Lyapunov stable homoclinic classes for smooth vector fields

Manseob Lee

doi:10.1515/math-2019-0068

Article Open Access

Lyapunov stable homoclinic classes for smooth vector fields

Manseob Lee

Published/Copyright: August 24, 2019

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

$Open Mathematics$

From the journal Open Mathematics Volume 17 Issue 1

Abstract

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

Keywords: homoclinic class; Lyapunov stable; shadowing; generic; hyperbolic

MSC 2010: 37C50; 37C10; 37C20; 37C29; 37D05

1 Introduction

Let M be a compact smooth Riemannian manifold, and let Diff(M) be the space of diffeomorphisms of M endowed with the C¹ topology. Hyperbolicity and stability have been important topics in differentiable dynamical systems since they were introduced by Smale [1]. For instance, a diffeomorphism f : M → M is structurally stable if and only if it satisfies Axiom A and the strong transversality condition. A diffeomorphism f : M → M satisfies Axiom A if the nonwandering set Ω(f) is P(f) and is hyperbolic, where P(f) is the set of all periodic points of f. A set of diffeomorphisms is generic (or residual) if it contains a countable intersection of dense open sets of Diff(M). Abraham and Smale [2] showed that the set of diffeomorphisms f : M → M satisfying Axiom A and the no-cycle condition is not dense in the space of Diff(M).

If a diffeomorphism f : M → M satisfies Axiom A, then from the work of Smale [1], the nonwandering set Ω(f) = ⋃i=1n 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 C¹ generic diffeomorphism 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 C¹, if a diffeomorphism f has the shadowing property on a locally maximal homoclinic class, then it is hyperbolic. Lee [4] proved that for generic C¹, if a diffeomorphism 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 C¹, 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 flows, that is, for a C¹ generic vector field X ∈ 𝔛(M), a condition under which we can obtain hyperbolicity of homoclinic classes. Unfortunately, we cannot use the same arguments as in the diffeomorphism case.

We say that a diffeomorphism f satisfies the star condition if there is a C¹ neighborhood 𝓤(f) ⊂ Diff(M) such that for any g ∈ 𝓤(f), every periodic point of g is hyperbolic. Aoki [16] and Hayashi [17] showed that if a diffeomorphism f satisfies the star condition, then it is Axiom A and the no-cycle condition, that is, Ω stable. We say that a flow X^t satisfies the star condition if there is a C¹ 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 satisfies the star condition, but it is not Ω-stable because the attractor contains a hyperbolic singular point. However, if a flow does not contain singularities and satisfies the star condition, then it is Ω stable (see [19]).

2 Basic notions and main theorem

Let M be a compact n ( ≥ 3)-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 C¹ vector fields on M endowed with the C¹ topology. Then, every X ∈ 𝔛(M) generates a C¹ flow X^t : M × ℝ → M; that is, a C¹ map such that X^t : M → M is a diffeomorphism satisfying X⁰(x) = x, and X^t+s(x) = X^t(X^s(x)) for all s, t ∈ ℝ and x ∈ M. The flow of X will be denoted by X^t, t ∈ ℝ. For X ∈ 𝔛(M), a point x ∈ M is singular of X if X(x) = 0. 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) > 0 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). For any δ > 0, a sequence {(x_i, t_i) : x_i ∈ M, t_i ≥ 1, and i ∈ ℤ} is a δ-pseudo-orbit of X if d(X^t_i(x_i), x_i+1) < δ for any i ∈ ℤ.

An increasing homeomorphism h : ℝ → ℝ with h(0) = 0 is called a reparametrization of ℝ. Denote by Rep(ℝ) the set of reparametrizations of ℝ. Fix ϵ > 0 and define Rep(ϵ) as follows:

Rep(ϵ)={h∈Rep:|h(t)t−1|<ϵ}.

For a closed X^t-invariant set Λ ⊂ M, we say that X has the shadowing property on Λ if for any ϵ > 0, there is δ > 0 satisfying the following property: given any δ-pseudo-orbit ξ = {(x_i, t_i) : t_i ≥ 1, i ∈ ℤ} with x_i ∈ Λ, there is a point y ∈ M and an increasing homeomorphism h ∈ Rep(ϵ) such that d(X^h(t)(y), X^t−s_i(x_i)) < ϵ for any s_i < t < s_i+1, where s_i is defined as

si=t0+t1+⋯+ti−1,ifi>00,ifi=0−t−1−t−2−⋯−ti,ifi<0.

