STRUCTURAL STABILITY OF ORBITAL INVERSE SHADOWING VECTOR FIELDS. 1. Introduction

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1 Trends in Mathematics Information Center for Mathematical Sciences Volume 7, Number 2, December, 200, Pages STRUCTURAL STABILITY OF ORBITAL INVERSE SHADOWING VECTOR FIELDS KEONHEE LEE, ZOONHEE LEE AND YONG ZHANG Abstract. In this paper we give a characterization of the structurally stable vector fields via the notion of orbital inverse shadowing. More precisely it is proved that the C 1 interior of the set of C 1 vector fields with the orbital inverse shadowing property coincides with the set of structurally stable vector fields on a compact smooth manifold. This fact improves the result obtained by K. Moriyasu, K. Sakai and N. Sumi in [13]. 1. Introduction Structurally stable systems (both diffeomorphisms and flows) were the main objects of interest in the global qualitative theory of dynamical systems in the last 30 years. Now we know that structural stability is equivalent to Axiom A combined with the strong transversality condition. One of the most important properties of a structurally stable system is the shadowing property (also known as the pseudo orbit tracing property). The shadowing property is the key of the analysis of such diffeomorphisms or flows. Long time ago, various approaches were applied to show that a structurally stable diffeomorphism has the shadowing property. But the fact that a structurally stable flow has the shadowing property was proved by Pilyugin recently (see [15]). The main difficulty of the shadowing problem for a structurally stable flow is created by the following fact specific for flows. Let p be a nonwandering point of a structurally stable flow. Then the trajectory of p is hyperbolic. Denote by S(p) and U(p) the corresponding stable and unstable subspaces of the hyperbolic structure. If p 1 is a rest point, and p 2 belongs to a nonsingular nonwandering trajectory, then dim(s(p 1 ) + U(p 1 )) dim(s(p 2 ) + U(p 2 )). Hence the hyperbolic structures near rest points and near nonsingular nonwandering trajectories are qualitatively different. Consequently the standard shadowing approaches do not work. In fact, it was proved by K. Sakai [18] that the C 1 interior of the set of diffeomorphisms with the shadowing property coincides with the set of structurally stable diffeomorphisms. However it is still open problem whether the 2000 Mathematics Subject Classification. Primary: 37C50; Secondary; 37D20. Key words and phrases. flow; method; shadowing; inverse shadowing; orbital inverse shadowing; structurally stable; vector fields. 213 c 200 Information Center for Mathematical Sciences

2 21 KEONHEE LEE, ZOONHEE LEE AND YONG ZHANG above results can be applied to the case of flows; i.e. is a flow in the C 1 interior of the set of flows (or vector fields) with the shadowing property structurally stable? For the case of flows, the only known result in this direction is that the C 1 interior of the set of topologically stable flows coincides with the set of structurally stable flows (see [1]). The concept of inverse shadowing for homeomorphisms dual to the shadowing was established by Corless and Pilyugin [3], and Kloeden et al [8, 9] redefined this property using the concept of a method. Generally speaking, a homeomorphism has the inverse shadowing property with respect to a class of methods if any trajectory can be uniformly approximated with given accuracy by a δ-pseudotrajectory generated by a method from the chosen class if δ > 0 is sufficiently small. An appropriate choice of the class of admissible pseudotrajectories is crucial here (see [, 10, 16]). It was shown by S. Pilyugin [16] that every structurally stable diffeomorphism on a compact smooth manifold has the inverse shadowing property with respect to the class of continuous methods. Recently K. Lee, Z. Lee [11]introduced the notion of inverse shadowing for flows and showed that every expansive flow with shadowing has the inverse shadowing proeprty. Recently Y. Han and K. Lee [6] proved that every structurally stable flow on a compact smooth manifold has the inverse shadowing property with respect to the class of continuous methods. In this paper we introduce the notion of orbital inverse shadowing for flows and prove that the C 1 interior of the set of vector fields with the orbital inverse shadowing property (or inverse shadowing property) coincides with the set of vector fields satisfying Axiom A and the strong transversality condition. 2. Preliminaries Let M be a compact smooth n-dimensional manifold with a Riemannian metric d. We denote by T x M the tangent space of M at x and by T M the tangent bundle of M. Consider a C 1 vector field X on M and the system of differential equations (1) ẋ = X(x). For two vector fields X and Y on M, we define d C 0(X, Y ) = max X(p) Y (p). p M In this setting, DX(p) is considered as the derivative of the mapping, and for p M we define DX(p) DY (p) = max DX(p)v DY (p)v. v T pm, v =1 Define the C 1 distance d C 1 between two vector fields X and Y on M by d C 1(X, Y ) = d C 0(X, Y ) + max DX(p) DY (p). p M We denote by X 1 (M) the space of C 1 vector fields on M with the topology induced by d C 1.

