Vector fields satisfying the barycenter property

Manseob Lee

doi:10.1515/math-2018-0040

Article Open Access

Vector fields satisfying the barycenter property

Manseob Lee

Published/Copyright: April 23, 2018

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$Open Mathematics$

From the journal Open Mathematics Volume 16 Issue 1

Abstract

We show that if a vector field X has the C¹ robustly barycenter property then it does not have singularities and it is Axiom A without cycles. Moreover, if a generic C¹-vector field has the barycenter property then it does not have singularities and it is Axiom A without cycles. Moreover, we apply the results to the divergence free vector fields. It is an extension of the results of the barycenter property for generic diffeomorphisms and volume preserving diffeomorphisms [1].

Keywords: Barycenter property; Singular point; Generic; Hyperbolic; Axiom A; Anosov

MSC 2010: 37D20; 37C75

1 Introduction

Let M be a closed 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 𝔛¹(M) the set of C¹-vector fields on M endowed with the C¹-topology. Then every X ∈ 𝔛¹(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 t, s ∈ ℝ and x ∈ M. Let X_t be the flow of X ∈ 𝔛¹(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 invariant continuous splitting and

∥DXt|Exs∥≤Ce−λtand∥DX−t|Exu∥≤Ce−λt

for t > 0 and x ∈ Λ, where 〈X(x)〉 is the subspace generated by X(x). If Λ = M, then we say that X is Anosov.

We say that p ∈ M is a periodic point if it is not a singularity and there exists T > 0 such that X_T(p) = p. Then the smallest positive T is the period ofp and is denoted by π(p). The orbit of a periodic point is called a periodic orbit. Denote the set of periodic orbits by P(X). Then the set of critical orbits of X is defined as the set of critical orbits of X is the set Crit(X) = Sing(X) ∪ P(X), where Sing(X) is the set of singularities of X. We say that X is transitive if there is a point x ∈ M such that ω(x) = M, where ω(x) is the omega limit set of x.

Remark 1.1

In the study of smooth dynamics, there are many results about the diffeomorphisms which also hold for vector fields without singularity but do not hold for vector fields with singularities (see [2,3,4,5,6]). For example, if a diffeomorphism is a star diffeomorphism then it isΩ-stable (see [2, 4]). However, we know that Lorenz attractor is a star flow, but its non-wandering set is not hyperbolic (see [7]).

Note that if a vector field X satisfies a star vector field and Sing(X) = ∅ then it satisfies both Axiom A and the no-cycle condition (see [3]). Recently, many authors have used the dynamical properties to control of singularities of vector fields (see [6, 8, 9]).

The stability theory is a main topic in differentiable dynamical systems. For instance, Mañé [10] proved that if a diffeomorphism f on a compact smooth manifold M with dim M = 2 is robustly transitive then it is Anosov. For vector fields, Doering [11] proved that if a vector field X on a compact smooth manifold M with dim M = 3 is robustly transitive then it is Anosov. For the types of the pseudo orbit tracing properties (shadowing property, specification property, limit shadowing property, …), there are close relations between these properties and structural stability hyperbolicity. Lee and Sakai [5] proved that if a vector field without singularities has the C¹ robustly shadowing property then it is structurally stable. Arbieto et al. [8] proved that if a vector field X has the C¹ robustly specification property then it is Anosov. Lee [6] proved that if a vector field X has the C¹ robustly limit shadowing property then it is Anosov.

For a compact invariant set Λ of a diffeomorphism f, we say that the set Λ is robustly transitive if there are a C¹ neighborhood 𝓤(f) of f and a neighborhood U of Λ such that for any g ∈ 𝓤(f), Λ_g(U) = ⋂_n∈ℤgⁿ(U) is transitive for g, where Λ_g(U) is the continuation of Λ. Firstly, Abdenur et al. [12] introduced the barycenter property and later, Tian and Sun [13] introduced a new type of the barycenter property. For any two periodic points p, q ∈ P(f), we say that p, q have the barycenter property if for any ϵ > 0 there exists an integer N = N(ϵ, p, q) > 0 such that for any two integers n₁ > 0, n₂ > 0 there is a point x ∈ M such that

d(fi(x),fi(p))<ϵ,−n1≤i≤0,andd(fi+N(x),fi(q))<ϵ,0≤i≤n2.