The point y ∈ M is said to be a shadowing point of ξ.

Let X^t be the flow 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 > 0, λ > 0 and a splitting T_xM = Exs ⊕ 〈X(x)〉 ⊕ Exu such that the tangent flow DX^t: TM → TM leaves the continuous splitting invariant and

∥DXt|Exs∥≤Ce−λtand∥DX−t|Exu∥≤Ce−λt

for t > 0 and x ∈ Λ. We say that X ∈ 𝔛(M) is Anosov if M is hyperbolic for X.

Let γ be a hyperbolic closed orbit of a vector field X ∈ 𝔛(M), and we define the stable and unstable manifolds of γ by

Ws(γ)={y∈M:ω(y)=γ}

and

Wu(γ)={y∈M:α(y)=γ}.

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

HX(γ)=Ws(γ)⋔Wu(γ)¯.

It is clear that H_X(γ) is a compact, transitive, and X^t-invariant set.

For two hyperbolic closed orbits γ₁ and γ₂ of X^t, we say that γ₁ and γ₂ are homoclinic related, denoted by γ₁ ∼ γ₂, if W^s(γ₁) ⋔ W^u(γ₂) = ∅ and W^u(γ₁) ⋔ W^s(γ₂) = ∅. It is clear that if γ₁ ∼ γ, then index(γ₁) = index(γ), where index(γ) = dimW^s(γ). Note that if γ is a hyperbolic closed orbit of X^t, then there exist a C¹ 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) = index(γ_Y).

A closed invariant set Λ is Lyapunov stable if for any neighborhood U of Λ, there is a neighborhood V of Λ such that X^t(V) ⊂ U for all t > 0. 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 C¹-generic if “P” holds for all vector fields that belong to some residual subset of 𝔛(M). We write for C¹ 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 flows.

Theorem

For C¹ generic X ∈ 𝔛(M), if a flow 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.

3 Proof of the Theorem

Let M be as previously, and let X ∈ 𝔛(M). We define the strong stable and unstable manifolds of a hyperbolic periodic point p respectively as follows:

Wss(p)={y∈M:d(Xt(y),Xt(p))→0ast→∞}

and

Ws(Orb(p)))=⋃t∈RWss(Xt(p)),

where Orb(p) is the orbit of p. If ϵ > 0, the local strong stable manifold is defined as

Wϵ(p)ss(p)={y∈M:d(Xt(y),Xt(p))<ϵ,ift≥0}.

By the stable manifold theorem, there is an ϵ = ϵ(p) > 0 such that

Wss(p)=⋃t≥0X−t(Wϵ(p)ss(Xt(p))).

We can define the unstable manifolds similarly. If σ is a hyperbolic singularity of X, then there exists an ϵ = ϵ(σ) > 0 such that

Wϵs(σ)={x∈M:d(Xt(x),σ)<ϵast≥0}

and

Ws(σ)=⋃t≥0Xt(Wϵs(σ)).

Analogous definitions hold for unstable manifolds.

3.1 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.

Lemma 3.1

Let γ be a hyperbolic closed orbit of X. If a flow X^t has the shadowing property on H_X(γ), then for every hyperbolic σ ∈ H_X(γ) ∩ Crit(X), we have

Ws(γ)∩Wu(σ)=∅andWu(γ)∩Ws(σ)=∅.

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 0 < 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 t₁ > 0 and t₂ > 0 such that X^t₁(x) ∈ B_δ(p) and X^t₂(x) ∈ B_δ(q). Assume that t₂ = t₁ + k for some k > 0. Then, the sequence

{p,Xt1(x),Xt1+1(x),…,Xt1+k−1(x),q}⊂HX(γ)

is a finite δ- pseudo-orbit of X. We construct a δ-pseudo-orbit {(x_i, t_i) : t_i ≥ 1, i ∈ ℤ} ⊂ H_X(γ) as follows:

X⁻ⁱ(p) = x_−i for i ≥ 0;
X^t₁+i(x) = x_i for i = 1, …, k − 1; and
Xⁱ(q) = x_k+i for all i ≥ 0.

