Orbital shadowing and stability for vector fields

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Orbital shadowing and stability for vector fields Ming Li School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, People’s Republic of China Received 10 May 2018; revised 25 October 2019; accepted 3 January 2020

Abstract The orbital shadowing property means that there exists a true orbit close to the approximate orbit in the sense of Hausdroff distance. In this paper, we show that in any dimension the C 1 interior of the set of vector fields having the orbital shadowing property is contained in the set of -stable vector fields. We also discuss the structural stability of vector fields having the orbital shadowing property. © 2020 Elsevier Inc. All rights reserved. MSC: 37C50; 37D20 Keywords: Vector field; Orbital shadowing; -stability; Structural stability

1. Introduction The shadowing theory is an important part of the global theory of dynamical systems. It is closely related to the hyperbolic theory. A diffeomorphism has the shadowing property in a neighborhood of a hyperbolic set [1,2], and a structurally stable diffeomorphism has the shadowing property globally [10,17,19]. Various types of shadowing property are studied. Besides the usual shaowing property, the orbital shadowing property is also widely used. It considers the shadowing of true and approximate orbits as two wholes instead of each corresponding points. Such shadowing property is closely related to Zeeman’s tolerance stability conjecture: the C 1 tolerance stability conjecture is equivalent to the C 1 genericity of the orbital shadowing property [3]. E-mail address: [email protected]. https://doi.org/10.1016/j.jde.2020.01.026 0022-0396/© 2020 Elsevier Inc. All rights reserved.

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Differences in the hyperbolicity, neither the shadowing property nor the orbital shadowing property is C 1 robustly. It led people to consider the C 1 interior of shadowing properties from the point view of hyperbolicity. It is well known that the C 1 interior of both the set of diffeomorphisms having the shadowing property and the set of diffeomorphisms having the orbital shadowing property coincide with the set of structurally stable diffeomorphisms [18,15]. One may find in [14] more recent results in this subject. Analogous questions were asked for vector fields. As same as diffeomorphisms, vector fields have the shadowing property in a neighborhood of a hyperbolic set [11,12], and structurally stable vector fields have the shadowing property globally [12,13]. For vector fields without singularities, the C 1 interior of both the set of vector fields having the shadowing property and the set of vector fields having the orbital shadowing property coincide with the set of structurally stable vector fields [9,16]. The situation changes completely when we consider vector fields with singularities. One of the main differences between diffeomorphisms and vector fields is the possibility of accumulation of regular recurrent points to a singularity in the latter case. Such phenomenon may be robustly. An invariant set containing such singularities can not be hyperbolic (for example, the Lorenz attractor). Non structural stable vector fields which are contained in the C 1 interior of the set of vector fields having the shadowing property were constructed in [16]. In fact, the C 1 interior of the set of vector fields having the shadowing property consists of some -stable vector fields [5], but doesn’t only consist of structural stable vector fields. For the orbital shadowing property for vector fields containing singularities, a natural question is, whether the same result holds as it for the usual shadowing property. On 3-dimensional manifolds, it was proved in [4] that the C 1 interior of the set of vector fields having orbital shadowing property is a subset of the set of -stable vector fields. In this paper, we will prove the same result as in dimension 3 for any dimension. Some arguments in [4] for dealing with singularities having complex eigenvalues are dependent on the 3-dimensional hypothesis for controlling the dimensions of stable/unstable manifolds of periodic orbits (see [4, Remark 5.7]). In this paper we will use a new method to create so called double homoclinic connections (for getting contradiction) under C 1 small perturbations for codimension 1 singularities. This method works even the singularity has complex eigenvalues. And it works in any dimension. For the higher co-dimensional case, we will create strong connections of singularities for getting contradiction. We assume that M is a closed smooth Riemannian manifold. Denote by X 1 (M) the set of C 1 vector fields on M endowed with the C 1 topology. For any X ∈ X 1 (M), X generates a C 1 flow φt = φX,t : M → M, t ∈ R. Denote by Orb(x) = OrbX (x) = φ(−∞,+∞) (x) the orbit of x for X. A map g : R → M (not necessarily continuous) is called a d-pseudo orbit of X, if dist(g(t + τ ), φτ (g(t))) < d for all t ∈ R and τ ∈ [0, 1]. We say that a vector field X ∈ X 1 (M) has the orbital shadowing property if for any ε > 0 there is d > 0 such that, for any d-pseudo orbit g of X there exists x ∈ M satisfying

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distH (g(R), Orb(x)) < ε, where distH (·, ·) is the Hausdorff distance and (·) is the closure. Denote by OrbSh(M) the set of vector fields having the orbital shadowing property. We say that a vector field has the C 1 robustly orbital shadowing property if it is contained in Int1 (OrbSh(M)), the C 1 interior of OrbSh(M). The following Theorem is the main result of this paper. Theorem 1.1. Let M be a compact manifold without boundary, then every C 1 vector field on M having the C 1 robustly orbital shadowing property is -stable. The paper is organised as follows. In Section 2 we recall some useful results and simplify Theorem 1.1 to two propositions. Those propositions are proved in Section 3 and 4. In Section 5, we discuss the structure stability of vector fields having the C 1 robustly orbital shadowing property. 2. Preliminaries In this section we will recall some results and simplify the main theorem. 2.1. Singularities of vector fields Singularities are zero points of vector fields. Denote by Sing(X) = {x ∈ M : X(x) = 0} the set of singularities of X. We say a point x ∈ M is regular if it is not a singularity. Denote by Per(X) the set of (regular) periodic points of X. Let X ∈ X 1 (M) and σ be a hyperbolic singularity of X. Denote by Re(λs ) ≤ · · · ≤ Re(λ2 ) ≤ Re(λ1 ) < 0 < Re(γ1 ) ≤ Re(γ2 ) ≤ · · · ≤ Re(γu ) the eigenvalues of DX(σ ), where s + u = dim M. The sum SV(σ ) = Re(λ1 ) + Re(γ1 ) is called the saddle value of σ . Denote by W s (σ ) and W u (σ ) the stable and unstable manifold of σ , which are tangent to the stable and unstable subspace Eσs and Eσu at σ respectively. Recall that a homoclinic connection of σ is the closure of a regular orbit which is contained in both W s (σ ) and W u (σ ). 2.2. Star fields We say that a vector field X ∈ X 1 (M) is a star vector field, denote by X ∈ X ∗ (M), if there is a C 1 neighborhood U of X such that for any Y ∈ U , all periodic orbits and singularities of Y are hyperbolic. A point x ∈ M is called a preperiodic point of X, if for any C 1 neighborhood U of X and any neighborhood U of x, there exists Y ∈ U such that Y has a regular periodic point contained in U . Denote by Per∗ (X) the set of preperiodic points of X. We will use the following results for star vector fields.

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Theorem 2.1. [6] Let X ∈ X ∗ (M) be a star vector field. If Sing(X) ∩ Per∗ (X) = ∅, then X is -stable. Lemma 2.2. [22] Let σ ∈ Sing(X) be a singularity of a star vector field X ∈ X ∗ (X) exhibiting a homoclinic connection. If SV(σ ) ≥ 0, then Eσs splits into a dominated splitting Eσs = Eσss ⊕ Eσcs , where dim Eσcs = 1. According to the Invariant Manifold Theorem [8], under the assumption of Lemma 2.2, there exists the strong stable manifold W ss (σ ) of σ which is tangent to Eσss at σ . We say that a homoclinic connection is a strong connection if it is contained in the intersection of the unstable manifold and the strong stable manifold of the singularity. Lemma 2.3. [20] There is no strong connection for star vector fields. 2.3. The C 1 connecting lemma Denote by Orb+ (x) = φ[0,+∞) (x), Orb− (x) = φ(−∞,0] (x) the positive, negative orbit of x respectively. Let B(x, δ) be the open ball centered at x of radius δ, and write B(A, δ) = ∪x∈A B(x, δ) the δ-neighborhood of A ⊂ M. We will use several times the following version of the C 1 connecting lemma, which extends Hayashi’s C 1 connecting lemma [7]. Lemma 2.4. [21] Let X ∈ X 1 (M) and z ∈ M be neither periodic nor singular of X. For any C 1 neighborhood U of X, there exist three numbers ρ > 1, L > 1 and δ0 > 0 such that for any 0 < δ < δ0 and any two points x, y outside the tube  = B(φ[−L,0] (z), δ), if both of Orb+ (x) and Orb− (y) intersect B(z, δ/ρ), then there exists Y ∈ U with Y = X outside  such that y ∈ Orb+ Y (x). The following lemma is proved in [5] by using the (uniform version) C 1 connecting lemma. Lemma 2.5. [5] Let X ∈ X 1 (M) and σ ∈ Sing(X) be a hyperbolic singularity of X which is preperiodic. Then for any C 1 neighborhood U ⊂ X 1 (M) of X, there exists Y ∈ U such that σY exhibiting a homoclinic connection, where σY ∈ Sing(Y ) is the continuation of σ . 2.4. Simplification of the main theorem Firstly, we have that the C 1 robustly orbital shadowing property implies the star condition. Lemma 2.6. [16] Int1 (OrbSh(M)) ⊂ X ∗ (M). The following theorem is the main step in our proof. Theorem 2.7. Every vector field having the C 1 robustly orbital shadowing property exhibits no homoclinic connection of singularity.

