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# Open Mathematics

### formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 16, Issue 1

# Vector fields satisfying the barycenter property

Manseob Lee
Published Online: 2018-04-23 | DOI: https://doi.org/10.1515/math-2018-0040

## 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].

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 Xt : M × ℝ → M; that is a C1-map such that Xt : MM is a diffeomorphism satisfying X0(x) = x and Xt+s(x) = Xt(Xs(x)) for all t, s ∈ ℝ and xM. Let Xt be the flow of X ∈ 𝔛1(M), and let Λ be a Xt-invariant compact set. The set Λ is called hyperbolic for Xt if there are constants C > 0, λ > 0 and a splitting TxM = $\begin{array}{}{E}_{x}^{s}\end{array}$ ⊕ 〈X(x)〉 ⊕ $\begin{array}{}{E}_{x}^{u}\end{array}$ such that the tangent flow DXt: TMTM leaves the invariant continuous splitting and

$∥DXt|Exs∥≤Ce−λtand∥DX−t|Exu∥≤Ce−λt$

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

We say that pM is a periodic point if it is not a singularity and there exists T > 0 such that XT(p) = p. Then the smallest positive T is the period of p 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 xM 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 is Ω-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), Λg(U) = ⋂n∈ℤgn(U) is transitive for g, where Λg(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, qP(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 xM 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, M satisfies the barycenter property) if the barycenter property holds for any two periodic points p, qP(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 pγ, qη 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 zM 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 C1 robustly 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 field X has the C1 robustly barycenter property, then Sing(X) = ∅ and X is 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 C1 generic 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

For C1 generic X ∈ 𝔛1(M), if a vector field X has the barycenter property then Sing(X) = ∅ and X is Axiom A without cycles.

## 2 Proof of Theorem A

Let M be as before, and let X ∈ 𝔛1(M). For pγP(X), the strong stable manifold Wss(p) of p and stable manifold Ws(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 $\begin{array}{}{W}_{\eta \left(p\right)}^{ss}\end{array}$ (p) of p and the local stable manifolds $\begin{array}{}{W}_{\eta \left(\gamma \right)}^{s}\end{array}$ (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 Ws(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

Let γ and η be hyperbolic critical points of X. If a vector field X has the barycenter property then Ws(γ) ∩ Wu(η) ≠ ∅ and Wu(γ) ∩ Ws(η) ≠ ∅.

#### Proof

First, we consider periodic orbits. Take pγ and qη 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 xtM such that d(Xs(xt), Xs(p)) ≤ ϵ for −ts ≤ 0 and d(XT+s(xt), Xs(q)) ≤ ϵ for 0 ≤ st. Since M is compact, there is a subsequence {xtn} ⊂ {xt} such that xtnx as tn → ∞(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$\begin{array}{}{W}_{ϵ}^{uu}\end{array}$ (p) and XT(x) ∈ $\begin{array}{}{W}_{ϵ}^{ss}\end{array}$ (q). Thus we have Wu(p) ∩ Ws(q) ≠ ∅. Similarly, we can show that Wu(γ) ∩ Ws(η) ≠ ∅. Consequently, Ws(γ) ∩ Wu(η) ≠ ∅ and Wu(γ) ∩ Ws(η) ≠ ∅.

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

A singularity σ is a sink if all eigenvalues of DσX 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 field X has the barycenter property then X dose not contains a sink and a source.

#### Proof

Suppose, by contradiction, that there are pCrit(X) and qCrit(X) such that p is a sink and q is a saddle. Since X has the barycenter property, by Lemma 2.1, Ws(p) ∩ Wu(q) ≠ ∅ and Wu(p) ∩ Ws(q) ≠ ∅. Since p is sink, Wu(p) ∩ Ws(q) = ∅. But, since X has the barycenter property, Wu(p) ∩ Ws(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 σCrit(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]). Let X ∈ 𝔛1(M) be a Kupka-Smale and let σ, ρCrit(X). If dim Ws(σ) + dim Wu(ρ) ≤ dim M then Ws(σ) ∩ Wu(ρ) = ∅.

Let σCrit(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 σY such that σY is the continuation of σ and index(σ) = index(σY) (see [16]).

