Lyapunov stable homoclinic classes for smooth vector fields

Abstract In this paper, we show that for generic C1, if a flow Xt has the shadowing property on a bi-Lyapunov stable homoclinic class, then it does not contain any singularity and it is hyperbolic.


Introduction
Let M be a compact smooth Riemannian manifold, and let Di (M) be the space of di eomorphisms of M endowed with the C topology. Hyperbolicity and stability have been important topics in di erentiable dynamical systems since they were introduced by Smale [ [2] showed that the set of di eomorphisms f : M → M satisfying Axiom A and the no-cycle condition is not dense in the space of Di (M).
If a di eomorphism f : M → M satis es Axiom A, then from the work of Smale [1], the nonwandering set Ω(f ) = n i= Λ i , where each Λ i is a basic set. If a basic set contains a hyperbolic periodic point, then it is a homoclinic class. In general, a homoclinic class is not hyperbolic even in a generic sense. For a C generic di eomorphism f : M → M, several extra conditions are imposed to obtain hyperbolicity of the homoclinic classes.
Let us give a short review of related results. Ahn et al. [3] proved that for generic C , if a di eomorphism f has the shadowing property on a locally maximal homoclinic class, then it is hyperbolic. Lee [4] proved that for generic C , if a di eomorphism f has the limit shadowing property on a locally maximal homoclinic class, then it is hyperbolic. Note that local maximality is quite a restrictive condition. Arbieto et al. [5] proved that for generic C , if a bi-Lyapunov stable homoclinic class is homogeneous and has the shadowing property, then it is hyperbolic. See [3,4,[6][7][8][9][10][11][12][13][14][15] for related results.
We want to extend some of the above results for ows, that is, for a C generic vector eld X ∈ X(M), a condition under which we can obtain hyperbolicity of homoclinic classes. Unfortunately, we cannot use the same arguments as in the di eomorphism case.
We say that a di eomorphism f satis es the star condition if there is a C neighborhood U(f ) ⊂ Di (M) such that for any g ∈ U(f ), every periodic point of g is hyperbolic. Aoki [16] and Hayashi [17] showed that if a di eomorphism f satis es the star condition, then it is Axiom A and the no-cycle condition, that is, Ω stable.
We say that a ow X t satis es the star condition if there is a C neighborhood U(X) ⊂ X(M) such that for any Y ∈ U(X), every critical point of Y is hyperbolic. From the results of Guchenheimer [18], the Lorenz attractor satis es the star condition, but it is not Ω-stable because the attractor contains a hyperbolic singular point. However, if a ow does not contain singularities and satis es the star condition, then it is Ω stable (see [19]).