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It is easy to see that Theorem 1.1 follows immediately from Theorem 2.1, 2.7 and Lemma 2.5, 2.6. We may simplify Theorem 2.7 to the following two propositions, which will be proved in the next two sections. Proposition 2.8. Let X ∈ Int1 (OrbSh(M)). If σ ∈ Sing(X) satisfying SV(σ ) > 0 and dim Eσs = 1, then σ has no homoclinic connection. Proposition 2.9. Let X ∈ Int1 (OrbSh(M)). If σ ∈ Sing(X) satisfying SV(σ ) > 0 and dim Eσs > 1, then σ has no homoclinic connection. Proof of Theorem 2.7. Suppose on the contrary that there exists X having the C 1 robustly orbital shadowing property and σ ∈ Sing(X) exhibiting a homoclinic connection. Up to an arbitrary C 1 small perturbation, we may get a vector field Y such that SV(σY ) = 0, and σY still exhibits a homoclinic connection, where σY is the continuation of σ . Since Int1 (OrbSh(M)) is open, we may assume that Y ∈ Int1 (OrbSh(M)). Note that −Y ∈ Int1 (OrbSh(M)). Consider −Y if necessary, we may assume that SV(σY ) > 0. Then the existence of homoclinic connection is contrary to Proposition 2.8 or 2.9. 2 3. Proof of Proposition 2.8: dim Eσs = 1 We will prove the proposition by contradiction. Suppose on the contrary that there exist X ∈ Int1 (OrbSh(M)) and σ ∈ Sing(X) such that SV(σ ) > 0, dim Eσs = 1, and σ exhibits a homoclinic connection . Let λ1 < 0 < Re(γ1 ) ≤ Re(γ2 ) ≤ · · · ≤ Re(γu ), be the eigenvalues of DX(σ ), where u = dim M − 1. Up to an arbitrarily C 1 small perturbation, we may assume that X is linear in a small neighborhood of σ , and it still exhibits a homoclinic connection (see [16] for more details of the perturbations). Changing the Riemannian metric if necessary, we assume that Eσs is orthogonal to Eσu . s (σ ) and W u (σ ) the local stable and unstable manifold of σ . For simplifying Denote by Wloc loc s (σ )/W u (σ ) in the linear neighthe notations, we ignore the difference between Eσs /Eσu and Wloc loc borhood, and denote the perturbed vector field by X again. That is, we assume that X satisfies Condition 3.1. • X is linear in B(σ, 2r) for some r > 0; • ∩ B(σ, 2r) ⊂ Eσs ∪ Eσu ; • Eσs ⊥ Eσu . According to the splitting Tσ M = Eσs ⊕ Eσu ,

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Fig. 1. The homoclinic connection.

we introduce an orthogonal coordinate system x = (x s , x u ) in B(σ, 2r), where x s and x u are the coordinates of x in Eσs and Eσu respectively. For any x ∈ B(σ, 2r) and t > 0 such that φ[0,t] (x) ⊂ B(σ, 2r) we have |φt (x)s | = |x s | exp(λ1 t) and |φt (x)u | ≥ |x u | exp(Re(γ1 )t),

(3.1)

where (φt (x)s , φt (x)u ) = φt (x). Take two points Os = (Oss , 0) ∈ ∩ Eσs and Ou = (0, Ouu ) ∈ ∩ Eσu such that dist(Os , σ ) = dist(Ou , σ ) = r. Let s , u ⊂ B(σ, 2r) be two cross sections which are orthogonal to Eσs and at Os and Ou respectively. Reduce u if necessary, we assume that r/2 ≤ dist(x, σ ) ≤ 3r/2, for any x ∈ u .

(3.2)

Denote by Q : u → s the Poincaré map generated by the flow and τQ (x) the minimal positive number t such that φt (x) = Q(x). Reduce u again if necessary, we assume that τQ is continuous and bounded on u . Denote by P : Dom(P ) ⊂ s → u the Poincaré map from s to u in B(σ, 2r), where Dom(P ) = {x ∈ s : there is T > 0 such that φT (x) ∈ u and φ[0,T ] (x) ⊂ B(σ, 2r)}. Write by τP (x) the minimal positive number t such that φt (x) = P (x). It is clear that τP (x) → +∞ as x → Os . Denote Lu = Eσu ∩ u = {x ∈ u : x s = 0}. Since dim Eσs = 1, it follows that Lu splits u to two connected components. One of them contains the image of P . We denote this component by u(+). See Fig. 1.

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Denote s (a) = B(Os , a) ∩ s , u (b) = B(Ou , b) ∩ u and u (b, +) = B(Ou , b) ∩ u (+). Since dim Eσs = 1, it is easy to see that u (b, +) is contained in the image of P when b is small enough. Take a0 > 0 and b0 > 0 such that Q( u (b0 )) ⊂ s (a0 ) and u (b0 , +) ⊂ P (Dom(P )). 3.1. Properties of the orbits near We firstly give some properties of the orbits near . The following lemma says that the positive orbit of points close to (but not contained in ) will be away from when t is large enough, because of the saddle value SV(σ ) is positive. The argument is similar to the 3-dimensional case [4]. We give the proof here for completeness. Lemma 3.2. There is b1 ∈ (0, b0 ) such that if x ∈ u (b1 ) and Q(x) ∈ Dom(P ), then dist(P ◦ Q(x), Ou ) ≥ 2dist(x, Ou ). Proof. By the C 1 continuity, there are ξ ∈ (0, b0 ) and K > 1 such that dist(Q(x), Os ) ≥

1 dist(x, Ou ) K

for all x ∈ u (ξ ). For any y = (Oss , y u ) ∈ Dom(P ), we obtain from (3.1) that dist(y, Os ) = dist((Oss , y u ), (Oss , 0)) = |y u |, dist(P (y), Ou ) = dist((P (y)s , P (y)u ), (0, Ouu )) ≥ |P (y)s | = |Oss | exp(λ1 τP (y)), and dist(P (y), σ ) = dist((P (y)s , P (y)u ), (0, 0)) ≥ |P (y)u | ≥ |y u | exp(Re(γ1 )τP (y)) = dist(y, Os ) exp(Re(γ1 )τP (y)). On the other hand, since P (y) ∈ u , from (3.2) we know that dist(P (y), σ ) ≤ 3r/2. Consequently, 2|Oss | dist(P (y), Ou ) |Oss | exp(λ1 τP (y)) = ≥ exp(SV(σ )τP (y)) → +∞ 3 dist(y, Os ) 3r r exp(−Re(γ1 )τP (y)) 2 as y → Os (which implies that τP (y) → +∞).

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So we may choose ζ ∈ (0, a0 ) such that dist(P (y), Ou ) ≥ 2Kdist(y, Os ) for any y ∈ s (ζ ) ∩ Dom(P ). Take b1 ∈ (0, ξ ) small enough so that Q( u (b1 )) ⊂ s (ζ ). It follows that dist(P ◦ Q(x), Ou ) ≥ 2Kdist(Q(x), Os ) ≥ 2dist(x, Ou ) for any x ∈ u (b1 ) satisfying Q(x) ∈ Dom(P ).

2

For negative orbits, we have Lemma 3.3. There is b2 ∈ (0, b0 /2) such that if x ∈ u (2b2 , +) then there exists y ∈ u (b0 ) satisfying P ◦ Q(y) = x. Moreover, we have dist(y, Ou ) ≤ dist(x, Ou )/2. That is, (P ◦ Q)−1 (x) ∈ u (b2 ). Proof. Let b1 be given by Lemma 3.2. Since u (b0 , +) ⊂ P (Dom(P )) and x (∈ u (b0 , +)) → Ou is equivalent to P −1 (x) → Os , there is b2 ∈ (0, b1 /2) such that P −1 ( u (2b2 , +)) ⊂ Q( u (b1 )). Therefore y = (P ◦ Q)−1 (x) ∈ u (b1 ) for any x ∈ u (2b2 , +). Moreover, by Lemma 3.2 we have that dist(y, Ou ) ≤ dist(x, Ou )/2.