Since X^t has the shadowing property on H_X(γ), there is y ∈ M and an increasing homeomorphism h : ℝ → ℝ with h(0) = 0 such that

d(Xh(t)(y),Xt−si(xi))<ϵandd(Xh(t)(y),Xt−s−i(x−i)<ϵ,

where s_i < t < s_i+1 and s_−i < t < s_−i+1 for all t ∈ ℝ and i ∈ ℤ. Then y ∈ Wϵu (p) and there is τ > 0 such that X^τ(y) ⊂ Wϵs (q). Thus, we have

Orb(y)∩Wu(γ)∩Ws(η)=∅.

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 0 < 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 t₁ > 0 and t₂ > 0 such that X^t₁(x) ∈ B_δ(σ) and X^t₂(x) ∈ B_δ(p). Assume that t₂ = t₁ + k for some k > 0. We can thus construct a δ-pseudo-orbit {(x_i, t_i) : t_i ≥ 1, i ∈ ℤ} ⊂ H_X(γ) as follows:

σ = x_−i, t_−i = 1 for i ≥ 0;
X^t₁+i(x) = x_i for i = 1, …, k − 1; and
Xⁱ(p) = x_k+i for all i ≥ 0.

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 fields. It is known that 𝓚𝓢 ⊂ 𝔛(M) is a residual subset (see [20]).

Lemma 3.2

There is a residual set 𝓖₁ ⊂ 𝔛(M) such that for any X ∈ 𝓖₁, if a flow X^t has the shadowing property on H_X(γ), then for all η ∈ H_X(γ) ∩ Crit(X), we have

Ws(γ)⋔Wu(η)=∅andWu(γ)⋔Ws(η)=∅.

Proof

Let X ∈ 𝓖₁ = 𝓚𝓢 and let η ∈ H_X(γ) ∩ Crit(X). Since a flow X^t has the shadowing property on H_X(γ), by Lemma 3.1, W^s(γ) ∩ W^u(η) = ∅ and W^u(γ) ∩ W^s(η) = ∅. Since X ∈ 𝓚𝓢, W^s(γ) ⋔ W^u(η) = ∅ and W^u(γ) ⋔ W^s(η) = ∅.□

Proposition 3.3

For any X ∈ 𝓖₁, if a flow X^t has the shadowing property on H_X(γ), then we have

HX(γ)∩Sing(X)=∅.

Proof

Let X ∈ 𝓚𝓢 and let γ be a hyperbolic periodic orbit of X in H_X(γ) with index j. Suppose that X has a hyperbolic singularity σ ∈ H_X(γ) with index i. If j < i, then dimW^u(σ) + dimW^s(γ ) ≤ dimM. Since X is a Kupka–Smale vector field, we have dimW^u(σ) + dim W^s(γ ) = dim M. By assumption, we can take x ∈ W^u(σ) ∩ W^s(γ). Then Orb(x) ∈ W^u(σ) ∩ W^s(γ ) and we can split

Tx(Wu(σ))=Tx(Orb(x))⊕E1andTx(Ws(γ))=Tx(Orb(x))⊕E2.

Thus, we know that

dim(Tx(Wu(σ))+Tx(Ws(γ)))<dimWu(σ)+dimWs(γ)=dimM.

This is a contradiction, because X is a Kupka–Smale vector field. If j ≥ i, then

dimWs(σ)+dimWu(γ)≤dimM.

By the previous arguments, we have a contradiction. Thus, H_X(γ) ∩ Sing(X) = ∅.□

3.2 Chain recurrent class and homoclinic class

For any x, y ∈ M, we say that x ⇝ y if for any δ > 0, there is a finite δ-pseudo-orbit {(x_i, t_i) : 0 ≤ i < n} with n > 1 such that x₀ = x and d(X^t_n−1(x_n−1), y) < δ and a δ-pseudo-orbit {(z_i, s_i) : 0 ≤ i < m} with m > 1 such that z₀ = y and d(X^s_m−1(z_m−1), 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

