . Furthermore we may perturbthat in a neighbourhood V of σ , the dynamic induced by X is expressed by the following differentialequation(6) ˙ x c ˙ x ss ˙ x u = A x c x ss x u = B B
00 0 C x c x ss x u , where B , B and C are preserving the splitting E c ⊕ E ss ⊕ E u . Here the eigenvalues of B and C have negative and positive real part, repectively, and the spectrum of B = { λ } . For more detailson these perturbations, see [4]. Since the dynamic on V is induced by the differential equation˙ x = Ax , we can express every point y in V by y = ( y c , y ss , y u ) based on the coordinates obtainedfrom E c ⊕ E ss ⊕ E u . Then we get(7) X t ( y ) = ( X t ( y c ) , X t ( y ss ) , X t ( y u )) = ( e B t y c , e B t y ss , e Ct y u ) . Next we are going to get some useful properties for Γ ∪ Γ ′ that helps us to complete the proof.Choose x ∈ Γ ′ and z , z ∈ Γ satisfying x, z ∈ W s ( σ ) , z ∈ W u ( σ ) , O + ( x ) ∪ O + ( z ) ⊂ V, and O − ( z ) ⊂ V. Fix r > y ∈ ˆ N x,r . Assume that there exists t > X t ( y ) ∈ ˆ N z ,r and X [0 ,t ] ⊂ V. For any y ∈ ˆ N x,r , denote by τ ( y ) the minimum of t with the above property (if such a t exists).Define a map P r by P r : Dom( P r ) ⊂ ˆ N x,r → ˆ N z ,r , P r ( y ) = X τ ( y ) ( y ) . We show that there is r > P r ) = ∅ for any r ∈ (0 , r ]. Fix r > [ t ≥ { B r ( X t ( x )) ∪ B r ( X − t ( z )) } ⊂ V. OBUSTLY SHADOWABLE CHAIN TRANSTIVE SETS AND HYPERBOLICITY 13
Let r ∈ (0 , r ], and take r ∈ (0 , r ] such that d ( X t ( y ) , x ) < r for y ∈ M and t >
0. Then there is t ′ ∈ [0 , t ] such that X t ′ ( y ) ∈ ˆ N x,r and X [ t ′ ,t ] ( y ) ⊂ B r ( x ) . If d ( X t ( y ) , z ) < r , then there is t ′ ∈ [ t, ∞ ) such that X t ′ ( y ) ∈ ˆ N z ,r and X [ t,t ′ ] ( y ) ⊂ B r ( z ) . Let δ > r obtained from the shadowing property of Λ. Let m ∈ N be such that d ( X m ( x ) , σ ) < δ d ( X − m ( z ) , σ ) < δ . Consider the following δ -pseudo orbit(8) ( x i , t i ) = ( ( x, i ≤ m, ( X − m ( z ) , i ≥ m + 1 . Then there are y ∈ M and h ∈ Rep such that d ( X h ( t ) ( y ) , x ∗ t ) < r . This implies that there are 0 ≤ t < t < t < ∞ such that d ( X [ h ( t ) ,h ( t )) ( y ) , X [0 ,m ] ( x )) < r d ( X [ h ( t ) ,h ( t )] ( y ) , X [ − m, ( z )) < r . Hence we have X [ h ( t ) ,h ( t )] ( y ) ⊂ V . Let t ′ , t ′ be constants corresponding to h ( t ) , h ( t ), respec-tively, obtained from the same way we get r . Then we get X t ′ ( y ) ∈ ˆ N x,r , X t ′ ( y ) ∈ ˆ N z ,r , and X [ t ′ ,t ′ ] ( y ) ⊂ exp σ ( T σ M (1)).Consequently, we have y ∈ Dom( P r ) and so Dom( P r ) = ∅ .Consider the following set L = { ( y c , y ss , y u ) | y ss = 0 } ⊂ ˆ N z ,r . We will show that for any ε > r > P r ( ˆ N x,r ) ⊂ C ε := { ( u, w ) ∈ N z | u ∈ L, w ∈ L ⊥ , k w k≤ ε k u k} . Let y ∈ Dom( P r ) ∩ ˆ N x,r . Since P r ( y ) ∈ ˆ N z ,r , we have0 < k z u k − r ≤ k P r ( y ) u k for sufficiently samll r >
0. Using (7), we get k P r ( y ) u k≤ e Cτ ( y ) k y u k . Hence τ ( y ) → + ∞ as k y u k→
0. On the other hand, we have(10) k P r ( y ) ss kk P r ( y ) c k ≤ e k B k τ ( y ) k y ss kk e B τ ( y ) y c k . Since x c = 0, we get y c y → x . In addition, because E c ⊕ E ss is a dominated splitting, theright side of (10) tends zero as τ ( y ) → + ∞ , and (9) is proved.Next we perturb X so that if z = X t ′ ( z ) then Ψ t ( L ) ∩ ∆ s = ∅ , where ∆ s = N z ∩ T z W s ( σ ).If Ψ t ′ ( L ) ∆ s we have nothing to prove. Otherwise, let u ∈ N z be such that u ∆ s . Fix α >
0, and denote u α = αu + (1 − α ) v , where Ψ t ′ ( L ) = Span { v } . Then there is a linear map H α : N z → N z such that H α ( v ) = u α and k H α k→ α → . Define a map Ψ ′ : N z → N z , Ψ ′ ( v ) = H α ◦ Ψ t ′ ( v ) . Choose α > t ′ with Ψ ′ . Then we getΨ ′ ( L ) ∩ ∆ s = Span { u α } ∩ ∆ s = { } . Since the Poincar´e map f z ,t : ˆ N z ,r → ˆ N z ,r is continuous, there is ε > f z ,t ( C ε ) ∩ W s ( σ ) ∩ ˆ N z ,r = { z } , where C ε is defined in (9). Let r > r satisfies (9) for ε , and let δ > ε ′ = min { r, ε, η } obtained from the shadowing property of Λ. Considerthe δ -pseudo orbit (8) we constructed in the above. Then there are y ∈ M and h ∈ Rep such that d ( X h ( t ) ( y ) , x ∗ t ) < ε ′ . This implies that there are constants 0 < t < t < t < t satisfying d ( X h ( t ) ( y ) , x ) < ε ′ , d ( X [ h ( t ) ,h ( t )) ( y ) , X [0 ,m ) ( x )) < ε ′ ,d ( X [ h ( t ) ,h ( t )) ( y ) , X [ − m, ( z )) < ε ′ , d ( X [ h ( t ) ,h ( t )] ( y ) , X [0 ,t ′ ] ( z )) < ε ′ , and d ( X [ h ( t ) , ∞ ) ( y ) , X [0 , ∞ ) ( z )) < ε ′ . Without loss of generality, we may assume that X h ( t ) ( y ) ∈ ˆ N x,r , X t ( y ) ∈ ˆ N z ,r , and X h ( t ) ( y ) ∈ ˆ N z , r .This means that X t ( y ) = P r ( X h ( t ) ( y )), and so we have X h ( t ) ( y ) ∈ C ε . Consequently, we get X h ( t ) W s ( σ ) ∩ ˆ N z ,r . This is a contradiction to the fact that d ( X [ h ( t ) , ∞ ) ( y ) , X [0 , ∞ ) ( z )) < η, and so completes the proof. (cid:3) Proposition 3.4.