3 ORBITAL INVERSE SHADOWING VECTOR FIELDS 215 Let Φ : R M M be the solution flow of system (1). We denote by DΦ(t, x) be the corresponding variational flow. For δ, T > 0 we say that a mapping φ : R M is a (δ, T )-pseudotrajectory of system (1) if, for any t R, for s T. For δ, T > 0 we say that a mapping d(φ(s, φ(t)), φ(s + t)) < δ, Ψ : R M M is a (δ, T )-method for system (1) if, for any x M, the map Ψ x : R M defined by Ψ x (t) = Ψ(t, x), t R, is a (δ, T )-pseudotrajectory of system (1). Ψ is said to be complete if Ψ(0, x) = x for x M. Note that a (δ, T )-method for system (1) can be considered as a family of (δ, T )- pseudotrajectories of system (1). A method Ψ for system (1) is said to be continuous if the map given by Ψ : M M R Ψ(x)(t) = Ψ(t, x), x M and t R, is continuous under the compact open topology on M R, where M R denotes the set of all functions from R into M. The set of all complete (δ, T )-methods [resp. complete continuous (δ, T )-methods] for system (1) will be denoted by T a (δ, T, Φ) [resp. T c (δ, T, Φ)]. It is clear that if Y X 1 (M) is another vector field which is sufficiently close to X in the C 0 topology then the system ẋ = Y (x) induces a complete continuous method for system (1). Let T h (δ, T, Φ)[resp. T d (δ, T, Φ)] be the set of all complete continuous (δ, T )- methods for system (1) which are induced by C 1 vector fields Y with d C 0(X, Y ) < δ [resp. d C 1(X, Y ) < δ ]. We say that the system (1) has the inverse shadowing property with respect to the class T α (α = a, c, h, d) if for any ε > 0 and T > 0 there exists δ > 0 such that for any (δ, T )-method Ψ T α (δ, T, Φ) and any point x M there are y M and α y Rep for which d(φ(t, x), Ψ(α y (t), y)) < ε, t R,

4 216 KEONHEE LEE, ZOONHEE LEE AND YONG ZHANG where, Rep denotes the set of increasing homeomorphisms α mapping R onto R with α(0) = Orbital Inverse Shadowing for vector fields DEFINITION 3.1. We say that the system (1) has the orbital inverse shadowing property with respect to the class T α (α = a, c, h, d) if for any ε > 0 and T > 0 there exists δ > 0 such that for any (δ, T )-method Ψ T α (δ, T, Φ) and any point x M there is y M for which d H (O(x, Φ), O(y, Ψ)) < ε, where, d H is Hausdorff metric and O(x, Φ) = {Φ(t, x) : t R}, O(y, Ψ) = {Ψ(t, y) : t R}. Remark 3.2. Let us denote IS α (M) by the set of vector fields with the inverse shadowing property with respect to the class T α, and OIS α (M) by the set of vector fields with the orbital inverse shadowing property with respect to the class T α, where α = a, c, h, d. We denote IS α(m) [resp. OIS α(m)] by the C 1 interior of the set IS α (M) [resp. OIS α (M)] in X 1 (M), where α = a, c, h, d. Clearly we have the following inclusions: IS a (M) IS c (M) IS h (M) IS d (M) OIS a (M) OIS c (M) OIS h (M) OIS d (M). A vector field X X 1 (M) is called structurally stable if there is a C 1 neighborhood U(X) of X X 1 (M) such that every Y U(X) is topologically conjugate to X. It is proved by C. Robinson [17] that if X satisfies Axiom A and the strong transversality condition, then X is structurally stable. The inverse implication is the famous stability conjecture which is proved for diffeomorphisms by R. Mañé [13] and is proved for flows by S. Hayashi [7], L. Wen [19] and S. Gan [5]. The purpose of this paper is to give a characterization of the structurally stable vector fields by making use of the notion of orbital inverse shadowing. The main result of this paper is the following one. For simplicity, we denote OIS d (M) by OIS(M). Main Theorem. The C 1 interior of OIS(M) is characterized as the set of vector fields satisfying Axiom A and the strong transversality condition. We say that X X 1 (M) is topologically stable in X 1 (M) if for any ε > 0, there is δ > 0 such that for any Y X 1 (M) with d C 0(X, Y ) < δ, there is a semiconjugacy (h, τ) from Y to X satisfying d(h(x), x) < ε for all x M. It is easy to see that a topologically stable vector field X X 1 (M) satisfies the orbital inverse shadowing property with respect to the class T h. Thus we have the following Corollary which is the main result obtained by Moriyasu et al in [13].

5 ORBITAL INVERSE SHADOWING VECTOR FIELDS 217 Corollary. The C 1 interior of the set of all topologically stable C 1 vector fields on M is characterized as the set of all vector fields satisfying Axiom A and the strong transversality condition. Before giving the proof of the main theorem, we are going to construct a smooth vector field X IS h (M) which is not topologically stable. This implies that the concept of topological stability for vector fields is surely stronger than that of inverse shadowing. Our construction is through the suspension. For any diffeomorphism f Diff 1 (M), there is a corresponding suspension flow S f X 1 (S M ) of f, where S M is the suspension of M; i.e., the cartesian product of M with the unit circle S 1. Then the followings are straightforward: (1) The suspension flow S f of f Diff 1 (M) is topologically stable if and only if for any ε > 0, there is δ > 0 such that for any g Diff 1 (M) with d C 0(f, g) < δ, there is a semiconjugacy h from g to f satisfying d(h(x), x) < ε for all x M. (2) The suspension flow S f of f Diff 1 (M) has the inverse shadowing property with respect to the class T h if and only if f has the inverse shadowing property with respect to the class T h (for the notion of the inverse shadowing property f, see [11]). The property for f in the previous result (1) is a little different from the original definition of topological stability for homeomorphisms. In the original definition, the perturbation should be a homeomorphism instead of a diffeomorphism. However these two notions are pairwise equivalent if the phase space is the unit circle S 1. More precisely, the following result can be proved by the techniques in [20]. Let f Diff 1 (S 1 ). Then the following are equivalent: (i) f is topologically conjugate to a Morse-Smale diffeomorphism. (ii) f is topologically stable. (iii) for any ε > 0, there is δ > 0 such that for any g Diff 1 (S 1 ) with d C 0(f, g) < δ, there is a semiconjugacy h from g to f satisfying d(h(x), x) < ε for all x S 1. Recently K. Lee and J. Park [11] proved that the shadowing property and the inverse shadowing property for homeomorphisms on the unit circle S 1 are pairwise equivalent. Now we give our construction. In [20], Yano constructed a homeomorphism f of S 1 which has the shadowing property but is not topologically stable. By a slight modification, f can be constructed as a diffeomorphism. By the result before, we know that the suspension flow S f of f is not topologically stable, but it has the inverse shadowing property with respect to the class T h. Let X (M) be the set of all X X 1 (M) with the property that there is a C 1 neighborhood U(X) X 1 (M) of X such that for every Y U(X), whose singularities and periodic orbits are hyperbolic. Denote X (M) all the systems X X (M) with the property that there is a C 1 neighborhood U(X) of X such that