We say that f has the barycenter property(or, Msatisfies the barycenter property) if the barycenter property holds for any two periodic points p, q ∈ P(f). The barycenter property is not equal to the specification property, and the shadowing property (see [1, Remark 1.1]). In this paper, we use the definition of Tian and Sun [13]. Tian and Sun [13] proved that if a robustly transitive diffeomorphism f on a compact smooth manifold has the barycenter property then it is hyperbolic. Very recently, Lee [1] proved that if a diffeomorphism f has the C¹ robustly barycenter property then it is Axiom A without cycles.

Using the barycenter property for vector fields, we sutdy a stability theory (Ω-stable) which is a very valuable subject by Remark 1.1. Now we introduce the barycenter property for vector fields. For any critical orbits γ and η, we say that p ∈ γ, q ∈ η have the barycenter property if for any ϵ > 0, there is T = T(ϵ, p, q) > 0 such that for any τ > 0 there is z ∈ M such that

d(Xt(z),Xt(p))<ϵfor−τ≤t≤0andd(XT+t(z),Xt(q))<ϵfor0≤t≤τ.

A vector field X has the barycenter property if the barycenter property holds for any critical orbits γ and η. We say that X ∈ 𝔛(M) has the C¹robustly barycenter property if there is a C¹-neighborhood 𝓤(X) of X such that for any Y ∈ 𝓤(X), Y has the barycenter property. Then we have the following:

Theorem A

If a vector fieldXhas theC¹robustly barycenter property, thenSing(X) = ∅ andXis Axiom A without cycles.

A subset 𝓖 ⊂ 𝔛¹(M) is called residual if it contains a countable intersection of open and dense subsets of 𝔛¹(M). A dynamic property is called C¹generic if it holds in a residual subset of 𝔛¹(M). Arbieto et al. [8] proved that C¹ generically, if a vector field X has the specification property then it is Anosov. Ribeiro [14] proved that C¹ generically, if a transitive vector field has the shadowing property then it is Anosov and if a vector field has the limit shadowing property then it is Anosov. For that, we have the following which is a result of the paper.

Theorem B

ForC¹genericX ∈ 𝔛¹(M), if a vector fieldXhas the barycenter property thenSing(X) = ∅ andXis Axiom A without cycles.

2 Proof of Theorem A

Let M be as before, and let X ∈ 𝔛¹(M). For p ∈ γ ∈ P(X), the strong stable manifold W^ss(p) of p and stable manifold W^s(γ) of γ are defined as follows:

Wss(p)={y∈M:d(Xt(y),Xt(p))→0ast→∞},

and

Ws(γ)=⋃t∈RWss(Xt(p)).

If η > 0 then the local strong stable manifold Wη(p)ss (p) of p and the local stable manifolds Wη(γ)s (γ) of γ are defined by

Wη(p)ss(p)={y∈M:d(Xt(y),Xt(p))<η(p),ift≥0},

and

Wη(γ)s(γ)={y∈M:d(Xt(y),Xt(γ))<η(γ),ift≥0}.

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

Wss(p)=⋃t≥0X−t(Wϵss(Xt(p))).

Analogously we can define the strong unstable manifold, unstable manifold, local strong unstable manifold and local unstable manifold. Denote by index(p) = dim W^s(p).

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}andWs(σ)=⋂t≥0Xt(Wϵs(σ)).

Analogous definitions hold for unstable manifolds.

Lemma 2.1

Letγandηbe hyperbolic critical points ofX. If a vector fieldXhas the barycenter property thenW^s(γ) ∩ W^u(η) ≠ ∅ andW^u(γ) ∩ W^s(η) ≠ ∅.

Proof

First, we consider periodic orbits. Take p ∈ γ and q ∈ η such that p and q are hyperbolic. Denote by ϵ(p) the size of the local strong unstable manifold of p and by ϵ(q) the size of the local strong unstable manifold of q. Let ϵ = min{ϵ(p), ϵ(q)} and let T = T(ϵ, p, q) be given by the barycenter property. For t > 0, there is x_t ∈ M such that d(X_s(x_t), X_s(p)) ≤ ϵ for −t ≤ s ≤ 0 and d(X_T+s(x_t), X_s(q)) ≤ ϵ for 0 ≤ s ≤ t. Since M is compact, there is a subsequence {x_{t_n}} ⊂ {x_t} such that x_{t_n} → x as t_n → ∞(n → ∞). Then we have that

d(X−s(x),X−s(p))≤ϵfor−s→∞andd(XT+s(x),Xs(q))≤ϵfors→∞.