#### Lemma 2.4

Let σ and τ be hyperbolic singular points and let 𝓤(X) be a C1 neighborhood of X. If a vector field X has the C1 robustly barycenter property, then for any Y ∈ 𝓤(X), we have index(σY) = index(τY), where σY and τY are the continuations of σ and τ, 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 σY and τY 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 ij. If j < i then dim Wu(σ) + dim Ws(τ) < dim M. Take Y ∈ 𝓚𝓢 ∩ 𝓤1(X). Then we have dim Wu(σY) + dim Ws(τY) < dim M. Since Y is Kupka-Smale, by Lemma 2.3 we know Wu(σY) ∩ Ws(τY) = ∅. Since X has the C1 robustly barycenter property, this is a contradiction by Lemma 2.1. If j > i then dim Ws(σ) + dim Wu(τ) < dim M As in the case of j < i, we can take Y ∈ 𝓚𝓢 ∩𝓤1(X). Then we have dim Ws(σY) + dim Wu(τY) < dim M. Since Y is Kupka-Smale, by Lemma 2.3 we know Ws(σY) ∩ Wu(τY) = ∅. 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

Let X ∈ 𝔛1(M) and σSing(X). Then for any C1 neighborhood 𝓤(X) of X there are δ0 > 0 and α > 0 such that if 𝓞(δ) : TσMTσM is a linear map with ∥𝓞(δ)-DσX∥ < δ < δ0 then there is Yδ ∈ 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(Yδ, Y0) →0 as δ → 0. Here Y0 is the vector field for 𝓞(0) = DσX and d0 is the C0 metric.

By Lemma 2.5, Y0|Bα/4(σ) is regarded as a linearization of X|Bα/4(σ) with respect to the exponential coordinates. If there is an interval I ⊂ℝ and integral curve ζ(t)(tI) of the linear vector field 𝓞(δ) in $\begin{array}{}{\mathrm{e}\mathrm{x}\mathrm{p}}_{\sigma }^{-1}\end{array}$ (Bα/4(σ)) ⊂ TσM then the composition expσζ:I(⊂ ℝ) → M is an integral curve of Yδ in Bα/4(σ) ⊂ M (see [17]).

#### Lemma 2.6

Let 𝓤(X) be a C1 neighborhood of X. If a singular point σ is not hyperbolic then there is Y ∈ 𝓤(X) such that Y has 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 DσX has an eigenvalue λ with Re(λ) = 0. By Lemma 2.5, there is Y ∈ 𝓤1(X) such that σY is a singular point of Y and μ is the only eigenvalue of DσYY with Re(μ) = 0. Then $\begin{array}{}{T}_{{\sigma }_{Y}}M={E}_{{\sigma }_{Y}}^{s}\oplus {E}_{{\sigma }_{Y}}^{c}\oplus {E}_{{\sigma }_{Y}}^{u}\end{array}$ where, $\begin{array}{}{E}_{\sigma }^{s}\end{array}$ is the eigenspace of DσY Y associated with real part less than zero $\begin{array}{}{E}_{{\sigma }_{Y}}^{u}\end{array}$ is the eigenspace of DσYY associated with real part greater than zero, and $\begin{array}{}{E}_{{\sigma }_{Y}}^{c}\end{array}$ is the eigenspace of DσYY associated to μ.

Note that if dim $\begin{array}{}{E}_{{\sigma }_{Y}}^{c}\end{array}$ = 2 then there are no singularities of Y nearby around σY in the neighborhood of σY (see [8, Theorem 6.2]).