2

We say that a singularity has double homoclinic connections if there are two different homoclinic connections with respect to this singularity. The following lemma is proved in [4] for dimension 3. However, the argument also works for higher dimensions. Proposition 3.4. If Y ∈ Int1 (OrbSh(M)), ς ∈ Sing(Y ) satisfying SV(ς) > 0 and dim Eςs = 1, then there is no double homoclinic connections of ς . Sketch of the proof. Suppose on the contrary that ς has two different homoclinic connections 1 and 2 . Up to an arbitrarily C 1 small perturbation, we assume that Y is linear in a small neighborhood of ς , still exhibits double homoclinic connections. e (ς) ∩ i , and four cross sections i which are orthogonal to Take four points Oei ∈ Wloc e i  : 1u ∪ 2u → 1s ∪ 2s and P  : Dom(P ) ⊂ at Oe respectively, e = s, u, i = 1, 2. Denote by Q

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 has two branches

1s ∪ 2s → 1u ∪ 2u the Poincaré maps generated by the flow. Note that Q  has four branches ( 1s to 1u and 2u , 2s to 1u and 2u ). ( 1u to 1s , 2u to 2s ) and P b > 0 such By using the same method in the proof of Lemma 3.2, one can prove that there is   ∈ Dom(P ), then u ( that if x ∈ b) = (B(Ou1 ,  b) ∩ 1u ) ∪ (B(Ou2 ,  b) ∩ 2u ) and Q(x)  ◦ Q(x),  dist(P {Ou1 , Ou2 }) ≥ 2dist(x, {Ou1 , Ou2 }). u ( Take ε > 0 small enough such that B( 1 ∪ 2 , ε) ∩ ( 1u ∪ 2u ) ⊂ b). Let d be the parameter with respect to ε of the orbital shadowing property. We may construct a d-pseudo orbit whose image is 1 ∪ 2 except a small jump near ς . Therefore there is x ∈ M such that its orbit shadows u ( b). It is clear that the pseudo orbit. Without loss of generality we may assume that x ∈ 1 2 x = Ou , Ou .  ∈ Dom(P ) and then Since OrbY (x) ⊂ B( 1 ∪ 2 , ε), we have Q(x)  ◦ Q(x),  dist(P {Ou1 , Ou2 }) ≥ 2dist(x, {Ou1 , Ou2 }).  ◦ Q(x)   ◦ Q)  n (x) ∈ u ( u ( Note that P ∈ b). Performing the same procedure, we obtain (P b) n 1 2  ◦ Q)  (x), {Ou , Ou }) → +∞. This leads to a contradiction. 2 for all n > 0 and dist((P 3.2. Another branch of W s (σ ) Denote Os− = (−Oss , 0). The singularity σ splits its 1-dimensional stable manifold to two branches \ {σ } and Orb(Os− ). In this subsection we will give some properties for Orb(Os− ). Denote by α(x) the set of α-limit points of x and write Sru = {x ∈ Eσu : dist(x, σ ) = r}. Firstly, we have the following lemma. Lemma 3.5. α(Os− ) ∩ Sru = ∅. Proof. Suppose on the contrary that α(Os− ) ∩ Sru = ∅. It follows that there exists ξ ∈ (0, r/8) such that Orb− (Os− ) ∩ B(Sru , ξ ) = ∅.

(3.3)

Choose ζ ∈ (0, a0 ) small enough such that the component of Orb+ (x) in B(σ, 2r) containing x intersects B(Sru , ξ/2) for any x ∈ s (ζ ) \ {Os }. Take η ∈ (0, b1 ) such that Q( u (η)) ⊂ s (ζ ), where b1 is given by Lemma 3.2. Denote ξ = ∩ B(Sru , ξ ). Note that ξ ∩ Sru = {Ou }. Thus by the continuous dependence on initial values, there is a small neighborhood V of ξ such that the component of Orb(x) in V containing x intersects u (η) for any x ∈ V . Let ε ∈ (0, ξ/2) small enough such that B( ξ , ε) ⊂ V , and d be the parameter of the orbital shadowing property with respect to ε. Take two points p ∈ Orb+ (Os− ) and q ∈ Orb− (Ou ) close enough to σ such that  g(t) =

φt (p), φt (q),

t ≤ 0; t >0

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is a d-pseudo orbit. Roughly speaking, the image of g is ∪ Orb(Os− ) except a small jump near σ . Then there exists z ∈ M such that distH (g(R), Orb(z)) < ε.

(3.4)

Since Ou ∈ g(R), there is z0 ∈ Orb(z) ∩ B(Ou , ε) ⊂ V . It implies that Orb(z) ∩ u (η) = ∅. Without loss of generality we may assume that z ∈ u (η). It follows that Q(z) ∈ s (ζ ). Note that Q(z) = Os because of distH (Orb(Os− ), ) > ε. So the positive orbit of Q(z) intersects B(Sru , ξ/2). If there is a intersect point z1 ∈ Orb+ (z) ∩ B(Sru , ξ/2) such that z1 ∈ / V , then we have dist(z1 , ) ≥ min(ξ/2, ε) ≥ ε. Thus by (3.4) there exists y ∈ Orb− (Os− ) such that dist(y, z1 ) < ε. Hence y ∈ Orb− (Os− ) ∩ B(Sru , ξ ). It contradicts to (3.3). So all the intersect points of Orb+ (z) and B(Sru , ξ/2) are contained in V . Therefore Q(z) ∈ Dom(P ) and P ◦ Q(z) ∈ u (η). Performing the same procedure, we obtain (P ◦ Q)n (z) ∈ u (η) for all n > 0. It contradicts to Lemma 3.2. 2 Furthermore, we have Lemma 3.6. α(Os− ) ∩ Sru = {Ou }. Proof. From Lemma 3.5 we know that α(Os− ) ∩ Sru = ∅. Suppose on the contrary that there / . Then by using the C 1 connecting exists x = Ou such that x ∈ α(Os− ) ∩ Sru . Note that x ∈ lemma, we may get another homoclinic connection of σ without destroying . It contradicts to Proposition 3.4. 2 The conclusion of Lemma 3.6 shows that the negative orbit of Os− does not approximate except . So we have the following lemma.

W u (σ )

Lemma 3.7. For any β > 0, there exists β  ∈ (0, β) such that if x ∈ Orb− (Os− ) ∩ u (β  ) then Q(x) ∈ Dom(P ) and P ◦ Q(x) ∈ u (β, +). Proof. By the continuous dependence on initial values, there is a neighborhood V of Ou such that the component of Orb(x) in V containing x intersects u (β) for any x ∈ V . According to Lemma 3.6, there exists ξ > 0 such that Orb− (Os− ) ∩ (B(Sru , ξ ) \ V ) = ∅. Choose ζ ∈ (0, a0 ) such that the positive orbit of x in B(σ, 2r) intersects B(Sru , ξ ) for any x ∈ s (ζ ) \ {Os }. By the continuity, there is β  > 0 such that Q( u (β  )) ⊂ s (ζ ). Then β  meets the needs of this lemma. 2 Let b2 be given by Lemma 3.3. According to Lemma 3.7, there exists b3 ∈ (0, b2 ) such that if x ∈ Orb− (Os− ) ∩ u (b3 ) then P ◦ Q(x) ∈ u (b2 , +). Lemma 3.8. For any integer n > 0 there is x ∈ Orb− (Os− ) ∩ u (b3 , +) such that P ◦ Q(x), (P ◦ Q)2 (x), . . . , (P ◦ Q)n (x) ∈ u (b3 , +) and (P ◦ Q)n+1 (x) ∈ u (b2 , +) \ u (b3 , +). Proof. For b3 , by Lemma 3.7, there is β1 ∈ (0, b3 ) such that if x ∈ Orb− (Os− ) ∩ u (β1 ) then P ◦ Q(x) ∈ u (b3 , +). Performing the same procedure, we may get β2 , β3 , . . . , βn+1 ∈ (0, b3 ) such that if x ∈ Orb− (Os− ) ∩ u (βj ) then P ◦ Q(x) ∈ u (βj −1 , +) for j = 2, 3, . . . , n + 1.