CX(γ)={x∈M:x⇝γandγ⇝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 𝓖₂ ⊂ 𝔛(M) such that every X ∈ 𝓖₂ satisfies:

the chain recurrence class C_X(γ) = H_X(γ) (see [21]);
if a closed orbit η ∈ H_X(γ), then H_X(γ) = H_X(η) (see [22]);
H_X(γ) = W^s(γ) ∩ W^u(γ) (see [22]);
W^s(γ) is Lyapunov stable for −X and W^u(γ) is Lyapunov stable for X (see [22]);
if H_X(γ) is Lyapunov stable for X, then there is a C¹ neighborhood 𝓤(X) of X such that for every Y ∈ 𝓤(X), H_Y(γ_Y) is Lyapunov stable (see [23]);
there exist a C¹ 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 fiber N_x at x ∈ M is the orthogonal linear subspace of 〈X(x) 〉 in T_x M, that is, N_x = 〈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

PtX(v)=π(DxXt(v)),

for v ∈ N_x 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 PtX -invariant splitting Δ^s ⊕ Δ^u and there exists an l > 0 such that

PlX|Δxs≤12andP−lX|ΔXl(x)u≤12,

for all x ∈ Λ. Then, Doering [25] proved the following result, which is a method of proof for hyperbolicity.

Proposition 3.5

Let Λ ⊂ M be a compact invariant set of X^t. Then, Λ is a hyperbolic set of X^t if and only if the linear Poincaré flow restriction on Λ has a hyperbolic splitting N_Λ = Δ^s ⊕ Δ^u, where N = ⋃_{x∈M_X}N_x.

3.3 Weak hyperbolic periodic points

An exponential map exp_p : T_pM(1) → M is well defined for all p ∈ M, where T_pM(δ) denotes the ball {v ∈ T_pM : ∥v∥ ≤ δ}. For every regular point x ∈ R(X), let N_x = 〈X(x)〉^⊥ ⊂ T_xM, and N_x(δ) be the δ-ball in N_x. Let 𝓝_x,r = exp_x(N_x(r)). Given any point x ∈ R(M) and t ∈ ℝ, there are r > 0 and a C¹ map τ: 𝓝_x,t → ℝ with τ(x) = t such that X^τ(y)(y) ∈ 𝓝_X^τ(x),1 for any y ∈ 𝓝_x,r. We define the Poincaré map as

fx,t:Nx,r→NXt(x),1y↦fx,t(y):=Xτ(y)(y).

Let X ∈ 𝔛(M), and suppose p ∈ γ ∈ Per(X)(X^π(p)(p) = p, where π(p) > 0 is the prime period. If f : 𝓝_p,r₀ → 𝓝_p is the Poincaré map (r₀ > 0), then f(p) = p. Note that γ is hyperbolic if and only if p is a hyperbolic fixed point of f.

The following lemma states that by perturbation of vector fields, 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 : 𝓝_p,r → 𝓝_p(r > 0) be the Poincaré map of X^t. For any δ > 0, if the eigenvalue λ of D_pf is 1 < λ < 1 + δ, then there is g that is C¹ close to f such that D_pg has an eigenvalue μ with μ ≤ 1 − δ, where g is the Poincaré map associated to Y.

Proof

Let Δ_p be the eigenspace corresponding to λ with index(p) = i, and let N_p = Δ_p ⊕ Δp⊥. For the splitting, we have

Df(p)=Df(p)|ΔpA1(f)OA2(f).

Applying Gourmelon’s result [26] (see also [5, Theorem 2.5]), we define the map T : [0, 1] → Γ_i as follows

T(t)=(1−t)Df(p)|Δp+t(1−δ1+δ)Df(p)|ΔpA1(f)OA2(f),

for t ∈ [0, 1]. Then, we have

Dpf=T(0)=Df(p)|ΔpA1(f)OA2(f)

and

Dpg=T(1)=(1−δ1+δ)Df(p)|ΔpA1(f)OA2(f).

Thus, one can see that

(1−δ1+δ)Df(p)|Δp≤(1−δ1+δ)(1+δ)=(1−δ).

The proof is complete.□

For any δ > 0, we say that a point p ∈ γ ∈ Per(X) is δ-weak hyperbolic periodic if there is an eigenvalue λ of D_pf such that (1 − δ) < ∣λ∣ < (1 + δ), where f : 𝓝_p,r → 𝓝_p is the Poincaré map associated to X^t.

Let X ∈ 𝓖₂ and let H_X(γ) be bi-Lyapunov stable with index(γ) = i(0 < i < dimM − 1). 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.