6 218 KEONHEE LEE, ZOONHEE LEE AND YONG ZHANG for each Y U(X), the stable manifolds and the unstable manifolds of singularities and periodic orbits of Y t are all transversal. Recently it is proved by S. Gan [5] that X X (M) if and only if X satisfies Axiom A and the strong transversality condition. Very recently Y. Han and K. Lee [6] proved that every structurally stable vector field satisfies the inverse shadowing property with respect to the class T c. Therefore we get the following inclusions: X (M) = SS(M) IS c (M) OIS c (M) OIS (M), where SS(M) denotes the set of structurally stable vector fields. To prove the main theorem, it is enough to show the following theorem: Theorem. OIS (M) X (M). The proof of the above theorem is completed by the following three propositions. The techniques in our proof are similar to those in [1]. Proposition A. OIS (M) X (M). Proposition B. Let X OIS (M), p Sing(X) and q Sing(X) P O(X t ). Then the stable manifold of p and the unstable manifold of q are transverse. Proposition C. Let X OIS (M) and let γ, γ P O(X t ). Then the stable manifold of γ and the unstable manifold of γ are transverse.. Some Lemmas For each vector field X X 1 (M), we denote the generated flows by X t. Throughout this paper, let Sing(X) be the set of all singularities of X, and let P O(X t ) be the set of all periodic orbits (which are not singularities) of the generated flow X t. We say that p Sing(X) is hyperbolic if the linear map D p X : T p M T p M has no eigenvalue λ with Re(λ) = 0. For a hyperbolic singularity p, we define the stable manifold W s (p, X t ) and the unstable manifold W u (p, X t ) of p as following; W s (p, X t ) = {x M : d(x t (x), p) 0, as t }, W u (p, X t ) = {x M : d(x t (x), p) 0, as t }. A point x M is called a non-wandering point of X if for any neighborhood U of x in M, there is t 1 such that X t (U) U. The set of all non-wandering points of X is denoted by Ω(X t ). Clearly, Sing(X) P O(X t ) Ω(X t ). Hereafter, we assume that the exponential map exp p : T p M(1) M is well defined for all p M, where T p M(1) = {v T p M : v 1}. Let B ε (x) = {y M : d(x, y) ε} (ε > 0). To prove the main theorem, we need the following results in [1].

7 ORBITAL INVERSE SHADOWING VECTOR FIELDS 219 LEMMA.1.([1, Lemma 1.1.]) Let X X 1 (M) and p Sing(X). Then for every C 1 neighborhood U(X) X 1 (M) of X, there are δ 0 > 0 and ε 0 > 0 such that if O δ : T p M T p M is a linear map with O δ D p X < δ < δ 0, then there is Y δ U(X) satisfying Y δ (x) = { (D exp 1 p (x) ) O δ exp 1 p (x) if x B ε0 /(p), X(x) if x / B ε0 (p). Furthermore, d C 1(Y δ, Y 0 ) 0 as δ 0. Here Y 0 is vector field for O δ = D p X. Let X X 1 (M). For every x M \ Sing(X), put ˆΠ x = (SpanX(x)) T x M, Π x,r = exp x ( ˆΠ x,r ) and Π x = Π x,1, where, ˆΠ x,r = {v ˆΠ x : v < r} for r > 0. Then, for given x = X t0 (x)(t 0 > 0), there are r 0 > 0 and a C 1 map τ : Π x,r0 R such that X τ(y) (y) Π x for y Π x,r0 ) and τ(x) = t 0. The flow X t uniquely defines the Poincaré map f : Π x,r0 Π x by f(y) = X τ(y) (y) for y Π x,r0. The map is C 1 embedding whose image is interior to Π x if r 0 is small. We denote the set of all C 1 embeddings from Π x,r to Π x (r > 0) by Emb 1 (Π x,r, Π x ) and topologize it by using the C 1 topology. If X t (x) x for 0 < t t 0 and r 0 is sufficiently small, then (t, y) X t (y) C 1 embeds for 0 < r r 0. The image {(t, y) R Π x,r : 0 t τ(y)} {X t (y) : y Π x,r and 0 t τ(y)} is called a t 0 -time length flow box and is denoted by F x (X t, r, t 0 ). For ε > 0, let N ε (Π x,r ) be the set of all diffeomorphisms ϕ : Π x,r Π x,r such that supp(ϕ) Π x, r and d 2 C1(ϕ, id) < ε. Here d C 1 is the usual C 1 metric, id : Π x,r Π x,r is the identity map and the support of ϕ is the closure of the set where it differs from id. LEMMA.2.([1, Lemma 1.2.]) Let X X 1 (M). Suppose X t (x) x for 0 < t t 0 (x / Sing(X)), and let f : Π x,r0 Π x (x = X t0 (x)) be the Poincaré map (r 0 > 0 is sufficiently small). Then, for every C 1 neighborhood U(X) X 1 (M) of X and 0 < r r 0, there is ε > 0 with the property that for every ϕ N ε (Π x,r ), there exists Y U(X) satisfying { Y (y) = X(y) if x / Fx (X t, r, t 0 ) f Y (y) = f ϕ(y) if y Π x,r. Here f Y : Π x,r Π x is the Poincaré map defined by Y t. Remark.3. Under the same notation and assumption of Lemma.2, let Y δ U(X) be given by Lemma.2 for ϕ δ N ε (Π x,r ) (δ > 0). If ϕ δ ϕ as