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From Lemma 3.6 we know that Ou ∈ α(Os− ). It implies that there exists x1 ∈ Orb− (Os− ) ∩

u (βn+1 ). Thus x2 = P ◦ Q(x1 ) ∈ u (βn , +), P ◦ Q(x2 ), (P ◦ Q)2 (x2 ), . . . , (P ◦ Q)n (x2 ) ∈

u (b3 , +) and (P ◦ Q)n+1 (x2 ) ∈ u (b2 , +). If (P ◦ Q)n+1 (x2 ) ∈ u (b2 , +) \ u (b3 , +), then we finish the proof. Otherwise, we have (P ◦ Q)n+1 (x2 ) ∈ u (b3 , +), which implies that (P ◦ Q)n+2 (x2 ) ∈ u (b2 , +). Repeating this process and according to Lemma 3.2, we obtain dist((P ◦ Q)n+j (x2 ), Ou ) ≥ 2n+j dist(x2 , Ou ) for all j > 0 such that (P ◦ Q)n+j (x2 ) ∈ u (b2 ). Hence there exists m > 0 such that (P ◦ Q)n+1 (x2 ), . . . , (P ◦ Q)n+m (x2 ) ∈ u (b3 , +) and (P ◦ Q)n+m+1 (x2 ) ∈ u (b2 , +) \ u (b3 , +). Then the conclusion holds for x = (P ◦ Q)m (x2 ). 2 3.3. Stay time According to Lemma 3.3, we have that (P ◦ Q)−1 (x) ∈ u (b2 ) for any x ∈ u (2b2 , +). If (P ◦Q)−1 (x) is still contained in u (2b2 , +), then (P ◦Q)−2 (x) ∈ u (b2 ). We use the following notation to denote how long a point “stays” in u(2b2 , +). We say that a positive integer ST(x) is the stay time of x ∈ u (2b2 , +), if (P ◦ Q)−1 (x), (P ◦ Q)−2 (x), . . . , (P ◦ Q)−ST(x)+1 (x) ∈ u (2b2 , +) and (P ◦ Q)−ST(x) (x) ∈ u (2b2 ) \ u (2b2 , +). If (P ◦ Q)−n (x) ∈ u (2b2 , +) for all n > 0, then we write ST(x) = +∞. We have the following properties for the stay time. Lemma 3.9. Let x ∈ u (2b2 , +). If ST(x) = +∞, then σ ∈ α(x) and dist(y, Ou ) ≤ dist(x, Ou )/2 for any y ∈ (Orb− (x) \ {x}) ∩ u (2b2 ). Proof. Since ST(x) = +∞, we know that (P ◦ Q)−n (x) ∈ u (2b2 , +) for all n > 0. Then by Lemma 3.3, we obtain dist((P ◦ Q)−n (x), Ou ) ≤ dist(x, Ou )/2n → 0. This inequality implies the conclusion of this lemma. 2 Lemma 3.10. Let x ∈ u (2b2 , +). If ST(x) < +∞ and x ∈ / W u (σ ), then there exists a neighborhood U of x in u (2b2 , +) such that ST(y) = ST(x) for any y ∈ U .

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Proof. Denote N = ST(x). According to Lemma 3.3 and noting that x ∈ / W u (σ ), we have (P ◦ Q)−1 (x), (P ◦ Q)−2 (x), . . . , (P ◦ Q)−N+1 (x) ∈ u (2b2 , +) and (P ◦ Q)−N (x) ∈ u (2b2 ) \ ( u (2b2 , +) ∪ Lu ). Take neighborhoods U1 , U2 , . . . , UN−1 of (P ◦ Q)−1 (x), (P ◦ Q)−2 (x), . . . , (P ◦ Q)−N+1 (x) in u (2b2 , +), and UN of (P ◦ Q)−N (x) in u (2b2 ) \ ( u (2b2 , +) ∪ Lu ). By the continuous dependence on initial values, there exists a neighborhood U of x in u (2b2 , +) such that (P ◦ Q)−n (y) ∈ Un for all y ∈ U and 1 ≤ n ≤ N . It implies that ST(y) = ST(x). 2 For the stay time of the points in Orb− (Os− ), we have the following. Lemma 3.11. There is a sequence of xn ∈ Orb− (Os− ) ∩ ( u (b2 , +) \ u (b3 , +)) (not necessarily different from each other) such that ST(xn ) → +∞ as n → +∞. Proof. It is enough to prove that for any integer k > 0, there is xk ∈ Orb− (Os− ) ∩ ( u (b2 , +) \

u (b3 , +)) such that ST(xk ) ≥ k. By using Lemma 3.8, it is easy to take such xk . 2 3.4. Ending of the proof of Proposition 2.8 From Lemma 3.8, we know that Orb− (Os− ) ∩ ( u (b2 , +) \ u (b3 , +)) = ∅. To finish the proof of Proposition 2.8, we first assume that dim M ≥ 3 and consider the following cases. Case 1: There is x0 ∈ Orb− (Os− ) ∩ ( u (b2 , +) \ u (b3 , +)) such that ST(x0 ) = +∞. Case 1.1: There is a neighborhood U of x0 in u (2b2 , +) such that ST(y) = +∞ for all y ∈ U. Choose y0 ∈ U and ξ > 0 such that • • • •

dist(x0 , Ou ) = dist(y0 , Ou ); B(x0 , ξ ) ∩ B(y0 , ξ ) = ∅; B(x0 , ξ ) ∩ u (2b2 , +) ⊂ U , B(y0 , ξ ) ∩ u (2b2 , +) ⊂ U ; and (l + ξ )/2 < l − ξ , where l = dist(x0 , Ou ).

Take ε ∈ (0, ξ ) small enough such that for any z in B(x0 , ε) or B(y0 , ε), the connected component of Orb(z) in B(x0 , ξ ) or B(y0 , ξ ) containing z intersects u (2b2 , +). Let d be the parameter with respect to ε of the orbital shadowing property. According to Lemma 3.9, ST(y0 ) = +∞ implies that σ ∈ α(y0 ). Then we may choose two points p ∈ Orb+ (Os− ) and q ∈ Orb− (y0 ) close enough to σ such that  g(t) =

φt (p), φt (q),

t ≤ 0; t >0

is a d-pseudo orbit. Then there exists z ∈ M such that distH (g(R), Orb(z)) < ε.

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Since both x0 and y0 are in the pseudo orbit, we know that Orb(z) intersect both B(x0 , ε) and B(y0 , ε). Thus there exist z1 ∈ Orb(z) ∩ (B(x0 , ξ ) ∩ u (2b2 , +)) ⊂ U and z2 ∈ Orb(z) ∩ (B(y0 , ξ ) ∩ u (2b2 , +)) ⊂ U . Note that ST(z1 ) = ST(z2 ) = +∞. / Orb− (z2 ) and z2 ∈ / Orb− (z1 ). It Then by Lemma 3.9 and (l + ξ )/2 < l − ξ , we have that z1 ∈ is a contradiction. Case 1.2: There is a sequence of yn ∈ u (2b2 , +) such that yn → x0 and ST(yn ) < +∞ for every n. Let U ⊂ Int1 (OrbSh(M)) be a C 1 neighborhood of X. By Lemma 2.4, there exist numbers ρ > 1, L > 1 and δ0 > 0 satisfying the property of the C 1 connecting lemma for x0 and U . Take 0 < δ < δ0 small enough such that  ∩ = ∅, where  = B(φ[−L,0] (x0 ), δ). Take n big enough such that yn ∈ B(x0 , δ/ρ). Denote by γ ⊂ u (2b2 , +) the segment connecting x0 and yn . According to Lemma 3.3, we have (P ◦ Q)−1 (γ ) ⊂ u (b2 ). We claim that there exists k > 0 such that (P ◦ Q)−k (γ ) ∩ Lu = ∅. If the claim is not true, from (P ◦ Q)−1 (γ ) ⊂ u (b2 ) and (P ◦ Q)−1 (x0 ) ∈ u (b2 , +) we know that (P ◦ Q)−1 (γ ) ⊂

u (b2 , +). It implies that (P ◦ Q)−2 (γ ) ⊂ u (b2 ). Performing the same procedure, we obtain / u (b2 , +) since ST(yn ) < (P ◦ Q)−k (γ ) ⊂ u (b2 , +) for all k > 0. But (P ◦ Q)−ST(yn ) (yn ) ∈ +∞. This contradiction proves the claim. It follows that γ ∩ W u (σ ) = ∅ since Lu ⊂ W u (σ ). Thus W u (σ ) ∩ B(x0 , δ/ρ) = ∅ because of γ ⊂ B(x0 , δ/ρ). Then by using the C 1 connecting lemma, we may get another homoclinic connection without destroying . It contradicts to Proposition 3.4. Case 2: ST(x) < +∞ for any x ∈ Orb− (Os− ) ∩ ( u (b2 , +) \ u (b3 , +)). According to Lemma 3.11, there exists a sequence of xn ∈ Orb− (Os− ) ∩ ( u (b2 , +) \