Basic notions and main theorem
Let M be a compact n (≥ )-dimensional smooth Riemannian manifold, and let d be the distance on M induced from a Riemannian metric · on the tangent bundle TM, and denote by X(M) the set of C vector elds on M endowed with the C topology. Then, every X ∈ X(M) generates a C ow X t : M × R → M; that is, a C map such that X t : M → M is a di eomorphism satisfying X (x) = x, and X t+s (x) = X t (X s (x)) for all s, t ∈ R and x ∈ M. The ow of X will be denoted by Denote by Sing(X) the set of all singular points of X. A point x ∈ M is regular if x ∈ M\Sing(X). Denote by R(M) the set of all regular points of X. A point p ∈ M is periodic if there is π(p) > such that X π(p) (p) = p, where π(p) is the prime period of p. Denote by Per(X) the set of all closed orbits of X. Let Crit(X) = Sing(X) ∪ Per(X).
An increasing homeomorphism h : R → R with h( ) = is called a reparametrization of R. Denote by Rep(R) the set of reparametrizations of R. Fix ϵ > and de ne Rep(ϵ) as follows: For a closed X t -invariant set Λ ⊂ M, we say that X has the shadowing property on Λ if for any ϵ > , there is δ > satisfying the following property: given any δ- The point y ∈ M is said to be a shadowing point of ξ .
Let X t be the ow of X ∈ X(M), and let Λ be a X t -invariant compact set. The set Λ is called hyperbolic for X t if there are constants C > , λ > and a splitting Tx M = E s x ⊕ X(x) ⊕ E u x such that the tangent ow DX t : TM → TM leaves the continuous splitting invariant and Let γ be a hyperbolic closed orbit of a vector eld X ∈ X(M), and we de ne the stable and unstable manifolds of γ by Let X ∈ X(M), and let γ be a hyperbolic closed orbit of X t . A point x ∈ W s (γ) W u (γ) is called a transversal homoclinic point of X t associated to γ. The closure of the transversal homoclinic points of X t associated to γ is called the homoclinic class of X t associated to γ, and it is denoted by It is clear that H X (γ) is a compact, transitive, and X t -invariant set.
For two hyperbolic closed orbits γ and γ of X t , we say that γ and γ are homoclinic related, denoted . Note that if γ is a hyperbolic closed orbit of X t , then there exist a C neighborhood U(X) of X and a neighborhood U of γ such that for any Y ∈ U(X), there exists a unique hyperbolic closed orbit γ Y that equals t∈R Y t (U). The hyperbolic closed orbit γ Y is called the continuation of γ with respect to Y , and index(γ We say that Λ is bi-Lyapunov stable if it is Lyapunov stable for X and for −X. We say that a subset G ⊂ X(M) is residual if G contains the intersection of a countable family of open and dense subsets of X(M). In this case G is dense in X(M). A property "P" is said to be C -generic if "P" holds for all vector elds that belong to some residual subset of X(M). We write for C generic X ∈ X(M) in the sense that there is a residual set G ⊂ X(M) for any X ∈ G. In this paper, we prove the following theorem, which is an extension of a result of Arbieto et al. [5] for ows.
Theorem. For C generic X ∈ X(M), if a ow X t has the shadowing property on a bi-Lyapunov stable homoclinic class H X (γ), then H X (γ) ∩ Sing(X) = ∅ and H X (γ) is hyperbolic.

Proof of the Theorem
Let M be as previously, and let X ∈ X(M). We de ne the strong stable and unstable manifolds of a hyperbolic periodic point p respectively as follows: where Orb(p) is the orbit of p. If ϵ > , the local strong stable manifold is de ned as By the stable manifold theorem, there is an ϵ = ϵ(p) > such that W ss (p) = t≥ X −t (W ss ϵ(p) (X t (p))).
We can de ne the unstable manifolds similarly. If σ is a hyperbolic singularity of X, then there exists an Analogous de nitions hold for unstable manifolds.

. Transversal intersection and the absence of singularities
The following lemma states that there are transversal intersections between invariant manifolds of hyperbolic closed orbits and singularities. Proof. First, we assume that η ∈ H X (γ) ∩ Per(X). Let p ∈ γ and q ∈ η. Take ϵ = min{ϵ(p), ϵ(q)} and let < δ ≤ ϵ be given by the shadowing property according to ϵ. Since H X (γ) is transitive, there is x ∈ H X (γ) such that ω(x) = H X (γ). Then, there are t > and t > such that X t (x) ∈ B δ (p) and X t (x) ∈ B δ (q). Assume that t = t + k for some k > . Then, the sequence Since X t has the shadowing property on H X (γ), there is y ∈ M and an increasing homeomorphism h : where s i < t < s i+ and s −i < t < s −i+ for all t ∈ R and i ∈ Z. Then y ∈ W u ϵ (p) and there is τ > such that X τ (y) ⊂ W s ϵ (q). Thus, we have The other case is similar. Now, we assume that σ ∈ H X (γ) ∩ Sing(X). Let p ∈ γ. Take ϵ = min{ϵ(p), ϵ(σ)} and let < δ ≤ ϵ be given by the shadowing property according to ϵ. Since H X (γ) is transitive, there is x ∈ H X (γ) such that ω(x) = H X (γ). Then, there are t > and t > such that X t (x) ∈ B δ (σ) and X t (x) ∈ B δ (p). Assume that t = t + k for some k > . We can thus construct a δ-pseudo-orbit {(x i , t i ) : t i ≥ , i ∈ Z} ⊂ H X (γ) as follows: (ii) X t +i (x) = x i for i = , . . . , k − ; and (iii) X i (p) = x k+i for all i ≥ . Since X t has the shadowing property on H X (γ), as in the proof of previous arguments, we have W u (σ)∩W s (γ) ≠ ∅. The other case is similar.
We say that X is Kupka-Smale if every σ ∈ Crit(X) is hyperbolic, and their invariant manifolds intersect transversally. Denote by KS the set of all Kupka-Smale vector elds. It is known that KS ⊂ X(M) is a residual subset (see [20]).
Thus, we know that This is a contradiction, because X is a Kupka-Smale vector eld. If j ≥ i, then By the previous arguments, we have a contradiction. Thus, H X (γ) ∩ Sing(X) = ∅.