This means that x ∈ Wϵuu (p) and X_T(x) ∈ Wϵss (q). Thus we have W^u(p) ∩ W^s(q) ≠ ∅. Similarly, we can show that W^u(γ) ∩ W^s(η) ≠ ∅. Consequently, W^s(γ) ∩ W^u(η) ≠ ∅ and W^u(γ) ∩ W^s(η) ≠ ∅.

Finally, we consider singular points. Let σ and τ be hyperbolic singular points of X. As in the first case, we have W^s(σ) ∩ W^u(τ) ≠ ∅ and W^u(σ) ∩ W^s(τ) ≠ ∅. □

A singularity σ is a sink if all eigenvalues of D_σX have a negative real part. A periodic point p is a sink if the eigenvalues of the derivative of the Poincaré map associated to p have absolute value less than one. A source is a sink for the vector field −X.

Lemma 2.2

If a vector fieldXhas the barycenter property thenXdose not contains a sink and a source.

Proof

Suppose, by contradiction, that there are p ∈ Crit(X) and q ∈ Crit(X) such that p is a sink and q is a saddle. Since X has the barycenter property, by Lemma 2.1, W^s(p) ∩ W^u(q) ≠ ∅ and W^u(p) ∩ W^s(q) ≠ ∅. Since p is sink, W^u(p) ∩ W^s(q) = ∅. But, since X has the barycenter property, W^u(p) ∩ W^s(q) ≠ ∅. This is a contradiction. If p is source then it is similar to the previous proof. Thus if X has the barycenter property then X has neither sinks nor sources. □

We say that X ∈ 𝔛¹(M) is Kupka-Smale if every σ ∈ Crit(X) is hyperbolic and their stable and unstable manifolds intersect transversally. Denote by 𝓚𝓢 the set of all Kupka-Smale vector fields. It is well-known that the set of Kupka-Smale vector fields is residual in 𝔛¹(M) (see [15]).

Lemma 2.3

([8, Lemma 3.4]). LetX ∈ 𝔛¹(M) be a Kupka-Smale and letσ, ρ ∈ Crit(X). If dim W^s(σ) + dim W^u(ρ) ≤ dim MthenW^s(σ) ∩ W^u(ρ) = ∅.

Let σ ∈ Crit(X) be hyperbolic. Then there exist a C¹-neighborhood 𝓤(X) of X and a neighborhood U of σ such that for any Y ∈ 𝓤(X), there is σ_Y such that σ_Y is the continuation of σ and index(σ) = index(σ_Y) (see [16]).

Lemma 2.4

Letσandτbe hyperbolic singular points and let 𝓤(X) be aC¹neighborhood ofX. If a vector fieldXhas theC¹robustly barycenter property, then for anyY ∈ 𝓤(X), we have index(σ_Y) = index(τ_Y), whereσ_Yandτ_Yare the continuations ofσandτ, respectively.

Proof

Let σ and τ be hyperbolic singular points, and let 𝓤(X) be a C¹-neighborhood of X. Then there is Y ∈ 𝓤₁(X) ⊂𝓤(X) such that σ_Y and τ_Y are the continuations of σ and τ, respectively. Since X has the barycenter property, by Lemma 2.2, X has neither sinks nor sources. Thus we may assume that σ has index i and τ has index j with i ≠ j. If j < i then dim W^u(σ) + dim W^s(τ) < dim M. Take Y ∈ 𝓚𝓢 ∩ 𝓤₁(X). Then we have dim W^u(σ_Y) + dim W^s(τ_Y) < dim M. Since Y is Kupka-Smale, by Lemma 2.3 we know W^u(σ_Y) ∩ W^s(τ_Y) = ∅. Since X has the C¹ robustly barycenter property, this is a contradiction by Lemma 2.1. If j > i then dim W^s(σ) + dim W^u(τ) < dim M As in the case of j < i, we can take Y ∈ 𝓚𝓢 ∩𝓤₁(X). Then we have dim W^s(σ_Y) + dim W^u(τ_Y) < dim M. Since Y is Kupka-Smale, by Lemma 2.3 we know W^s(σ_Y) ∩ W^u(τ_Y) = ∅. Since X has C¹ robustly the barycenter property, this is a contradiction by Lemma 2.1. □

The following was proved by [17, Lemma 1.1] which is Franks’ lemma for singular points.