u (b3 , +)) such that ST(xn ) → +∞ as n → +∞. Take a subsequence if necessary, we may assume that xn → y0 ∈ u (b2 , +) \ u (b3 , +). If y0 ∈ W u (σ ), then by using the C 1 connecting lemma, we may connect the orbits of some  y0 ∈ Orb− (y0 ) and xn ∈ Orb+ (xn ) for some n sufficiently big without destroying . The existence of double homoclinic connections contradicts to Proposition 3.4. Below we assume that / W u (σ ). Hence y0 ∈ u (2b2 , +). y0 ∈ We claim that ST(y0 ) = +∞. Otherwise, by Lemma 3.10, there is a neighborhood U of y0 in

u (2b2 , +) such that ST(x) = ST(y0 ) for any x ∈ U . It contradicts to ST(xn ) → +∞. Similar to the argument in Case 1.2, we have that the unstable manifold W u(σ ) intersects the segment connecting y0 and xn for all n. Thus W u (σ ) ∩ B(y0 , ) = ∅ for any  > 0. / Per(X). For y0 and a neighborAccording to Lemma 3.9, ST(y0 ) = +∞ implies that y0 ∈ hood U ⊂ Int1 (OrbSh(M)) of X, by Lemma 2.4, there exist numbers ρ  > 1, L > 1 and δ0 > 0 satisfying the property of the C 1 connecting lemma. Take 0 < δ  < δ0 small enough such that  ∩ = ∅, where  = B(φ[−L ,0] (y0 ), δ  ). Since both of W u (σ ) and Orb− (Os− ) intersect B(y0 , δ  /ρ  ), by using the C 1 connecting lemma, we may get another homoclinic connection without destroying . This leads to a contradiction. Now we assume that dim M = 2. u (b0 ) is a segment in this case.

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If is non-orientable, it is easy to see that if Ou ∈ is accumulated by a sequence {zn }, then every point p ∈ W u (σ ) \ is accumulated by some sequence {zn } such that zn ∈ Orb+ (zn ). Thus Ou ∈ α(Os− ) implies that W u (σ ) ⊂ α(Os− ). It contracts to Lemma 3.6. If is orientable, we have that ST(y) = +∞ for any y ∈ u (2b2 , +). Note that the argument in the proof of Case 1.1 for dim M ≥ 3 also works when dim M = 2. We may get a contradiction in the same way. We complete the proof of Proposition 2.8. 2 4. Proof of Proposition 2.9: dim Eσs > 1 Suppose on the contrary, there is a singularity σ of X ∈ Int1 OrbSh(M) satisfying SV(σ ) > 0, dim Eσs > 1 and having a homoclinic connection . In this case, there exists the strong stable manifold W ss (σ ) of σ (Lemma 2.2). We will prove s (σ ) \ W ss (σ ), the α-limit set of x intersects W u (σ ). Then by using the C 1 that for any x ∈ Wloc loc loc connecting lemma, we may get a strong connection of σ . It contradicts to Lemma 2.3. Denote by Re(λs ) ≤ · · · ≤ Re(λ2 ) ≤ Re(λ1 ) < 0 < Re(γ1 ) ≤ Re(γ2 ) ≤ · · · ≤ Re(γu ), the eigenvalues of DX(σ ), where s + u = dim M. Since SV(σ ) > 0, we know from Lemma 2.2 that there is a dominated splitting Eσs = Eσss ⊕ Eσcs , where Eσcs is the 1-dimensional eigenspace with respect to λ1 , and Eσss is the eigenspace with respect to λ2 , . . . , λs . By using the same perturbations as in Section 3, we may assume that X satisfies Condition 4.1. • X is linear in B(σ, 2r) for some r > 0; • ∩ B(σ, 2r) ⊂ Eσs ∪ Eσu ; • Eσss , Eσcs and Eσu are mutually orthogonal. We introduce an orthogonal coordinate system x = (x ss , x cs , x u ) in B(σ, 2r) with respect to the splitting Tσ M = Eσss ⊕ Eσcs ⊕ Eσu , where x ss , x cs , x u are the coordinates of x in Eσss , Eσcs , Eσu respectively. For any x ∈ B(σ, 2r) and t > 0 such that φ[0,t] (x) ⊂ B(σ, 2r), we have |φt (x)cs | = |x cs | exp(λ1 t), |φt (x)ss | ≤ |x ss | exp(Re(λ2 )t), and |φt (x)u | ≥ |x u | exp(Re(γ1 )t), where (φt (x)ss , φt (x)cs , φt (x)u ) = φt (x). Take Os = (Osss , Oscs , 0) ∈ ∩ Eσs and Ou = (0, 0, Ouu ) ∈ ∩ Eσu such that dist(Os , σ ) = dist(Ou , σ ) = r. According to Lemma 2.3, we have ⊂ W ss (σ ) and then Oscs = 0. Let s , u ⊂

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B(σ, 2r) be two cross sections which are orthogonal to Eσcs and at Os and Ou respectively. Reduce u if necessary, we assume that r/2 ≤ dist(x, σ ) ≤ 3r/2, for any x ∈ u . Similarly, denote by Q : u → s and P : Dom(P ) ⊂ s → u the Poincaré map generated by the flow, where Dom(P ) = {x ∈ s : there is T > 0 such that φT (x) ∈ u and φ[0,T ] (x) ⊂ B(σ, 2r)}. Let τQ (x) and τP (x) be the minimal positive number t such that φt (x) = Q(x) and φt (x) = P (x). Reduce u if necessary, we assume that τQ is continuous and bounded on u . It is easy to see that τP (x) → +∞ as x → Eσs ∩ s . Denote s (a) = B(Os , a) ∩ s and u (b) = B(Ou , b) ∩ u . Fix a0 > 0 and b0 > 0 such that Q( u (b0 )) ⊂ s (a0 ). Denote Lu = {x ∈ u : x ss = 0} and Ls = {x ∈ s : x u = 0} = Eσs ∩ s . Since Lu is a u-dimensional disc in u and Ls is a (s − 1)-dimensional disc in s , up to an arbitrarily C 1 small perturbation, we may further assume that Condition 4.2. • Q(Lu ) is transverse to Ls at Os in s . Denote Cu (ϑ) = {x ∈ u : |x ss | ≤ ϑ|x cs |}, and Cs (ϑ) = {x ∈ s : |x u | ≤ ϑ|x ss − Osss |} two cones in u and s respectively. Since Q(Lu ) and Ls are transverse at Os and Cu (0) = Lu , Cs (0) = Ls , the following lemma follows immediately from the C 1 continuity. Lemma 4.3. There exists θ > 0 and b1 ∈ (0, b0 ) such that Q(Cu (θ ) ∩ u (b1 )) ∩ Cs (θ ) = {Os }.

2

Note that in this case we don’t have the exponential growth of the distance between P ◦ Q(x) and Ou for every x ∈ u close to Ou satisfying Q(x) ∈ Dom(P ), because of the coordinate of Eσss . However, such property holds for points having relatively small Eσss coordinate. See [4, Lemma 4.6] for the 3-dimensional case. Lemma 4.4. There is b2 ∈ (0, b0 ) such that if x ∈ u (b2 ) and Q(x) ∈ Dom(P ) \ Cs (θ ), then dist(P ◦ Q(x), Ou ) ≥ 2dist(x, Ou ).

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Proof. The proof of this lemma is similar to that of Lemma 3.2. We only need to note that here √ dist(y, Os ) = (|y

ss

− Osss |2

+ |y | )

u 2 1/2

≤

1 + θ2 u |y | θ

for any y = (y ss , Oscs , y u ) ∈ s \ Cs (θ ). Then since dist(P (y), Ou ) ≥ |P (y)cs | = |Oscs | exp(λ1 τP (y)), dist(P (y), σ ) ≥ |P (y)u | ≥ |y u | exp(Re(γ1 )τP (y))

 ≥ dist(y, Os ) exp(Re(γ1 )τP (y))θ/ 1 + θ 2 , and dist(P (y), σ ) ≤ 3r/2, we obtain dist(P (y), Ou ) |Oscs | exp(λ1 τP (y)) 2θ |O cs | exp(SV(σ )τP (y)) → +∞ ≥ = √ s √ dist(y, Os ) 3 1 + θ2 3r 1 + θ 2 r exp(−Re(γ1 )τP (y)) 2 θ as y → Os (which implies that τP (y) → +∞). By the C 1 continuity, there are ξ ∈ (0, b0 ) and K > 1 such that dist(Q(x), Os ) ≥

