Let *M* be as before, and let *X* â đ^{1}(*M*). For *p* â *Îł* â *P*(*X*), the strong stable manifold *W*^{ss}(*p*) of *p* and stable manifold *W*^{s}(*Îł*) of *Îł* are defined as follows:

$$\begin{array}{}{\displaystyle {W}^{ss}(p)=\{y\xe2\x88\x88M:d({X}_{t}(y),{X}_{t}(p))\xe2\x86\x920\phantom{\rule{thinmathspace}{0ex}}\text{as}\phantom{\rule{thinmathspace}{0ex}}t\xe2\x86\x92\mathrm{\xe2\x88\x9e}\},}\end{array}$$

and

$$\begin{array}{}{\displaystyle {W}^{s}(\mathrm{\xce\u0142})=\underset{t\xe2\x88\x88\mathbb{R}}{\xe2\x8b\x83}{W}^{ss}({X}_{t}(p)).}\end{array}$$

If *Î·* > 0 then the local strong stable manifold
$\begin{array}{}{\displaystyle {W}_{\mathrm{\xce\xb7}(p)}^{ss}}\end{array}$
(*p*) of *p* and the local stable manifolds
$\begin{array}{}{\displaystyle {W}_{\mathrm{\xce\xb7}(\mathrm{\xce\u0142})}^{s}}\end{array}$
(*Îł*) of *Îł* are defined by

$$\begin{array}{}{\displaystyle {W}_{\mathrm{\xce\xb7}(p)}^{ss}(p)=\{y\xe2\x88\x88M:d({X}_{t}(y),{X}_{t}(p))<\mathrm{\xce\xb7}(p),\text{if}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}t\xe2\x89\u201e0\},}\end{array}$$

and

$$\begin{array}{}{\displaystyle {W}_{\mathrm{\xce\xb7}(\mathrm{\xce\u0142})}^{s}(\mathrm{\xce\u0142})=\{y\xe2\x88\x88M:d({X}_{t}(y),{X}_{t}(\mathrm{\xce\u0142}))<\mathrm{\xce\xb7}(\mathrm{\xce\u0142}),\phantom{\rule{thinmathspace}{0ex}}\text{if}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}t\xe2\x89\u201e0\}.}\end{array}$$

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

$$\begin{array}{}{\displaystyle {W}^{ss}(p)=\underset{t\xe2\x89\u201e0}{\xe2\x8b\x83}{X}_{\xe2\x88\x92t}({W}_{\mathrm{\xcf\u201d}}^{ss}({X}_{t}(p))).}\end{array}$$

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 *Ï* is a hyperbolic singularity of *X* then there exists an *Ï”* = *Ï”*(*Ï*) > 0 such that

$$\begin{array}{}{\displaystyle {W}_{\mathrm{\xcf\u201d}}^{s}(\mathrm{\xcf\x83})=\{x\xe2\x88\x88M:d({X}^{t}(x),\mathrm{\xcf\x83})\xe2\x89\u20ac\mathrm{\xcf\u201d}\phantom{\rule{thinmathspace}{0ex}}\text{as}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}t\xe2\x89\u201e0\}\phantom{\rule{thinmathspace}{0ex}}\text{and}}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}{W}^{s}(\mathrm{\xcf\x83})=\underset{t\xe2\x89\u201e0}{\xe2\x8b\x82}{X}^{t}({W}_{\mathrm{\xcf\u201d}}^{s}(\mathrm{\xcf\x83})).}\end{array}$$

Analogous definitions hold for unstable manifolds.

#### Lemma 2.1

*Let* *Îł* *and* *Î·* *be hyperbolic critical points of* *X*. *If a vector field* *X* *has the barycenter property then* *W*^{s}(*Îł*) â© *W*^{u}(*Î·*) â â
*and* *W*^{u}(*Îł*) â© *W*^{s}(*Î·*) â â
.

#### Proof

First, we consider periodic orbits. Take *p* â *Îł* and *q* â *Î·* such that *p* and *q* are hyperbolic. Denote by *Ï”*(*p*) the size of the local strong unstable manifold of *p* and by *Ï”*(*q*) the size of the local strong unstable manifold of *q*. Let *Ï”* = min{*Ï”*(*p*), *Ï”*(*q*)} and let *T* = *T*(*Ï”*, *p*, *q*) be given by the barycenter property. For *t* > 0, there is *x*_{t} â *M* such that *d*(*X*_{s}(*x*_{t}), *X*_{s}(*p*)) â€ *Ï”* for â*t* â€ *s* â€ 0 and *d*(*X*_{T+s}(*x*_{t}), *X*_{s}(*q*)) â€ *Ï”* for 0 â€ *s* â€ *t*. Since *M* is compact, there is a subsequence {*x*_{tn}} â {*x*_{t}} such that *x*_{tn} â *x* as *t*_{n} â â(*n* â â). Then we have that