Let Λ be a chain transitive set. If Λ is robustly shadowable, then it admits ahomogeneous dominated splitting for Ψ t .Proof. If Λ is a periodic orbit, then it admits a dominated splitting for Ψ t by Proposition 3.2.Hence we suppose Λ is not a periodic orbit, and take a point x ∈ Λ be such that ω ( x ) = Λ. By OBUSTLY SHADOWABLE CHAIN TRANSTIVE SETS AND HYPERBOLICITY 15 applying the Pugh’s closing lemma (see [11]), we can select a sequence { Y n } n ∈ N ⊂ U convergingto X such that each Y n has a periodic point p n converging to x ; and for each t >
0, the sequence φ n : [0 , t ] → M given by φ n ( s ) = Y ns ( p n ) converges to φ : [0 , t ] → M, φ ( s ) = X s ( x ). Note thathere O ( p n ) is hyperbolic for Y nt for every n . Moreover we can see that the period of p n tends to ∞ as n → ∞ . By applying Proposition 2.6, we can take l > Y n over O ( p n ) admits an l -dominated splitting. By taking a subsequence, if necessary, we mayassume that there is k ∈ N such that ind ( p n ) = k for all n ∈ N .Let { x k } be a sequence in Λ converging to x , and let E ( x k ) be an m -dimensional subspace of T x k M . We say that E ( x k ) converges to E ( x ) if, for each k, there is a basis { e k , . . . , e mk } of E ( x k )and a basis { e , . . . , e m } of E ( x ) such that e ik → e i for each i = 1 , · · · , m .Put lim n →∞ E sn ( p n ) = ∆ s ( x ) and lim n →∞ E un ( p n ) = ∆ u ( x ) . For each t >
0, we denote bylim n →∞ E sn ( Y nt ( p n )) = ∆ sn ( X t ( x )) and lim n →∞ E un ( Y nt ( p n )) = ∆ un ( X t ( x )) , where T Y nt ( p n ) M = E sn ( Y nt ( p n )) ⊕ E un ( Y nt ( p n )). Then we have∆ s ( X t ( x )) = lim n →∞ ∆ sn ( Y nt ( p n )) = lim n →∞ Ψ nt (∆ sn ( p n )) = Ψ t (∆ s ( X t ( x ))) , and∆ u ( X t ( x )) = lim n →∞ ∆ un ( Y nt ( p n )) = lim n →∞ Ψ nt (∆ un ( p n )) = Ψ t (∆ u ( X t ( x ))) , where Ψ nt is the linear Poincar´e flow for Y n . This means that the splitting ∆ s ( x ) ⊕ ∆ u ( x ) is Ψ t invariant, and we have N x = ∆ s ( x ) ⊕ ∆ u ( x ). If t is sufficiently large, then we can see that k Ψ t | ∆ s ( x ) k · k Ψ − t | ∆ u ( X t ( x )) k = lim n →∞ k Ψ nt | ∆ sn ( x ) k · k Ψ n − t | ∆ un ( X t ( x )) k≤ . This means that the orbit O ( x ) admits a dominated splitting for Ψ t , and so Λ = O ( x ) also has adominated splitting for Ψ t , (cid:3) From dominated splitting to hyperbolicity
Lemma 4.1.
If a chain transitive set Λ of X t is robustly shadowable, then it admits a hyperbolicperiodic orbit.Proof. Let ∆ s ⊕ ∆ u be the l -dominated splitting of ( T Λ M, Ψ t | N Λ ) obtained in Proposition 3.4. Byusing lemma 3 . dim (∆ s ) ≤ dim M −
2. Denote by α = min {k Ψ t | N z k| z ∈ Λ , t ∈ [ − , } . For any ε >
0, choose ε ′ ∈ (0 , α ), δ ′ >
0, and Y ∈ U having a periodic point p such that(11) log( s + ε ′ ) ≤ log( s ) + ε, ∀ s ∈ [ α , ∞ ) , log( s − ε ′ ) ≥ log( s ) − ε, ∀ s ∈ [ α , ∞ ) , | k Ψ t | ∆ s ( u ) ( z ) k − k Ψ ′ t | ∆ s ( u ) Y ( y ) k | < ε ′ , ∀ t ∈ [ − , , d ( z, y ) < δ ′ , z ∈ Λ , y ∈ O ( p ) ,d H ( O ( p ) , Λ) < δ ′ , where Ψ and Ψ ′ are linear Poincar´e flows of X and Y , respectively. Since p is a hyperbolic periodicpoint of Y t , there are C > λ ∈ (0 ,