1 dist(x, Ou ) K

for all x ∈ u (ξ ). Take ζ ∈ (0, a0 ) such that dist(P (y), Ou ) ≥ 2Kdist(y, Os ) for any y ∈ s (ζ ) ∩ (Dom(P ) \ Cs (θ )). Then a small number b2 ∈ (0, ξ ) satisfying Q( u (b2 )) ⊂ s (ζ ) meets the needs of this lemma. 2 s ∈ (Eσs \ Eσss ) ∩ B(σ, 2r), take a small cross section s ⊂ B(σ, 2r) which is orFor any O cs   s to u in B(σ, 2r). thogonal to Eσ at Os . We may also consider the Poincaré map P from   Define Dom(P ) and τP for P similarly, and let s = {x ∈ s : x u = 0} = Eσs ∩ s . L (y)ss |/|P (y)cs | Since the contracting rate of Eσss is larger than that of Eσcs , the quotient |P   tends to 0 as y ∈ Dom(P ) tends to Os . Therefore s ∈ (Eσs \ Eσss ) ∩ B(σ, 2r), there exists  a > 0 such that Lemma 4.5. [5, Lemma 5.4] For any O (y) ∈ Cu (θ ) P ). s ( for any y ∈ a ) ∩ Dom(P

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Denote Sru = {x ∈ Eσu : dist(x, σ ) = r}. Similar to Lemma 3.5, we have the following lemma. s ∈ (Eσs \ Eσss ) ∩ B(σ, 2r). s ) ∩ Sru = ∅ for any O Lemma 4.6. α(O s ∈ (Eσs \ Eσss ) ∩ B(σ, 2r). It s ) ∩ Sru = ∅ for some O Proof. Suppose on the contrary that α(O implies that there is ξ > 0 such that s ) ∩ B(Sru , ξ ) = ∅. Orb− (O

(4.5)

s , σ ) = r and ξ < r/8. Without loss of generality we assume that dist(O s , the component of Orb+ (x) s (ζ ) \ L Take ζ ∈ (0, a0 ) such that for any x ∈ s (ζ ) \ Ls or u in B(σ, 2r) containing x intersects B(Sr , ξ/2). According to Lemma 4.5, reduce ζ if necessary, ), one has P (y) or P (y) ∈ s (ζ ) ∩ Dom(P we may assume that for any y ∈ s (ζ ) ∩ Dom(P ) or Cu (θ ). Let η ∈ (0, min(b1 , b2 )) be a small number such that Q( u (η)) ⊂ s (ζ ), where b1 and b2 is given by Lemma 4.3 and 4.4. Denote ξ = ∩ B(Sru , ξ ). By the continuous dependence on initial values, there are small  of O s such that for any x ∈ V , U , or U , the component neighborhoods V of ξ , U of Os , and U  containing x intersects u (η), s (ζ ), or s (ζ ) respectively. of Orb(x) in V , U , or U s , ε) ⊂ U . Take ε ∈ (0, ξ/2) small enough such that B( ξ , ε) ⊂ V , B(Os , ε) ⊂ U , and B(O Let d be the parameter of the orbital shadowing property with respect to ε. Choose two points s ), q ∈ Orb− (Ou ) close enough to σ such that p ∈ Orb+ (O  g(t) =

φt (p), φt (q),

t ≤ 0; t >0

is a d-pseudo orbit. Then there exists Orb(z) orbitally shadows g(R). Without loss of generality s (ζ ). we assume that z ∈ s . Case 1. z ∈ /L ) and P (z) ∈ u (η). Otherwise, we have that the positive orbit of We claim that z ∈ Dom(P z intersects B(Sru , ξ/2) at some point z1 outside V . Then by the orbital shadowing property, s ) such that dist(y, z1 ) < ε. It follows that y ∈ B(Sru , ξ ), we obtain that there exists y ∈ Orb− (O which contradicts to (4.5). This contradiction proves the claim. (z) ∈ Cu (θ ) ∩ u (η). By Lemma 4.3, it By the choice of ζ and Lemma 4.5, we obtain z0 = P  implies that Q(z0 ) ∈ s (ζ ) \ Cs (θ ). Replacing z, P with Q(z0 ), P and using the same argument in the above claim, we have Q(z0 ) ∈ Dom(P ) and P ◦ Q(z0 ) ∈ u (η). Note that we have dist(P ◦ Q(z0 ), Ou ) ≥ 2dist(z0 , Ou ) according to Lemma 4.4. Performing the same procedure, we obtain (P ◦ Q)n (z0 ) ∈ u (η) and dist((P ◦ Q)n (z0 ), Ou ) ≥ 2n dist(z0 , Ou ) for all n > 0. This leads to a contradiction. s . Case 2. z ∈ L Since Os ∈ g(R), we have that there is z ∈ Orb(z) ∩ B(Os , ε). Without loss of generality we  with z , P in the claim assume that z ∈ s (ζ ). Note that z ∈ / Ls in this case. So replacing z, P     in Case 1, we can get z ∈ Dom(P ) and P (z ) ∈ u (η). Let z0 = P (z ). Then following the same lines as in Case 1 (replace z0 with z0 ), we may get a contradiction. 2 Now we may finish the proof of Proposition 2.9. Take a C 1 neighborhood U ⊂ Int1 (OrbSh(M)) of X, and Oss ∈ Eσss ∩ B(σ, 2r) such that Oss = σ . For Oss and U , by Lemma 2.4, there exist numbers ρ1 > 1, L1 > 1 and δ0,1 > 0 with

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the property of the C 1 connecting lemma. Since Oss ∈ / W u (σ ), we may take δ1 ∈ (0, δ0,1 ) small enough such that u 1 ∩ Wloc (σ ) = ∅,

where 1 = B(φ[−L1 ,0] (Oss ), δ1 ). s ∈ (Eσs \ Eσss ) ∩ B(σ, 2r) such that dist(O s , Oss ) < δ1 /ρ1 . By Lemma 4.6, we have Take O u u   α(Os ) ∩ Sr = ∅. Take y ∈ α(Os ) ∩ Sr . For y and U , there exist numbers ρ2 > 1, L2 > 1 and δ0,2 > 0 satisfying the C 1 connecting lemma. Take δ2 ∈ (0, δ0,2 ) small enough such that s 2 ∩ 1 = ∅ and 2 ∩ Wloc (σ ) = ∅,

where 2 = B(φ[−L2 ,0] (y), δ2 ). Take zu ∈ Orb− (y) and zss ∈ Orb+ (Oss ) such that zu ∈ / 2 and zss ∈ / 1 . Since both of s ) intersect B(y, δ2 /ρ2 ), by using the C 1 connecting lemma, we can get Orb+ (zu ) and Orb− (O s ∈ Orb+ (zu ). Note that Oss is still on the negY ∈ U that differs from X only in 2 such that O Y − ative orbit of zss respecting Y . Thus both of Orb+ Y (zu ) and OrbY (zss ) intersect B(Oss , δ1 /ρ1 ). 1 Then by using the C connecting lemma, we can get Z that differs from Y only in 1 such that zss ∈ Orb+ Z (zu ). Since Z differs from X only in 1 ∪ 2 , we have that zss and zu still belong to the local strong stable manifold and local unstable manifold of σ respecting Z. Thus the closure of OrbZ (zss ) is a strong connection of σ . But from Lemma 2.3 we know that there is no strong connection for star vector fields. This contradiction finishes the proof. 2 5. Structure stability In this section we will consider the relationship between the C 1 robustly orbital shadowing property and C 1 structural stability. We say that a vector field X has the oriented shadowing property if for any ε > 0 there exists d > 0 such that, for any d-pseudo orbit g of X, there is x ∈ M and an increasing homeomorphism h of R satisfying dist(g(t), φh(t) (x)) < ε, ∀t ∈ R. Denote by OriSh(M) the set of vector fields having the oriented shadowing property. A special class B was introduced in [16] of vector fields satisfying: • the vector field has two hyperbolic singularities σ1 and σ2 , the eigenvalues of DX(σj ) is denoted by j

j

j

j

j

j

Re(λs ) ≤ · · · ≤ Re(λ2 ) ≤ Re(λ1 ) < 0 < Re(γ1 ) ≤ Re(γ2 ) ≤ · · · ≤ Re(γu ); • λ11 and λ12 are conjugate non-real complex numbers, Re(λ13 ) < Re(λ12 ); • γ12 and γ22 are conjugate non-real complex numbers, Re(γ22 ) < Re(γ32 ); • the stable manifold of σ2 and the unstable manifold of σ1 have a non-transverse intersection.

