Let *M* be as before, and let *X* ∈ 𝔛^{1}(*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:

$$\begin{array}{}{\displaystyle {W}^{ss}(p)=\{y\in M:d({X}_{t}(y),{X}_{t}(p))\to 0\phantom{\rule{thinmathspace}{0ex}}\text{as}\phantom{\rule{thinmathspace}{0ex}}t\to \mathrm{\infty}\},}\end{array}$$

and

$$\begin{array}{}{\displaystyle {W}^{s}(\gamma )=\bigcup _{t\in \mathbb{R}}{W}^{ss}({X}_{t}(p)).}\end{array}$$

If *η* > 0 then the local strong stable manifold
$\begin{array}{}{\displaystyle {W}_{\eta (p)}^{ss}}\end{array}$
(*p*) of *p* and the local stable manifolds
$\begin{array}{}{\displaystyle {W}_{\eta (\gamma )}^{s}}\end{array}$
(*γ*) of *γ* are defined by

$$\begin{array}{}{\displaystyle {W}_{\eta (p)}^{ss}(p)=\{y\in M:d({X}_{t}(y),{X}_{t}(p))<\eta (p),\text{if}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}t\ge 0\},}\end{array}$$

and

$$\begin{array}{}{\displaystyle {W}_{\eta (\gamma )}^{s}(\gamma )=\{y\in M:d({X}_{t}(y),{X}_{t}(\gamma ))<\eta (\gamma ),\phantom{\rule{thinmathspace}{0ex}}\text{if}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}t\ge 0\}.}\end{array}$$

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

$$\begin{array}{}{\displaystyle {W}^{ss}(p)=\bigcup _{t\ge 0}{X}_{-t}({W}_{\u03f5}^{ss}({X}_{t}(p))).}\end{array}$$

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

$$\begin{array}{}{\displaystyle {W}_{\u03f5}^{s}(\sigma )=\{x\in M:d({X}^{t}(x),\sigma )\le \u03f5\phantom{\rule{thinmathspace}{0ex}}\text{as}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}t\ge 0\}\phantom{\rule{thinmathspace}{0ex}}\text{and}}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}{W}^{s}(\sigma )=\bigcap _{t\ge 0}{X}^{t}({W}_{\u03f5}^{s}(\sigma )).}\end{array}$$

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* *W*^{s}(*γ*) ∩ *W*^{u}(*η*) ≠ ∅ *and* *W*^{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*_{tn}} ⊂ {*x*_{t}} such that *x*_{tn} → *x* as *t*_{n} → ∞(*n* → ∞). Then we have that

$$\begin{array}{}{\displaystyle d({X}_{-s}(x),{X}_{-s}(p))\le \u03f5\phantom{\rule{thinmathspace}{0ex}}\text{for}\phantom{\rule{thinmathspace}{0ex}}-s\to \mathrm{\infty}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}}\\ {\displaystyle \phantom{\rule{1em}{0ex}}d({X}_{T+s}(x),{X}_{s}(q))\le \u03f5\phantom{\rule{thinmathspace}{0ex}}\text{for}\phantom{\rule{thinmathspace}{0ex}}s\to \mathrm{\infty}.}\end{array}$$

This means that *x* ∈
$\begin{array}{}{\displaystyle {W}_{\u03f5}^{uu}}\end{array}$
(*p*) and *X*_{T}(*x*) ∈
$\begin{array}{}{\displaystyle {W}_{\u03f5}^{ss}}\end{array}$
(*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*_{σ}*X* 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 *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* ∈ 𝔛^{1}(*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 𝔛^{1}(*M*) (see [15]).

#### Lemma 2.3

([8, Lemma 3.4]). *Let* *X* ∈ 𝔛^{1}(*M*) *be a Kupka*-*Smale and let* *σ*, *ρ* ∈ *Crit*(*X*). *If* dim *W*^{s}(*σ*) + dim *W*^{u}(*ρ*) ≤ dim *M* *then* *W*^{s}(*σ*) ∩ *W*^{u}(*ρ*) = ∅.

Let *σ* ∈ *Crit*(*X*) be hyperbolic. Then there exist a *C*^{1}-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

*Let* *σ* *and* *τ* *be hyperbolic singular points and let* 𝓤(*X*) *be a* *C*^{1} *neighborhood of* *X*. *If a vector field* *X* *has the* *C*^{1} *robustly barycenter property*, *then for any* *Y* ∈ 𝓤(*X*), *we have* index(*σ*_{Y}) = index(*τ*_{Y}), *where* *σ*_{Y} *and* *τ*_{Y} *are the continuations of* *σ* *and* *τ*, *respectively*.