8 220 KEONHEE LEE, ZOONHEE LEE AND YONG ZHANG δ 0 with respect to the C 1 topology, then by the construction of Y δ, we have d C 1(Y δ, Y ) 0 as δ 0. Let X X 1 (M) and suppose p γ P O(X t )(X T (p) = p, T > 0). If f : Π p,r0 Π p is the Poincaré map (r 0 > 0), then f(p) = p. We say that γ is hyperbolic if p is a hyperbolic fixed point of f. If γ P O(X t ) is hyperbolic, then the stable manifold W s (γ, X t ) and the unstable manifold W u (γ, X t ) of γ are defined by the usual way. Let γ, γ P O(X t ) be hyperbolic. We say that γ is transverse to γ if for any x W s (γ, X t ) W u (γ, X t ), T x M = T x W s (γ, X t ) + T x W u (γ, X t ). LEMMA..([1, Lemma 1.3.])Let X X 1 (M), p γ P O(X t ) (X T (p) = p) and f : Π p,r0 Π p be as above, and let U(X) X 1 (M) be a C 1 neighborhood of X and 0 < r r 0 be given. Then there are δ 0 > 0 and 0 < ε 0 < r 2 such that for a linear isomorphism O δ : ˆΠ p ˆΠ p with O δ D p f < δ < δ 0, there is Y δ U(X) satisfying (i) Y δ (x) = X(x) if x / F p (X t, r, T ), (ii) p γ P O(Yt δ ), (iii) { expp O g Y δ(x) = δ exp 1 p (x) if x B ε0 /(p) Π p,r f(x) if x / B ε0 (p) Π p,r, where g Y δ : Π p,r Π p is the Poincaré map of Yt δ. Furthermore, let Y 0 be the vector field for O δ = D p f. Then we have (iv) d C 1(Y δ, Y 0 ) 0 as δ Proof of Main Theorem In this section, we will prove Propositions A-C. Proof of Proposition A. The proof is divided into two cases. Case 1. we prove Proposition A for singularities. Let X OIS (M). Suppose that there is an eigenvalue λ of D p X with Re(λ) = 0 for some p Sing(X). By Lemma.1, for any C 1 neighborhood U(X) OIS (M) of X, there are δ 0 > 0 and ε 0 > 0 such that for every linear isomorphism O δ : T p M T p M with O δ D p X < δ < δ 0, there is Y δ U(X) satisfying Y δ (x) = { (D exp 1 p (x) exp p) O δ exp 1 p (x) if x B ε0 /(p) X(x) if x / B ε0 (p). Let Y 0 U(X) be as above for O δ = D p X and denote Y 0 by Y. For 0 < ε < ε 0 16, let 0 < δ < min{δ 0, ε} be as in the definition of OIS(M) of Y t. Pick 0 < δ < δ and a linear isomorphism O δ : T p M T p M whose any eigenvalue has a non-zero real part such that if O δ D p X < δ then d C 1(Y δ, Y ) < δ. Then Yt δ T d (δ, 1, Y ) and p is a hyperbolic singularity of Y δ. The restriction Yt δ Bε0 /(p) can be regarded

9 ORBITAL INVERSE SHADOWING VECTOR FIELDS 221 as the flow induced from the hyperbolic linear vector field O δ exp 1 p (B ε0 /(p)) with respect to the exponential coordinates. Since d C 1(Y, Y δ ) < δ, that is (δ, 1)-method, therefore for any x M, there is a y M such that d H (O(x, Y t ), O(y, Y δ t )) < ε. By the existence of the λ, we can take z M such that p / B ε (O(z, Y t )) B ε0 /8(p) (by reducing ε if necessary). Here B ε (A) = x A B ε(x) for A M. Let w be an orbital inverse shadowing point of O(z, Y t ). We have O(w, Yt δ ) B ε (O(z, Y t )). This is a contradiction since Y δ t hyperbolic linear vector field O δ exp 1 p Bε0 /(p) is regarded as the flow induced from the (B ε0 /(p)) ; i.e., p N ε (O(w, Y δ t )) but p / N ε (O(z, Y t )). Case 2. we prove Proposition A for periodic orbits. We prove for periodic orbits. Let U(X) OIS (M) be a C 1 neighborhood of X and pick p γ P O(X t )(X T (p) = p, T > 0). The flow X t defines the Poincaré map f : Π p,r0 Π p, (for some r 0 > 0). By assuming that there is an eigenvalue λ of D p f with λ = 1, we shall derive a contradiction. Let δ 0 > 0 and 0 < ε 0 < r 0 be given by Lemma. for the U(X). Then, for every linear isomorphism O δ : ˆΠ p ˆΠ p with O δ D p f < δ < δ 0, there is Y δ U(X) such that Y δ (x) = X(x) if x / F p (X t, r 0, T ), g Y δ(x) = { d C 1(Y δ, Y 0 ) 0 as δ 0. exp p O δ exp 1 p (x) if x B ε 0 (p) Π p,r0 f(x) if x / B ε0 (p) Π p,r0, Denote Y 0 by Y. For 0 < ε < ε0 16, let 0 < δ < min{δ 0, ε} be as in the definition of OIS(M) of Y t. Take a 0 < δ < δ and a hyperbolic linear isomorphism O δ : ˆΠ p ˆΠ p such that if O δ D p X < δ then d C 1(Y, Y δ ) < δ. Then Yt δ T d (δ, 1, Y ) and γ is a hyperbolic periodic orbit of Yt δ. Therefore for any x M, there is a y M such that d H (O(x, Y t ), O(y, Y δ t )) < ε. Note that g Y δ (p) = p and the restriction g Y δ B ε0 /(p) Π p,r0 is regarded as the hyperbolic linear isomorphism O δ exp 1 p (B ε0 /(p)) Π p,r 0 with respect to the exponential coordinates. Since λ = 1, we may take z Π p,r0 such that (2) p / B 2ε ({g i Y (z) : i Z}) B ε0 /(p) Π p,r0. Let w be an orbital inverse shadowing point of O(z, Y t ). Set w = Yt δ (w) Π p,r 0, where t = min{ t : Yt δ (w) Π p,r0 }. Since d H (O(z, Y t ), O(w, Y δ t )) < ε.