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Denote by SS ⊂ X 1 (M) the set of C 1 structural stable vector fields. The following theorem is proved in [16]. Theorem 5.1. [16] • Int1 (OriSh(M) \ B) = SS; • Int1 (OriSh(M)) ∩ B = ∅. Let X be a vector field having the C 1 robustly orbital shadowing property. According to Theorem 1.1, we know that X is -stable. If X ∈ / B, then by using the same method as in the proof of [16, Theorem 3] (also see [15, Theorem 4] for more details), one may prove that X has no non-transverse intersection between the stable and unstable manifolds of periodic orbits or singularities. It implies that X is structural stable. Thus we have the following theorem: Theorem 5.2. Int1 (OrbSh(M) \ B) = SS. Sketch of the proof. It is easy to see that OriSh(M) ⊂ OrbSh(M). Thus by Theorem 5.1 we have SS ⊂ Int1 (OrbSh(M) \ B). So we only need to prove the converse. For any X ∈ Int1 (OrbSh(M)), by Lemma 2.6, all periodic orbits and singularities of X are hyperbolic. If all intersections between the stable and unstable manifolds of periodic orbits or singularities are transverse, then X is a so-called Kupka-Smale vector field. Since the C 1 interior of the set of Kupka-Smale vector fields coincides with the set of structurally stable vector fields. It suffices to prove that there is no non-transverse intersection between those for all Int1 (OrbSh(M) \ B). Suppose on the contrary that there is X ∈ Int1 (OrbSh(M) \ B) and a non-transverse intersection point z ∈ W s (ς) ∩ W u (σ ), where ς and σ are periodic orbits or singularities of X. Note that ς and σ are contained in two different basic sets because of the -stability. If σ is a singularity, then there is a hyperbolic splitting Tσ M = Eσs ⊕ Eσu with respect to X. If σ is a periodic orbit, then take a point w ∈ σ and a cross section transverse to σ at w. Let R be s ⊕ E u with respect the Poincaré return map on . There exists a hyperbolic splitting Tw = Ew w to R. Up to an arbitrarily C 1 small perturbation, we may assume that • if σ is a singularity, then there exists one real eigenvalue or two conjugate complex eigenvalues of DX(σ )|Eσs whose real part is bigger than the real part of any other eigenvalue of DX(σ )|Eσs , thus there is a dominated splitting Eσs = Eσss ⊕ Eσcs with dim Eσcs = 1 or 2; • if σ is a periodic orbit, then there exists one real eigenvalue or two conjugate complex eigenvalues of DR(w)|Ews whose norm is bigger than the norm of any other eigenvalue of s = E ss ⊕ E cs with dim E cs = 1 or 2. DR(w)|Ews , thus there is a dominated splitting Ew w w w Case 1: σ is a singularity and dim Eσcs = 1. Using the same perturbations as in Section 3 and changing the Riemannian metric if necessary, we may assume that X is linear in B(σ, r) for some r > 0, and Eσss , Eσcs , Eσu are mutually orthogonal. We introduce an orthogonal coordinate system x = (x ss , x cs , x u ) in B(σ, r) with respect to the splitting Tσ M = Eσss ⊕ Eσcs ⊕ Eσu ,

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where x ss , x cs , x u are the coordinates of x in Eσss , Eσcs , Eσu respectively. Take Ou ∈ Orb− (z) ∩ B(σ, r) and a cross section u orthogonal to Orb(Ou ) at Ou . Denote u (σ ) ∩ two affine spaces centered at O . Since O by Es = TOu (W s (ς) ∩ u ) and Eu = Wloc u u u is a non-transverse intersection, we have that TOu u = Es + Eu . Let E  be the projection of Es to Eσs parallel to Eσu . The non-transverse means that E  = Eσs . Up to a C 1 small perturbation, we may assume that Eσcs ∩ E  = {σ }. It implies that Lu ∩ Es ⊂ u (σ ), where L = {x ∈ : x ss = 0}. Wloc u u Moreover, by using the perturbations in [16], we may assume that there is a small neighborhood Uu of Ou in u such that the component of W s (ς) ∩ Uu containing Ou coincides with Es in Uu . We first consider a easy case. Case 1.1: dim Eσs = 1. u (σ ). In this case we have Es ⊂ Eu ⊂ Wloc Since X is -stable, we know that X satisfies the no-cycle condition C 1 robustly. Recall that σ and ς are contained in two different basic sets and W s (ς) ∩ W u (σ ) = ∅. Thus the distance between W s (σ ) and W u (ς) is positive, denote it by ξ1 > 0. u (ς) ∩ ∂B(ς, ) for a small number  > 0, where ∂B is the boundary of Denote Su (ς) = Wloc B. Let ξ2 = dist(Orb(Ou ), Su (ς)) and ξ = min(ξ1 , ξ2 ). By the continuity, there is 0 < ε < ξ/100 such that B(Ou , ε) ∩ u ⊂ Uu and for any x ∈ B(Ou , ε) \ Es , the positive orbit of x intersects B(Su (ς), ξ/100). s (σ ) \ {σ }. Let d be the parameter with respect to ε of the orbital shadowing Take Os ∈ Wloc property. Take two points p ∈ Orb+ (Os ), q ∈ Orb− (Ou ) close enough to σ such that  g(t) =

φt (p), φt (q),

t ≤ 0; t >0

(5.6)

is a d-pseudo orbit. Then there exists x ∈ Uu such that distH (g(R), Orb(x)) < ε.

(5.7)

We claim that x ∈ Es ∩ Uu ⊂ W s (ς). If x ∈ / Es , then there exists y ∈ Orb+ (x) ∩ u B(S (ς), ξ/100). This contradicts to the fact dist(g(R), Su (ς)) ≥ ξ . u (σ ) in this case. It is a contradiction because that the negative So we have x ∈ Es , thus x ∈ Wloc orbit of Os can not be orbitally shadowed by Orb(x). Case 1.2: dim Eσs > 1. Take Os ∈ Eσcs ∩ B(σ, r), Os = σ . An analog of the claim in Case 1.1 can be proved in the same way as shown above. That is, there is ε > 0 such that if the orbit of some x ∈ Uu ε-orbitally shadows the pseudo orbit (5.6) then x ∈ Es . Let s be a cross section which is orthogonal to Eσcs at Os and P : Dom(P ) ⊂ s → u be the Poincaré map. We still denote Cu (ϑ) = {x ∈ u : |x ss | ≤ ϑ|x cs |} u (σ ). Thus there is θ > 0 small such that a cone in u . Recall that Lu ∩ Es ⊂ Wloc u Cu (θ ) ∩ Es ⊂ Wloc (σ ).

(5.8)

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By Lemma 4.5, there is a neighborhood Us of Os in s such that P (y) ∈ Cu (θ ) for any y ∈ Us ∩ Dom(P ). Note that Os is a wandering point since singularities are isolated in the non-wandering set of X. So the C 1 robust no-cycle condition implies that Orb− (Os ) ∩ W u (σ ) = ∅. Reduce r if necessary, we may assume that s Orb(Os ) ∩ B(σ, r) ⊂ Wloc (σ ).

(5.9)