. Chain recurrent class and homoclinic class
For any x, y ∈ M, we say that x y if for any δ > , there is a nite δ-pseudo-orbit {(x i , t i ) : ≤ i < n} with n > such that x = x and d(X tn− (x n− ), y) < δ and a δ-pseudo-orbit {(z i , s i ) : ≤ i < m} with m > such that z = y and d(X sm− (z m− ), x) < δ. It is easy to see that gives an equivalent relation on the chain recurrent set CR(X). We denoted the equivalence class as C X (γ) = {x ∈ M : x γ and γ x} and called the chain recurrence class associated to γ. It is known that H X (γ) ⊂ C X (γ), but the converse is not true in general. We now summarize some results about homoclinic classes and chain recurrence classes.

Lemma 3.4.
There is a residual set G ⊂ X(M) such that every X ∈ G satis es: (a) the chain recurrence class C X (γ) = H X (γ) (see [21]); (b) if a closed orbit η ∈ H X (γ), then H X (γ) = H X (η) (see [22]); (c) H X (γ) = W s (γ) ∩ W u (γ) (see [22]); (d) W s (γ) is Lyapunov stable for −X and W u (γ) is Lyapunov stable for X (see [22]); (e) if H X (γ) is Lyapunov stable for X, then there is a C neighborhood U(X) of X such that for every Y ∈ U(X), is Lyapunov stable (see [23]); (f) there exist a C neighborhood U(X) of X and an interval of natural numbers [α, β] such that for every Y ∈ U(X), H Y (γ Y ) has closed orbits of every index in [α, β]; moreover, every closed orbit in H Y (γ Y ) has its index in that interval (see [24]).
Let X ∈ X(M) have no singularities and let N ⊂ TM be the sub-bundle such that the ber Nx at x ∈ M is the orthogonal linear subspace of X(x) in Tx M, that is, Nx = X(x) ⊥ . Here X(x) is the linear subspace spanned by X(x) for x ∈ M. Let π : TN → N be the projection along X, and let for v ∈ Nx and x ∈ M. Let Λ be a closed X t -invariant regular set. We say that Λ is hyperbolic if the bundle N Λ has a P X t -invariant splitting ∆ s ⊕ ∆ u and there exists an l > such that for all x ∈ Λ. Then, Doering [25] proved the following result, which is a method of proof for hyperbolicity.  (Nx(r)). Given any point x ∈ R(M) and t ∈ R, there are r > and a C map τ : N x,t → R with τ(x) = t such that X τ(y) (y) ∈ N X τ (x), for any y ∈ Nx,r . We de ne the Poincaré map as Let X ∈ X(M), and suppose p ∈ γ ∈ Per(X)(X π(p) (p) = p, where π(p) > is the prime period. If f : Np,r → Np is the Poincaré map (r > ), then f (p) = p. Note that γ is hyperbolic if and only if p is a hyperbolic xed point of f .
The following lemma states that by perturbation of vector elds, we can gain some control on eigenvalues of the Poincaré map. Lemma 3.6. Let p ∈ η ∈ H X (γ) ∩ Per(X) and let f : Np,r → Np(r > ) be the Poincaré map of X t . For any δ > , if the eigenvalue λ of Dp f is < λ < + δ, then there is g that is C close to f such that Dp g has an eigenvalue µ with µ ≤ − δ, where g is the Poincaré map associated to Y.