Thus we consider dim $\begin{array}{}{E}_{{\sigma }_{Y}}^{c}\end{array}$ = 1. Then there is r > 0 such that for all v$\begin{array}{}{E}_{{\sigma }_{Y}}^{c}\end{array}$ (r), Y(expσY) = 0, where $\begin{array}{}{E}_{{\sigma }_{Y}}^{c}\end{array}$ (r) = $\begin{array}{}{E}_{{\sigma }_{Y}}^{c}\end{array}$TσYM(r). We can take τ ∈ expσY( $\begin{array}{}{E}_{{\sigma }_{Y}}^{c}\end{array}$ (r)) − {σy} such that τ is sufficiently close to σY and τ is not a hyperbolic singularity for Y. We assume that index(σY) = index(τ) = j. Then we can make a hyperbolic singular point which index is different from index(σY) = index(τ) = j. By Lemma 2.5, take 0 < α < d(σY, τ)/2, 0 < δ < δ0 and a linear map 𝓞 : TσYMTσM such that 𝓞(v) = −δ v, for all v$\begin{array}{}{E}_{\sigma }^{c}\end{array}$ , and 𝓞(v) = DσYY(v), for all v$\begin{array}{}{E}_{{\sigma }_{Y}}^{s}\oplus {E}_{{\sigma }_{Y}}^{u}\end{array}$ . 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 σZ such that σZ is hyperbolic and index(σZ) = j + 1. Since Z(x) = Y(x) for all xBα(σY), τ is a non-hyperbolic singular point for Z which index is j. Using Lemma 2.5, there are W C1 close to Z(W ∈ 𝓤1(X)) and a linear map L : TτMTτM such that for some 0 < α1α, W(x) = $\begin{array}{}\left({D}_{\mathrm{e}\mathrm{x}{\mathrm{p}}_{\mathrm{\varnothing }}^{-1}\left(\mathrm{x}\right)}{\mathrm{e}\mathrm{x}\mathrm{p}}_{\tau }\right)\circ L\circ {\mathrm{e}\mathrm{x}\mathrm{p}}_{\tau }^{-1}\left(x\right),\end{array}$ if xBα1/4(τ), and W(x) = Z(x) if x ≠ ∈ Bα1(τ). Then τ is hyperbolic singularity for W which index is j. Thus the vector filed W has two hyperbolic singular points σZ with index(σZ) = j + 1 and τ with index(τ) = j.□

#### Proposition 2.7

If a vector field X ∈ 𝔛1(M) has the C1 robustly barycenter property then every singular points is hyperbolic.

#### Proof

Suppose, by contradiction, that there is a σSig(X) such that σ is not hyperbolic. By lemma 2.6, there is Y C1 close to X such that Y has two hyperbolic singular points σY 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

Let γ and η be hyperbolic periodic orbits and let 𝓤(X) be a C1 neighborhood of X. If a vector field X has the C1 robustly barycenter property, then for any Y ∈ 𝓤(X), we have index(γY) = index(ηY), where γY and ηY are the continuations of γ and η, 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 γY and ηY are the continuations of γ and η, respectively.

Suppose that γ has index i and η has index j with ij. If j < i then dim Wu(γ) + dim Ws(η) ≤ dim M. Take Y ∈ 𝓚𝓢 ∩ 𝓤1(X). Then we have dim Wu(γY) + dim Ws(ηY) ≤ dim M. Since Y is Kupka-Smale, by Lemma 2.3 we know Wu(γY) ∩ Ws(ηY) = ∅. 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 a C1 neighborhood of X and let γ be a periodic orbit of X. If a periodic point pγ is not hyperbolic then there is Y ∈ 𝓤(X) such that Y has two hyperbolic periodic orbits with different indices.

#### Proposition 2.10

Let 𝓤(X) be a C1 neighborhood of X. Suppose that X has the C1 robustly barycenter property. Then for any Y ∈ 𝓤(X), every periodic orbits of Y is 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 field X has the C1 robustly barycenter property then Sing(X) = ∅.

#### Proof

Let 𝓤(X) be a C1 neighborhood of X. Suppose that Sing(X) ≠ ∅. Then there are a hyperbolic σSing(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 σY and γY of σ and γ, respectively. Thus we know that dim Ws(σ) = dim Ws(σY), dim Wu(σ) = dim Wu(σY), dim Ws(γ) = dim Ws(γY) and dim Wu(γ) = dim Wu(γY).

If j < i then dim Wu(σ) + dim Ws(γ) ≤ dim M. Take a vector field Z ∈ 𝓚𝓢 ∩ 𝓤1(X) such that dim Wu(σZ) + dim Ws(γZ) ≤ dim M. By Lemma 2.3, Wu(σZ) ∩ Ws(γZ) = ∅. This is a contradiction by Lemma 2.1.