Lemma 2.5

LetX ∈ 𝔛¹(M) andσ ∈ Sing(X). Then for anyC¹neighborhood 𝓤(X) ofXthere areδ₀ > 0 andα > 0 such that if 𝓞(δ) : T_σM → T_σMis a linear map with ∥𝓞(δ)-D_σX∥ < δ < δ₀then there isY^δ ∈ 𝓤(X) satisfying

Yδ(x)=(Dexpσ−1(x)expσ)∘O(δ)∘expσ−1(x),ifx∈Bα/4(x)X(x),ifx∉Bα(x).

Furthermore, d₀(Y^δ, Y⁰) →0 asδ → 0. HereY⁰is the vector field for 𝓞(0) = D_σXandd₀is theC⁰metric.

By Lemma 2.5, Y⁰|_{B_α/4}(σ) is regarded as a linearization of X|_{B_α/4}(σ) with respect to the exponential coordinates. If there is an interval I ⊂ℝ and integral curve ζ(t)(t ∈ I) of the linear vector field 𝓞(δ) in expσ−1 (B_α/4(σ)) ⊂ T_σM then the composition exp_σ ∘ ζ:I(⊂ ℝ) → M is an integral curve of Y^δ in B_α/4(σ) ⊂ M (see [17]).

Lemma 2.6

Let 𝓤(X) be aC¹neighborhood ofX. If a singular pointσis not hyperbolic then there isY ∈ 𝓤(X) such thatYhas two hyperbolic singular points with different indices.

Proof

Let 𝓤₁(X) ⊂ 𝓤(X) be a C¹ neighborhood of X. Since a singular point σ is not hyperbolic we have that D_σX has an eigenvalue λ with Re(λ) = 0. By Lemma 2.5, there is Y ∈ 𝓤₁(X) such that σ_Y is a singular point of Y and μ is the only eigenvalue of D_{σ_Y}Y with Re(μ) = 0. Then TσYM=EσYs⊕EσYc⊕EσYu where, Eσs is the eigenspace of D_{σ_Y}Y associated with real part less than zero EσYu is the eigenspace of D_{σ_Y}Y associated with real part greater than zero, and EσYc is the eigenspace of D_{σ_Y}Y associated to μ.

Note that if dim EσYc = 2 then there are no singularities of Y nearby around σ_Y in the neighborhood of σ_Y (see [8, Theorem 6.2]).

Thus we consider dim EσYc = 1. Then there is r > 0 such that for all v ∈ EσYc (r), Y(exp_{σ_Y}) = 0, where EσYc (r) = EσYc ∩ T_{σ_Y}M(r). We can take τ ∈ exp_{σ_Y}( EσYc (r)) − {σ_y} such that τ is sufficiently close to σ_Y and τ is not a hyperbolic singularity for Y. We assume that index(σ_Y) = index(τ) = j. Then we can make a hyperbolic singular point which index is different from index(σ_Y) = index(τ) = j. By Lemma 2.5, take 0 < α < d(σ_Y, τ)/2, 0 < δ < δ₀ and a linear map 𝓞 : T_{σ_Y}M → T_σM such that 𝓞(v) = −δv, for all v ∈ Eσc , and 𝓞(v) = D_{σ_Y}Y(v), for all v ∈ EσYs⊕EσYu . By Lemma 2.5, there is Z ∈ 𝓤₁(X) such that

Z(x)=(DexpœY−1(x)expσY)∘O∘expσY−1(x),ifx∈Bα/4(σY).

Then there is the singular point σ_Z such that σ_Z is hyperbolic and index(σ_Z) = j + 1. Since Z(x) = Y(x) for all x ∈ B_α(σ_Y), τ is a non-hyperbolic singular point for Z which index is j. Using Lemma 2.5, there are WC¹ close to Z(W ∈ 𝓤₁(X)) and a linear map L : T_τM → T_τM such that for some 0 < α₁ ≤ α, W(x) = (Dexp∅−1(x)expτ)∘L∘expτ−1(x), if x ∈ B_{α_1/4}(τ), and W(x) = Z(x) if x ≠ ∈ B_α₁(τ). Then τ is hyperbolic singularity for W which index is j. Thus the vector filed W has two hyperbolic singular points σ_Z with index(σ_Z) = j + 1 and τ with index(τ) = j.□

Proposition 2.7

If a vector fieldX ∈ 𝔛¹(M) has theC¹robustly barycenter property then every singular points is hyperbolic.