Proof. Let ∆p be the eigenspace corresponding to λ with index(p) = i, and let Np = ∆p ⊕ ∆ ⊥ p . For the splitting, we have Applying Gourmelon's result [26] (see also [5, Theorem 2.5]), we de ne the map T : [ , ] → Γ i as follows for t ∈ [ , ]. Then, we have and Thus, one can see that − δ The proof is complete.
For any δ > , we say that a point p ∈ γ ∈ Per(X) is δ-weak hyperbolic periodic if there is an eigenvalue λ of Dp f such that ( − δ) < |λ| < ( + δ), where f : Np,r → Np is the Poincaré map associated to X t .
Let X ∈ G and let H X (γ) be bi-Lyapunov stable with index(γ) = i( < i < dimM − ). Then, there is U(X) of X such that for any Y ∈ U(X), H Y (γ Y ) is bi-Lyapunov stable and every closed orbit in H Y (γ Y ) has the same index i. From this fact, we have the following result.
for any δ > there is p ∈ η ∈ H X (γ) ∩ Per(X) such that p is a δ weak hyperbolic periodic point. Then, there is an eigenvalue λ of Dp f such that − δ < |λ| < + δ, where f : Np,r → Np is the Poincaré map corresponding to the ow X t . Assume that < λ < + δ (the other case is similar). Let p ∈ γ and q ∈ η ∈ H X (γ) ∩ Per(X). Take x ∈ W ss (p) ∩ W uu (q) and choose a neighborhood U of q such that: Then, by [5, Theorem 2.5] and Lemma 3.6, there is g C close to f such that: (ii) it preserves the i strong stable manifold of qg ∈ η Y outside U; and (iii) W uu (pg) ∩ W ss (qg) ≠ ∅; where γ Y is the continuation of γ, η Y is the continuation of η, Y t is the ow corresponding to g, and pg ∈ γ Y . Using the λ-lemma, we have qg ∈ W uu (pg). Since H Y (γ Y ) is Lyapunov stable for Y, we have W u (γ Y ) ⊂ H Y (γ Y ), and so qg ∈ η Y ⊂ H Y (γ Y ). This is a contradiction. Since X ∈ G , if every η ∈ H X (γ) ∩ Per(X) has index i, then by Lemma 3.4, every η Y ∈ H Y (γ Y ) ∩ Per(Y) has index i.
By Proposition 3.3, H X (γ) ∩ Sing(X) = ∅. Then, we have the following lemma, which is a ow version of the result proved by Wang [27].

Lemma 3.8.
There is a residual set G ⊂ X(M) such that for any X ∈ G , if a homoclinic class H X (γ) is not Hyperbolic, then for any δ > , there is a periodic point q ∈ η ⊂ H X (γ) ∩ Per(X) such that η ∼ γ and q is a δ-weak hyperbolic periodic point.
Proof of the Theorem. Let X ∈ G ∩ G have the shadowing property on H X (γ). Suppose, by contradiction, that H X (γ) is not hyperbolic. Since X has the shadowing property on H X (γ), by Lemma 3.2 we have η ∼ γ, for all η ∈ H X (γ) ∩ Per(X). Then, by Lemma 3.8, for any δ > there is q ∈ η ∈ H X (γ) ∩ Per(X) such that q is a weak hyperbolic periodic point. This is a contradiction by Lemma 3.7.