#### Proof

Let *σ* and *τ* be hyperbolic singular points, and let 𝓤(*X*) be a *C*^{1}-neighborhood of *X*. Then there is *Y* ∈ 𝓤_{1}(*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 ∩ 𝓤_{1}(*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 *C*^{1} 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 ∩𝓤_{1}(*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 *C*^{1} 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* ∈ 𝔛^{1}(*M*) *and* *σ* ∈ *Sing*(*X*). *Then for any* *C*^{1} *neighborhood* 𝓤(*X*) *of* *X* *there are* *δ*_{0} > 0 *and* *α* > 0 *such that if* 𝓞(*δ*) : *T*_{σ}M → *T*_{σ}M *is a linear map with* ∥𝓞(*δ*)-*D*_{σ}X∥ < *δ* < *δ*_{0} *then there is* *Y*^{δ} ∈ 𝓤(*X*) *satisfying*

$$\begin{array}{}{\displaystyle {Y}^{\delta}(x)=\left\{\begin{array}{}({D}_{{\mathrm{e}\mathrm{x}\mathrm{p}}_{\sigma}^{-1}(x)}{\mathrm{e}\mathrm{x}\mathrm{p}}_{\sigma})\circ \mathcal{O}(\delta )\circ {\mathrm{e}\mathrm{x}\mathrm{p}}_{\sigma}^{-1}(x),& if\phantom{\rule{thinmathspace}{0ex}}x\in {B}_{\alpha /4}(x)\\ X(x),& if\phantom{\rule{thinmathspace}{0ex}}x\notin {B}_{\alpha}(x).\end{array}\right.}\end{array}$$

*Furthermore*, *d*_{0}(*Y*^{δ}, *Y*^{0}) →0 *as* *δ* → 0. *Here* *Y*^{0} *is the vector field for* 𝓞(0) = *D*_{σ}X *and* *d*_{0} *is the* *C*^{0} *metric*.

By Lemma 2.5, *Y*^{0}|_{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*)(*t* ∈ *I*) of the linear vector field 𝓞(*δ*) in
$\begin{array}{}{\displaystyle {\mathrm{e}\mathrm{x}\mathrm{p}}_{\sigma}^{-1}}\end{array}$
(*B*_{α/4}(*σ*)) ⊂ *T*_{σ}M 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 a* *C*^{1} *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 𝓤_{1}(*X*) ⊂ 𝓤(*X*) be a *C*^{1} neighborhood of *X*. Since a singular point *σ* is not hyperbolic we have that *D*_{σ}X has an eigenvalue λ with *Re*(λ) = 0. By Lemma 2.5, there is *Y* ∈ 𝓤_{1}(*X*) such that *σ*_{Y} is a singular point of *Y* and *μ* is the only eigenvalue of *D*_{σY}*Y* with *Re*(*μ*) = 0. Then
$\begin{array}{}{\displaystyle {T}_{{\sigma}_{Y}}M={E}_{{\sigma}_{Y}}^{s}\oplus {E}_{{\sigma}_{Y}}^{c}\oplus {E}_{{\sigma}_{Y}}^{u}}\end{array}$
where,
$\begin{array}{}{\displaystyle {E}_{\sigma}^{s}}\end{array}$
is the eigenspace of *D*_{σY} *Y* associated with real part less than zero
$\begin{array}{}{\displaystyle {E}_{{\sigma}_{Y}}^{u}}\end{array}$
is the eigenspace of *D*_{σY}*Y* associated with real part greater than zero, and
$\begin{array}{}{\displaystyle {E}_{{\sigma}_{Y}}^{c}}\end{array}$
is the eigenspace of *D*_{σY}*Y* associated to *μ*.