10 222 KEONHEE LEE, ZOONHEE LEE AND YONG ZHANG we have g i Y δ (w ) N ε ({gy i (z) : i Z}). But g Y δ B ε0 /(p) Π p,r0 is hyperbolic linear isomorphism, we get p N ε ({g i Y δ (w )}). This fact contradicts to (2). Proof of Proposition B. Before the proof of Proposition B, we give the following lemma. Lemma 5.1. Let X X 1 (M). Suppose that p Sing(X) P O(X t ) and q Sing(X) P O(X t ) are hyperbolic and p q. Let x W s (p, X t ) W u (q, X t ). Then there exist ε > 0 and a neighborhood B η (x) of x satisfying: for any y B η (x) s [resp. y B η (x) u ], there exists t > 0 [resp. t < 0] such that d H (X t (y), O(x, X t ) ε. Here s [resp. u )] is the connected component of W s (p, X t ) [resp. W u (q, X t ))] in B η (x) containing x. Proof. We will prove the case that q and p are singularities. The proof of the other cases are similar. Let r 0 > 0 be a small constant such that B r0 (p) is a local chart about p. Let C u (r 0 /2) = B r0/2(p) Wloc u (p). Since x W s (p, X t ) W u (q, X t ), we have Thus C u (r 0 /2) O(x, X t ) =. d H (C u (r 0 /2), O(x, X t )) > 0. Set ε 1 = d H (C u (r 0 /2), O(x, X t ))/2. By the hyperbolicity of p, there exists a neighborhood B η1 (x) of x such that for any y B η1 (x) s, there exists t > 0 with X t (y) N ε1 (C u (r 0 /2)). Thus we have d H (X t (y), O(x, X t )) d H (C u (r 0 /2), O(x, X t )) d H (X t (y), C u (r 0 /2)) 2ε 1 ε 1 = ε 1. Similarly we can choose ε 2 > 0 and η 2 > 0 such that for any y B η2 (x) u, there exists t < 0 satisfying d H (X t (y), O(x, X t ) ε 2. Let ε = min{ε 1, ε 2 } and η = min{η 1, η 2 }. It is easy to see that ε and η satisfy the conclusion. Let us start the proof of Proposition B. Let X OIS (M). Suppose that x W s (p, X t ) W u (q, X t ), p Sing(X), q Sing(X) P O(X t ) and T x M T x W s (p, X t ) + T x W u (q, X t ). We may assume that x is very near p. Take r 0 > 0 small enough so that there are the Poincaré maps f : Π x,r0 Π X1(x) and f : Π X 1(x),r 0 Π x, {X t (x) : t < 0} Π x,r0 = and {X t : t < 1} Π X 1(x),r 0 =, W u 2r 0 (q, X t ) (Π x,r0 Π X 1 (x),r 0 ) =,

11 ORBITAL INVERSE SHADOWING VECTOR FIELDS 223 where W u 2r 0 (q, X t ) is the local unstable manifold of q. Let V s (x) be the connected component of W s (p, X t ) Π x containing x, and V u (x) be the connected component of W u (q, X t ) Π x containing x. Clearly, we have 0 dim V s (x) dim Π x and 0 dim V u (x) dim Π x. If dim V s (x) = dim Π x or dim V u (x) = dim Π x, then there is nothing to prove. Notice that ˆΠ x T x V s (x)+t x V u (x). We shall divide the proof into the following two cases: Case (1.1) dim V s (x) + dim V u (x) < dim Π x, Case (1.2) dim V s (x) + dim V u (x) dim Π x. Proof for case (1.1). Put Vr u (x) be the connected component of V u (x) B r (x) containing x, and Vr s (x) be the connected component of V s (x) B r (x) containing x for r > 0. Choose 0 < r < r 0 with the following properties: for every δ > 0, there exists ϕ δ N ε(δ) (Π x,r0 ) satisfying (3) ϕ δ (V u r (x)) V s (x) =, Vr u(x) = [ 0 t t 1 X t (W2r u 0 (q, X t ))] Π x,r0 B r (x), where t 1 > 0 such that X t1 (x) Wr u 0 (q, X t ), Vr s (x) = [ 0 t t 2 X t (W2r s 0 (p, X t ))] Π x,r0 B r (x), where t 2 > 0 such that X t2 (x) Wr s 0 (p, X t ), B r ({X t (x) : t 0}) Π x,r0 = B r (x) Π x,r. For any small enough δ > 0, we can choose a vector field Y δ given by Lemma.2 from (3); i.e., Y δ satisfies the followings: Y δ (y) = X(y) if y / F x (X t, r 0, 1), g(y) = f ϕ δ (y) if y Π x,r0, d C 1(X, Y δ ) < δ, where g : Π x,r0 Π X1(x) is the Poincaré map induced by Yt δ. We may assume that F x (Yt δ, r 0, 1) Wr s 0 (p, Yt δ ) = for sufficiently small r 0. For vector field X and point x W s (p, X t ) W u (q, X t ), we have the numbers ε > 0 and η > 0 satisfying the conclusion of Lemma 5.1. Take ε > 0 with ε < min{ ε 2, η 2, r 2 }. Let 0 < δ = δ(ε) < ε be the corresponding number with repect to ε in the definition of OIS(M) of X. So Y δ constructed as the above is an element of T d (δ, 1, X t ). For simplicity, we denote Y δ by Y. So, there is w M such that d H (O(x, X t ), O(w, Y t )) < ε. Since X(y) = Y (y) if y / F x (X t, r 0, 1), p is singularity and q is also singularity or periodic orbit if Y t. Thus O(w, Y t ) through B η (x). Take w in the orbit of O(w, Y t ) which is in B η (x) Π x,r0. If w Vr u (x), then by the properties in the construction of Y and by Lemma 5.1, d H (O(x, X t ), O(w, Y t )) ε > ε.