Lemma 3.7

There is a residual set 𝓖₃ ⊂ 𝔛(M) such that for any X ∈ 𝓖₃, if a homoclinic class H_X(γ) is bi-Lyapunov stable, then H_X(γ) does not contain a δ- weak hyperbolic periodic point.

Proof

Let X ∈ 𝓖₃ = 𝓖₁ ∩ 𝓖₂ have the shadowing property on H_X(γ). Since X has the shadowing property on H_X(γ), by Lemmas 3.1 and 3.2, we have η ∼ γ for every η ∈ H_X(γ) ∩ Per(X). Suppose, by contradiction, that for any δ > 0 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 D_pf such that

1−δ<|λ|<1+δ,

where f : 𝓝_p,r → 𝓝_p is the Poincaré map corresponding to the flow X^t. Assume that 1 < λ < 1 + 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:

U ∩ {γ} = ∅;
U ∩ Orb⁺(x) = ∅; and
Orb⁻(x) ⊂ U.

Then, by [5, Theorem 2.5] and Lemma 3.6, there is g C¹ close to f such that:

index(η_Y) > index(γ_Y);
it preserves the i strong stable manifold of q_g ∈ η_Y outside U; and
W^uu(p_g) ∩ W^ss(q_g) = ∅;

where γ_Y is the continuation of γ, η_Y is the continuation of η, Y^t is the flow corresponding to g, and p_g ∈ γ_Y. Using the λ-lemma, we have q_g ∈ W^uu(p_g). Since H_Y(γ_Y) is Lyapunov stable for Y, we have W^u(γ_Y) ⊂ H_Y(γ_Y), and so q_g ∈ η_Y ⊂ H_Y(γ_Y). This is a contradiction. Since X ∈ 𝓖₃, 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 flow version of the result proved by Wang [27].

Lemma 3.8

There is a residual set 𝓖₄ ⊂ 𝔛(M) such that for any X ∈ 𝓖₄, if a homoclinic class H_X(γ) is not Hyperbolic, then for any δ > 0, 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 ∈ 𝓖₃ ∩ 𝓖₄ 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 > 0 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.□

Acknowledgments

This work is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (No. 2017R1A2B4001892).

References

[1] Smale S., Differentiable dynamical systems, Bull. Amer. Math. Soc., 1967, 73, 747–817.10.1090/S0002-9904-1967-11798-1Search in Google Scholar

[2] Abraham R., Smale S., Nongenericity of Ω-stability, Global Analysis, Proc. Sympos. Pure Math., 1970, 14, 5.8, A.M.S., Providence, R.I.10.1090/pspum/014/0271986Search in Google Scholar

[3] Ahn J., Lee K., Lee M., Homoclinic classes with shadowing, J. Inequal. Appl., 2012, 97.10.1186/1029-242X-2012-97Search in Google Scholar

[4] Lee M., Usual limit shadowable homoclinic classes of generic diffeomorphisms, Adv. Difference Equ., 2012, 2012:91.10.1186/1687-1847-2012-91Search in Google Scholar

[5] Arbieto A., Carvalho B., Cordeiro W., Obata D.J., On bi-Lyapunov stable homoclinic classes, Bull. Braz. Math. Soc. New Ser., 2013, 44, 105–127.10.1007/s00574-013-0005-ySearch in Google Scholar

[6] Lee K., Lee M., Shadowable chain recurrence classes for generic diffeomorphisms, Taiwan J. Math., 2016, 20, 399–409.10.11650/tjm.20.2016.5815Search in Google Scholar

[7] Lee M., Asymptotic orbital shadowing property for diffeomorphisms, Open Math., 2019, 17, 191–201.10.1515/math-2019-0002Search in Google Scholar

[8] Lee M., Vector fields satisfying the barycenter property, Open Math., 2018, 16, 429–436.10.1515/math-2018-0040Search in Google Scholar

[9] Lee M., Locally maximal homoclinic classes for generic diffeomorphisms, Balkan J. Geom. Its Appl., 2017, 22, 44–49.Search in Google Scholar

[10] Lee M., The barycenter property for robust and generic diffeomorphisms, Acta Math. Sin. Engl. Ser., 2016, 32, 975–981.10.1007/s10114-016-5123-1Search in Google Scholar