Take ξ  > 0 small such that B(Os , ξ  ) ∩ s ⊂ Us . Denote r  = dist(Os , σ ) and Srs (σ ) = s (σ ) ∩ ∂B(σ, r  ). Reduce ε if necessary, we assume that for any x ∈ B(O , ε) \ W u (σ ), the Wloc u loc negative orbit of x intersects B(Srs (σ ), ξ  /100). Then consider the d-pseudo orbit (5.6) which is orbitally shadowed by the orbit of some x ∈ Uu . Recall that we get x ∈ Es above. u (σ ), then there is a contradiction that Orb(x) doesn’t orbitally shadow the negative If x ∈ Wloc u (σ ). It is easy to see that x ∈ P (U ). Otherwise, there is some orbit of Os . So we have x ∈ / Wloc s − s  y ∈ Orb (x) ∩ B(Sr  (σ ), ξ /100) with dist(y, Os ) ≥ ξ  . It contradicts to (5.7) and (5.9) when ε > u (σ ) by (5.8). This contraction 0 is small enough. So we have x ∈ Cu (θ ), which implies x ∈ Wloc finish the proof in Case 1. When σ is a periodic orbit, the proof is similar to it for diffeomorphisms, no matter whether ς is a periodic orbit or not. We will only outline the idea of Case 2 and 3. cs = 1. Case 2: σ is a periodic orbit and dim Ew In this case, we can linearize the Poincaré return map R in a neighborhood B(w, r) ∩ of w in , and use the same idea as in Case 1 to get a contradiction. Here we only point out the differences and refer the reader to [16, Theorem 3] and [15, Theorem 4] for the rigorous proof. u = 1. In the case dim E u > 1, now O ∈ E cs ∩ B(w, r) may be It is also a easy case if dim Ew s w w not wandering. If Os is wandering, then we may use the same argument as in Case 1.2 to get a contradiction. If Os is non-wandering, then the basic set containing σ , denote by , is non-trivial and Os is contained in the unstable manifold of some point υ ∈ . Since the periodic orbits in ss close to O such  are dense, there is a periodic point υ  ∈  close to υ and a point Os ∈ / Ew s −   u  s  u / σ and Os ∈ W (υ ) ∩ Ew . It is clear that Orb (Os ) ∩ W (σ ) = ∅. Thus by replacing that υ ∈ Os with Os in the proof of Case 1.2, we may also get a contradiction. cs = 2. Case 3: σ is a periodic orbit and dim Ew cs . Up to a C 1 Denote by λ1,2 = λ(cos ϕ + i sin ϕ) the conjugate complex eigenvalues of Ew small perturbation, we assume that π/ϕ is rational. Similar to Case 1 and 2, we linearize the Poincaré return map R in a neighborhood B(w, r) ∩ of w in , and introduce an orthogonal coordinate system x = (x ss , x cs , x u ) in B(w, r) with respect to the splitting ss cs u Tw = E w ⊕ Ew ⊕ Ew . u (σ ) ∩ two Take Ou ∈ Orb− (z) ∩ B(w, r). Denote by Es = TOu (W s (ς) ∩ ) and Eu = Wloc affine spaces centered at Ou . The non-transverse implies that TOu = Es + Eu . Let E  be the s parallel to E u . Then we have E  = E s . Up to a C 1 small perturbation, projection of Es to Ew w w cs ⊂ E  and there is a small neighborhood U of O in such that the we may assume that Ew u u component of W s (ς) ∩ Uu containing Ou coincides with Es in Uu . cs ∩ E  = ∅, then the proof is same as it in Case 2. Now we assume that E = E cs ∩ E  is If Ew 1 w 1-dimensional.

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Since π/ϕ is rational, there is a natural number m such that (cos ϕ + i sin ϕ)m = 1. So there are m − 1 1-dimensional subspaces E2 , E3 , . . . , Em such that R(Ej ) = Ej +1 for 1 ≤ j ≤ m, where Em+1 = E1 . cs ∩B(w, r)) \∪m E . Let E  be the 1-dimensional subspace in E cs containing Take Os ∈ (Ew w j =1 j 1    be 1-dimensional subspaces such that R(E  ) = E  Os , and E2 , E3 , . . . , Em for 1 ≤ j ≤ m, j j +1  where Em+1 = E1 . Take 0 < r1  r. Denote ζ = dist(∂B(w, r1 )



∪m j =1 Ej , ∂B(w, r1 )



 ∪m j =1 Lj )).

By using the same method in the above cases, we may prove that • there is 0 < ε < ζ /100, such that for the pseudo orbit (5.6) there exists x = (x ss , x cs , x u ) ∈ Es ∩ Uu satisfying (5.7); and • reduce r if necessary, there are two integers k1 < k2 such that Orb(Os ) ∩ (B(w, r) \ B(w, r1 )) ∩ = {R n (Os ) : k1 ≤ n ≤ k2 }. u (σ ), which contradicts to (5.7). If x cs = 0 and If x ss = x cs = 0, then we have x ∈ W s (ς) ∩ Wloc −n ss u ss increase

= 0, then we have that R (x) ∈ Ew ⊕ Ew and the coordinate of R −n (x) in Ew −1 under the iteration R . It is no hard to see this is contrary to (5.7) and the fact Orb(Os ) ∩ cs . B(w, r) ∩ ⊂ Ew cs parallel to Now we assume that x cs = 0. Denote by  the projection from to Ew m ss u −n Ew ⊕ Ew . x ∈ Es implies that (R (x)) ∈ ∪j =1 Ej for all non-negative n = 0, 1, . . . until R −n (x) leaves B(w, r). Thus there exist some R −n0 (x) ∈ ∩ (B(w, r) \ B(w, r1 )) such m   n that dist(R −n0 (x), ∪m j =1 Ej ) ≥ ζ . It contradicts to (5.7) and the fact R (Os ) ∈ ∪j =1 Ej for all k1 ≤ n ≤ k2 . We refer the reader to the prove of the case of periodic point with complex eigenvalues for diffeomorphisms in [15, Theorem 4]. Case 4: σ is a singularity and dim Eσcs = 2. / B, we have that ς is The argument in Case 3 doesn’t work in this case. However, since X ∈ either a periodic orbit or a singularity whose weakest expanding eigenvalues are real numbers. Thus we may consider ς and the inverse vector field −X. Up to an arbitrarily C 1 small perturbation, we have that ς is belongs to Case 1, 2 or 3 above for −X. The problem is reduced to the previous cases. 2

x ss

It follows immediately from Theorem 5.1 and 5.2 that Int1 (OrbSh(M) \ B) = Int1 (OriSh(M) \ B). It is well known that the C 1 interior of the set of diffeomorphisms having the orbital shadowing property coincides with the C 1 interior of the set of diffeomorphisms having the shadowing property. The same question for vector fields is still open: Question. Whether the C 1 interior of the set of vector fields having the orbital shadowing property coincides with the C 1 interior of the set of vector fields having the oriented shadowing property?

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Acknowledgment I would like to thank the referee for his/her careful reading of this paper and for the useful comments. Ming Li is supported by the National Natural Science Foundation of China (Grant No. 11571188), and the LPMC of Nankai University. References [1] D.V. Anosov, On a class of invariant sets of smooth dynamical systems, in: Proc. 5th Int. Conf. on Nonlin. Oscill., vol. 2, Kiev, 1970, pp. 39–45. [2] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math., vol. 470, Springer, Berlin, 1975. [3] S. Crovisier, Periodic points and chain-transitive sets of C 1 -diffeomorphisms, Publ. Math. Inst. Hautes Études Sci. 104 (2006) 87–141. [4] S. Gan, M. Li, Orbital shadowing for 3-flows, J. Differ. Equ. 262 (2017) 5022–5051. [5] S. Gan, M. Li, S.B. Tikhomirov, Oriented shadowing property and -stability for vector fields, J. Dyn. Differ. Equ. 28 (2014) 1–13. [6] S. Gan, L. Wen, Nonsingular star flows satisfy Axiom A and the nocycle condition, Invent. Math. 164 (2006) 279–315. [7] S. Hayashi, Connecting invariant manifolds and the solution of the C 1 -stability and -stability conjectures for flows, Ann. Math. 145 (1997) 81–137. [8] M.W. Hirsch, C.C. Pugh, M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, vol. 583, Springer-Verlag, 1977. [9] K. Lee, K. Sakai, Structural stability of vector fields with shadowing, J. Differ. Equ. 232 (2007) 303–313. [10] A. Morimoto, The Method of Pseudo-Orbit Tracing and Stability of Dynamical Systems, Sem. Note, vol. 39, Tokyo Univ., 1979. [11] K.J. Palmer, Shadowing in Dynamical Systems: Theory and Applications, Kluwer, 2000. [12] S.Yu. Pilyugin, Shadowing in Dynamical Systems, Lecture Notes in Math., vol. 1706, Springer, 1999. [13] S.Yu. Pilyugin, Shadowing in structurally stable flows, J. Differ. Equ. 140 (2) (1997) 238–265. [14] S.Yu. Pilyugin, Theory of pseudo-orbit shadowing in dynamical systems, Differ. Equ. 47 (13) (2011) 1929–1938. [15] S.Yu. Pilyugin, A.A. Rodionova, K. Sakai, Orbital and weak shadowing properties, Discrete Contin. Dyn. Syst. 9 (2003) 287–308. [16] S.Yu. Pilyugin, S.B. Tikhomirov, Vector fields with the oriented shadowing property, J. Differ. Equ. 248 (2010) 1345–1375. [17] C. Robinson, Stability theorems and hyperbolicity in dynamical systems, Rocky Mt. J. Math. 7 (1977) 425–437. [18] K. Sakai, Pseudo orbit tracing property and strong transversality of diffeomorphisms on closed manifolds, Osaka J. Math. 31 (1994) 373–386. [19] K. Sawada, Extended f -orbits are approximated by orbits, Nagoya Math. J. 79 (1980) 33–45. [20] Yi Shi, Shaobo Gan, Lan Wen, On the singular-hyperbolicity of star flows, J. Mod. Dyn. 8 (2) (2014) 191–219. [21] L. Wen, Z. Xia, C 1 connecting lemmas, Trans. Am. Math. Soc. 352 (2000) 5213–5230. [22] S. Zhu, S. Gan, L. Wen, Indices of singularities of robustly transitive sets, Discrete Contin. Dyn. Syst. 21 (2008) 945–957.