12 22 KEONHEE LEE, ZOONHEE LEE AND YONG ZHANG If w is not in V u r 0 (x), then similarly we have d H (O(x, X t ), O(w, Y t )) ε > ε. Consequently we get a contradiction, and so complete of the proof of case (1.1). Proof for case (1.2). Take a C 1 neighborhood U(X) OIS(M). Let V s (X 1 (x)) [resp. V u (X 1 (x))] be the connected component of W s (p, X t ) Π X 1 (x) [resp. W u (q, X t ) Π X 1 (x)] containing X 1 (x). Let f : Π X 1 (x),r 0 Π x be the Poincaré map. For the above U(X), let µ = µ(u(x)) > 0 be given by Lemma.2. Since ˆΠ x T x V s (x) + T x V u (x) and dim V s (x) + dim V u (x) dim Π x, there are 0 < r 1 < r 0, ϕ N µ (Π X 1(x),r 0 ) and a submanifold V (X 1 (x)) such that Π X 1 (x),r 0 V s (X 1 (x)) B r1 (X 1 (x)) V (X 1 (x)), ϕ(v u (X 1 (x)) B r1 (X 1 (x))) V (X 1 (x)) and ϕ(x 1 (x)) = X 1, dim V s (x) + dim V u (x) dim Π x < dim V (X 1 (x)) < dim Π X 1 (x). Let Y U(X) and g = f ϕ : Π X 1 (x),r 0 Π x (since g(x 1 (x)) = x) be given by Lemma.2. Let V s (x, Y t ) [resp. V u (x, Y t )] be the connected component of W s (p, Y t ) Π x [resp. W u (q, Y t ) Π x ] containing x. If we put V (x, Y t ) = f (V (X 1 (x))), then we get dim V (x, Y t ) < dim Π x. It is easy to choose 0 < r 2 < r 0 satisfying V u (x, Y t ) B r2 (x) V (x, Y t ) and V s (x, Y t ) B r2 (x) V (x, Y t ). By the choice of r 0, we can see that the map f : Π x,r0 Π X1(x) is also a Poincaré map for Y t, q Sing(Y ) P O(Y t ), X t (x) = Y t (x) for t 0. Since Y (y) = X(y) for y / F X 1(x)(X t, r 0, 1), we have W s 2r 0 (p, X t ) = W s 2r 0 (p, Y t ) and W u 2r 0 (q, X t ) = W u 2r 0 (q, Y t ). As in the proof of proposition B in [1], by applying Lemma.2 for Y and f, for any δ > 0 we can obtain Z δ U(X) and r < r 2 such that Z δ is perturbation of ϕ δ N ε(δ) (Π x,r0 ) satisfying ϕ δ (V u r (x, Y t)) V (x, Y t ) =, Z δ (y) = Y (y) if y / F x (Y t, r 0, 1), g (y) = f ϕ δ (y) if y Π x,r0, d C 1(Y, Z δ ) < δ, where g : Π x,r0 Π Y1 (x) is the Poincaré map induced by Zt δ. For C 1 vector field Y and x W s (p, Y t ) W u (q, Y t ), we have numbers ε > 0 and η satisfying Lemma 5.1. Choose a sufficiently small 0 < ε < min{ ε 2, r 2, η 2, µ 2 }, let 0 < δ < ε be as in the definition of OIS(M) of Y t. So Z δ constructed as the above is an element of T d (δ, 1, Y t ). For simplicity, we denote Z δ by Z. By the