Proof

Suppose, by contradiction, that there is a σ ∈ Sig(X) such that σ is not hyperbolic. By lemma 2.6, there is YC¹ close to X such that Y has two hyperbolic singular points σ_Y and τ with different indices which is a contradiction by Lemma 2.4. Thus if a vector filed X has the C¹ robustly barycenter property then every singular points are hyperbolic □

Lemma 2.8

Letγandηbe hyperbolic periodic orbits and let 𝓤(X) be aC¹neighborhood ofX. If a vector fieldXhas theC¹robustly barycenter property, then for anyY ∈ 𝓤(X), we have index(γ_Y) = index(η_Y), whereγ_Yandη_Yare the continuations ofγandη, respectively.

Proof

Let γ and η be hyperbolic closed orbits, and let 𝓤(X) be a C¹ neighborhood of X. Then there is Y ∈ 𝓤₁(X) ⊂𝓤(X) such that γ_Y and η_Y are the continuations of γ and η, respectively.

Suppose that γ has index i and η has index j with i ≠ j. If j < i then dim W^u(γ) + dim W^s(η) ≤ dim M. Take Y ∈ 𝓚𝓢 ∩ 𝓤₁(X). Then we have dim W^u(γ_Y) + dim W^s(η_Y) ≤ dim M. Since Y is Kupka-Smale, by Lemma 2.3 we know W^u(γ_Y) ∩ W^s(η_Y) = ∅. Since X has the C¹ robustly barycenter property, this is a contradiction by Lemma 2.1. Other case is similar. □

The following was proved by [8, Theorem 4.3]. They used the C¹ robustly specification property. But, the result can be obtained similarly without any properties.

Lemma 2.9

Let 𝓤(X) be aC¹neighborhood ofXand letγbe a periodic orbit ofX. If a periodic pointp ∈ γis not hyperbolic then there isY ∈ 𝓤(X) such thatYhas two hyperbolic periodic orbits with different indices.

Proposition 2.10

Let 𝓤(X) be aC¹neighborhood ofX. Suppose thatXhas theC¹robustly barycenter property. Then for anyY ∈ 𝓤(X), every periodic orbits ofYis hyperbolic.

Proof

Let 𝓤(X) be a C¹ neighborhood of X. To derive a contradiction, we may assume that there is Y ∈ 𝓤(X) such that Y has not hyperbolic periodic orbits. By Lemma 2.9, Y has two hyperbolic periodic orbits with different indices. Since X has the C¹ robustly barycenter property, this is a contradiction by Lemma 2.8. □

Theorem 2.11

If a vector fieldXhas theC¹robustly barycenter property thenSing(X) = ∅.

Proof

Let 𝓤(X) be a C¹ neighborhood of X. Suppose that Sing(X) ≠ ∅. Then there are a hyperbolic σ ∈ Sing(X) with index i and a hyperbolic periodic orbit γ with index j. Then there is a C¹ neighborhood 𝓤₁(X) ⊂ 𝓤(X) of X such that for any Y ∈ 𝓤₁(X), there are the continuations σ_Y and γ_Y of σ and γ, respectively. Thus we know that dim W^s(σ) = dim W^s(σ_Y), dim W^u(σ) = dim W^u(σ_Y), dim W^s(γ) = dim W^s(γ_Y) and dim W^u(γ) = dim W^u(γ_Y).

If j < i then dim W^u(σ) + dim W^s(γ) ≤ dim M. Take a vector field Z ∈ 𝓚𝓢 ∩ 𝓤₁(X) such that dim W^u(σ_Z) + dim W^s(γ_Z) ≤ dim M. By Lemma 2.3, W^u(σ_Z) ∩ W^s(γ_Z) = ∅. This is a contradiction by Lemma 2.1.