1) such that k Ψ ′ t | ∆ sY ( y ) k≤ Cλ t and k Ψ ′− t | ∆ uY ( y ) k≤ Cλ t for all t ≥ y ∈ O ( p ). Denote by C ′ = max { C, C − } , and let δ be a constant as in Proposition2.7 for the triple ( ε, T, η ) = ( ε, , − ( log ( c ′ ) + ε )) . Because x is a nonwandering point, there is t ′ > d ( X t ′ ( x ) , x ) < δ . Let T , ..., T m ∈ R be such that0 = T < T < T < .... < T m = t ′ is a partition for [0 , t ′ ] with T i +1 − T i ∈ [1 , p , ..., p m ∈ O ( p ) be such that d ( p j , X T j ( x )) < δ ′ for j = 0 , ..., m . We show that X [0 ,t ′ ] ( x ) is an ( ε, T, η )-quasi hyperbolic arc. By using (11) we have T k k X j =1 log k Ψ T j − T j − | ∆ s ( X Tj − ( x )) k≤ T k k X j =1 log( k Ψ ′ T j − T j − | ∆ sY ( p j ) k + ε ′ ) ≤ T k k X j =1 (log( k Ψ ′ T j − T j − | ∆ sY ( p j ) k ) + ε ) ≤ T k k X j =1 log( C ′ λ T j − T j − ) + kT k ε ≤ T k k X j =1 log( C ′ T j − T j − λ T j − T j − ) + kT k ε ≤ log( C ′ ) + ε = − η. For the first and second inequality, we used the properties in (11); for the third inequality, we usedthe hyperbolicty of O ( p ); and for the fourth and fifth inequality, we used the property T j − T j − ≥ m (Ψ ′ t | ∆ uY ( y ) ) = 1 k Ψ ′− t | ∆ uY ( Y t ( y )) k ≥ C − λ − t ≥ C ′− λ − t . Hence we get T m − T k − m X j = k log m (cid:16) Ψ T j − T j − | ∆ u ( X Tj − ( x )) (cid:17) = T m − T k − k X j =1 log( 1 k Ψ T j − − T j | ∆ u ( X Tj ( x )) k ) ≥ T m − T k − k X j =1 log( 1 k Ψ ′ T j − − T j | ∆ uY ( p j ) k − ε ′ ) ≥ T m − T k − k X j =1 (cid:16) log( 1 k Ψ ′ T j − − T j | ∆ uY ( p j ) k )) − ε (cid:17) ≥ T m − T k − m X j = k (cid:16) ( T j − − T j )(log( C ′ ) + log( λ )) (cid:17) − m − k + 1 T m − T k − ε ≥ − log( C ′ ) − ε − log( λ ) ≥ − (log( C ′ ) + ε ) = η. OBUSTLY SHADOWABLE CHAIN TRANSTIVE SETS AND HYPERBOLICITY 17
Similarly we obtainlog k Ψ T k − T k − | ∆ s ( X Tk − ( x )) k − log m (cid:16) Ψ T k − T k − | ∆ u ( X Tk − ( x ) ) (cid:17) ≤ log( C ′ ) + ( T k − T k − )log( λ ) + ε − (cid:0) − log( C ′ ) + ( − T k + T k − )log( λ ) − ε (cid:1) = 2log ( C ′ ) + 2 ε + 2( T k − T k − )log( λ ) ≤ C ′ ) + 2 ε = − η, for all k ∈ { , ..., m } . Consequently we can see that Λ contains a hyperbolic periodic orbit byProposition 2.7. (cid:3) End of proof of main theorem.
Let Λ be a chain transitive set, and suppose it is robustly shadow-able. Then Λ contains a hyperbolic periodic orbit, say γ , by Lemma 4.1. Since Λ is transitive,we see that Λ ⊂ C X ( γ ) and also Λ ⊂ H X ( γ ). Since Λ is compact and the periodic points aredense in Λ, we may assume that for any T > p in Λ whose period isbigger than T . Then by using the results and techniques in Section 5 of [7], we can show that thedominated splitting N Λ = ∆ s ⊕ ∆ u is a hyperbolic spliting for Ψ t . Consequently we can see thatΛ is hyperbolic for X t by applying Proposition 2.5.The converse is clear by the robust property of hyperbolic sets and the shadowability of thehyperbolic sets, and so completes the proof of our main theorem. (cid:3) Acknowledgement.
The second author was supported by the NRF grant funded by the Koreagovernment (MSIP) (No. NRF-2015R1A2A2A01002437).
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Behin Andishan Aayyar research group, Sarparast st, Taleghani st, Tehran 1616893131, Iran.
E-mail address : [email protected] Keonhee Lee, Department of Mathematics, Chungnam National University, Daejeon 305-764, Korea.
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