[11] Lee M., Robustly chain transitive diffeomorphisms, J. Inequal. Appl., 2015, 230, 1–6.10.1186/s13660-015-0752-ySearch in Google Scholar

[12] Lee M., Vector fields with stably limit shadowing, Adv. Diff. Equat., 2013, 255, 1–6.10.1186/1687-1847-2013-255Search in Google Scholar

[13] Lee M., C¹-stable inverse shadowing chain components for generic diffeomorphisms, Commun. Korean Math. Soc., 2009, 24, 127–144.10.4134/CKMS.2009.24.1.127Search in Google Scholar

[14] Lee M., Lee S., Robustly chain transitive vector fields, Asian-European J. Math., 2015, 08, 1550026.10.1142/S1793557115500266Search in Google Scholar

[15] Lee M., Lee S., Park J., Shadowable chain components and hyperbolicity, Bull. Korean Math. Soc., 2015, 52, 149–157.10.4134/BKMS.2015.52.1.149Search in Google Scholar

[16] Aoki N., The set of axiom A diffeomorphisms with no cycles, Bol. Soc. Bras. Mat., 1992, 23, 21–65.10.1007/BF02584810Search in Google Scholar

[17] Hayashi S., Diffeomorphisms in 𝓕¹(M) satisfy axiom A, Ergodic Th. Dynam. Syst., 1992, 12, 233–253.10.1017/S0143385700006726Search in Google Scholar

[18] Guchenheimer J., A strange, strange attractor, The Hopf bifurcation theorems and its applications, Applied Mathematical Series, Springer, 1976, 19, 368–381.10.1007/978-1-4612-6374-6_25Search in Google Scholar

[19] Gand S., Wen L., Nonsingular star flows satisfy axiom A and the no-cycle condition, Invent. Math., 2006, 164, 279–315.10.1007/s00222-005-0479-3Search in Google Scholar

[20] Kupka I., Contribution à la théorie des champs génériques, Contrib. Diff. Equ., 1963, 2, 457–484 and 1964, 3, 411–420.Search in Google Scholar

[21] Bonatti C., Coriviser S., Récurrence et généricité, Invent. Math., 2004, 158, 33–104.10.1007/s00222-004-0368-1Search in Google Scholar

[22] Carballo C., Morales C.A., Pacifico M.J., Homoclinic classes for C¹ generic vector fields, Ergodic Th. Dynam. Syst., 2003, 23, 403–415.10.1017/S0143385702001116Search in Google Scholar

[23] Potrie R., Generic bi-Lyapunov stable homoclinic classes, Nonlinearity, 2010, 23, 1631–1649.10.1088/0951-7715/23/7/006Search in Google Scholar

[24] Abdenur F., Bonatti C., Crovisier S., Díaz L.J., Wen L., Periodic points and homoclinic classes, Ergodic Th. Dynam. Syst., 2007, 27, 1–22.10.1017/S0143385706000538Search in Google Scholar

[25] Doering C., Persistently transitive vector fields on three manifolds, Dynam. Syst. Biff. Theory. Pitman Res. Notes, 1987, 160, 59–89.Search in Google Scholar

[26] Gourmelon N., A Franks’ lemma that preserves invariant manifolds, Ergodic Th. Dynam. Syst., 2016, 36, 1167–1203.10.1017/etds.2014.101Search in Google Scholar

[27] Wang X., Hyperbolicity versus weak periodic orbits inside homoclinic classes, arXiv:1504.03153.Search in Google Scholar

[28] Gourmelon N., Generation of homoclinic tangencies by C¹ perturbations, Disc. Contin. Dynam. Syst., 2010, 26, 1–42.10.3934/dcds.2010.26.1Search in Google Scholar

[29] Palis J., Melo W., Geometric theory of dynamical systems: An introduction, Springer-Verlag, New York-Berlin, 1982.10.1007/978-1-4612-5703-5Search in Google Scholar

Received: 2019-01-27

Accepted: 2019-06-17

Published Online: 2019-08-24

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Articles in the same Issue

https://doi.org/10.1515/math-2019-0068

Keywords for this article

homoclinic class; Lyapunov stable; shadowing; generic; hyperbolic

Creative Commons

BY 4.0