If j ≥ i then dim W^s(σ)+dim W^u(γ) ≤ dim M. As in the case j < i, we can take a vector field Y ∈ 𝓚𝓢 ∩ 𝓤₁(X) such that dim W^s(σ_Y)+dim W^u(γ_Y) ≤ dim M. By Lemma 2.3, W^s(σ_Y) ∩ W^u(γ_Y) = ∅. This is a contradiction. Thus if a vector field X has the C¹ robustly barycenter property then X has no singularities. □

Proof of Theorem A

Suppose that X has the C¹ robustly barycenter property. Then by Theorem 2.11, Sing(X) = ∅. By Lemma 2.8 and Proposition 2.10, every periodic orbit of X is hyperbolic. Then by Gan and Wen [3], X satisfies Axiom A without cycles. □

If a vector field X is transitive, then it is clear that Ω(X) = M. Thus if a nonsingular vector field satisfies Axiom A then it is Anosov. Then we have the following:

Corollary 2.12

If a transitive vector fieldXhas theC¹robustly barycenter property thenXis Anosov.

3 Proof of Theorem B

In this section, we are going to prove that C¹ generically, if a vector field has the barycenter property, then we show that the vector field satisfies Axiom A and does not contain singularities.

Theorem 3.1

There is a residual set 𝓖₀ ⊂𝔛¹(M) such that for anyX ∈ 𝓖₀, if a vector fieldXhas the barycenter property thenSing(X) = ∅.

Proof

Let X ∈ 𝓖₀ = 𝓚𝓢 have the barycenter property. Suppose, by contradiction, that Sing(x) ≠ ∅. Then as in the proof of Theorem 2.11, there exist a hyperbolic σ ∈ Sing(X) with index i and a hyperbolic periodic orbit γ with index j. If j < i then dim W^u(σ) + dim W^s(γ) ≤ dim M. Since X ∈ 𝓚𝓢, by Lemma 2.3

Wu(σ)∩Wu(γ)=∅,

which is a contradiction by Lemma 2.1. If j ≥ i then dim W^s(σ) + dim W^u(γ) ≤ dim M. By the previous argument, we get a contradiction. □

Lemma 3.2

There is a residual set 𝓖₀ ⊂𝔛¹(M) such that for anyX ∈ 𝓖₀, if a vector fieldXhas the barycenter property then for any hyperbolic periodic orbitsγandη, index(γ) = index(η).

index(γ)=index(η).

Proof

Let X ∈ 𝓖₀ = 𝓚𝓢 have the barycenter property, and let γ be a hyperbolic periodic orbit with index i and η be a hyperbolic periodic orbit with index j. Assume that i ≠ j. If j < i then dim W^u(γ) + dim W^s(η) ≤ dim M. Since X ∈ 𝓚𝓢, by Lemma 2.3, we have W^u(γ) ∩ W^s(η) = ∅. This is a contradiction by Lemma 2.1. If j ≥ i then dim W^s(γ) + dim W^u(η) ≤ dim M. Then the previous argument, we get a contradiction. □

Lemma 3.3

([8, Lemma 5.1]). There is a residual set 𝓖₁ ⊂𝔛¹(M) such that for anyX ∈ 𝓖₁, if for anyC¹neighborhood 𝓤(X) ofXthere isY ∈ 𝓤(X) such thatYhas two distinct hyperbolic periodic orbits with different indices thenXhas two distinct hyperbolic periodic orbits with different indices.

We say that a point p in a hyperbolic periodic orbit of X has a δ-weak hyperbolic eigenvalue if there is a characteristic multiplier λ of the orbit of p such that

(1−δ)<|λ|<(1+δ).

Proposition 3.4

There is a residual set 𝓖₂ ⊂𝔛¹(M) such that for anyX ∈ 𝓖₂, if a vector fieldXhas the barycenter property then there isδ > 0 such thatXhas noδ-weak hyperbolic eigenvalue.

Proof

Let X ∈ 𝓖₂ = 𝓖₀ ∩ 𝓖₁ have the barycenter property. To derive a contradiction, we may assume that for any δ > 0 there is a periodic orbit γ of X such that γ has a δ-weak hyperbolic eigenvalue. Then there is YC¹ close to X such that Y has a non hyperbolic periodic orbit η. By Lemma 2.9, there is ZC¹ close to Y(C¹ close to X) such that Z has two distinct hyperbolic periodic orbits with different indices. By Lemma 3.3, X has two distinct hyperbolic periodic orbits with different indices. This is a contradiction by Lemma 3.2. □

Lemma 3.5

([8, Lemma 5.3]). There is a residual set 𝓖₃ ⊂ 𝔛¹(M) such that for anyX ∈ 𝓖₃, if for anyδ > 0 and for anyC¹-neighborhood 𝓤(X) ofXthere isY ∈ 𝓤(X) such thatYhas a hyperbolic periodic orbitγwhich has aδ-weak hyperbolic eigenvalue thenXhas a hyperbolic periodic orbitηwhich has a 2δ-weak hyperbolic eigenvalue.