13 ORBITAL INVERSE SHADOWING VECTOR FIELDS 225 definition of orbital inverse shadowing for the flow Y t with respect to the point x, there is a y M such that d H (O(x, Y t ), O(y, Z t )) < ε. We can derive a contradiction as in the case (1.1), by applying Lemma 5.1. This completes the proof of Proposition B. Proof of Proposition C. Suppose that γ, γ P O(X t ) are hyperbolic and x W s (γ, X t ) W u (γ, X t ). Fix p γ (X T (p) = p), and let r 0 > 0 be sufficiently small so that we can define the Pioncaré map f : Π p,r0 Π p. Since p is hyperbolic, there are a Df-invariant splitting ˆΠ p = E s p E u p and two constants C > 0, 0 < λ < 1 such that Df m E s p < Cλ m and Df m E < Cλ m for all m 0. Let W σ p u r (p, f) be the connected component of W σ (γ, X t ) Π p,r containing p for σ = s, u and 0 < r r 0. Suppose that x Wr s (p, f) \ intw s 0/2 r (p, f), and let T 0/2 > 0 be the number with f(x) = X T (x) and take 0 < r 1 < r0 such that F p(x t, r 1, T ) F x (X t, r 1, T ) =. Before continuing our proof, we will cite the following result from [1]. Lemma 5.2. ([1, Lemma.1.]) Under the above notation, for every C 1 neighborhood U(X) of X, there are 0 < ε 0 < r 0 and Y U(X) satisfying (i) Y (y) = X(y) if y / F p (X t, r 1, T ) F x (X t, r 1, T ), (ii) γ, γ P O(Y t ) and Y T (p) = p γ, (iii) { expp D g(y) = p f exp 1 p (y) if y B ε0/(p) Π p,r0 f(y) if y / B ε0 (p) Π p,r0, (iv) g(p) = p, x W s r 0 (p, g) and T x W s r 0 (p, g) = T x W s r 0 (p, f), (v) T x W u (γ, Y t ) = T x W u (γ, X t ). Here g : Π p,r0 Π p is the Poincaré map of Y t and Wr σ 0 (p, g) is the connected component of W σ (γ, Y t ) Π p,r0 containing p (σ = s, u). Put Ex σ (ε) = {v Ex σ v ε} for ε > 0 (σ = s, u), and g Emb 1 (Π p,r0, Π p ), p = g(p) Π p and ε 0 > 0 be given by Lemma 5.2. Then exp p (Ep σ ( ε 0 )) Wr σ 0 (p, g) and dim exp p (Ep σ ( ε 0 )) = dim Wr σ 0 (p, g) for σ = s, u since ε 0 is small. For convenience, we denote exp(ep σ (ε)) by Wε σ (p, g) for σ = s, u and 0 < ε ε0. Let X OIS (M), suppose that γ, γ P O(X t ) are hyperbolic and x W s (γ, X t ) W u (γ, X t ). Let p γ(x T (p) = p, T > 0) and f : Π p,r0 Π p (r 0 > 0) be as before. We may assume that x Wr s (p, f) \ intw s 0/2 r 0/2 (p, f), W2r u 0 (γ, X t ) Π p,r0. Here, Wr u 0 (γ, X t ) is the local unstable manifold of γ. Fix a C 1 neighborhood U(X) OIS (M) of X, and let 0 < ε 0 < r 0, Y U(X) and g be given by Lemma 5.2. Thus T x Wr s 0 (p, g) = T x Wr s 0 (p, f), W2r u 0 (γ, X t ) = W2r u 0 (γ, Y t ), and X t (x) = Y t (x) for all t 0. Clearly, 1 dim Wr s 0 (p, g) dim Π p and 1 dim(wr u 0 (γ, Y t ) Π p ) dim Π p. If dim Wr s 0 (p, g) = dim Π p or dim(wr u 0 (γ, Y t ) Π p = dim Π p,

14 226 KEONHEE LEE, ZOONHEE LEE AND YONG ZHANG then the conclusion is clear. Pick l > 0 so large that g l 1 (x) Wε s 0/8 (p, g) and set C u (g l (x)) be the connected coponent of W u (γ, Y t ) Π p containing g l (x). To simplify the notation, denote g l (x) by x. Then we get exp 1 p (C u (x)) ˆΠ p and T x C u (x) = T x (W u (γ, Y t ) Π p ). For a linear subspace E of ˆΠ p and ν > 0, let E ν (x) = {v + exp 1 p (x) v E with v ν} be a piece of an affine space running parallel to E. Let T, T > 0 be numbers with Y T (g 1 (x)) = (x) and Y T (x) = g(x), respectively. Choose a linear subspace E ˆΠ p and 0 < ν 0 < ε 0 /8 such that for every 0 < ν ν 0, exp p (E ν(x)) B ε0 /(p), T x exp p (E ν 0 (x)) = T x C u (x), (F g 1 (x)(y t, ν 0, T ) F x (Y t, ν 0, T )) γ =, {Y t (g 1 (x)) : t < 0} F g 1 (x)(y t, ν 0, T ) = and {Y t (x) : t < 0} F x (Y t, ν 0, T ) =, B ν0 ({Y t (x) : t 0}) F x (Y t, ν 0, T ) Π p,r0 = B ν0 (x) F x (Y t, ν 0, T ) Π p,ro, g i (W s r 0 (p, g) B ν0 (g 1 (x))) B ν0 (g 1 (x)) = for i 1, Y t (C u (g 1 (x))) B ν0 (g 1 (x)) = for all t > 0. Here C u (g 1 (x)) is the connected coponent of W u (γ, Y t ) Π p B ν0 (g 1 (x)) containing g 1 (x). Lemma 5.3. ([13, Lemma.2.]) Fix a C 1 neighborhood U(Y ) U(X) of Y. Then there are 0 < ν 1 < ν0 and Y U(Y ) such that (i) Y (y) = Y (y) if y / F g 1 (x)(y t, ν 0, T ) (ii) Y T (g 1 (x)) = x, (iii) exp p (E ν 1 (x)) W u (γ, Y t ) Π p and T x exp p (E ν 1 (x)) = T x (W u (γ, Y t ) Π p ). Remark 5.. (i) We see that W2r u 0 (γ, Y t ) = W2r u 0 (γ, Y t ) and Y t (x) = Y t (x) for t 0. (ii) Let g : Π p,r0 Π p be the Pioncaré map induced by Y t. Then by Lemma 5.3(i), γ P O(X t ) and g (y) = g(y), if y Π p,r0 \ B ν0 (g 1 (x)). Thus g (p) = g(p). By lemma 5.3(ii), g i (x) = g(x) for all i 0. (iii) By the perturbation used in the above lemma, Wε s 0 /(p, g) may be deformed near g j (x) Wε s 0 /(p, g) for some j > 0. We denote the deformed manifold by Wε s 0 / (p, g ). Then x Wε s 0 / (p, g ) and T x Wε s 0 / (p, g ) = T x Wε s 0 /(p, g). If exp p (E ν 1 (x)) does not meet Wε s (p, 0/ g ) trasverse at x, then the piece E ν(x) of the affine space is not transversal with respect to the local linear stable manifold Ep( s ε 0 ) of p. To simplify the notation, denote Y, g and Wε s 0 / (p, g ) by Y, g, and Wε s (p, g), respectively. If exp 0/ p(e ν 1 (x)) dose not meet Wε/ s (p) transversely at x then there is 0 < r < ν 1 2 such that