Note that if dim
$\begin{array}{}{\displaystyle {E}_{{\sigma}_{Y}}^{c}}\end{array}$
= 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
$\begin{array}{}{\displaystyle {E}_{{\sigma}_{Y}}^{c}}\end{array}$
= 1. Then there is *r* > 0 such that for all *v* ∈
$\begin{array}{}{\displaystyle {E}_{{\sigma}_{Y}}^{c}}\end{array}$
(*r*), *Y*(exp_{σY}) = 0, where
$\begin{array}{}{\displaystyle {E}_{{\sigma}_{Y}}^{c}}\end{array}$
(*r*) =
$\begin{array}{}{\displaystyle {E}_{{\sigma}_{Y}}^{c}}\end{array}$
∩ *T*_{σY}*M*(*r*). We can take *τ* ∈ exp_{σY}(
$\begin{array}{}{\displaystyle {E}_{{\sigma}_{Y}}^{c}}\end{array}$
(*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* < *δ*_{0} and a linear map 𝓞 : *T*_{σY}*M* → *T*_{σ}M such that 𝓞(*v*) = −*δ* *v*, for all *v* ∈
$\begin{array}{}{\displaystyle {E}_{\sigma}^{c}}\end{array}$
, and 𝓞(*v*) = *D*_{σY}*Y*(*v*), for all *v* ∈
$\begin{array}{}{\displaystyle {E}_{{\sigma}_{Y}}^{s}\oplus {E}_{{\sigma}_{Y}}^{u}}\end{array}$
. By Lemma 2.5, there is *Z* ∈ 𝓤_{1}(*X*) such that

$$\begin{array}{}{\displaystyle Z(x)=({D}_{\mathrm{e}\mathrm{x}{\mathrm{p}}_{{\text{\u0153}}_{\mathrm{Y}}}^{-1}(\mathrm{x})}{\mathrm{e}\mathrm{x}\mathrm{p}}_{{\sigma}_{Y}})\circ \mathcal{O}\circ {\mathrm{e}\mathrm{x}\mathrm{p}}_{{\sigma}_{Y}}^{-1}(x),\phantom{\rule{thinmathspace}{0ex}}\text{if}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}x\in {B}_{\alpha /4}({\sigma}_{Y}).}\end{array}$$

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 *W* *C*^{1} close to *Z*(*W* ∈ 𝓤_{1}(*X*)) and a linear map *L* : *T*_{τ}M → *T*_{τ}M such that for some 0 < *α*_{1} ≤ *α*, *W*(*x*) =
$\begin{array}{}{\displaystyle ({D}_{\mathrm{e}\mathrm{x}{\mathrm{p}}_{\mathrm{\varnothing}}^{-1}(\mathrm{x})}{\mathrm{e}\mathrm{x}\mathrm{p}}_{\tau})\circ L\circ {\mathrm{e}\mathrm{x}\mathrm{p}}_{\tau}^{-1}(x),}\end{array}$
if *x* ∈ *B*_{α1/4}(*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 field* *X* ∈ 𝔛^{1}(*M*) *has the* *C*^{1} *robustly 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 *Y* *C*^{1} 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 *C*^{1} 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* *C*^{1} *neighborhood of* *X*. *If a vector field* *X* *has the* *C*^{1} *robustly barycenter property*, *then for any* *Y* ∈ 𝓤(*X*), *we have* index(*γ*_{Y}) = index(*η*_{Y}), *where* *γ*_{Y} *and* *η*_{Y} *are the continuations of* *γ* *and* *η*, *respectively*.

#### Proof

Let *γ* and *η* be hyperbolic closed orbits, and let 𝓤(*X*) be a *C*^{1} neighborhood of *X*. Then there is *Y* ∈ 𝓤_{1}(*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 ∩ 𝓤_{1}(*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 *C*^{1} 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 *C*^{1} robustly specification property. But, the result can be obtained similarly without any properties.

#### Lemma 2.9

*Let* 𝓤(*X*) *be a* *C*^{1} *neighborhood of* *X* *and let* *γ* *be a periodic orbit of* *X*. *If a periodic point* *p* ∈ *γ* *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* *C*^{1} *neighborhood of* *X*. *Suppose that* *X* *has the* *C*^{1} *robustly barycenter property*. *Then for any* *Y* ∈ 𝓤(*X*), *every periodic orbits of* *Y* *is hyperbolic*.

#### Proof

Let 𝓤(*X*) be a *C*^{1} 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 *C*^{1} robustly barycenter property, this is a contradiction by Lemma 2.8. □

#### Theorem 2.11

*If a vector field* *X* *has the* *C*^{1} *robustly barycenter property then* *Sing*(*X*) = ∅.

#### Proof

Let 𝓤(*X*) be a *C*^{1} 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 *C*^{1} neighborhood 𝓤_{1}(*X*) ⊂ 𝓤(*X*) of *X* such that for any *Y* ∈ 𝓤_{1}(*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 ∩ 𝓤_{1}(*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 ∩ 𝓤_{1}(*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 *C*^{1} robustly barycenter property then *X* has no singularities. □

#### Proof of Theorem A

Suppose that *X* has the *C*^{1} 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* *C*^{1} *robustly barycenter property then* *X* *is Anosov*.

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