Proof of Theorem B

Let X ∈ 𝓖₂ ∩𝓖₃. Suppose that X has the barycenter property. By Lemma 3.1, Sing(X) = ∅. By the result of Gan and Wen [3], we show that every periodic orbits of X is hyperbolic. Assume that there is a periodic orbit γ of X such that for any δ > 0, γ has a δ/2-weak hyperbolic eigenvalue. Since X ∈ 𝓖₃, X has a hyperbolic periodic orbit η which has a 2δ-weak hyperbolic eigenvalue. Since X has the barycenter property, X has no δ-weak hyperbolic eigenvalue. This is a contradiction by Proposition 3.4. Since Sing(X) = ∅ and every periodic orbits of X is hyperbolic, by Gan and Wen [3], X is Axiom A without cycle. □

Corollary 3.6

ForC¹genericX ∈ 𝔛¹(M), if a transitive vector fieldXhas the barycenter property thenXis Anosov.

Let M be a closed, connected and smooth n(≥ 3)-dimensional Riemannian manifold endowed with a volume form, which has a measure μ, called the Lebesgue measure. Given a ^r(r ≥ 1) vector field X : M → TM the solution of the equation x^′ = X(x) generates a C^r flow, X_t; by the other side given a C^r flow we can define a C^r−1 vector field by considering X(x) = dXt(x)dt|t=0. We say that X is divergence-free if its divergence is equal to zero. Note that, by Liouville formula, a flow X_t is volume preserving if and only if the corresponding vector field, X, is divergence free. Let Xμ1 (M) denote the space of C^r divergence free vector fields and we consider the usual C¹ Whitney topology on this space. A vector field X ∈ Xμ1 (M) is a divergence-free star vector field if there exists a C¹ neighborhood 𝓤(X) of X in X ∈ Xμ1 (M) such that if Y ∈ 𝓤(X) then every point in P(X) ∪ Sing(X) is hyperbolic.

Theorem 3.7

If a divergence-free vector fieldX ∈ Xμ1 (M) has theC¹robustly barycenter property thenSing(X) = ∅ andXis Anosov.

Proof

By Ferreira [18, Theorem 1], if a divergence free vector field X ∈ Xμ1 (M) satisfies star vector fields then Sing(X) = ∅ and it is Anosov. To prove Theorem 3.7 we show that a divergence free vector field X satisfies a star condition. It is almost similar to prove of Theorem A. Thus as in the proof of Theorem A, if a divergence free vector field X has the C¹ robustly barycenter property then X is Axiom A without cycles, that is, X satisfies a star condition. This is a proof of Theorem 3.7. □

Bessa et al [19] proved that C¹ generically, if a divergence free vector field X has the shadowing property(expansive, specification property) then it is Anosov. From the results, we are going to prove C¹ generic divergence free vector fields when it has the barycenter property.

Theorem 3.8

ForC¹genericX ∈ Xμ1 (M), ifXhas the barycenter property thenSing(X) = ∅ andXis Anosov.

Proof

By Ferreira [18, Theorem 1], if a divergence-free vector field X ∈ Xμ1 (M) satisfies star vector fields then Sing(X) = ∅ and it is Anosov. By Bessa [20], C¹ generically, a divergence free vector field X ∈ Xμ1 (M) is transitive. Therefore, we show that C¹ generically, if a divergence-free vector field X has the barycenter property then X satisfies star vector fields. Thus as in the proof of Theorem B, C¹ generically, if a divergence free vector field X ∈ Xμ1 (M) has the barycenter property then it satisfies a star vector field, and so, Sing(X) = ∅ and it is Anosov. □

Acknowledgement

The author wishes to express their deepest appreciation to the referee for his/her useful comments and valuable suggestions.

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).

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Received: 2017-06-22

Accepted: 2018-02-28

Published Online: 2018-04-23

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Articles in the same Issue

https://doi.org/10.1515/math-2018-0040

Keywords for this article

Barycenter property; Singular point; Generic; Hyperbolic; Axiom A; Anosov

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