If ji then dim Ws(σ)+dim Wu(γ) ≤ dim M. As in the case j < i, we can take a vector field Y ∈ 𝓚𝓢 ∩ 𝓤1(X) such that dim Ws(σY)+dim Wu(γY) ≤ dim M. By Lemma 2.3, Ws(σY) ∩ Wu(γY) = ∅. 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 field X has the C1 robustly barycenter property then X is 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 any X ∈ 𝓖0, if a vector field X has the barycenter property then Sing(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 Wu(σ) + dim Ws(γ) ≤ dim M. Since X ∈ 𝓚𝓢, by Lemma 2.3

$Wu(σ)∩Wu(γ)=∅,$

which is a contradiction by Lemma 2.1. If ji then dim Ws(σ) + dim Wu(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 any X ∈ 𝓖0, if a vector field X has the barycenter property then for any hyperbolic periodic orbits γ and η, 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 ij. If j < i then dim Wu(γ) + dim Ws(η) ≤ dim M. Since X ∈ 𝓚𝓢, by Lemma 2.3, we have Wu(γ) ∩ Ws(η) = ∅. This is a contradiction by Lemma 2.1. If ji then dim Ws(γ) + dim Wu(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 any X ∈ 𝓖1, if for any C1 neighborhood 𝓤(X) of X there is Y ∈ 𝓤(X) such that Y has two distinct hyperbolic periodic orbits with different indices then X has 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 any X ∈ 𝓖2, if a vector field X has the barycenter property then there is δ > 0 such that X has no δ-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 Y C1 close to X such that Y has a non hyperbolic periodic orbit η. By Lemma 2.9, there is Z C1 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 any X ∈ 𝓖3, if for any δ > 0 and for any C1-neighborhood 𝓤(X) of X there is Y ∈ 𝓤(X) such that Y has a hyperbolic periodic orbit γ which has a δ-weak hyperbolic eigenvalue then X has a hyperbolic periodic orbit η which 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

For C1 generic X ∈ 𝔛1(M), if a transitive vector field X has the barycenter property then X is 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 : MTM the solution of the equation x = X(x) generates a Cr flow, Xt; by the other side given a Cr flow we can define a Cr−1 vector field by considering X(x) = $\begin{array}{}\frac{d{X}_{t}\left(x\right)}{dt}{|}_{t=0}.\end{array}$ We say that X is divergence-free if its divergence is equal to zero. Note that, by Liouville formula, a flow Xt is volume preserving if and only if the corresponding vector field, X, is divergence free. Let $\begin{array}{}{\mathfrak{X}}_{\mu }^{1}\end{array}$ (M) denote the space of Cr divergence free vector fields and we consider the usual C1 Whitney topology on this space. A vector field X$\begin{array}{}{\mathfrak{X}}_{\mu }^{1}\end{array}$ (M) is a divergence-free star vector field if there exists a C1 neighborhood 𝓤(X) of X in X$\begin{array}{}{\mathfrak{X}}_{\mu }^{1}\end{array}$ (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 field X$\begin{array}{}{\mathfrak{X}}_{\mu }^{1}\end{array}$ (M) has the C1 robustly barycenter property then Sing(X) = ∅ and X is Anosov.

#### Proof

By Ferreira [18, Theorem 1], if a divergence free vector field X$\begin{array}{}{\mathfrak{X}}_{\mu }^{1}\end{array}$ (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

For C1 generic X$\begin{array}{}{\mathfrak{X}}_{\mu }^{1}\end{array}$ (M), if X has the barycenter property then Sing(X) = ∅ and X is Anosov.

#### Proof

By Ferreira [18, Theorem 1], if a divergence-free vector field X$\begin{array}{}{\mathfrak{X}}_{\mu }^{1}\end{array}$ (M) satisfies star vector fields then Sing(X) = ∅ and it is Anosov. By Bessa [20], C1 generically, a divergence free vector field X$\begin{array}{}{\mathfrak{X}}_{\mu }^{1}\end{array}$ (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$\begin{array}{}{\mathfrak{X}}_{\mu }^{1}\end{array}$ (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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Accepted: 2018-02-28

Published Online: 2018-04-23

Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 429–436, ISSN (Online) 2391-5455,

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