15 ORBITAL INVERSE SHADOWING VECTOR FIELDS 227 such that for every δ > 0, there is ψ δ N ε(δ) (Π x,2ν1 ) satisfying { ψδ (exp p (E ν 1 (x)) B r (x)) W s r 0 (p) = ψ δ (y) = y if y / Π x,ν1 exp p (E ν 1 (x)) B r (x) = [ 0 t t 2 Y t (W u 2r 0 (γ, Y t ))] Π p,r0 B r (x), where Y t2 (x) Wr u 0 (γ, Y t ). For C 1 vector field Y and x W s (γ, Y t ) W u (γ, Y t ), we have numbers ε > 0 and η > 0 satisfying the conclusion of Lemma 5.1. Choose a sufficiently small 0 < ε < min{ ε 2, r }, let 0 < δ < ε be the corresponding number with respect 2, η 2 to ε in the definition of OIS(M) of Y t. Then by Lemma.2, we can construct Z U(Y ) for the above perturbation such that Z(z) = Y (z) if z / F x (Y t, 2ν 1, T ), g(z) = g ψ δ (z) if z Π x,2ν1, d C 1(Y, Z) < δ. Here g : Π x,2ν1 Π g(x) is the Poincaré map induced by Z t. Because Z is in T d (δ, 1, Y ), there is y M such that d H (O(x, Y t ), O(y, Z t ) < ε. Because Z(y) = Y (y) for y / F x (Y t, 2ν 1, T ), W s 2r 0 (γ, Z t ) = W s 2r 0 (γ, Y t ) and W u 2r 0 (γ, Z t ) = W u 2r 0 (γ, Y t ). Similar to the proof of Proposition B, we can get a contradiction. This completes the proof of Proposition C. Remark 5.5. The notion of weak inverse shadowing for homeomorphisms was introduced in [1] and it was shown that the weak shadowing property is C 0 generic in the space of homeomorphisms with the C 0 topology. We can introduce the notion of weak inverse shadowing property for flows as follows : we say that the system (1) has the weak inverse shadowing property with respect to the class T α (α = a, c, h, d) if for any (δ, T )-method Ψ T α (δ, T, Φ) and any point x M, there is a y M for which {Ψ t (y) : t R} N ε ({Φ t (x) : t R}). It is proved in [2] that the C 1 interior of the set of diffeomorphisms with the weak inverse shadowing property (with respect to the class T d ) coincides with the set of Ω-stable diffeomorphisms. We can easily show that the C 1 interior of the set of vector fields with the weak inverse shadowing property (with respect to the class T d ) is contained in the set of Ω-stable vector fields. We can easily see that every irrational flows on the 2-dimensional torus has the weak inverse shadowing property (with respect to the class T d ), but it is not Ω-stable. However we do not know yet whether the C 1 interior of the set of vector fields with the weak inverse shadowing property (with respect to the class T d ) coincides with the set of Ω-stable vector fields.

16 228 KEONHEE LEE, ZOONHEE LEE AND YONG ZHANG References [1] T. Choi, S. Kim and K. Lee, Weak inverse shadowing and genericity, preprint. [2] T. Choi, K. Lee and Y. Zhang,Characterizations of Ω-stability and structural stability via inverse shadowing, preprint. [3] R. Corless and S. Pilyugin, Approximate and real trajectories for generic dynamical systems, J. Math. Anal. Appl. 189 (1995), [] P. Diamond, Y. Han and K. Lee, Bishadowing and hyperbolicity, International J. of Bifurcation and Chaos 12 (2002), [5] S. Gan, Another proof for C 1 stability conjecture for flows, Sci. China Ser. A,1 (1998), [6] Y. Han and K. Lee, Inverse shadowing for structurally stable flows, Dynamical Systems: An International Journal 20 (2005), to appear. [7] S. Hayashi, On the solution of C 1 stability conjecture for flows, Ann. Math. 353 (1997), [8] P. Kloeden and J. Ombach, Hyperbolic homeomorphisms and bishadowing, Ann. Polon. Math. XLV (1997), [9] P. Kloeden, J. Ombach and A. Pokrovskii, Continuous and inverse shadowing, Functional Differential Equations 6 (1999) [10] K. Lee, Continuous inverse shadowing and hyperbolicity, Bull. Austral. Math. Soc. 67 (2003), [11] K. Lee and Z. Lee, Inverse shadowing for expansive flows, Bull. Korean. Math. Soc. 0 (2003), [12] K. Lee and J. Park, Inverse shadowing of cirlce maps, Bull. Austral. Math. Soc. 69 (200), [13] R. Mañé, A proof of the C 1 stability conjecture, Publ. Math. I.H.E.S., 66 (1988), [1] K. Moriyasu, K. Sakai and N. Sumi, Vector fields with topological stability, Trans.Amer.Math.Soc.353(2001), [15] S. Pilyugin, Shadowing in structurally stable flows, J. Differential Equations 10 (1997), [16] S. Pilyugin, Inverse shadowing by continuous methods, Discrete and Continuous Dynamical Systems 8 (2002), [17] C. Robinson, Structural stability of vector fields, Ann. of Math. 99 (197), [18] K. Sakai, Pseudo orbit tracing property and strong transversality of diffeomorphisms on closed manifolds, Osaka J. Math. 31 (199), [19] L. Wen, On the C 1 stability conjecture for flows, J. Differential Equations 129 (1996), [20] K. Yano, Topologically stable homeomorphisms of the circle, Nagoya Math. J. 79 (1980), Keonhee Lee: Department of Mathematics, Chungnam National University, Daejeon, , Korea address: khlee@math.cnu.ac.kr Zoonhee Lee: Department of Mathematics, Chungnam National University, Daejeon, , Korea address: ljh3967@hanmail.com Yong Zhang: Department of Mathematics, Chungnam National University, Daejeon, , Korea; Department of Mathematics, Suzhou University, Suzhou, China address: yzhang@math.chungnam.ac.kr; yongzhang@suda.edu.cn