$$\begin{array}{}{\displaystyle d({X}_{\xe2\x88\x92s}(x),{X}_{\xe2\x88\x92s}(p))\xe2\x89\u20ac\mathrm{\xcf\u201d}\phantom{\rule{thinmathspace}{0ex}}\text{for}\phantom{\rule{thinmathspace}{0ex}}\xe2\x88\x92s\xe2\x86\x92\mathrm{\xe2\x88\x9e}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}}\\ {\displaystyle \phantom{\rule{1em}{0ex}}d({X}_{T+s}(x),{X}_{s}(q))\xe2\x89\u20ac\mathrm{\xcf\u201d}\phantom{\rule{thinmathspace}{0ex}}\text{for}\phantom{\rule{thinmathspace}{0ex}}s\xe2\x86\x92\mathrm{\xe2\x88\x9e}.}\end{array}$$

This means that *x* â
$\begin{array}{}{\displaystyle {W}_{\mathrm{\xcf\u201d}}^{uu}}\end{array}$
(*p*) and *X*_{T}(*x*) â
$\begin{array}{}{\displaystyle {W}_{\mathrm{\xcf\u201d}}^{ss}}\end{array}$
(*q*). Thus we have *W*^{u}(*p*) â© *W*^{s}(*q*) â â
. Similarly, we can show that *W*^{u}(*Îł*) â© *W*^{s}(*Î·*) â â
. Consequently, *W*^{s}(*Îł*) â© *W*^{u}(*Î·*) â â
and *W*^{u}(*Îł*) â© *W*^{s}(*Î·*) â â
.

Finally, we consider singular points. Let *Ï* and *Ï* be hyperbolic singular points of *X*. As in the first case, we have *W*^{s}(*Ï*) â© *W*^{u}(*Ï*) â â
and *W*^{u}(*Ï*) â© *W*^{s}(*Ï*) â â
. âĄ

A singularity *Ï* is a *sink* if all eigenvalues of *D*_{Ï}*X* have a negative real part. A periodic point *p* is a *sink* if the eigenvalues of the derivative of the PoincarĂ© map associated to *p* have absolute value less than one. A source is a sink for the vector field â*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* â *Crit*(*X*) and *q* â *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*) â© *W*^{u}(*q*) â â
and *W*^{u}(*p*) â© *W*^{s}(*q*) â â
. Since *p* is sink, *W*^{u}(*p*) â© *W*^{s}(*q*) = â
. But, since *X* has the barycenter property, *W*^{u}(*p*) â© *W*^{s}(*q*) â â
. 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. âĄ

We say that *X* â đ^{1}(*M*) is *Kupka*-*Smale* if every *Ï* â *Crit*(*X*) is hyperbolic and their stable and unstable manifolds intersect transversally. Denote by đđą the set of all Kupka-Smale vector fields. It is well-known that the set of Kupka-Smale vector fields is residual in đ^{1}(*M*) (see [15]).

#### Lemma 2.3

([8, Lemma 3.4]). *Let* *X* â đ^{1}(*M*) *be a Kupka*-*Smale and let* *Ï*, *Ï* â *Crit*(*X*). *If* dim *W*^{s}(*Ï*) + dim *W*^{u}(*Ï*) â€ dim *M* *then* *W*^{s}(*Ï*) â© *W*^{u}(*Ï*) = â
.

Let *Ï* â *Crit*(*X*) be hyperbolic. Then there exist a *C*^{1}-neighborhood đ€(*X*) of *X* and a neighborhood *U* of *Ï* such that for any *Y* â đ€(*X*), there is *Ï*_{Y} such that *Ï*_{Y} is the continuation of *Ï* and index(*Ï*) = index(*Ï*_{Y}) (see [16]).

#### Lemma 2.4

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

#### Proof

Let *Ï* and *Ï* be hyperbolic singular points, and let đ€(*X*) be a *C*^{1}-neighborhood of *X*. Then there is *Y* â đ€_{1}(*X*) âđ€(*X*) such that *Ï*_{Y} and *Ï*_{Y} are the continuations of *Ï* and *Ï*, respectively. Since *X* has the barycenter property, by Lemma 2.2, *X* has neither sinks nor sources. Thus we may assume that *Ï* has index *i* and *Ï* has index *j* with *i* â *j*. If *j* < *i* then dim *W*^{u}(*Ï*) + dim *W*^{s}(*Ï*) < dim *M*. Take *Y* â đđą â© đ€_{1}(*X*). Then we have dim *W*^{u}(*Ï*_{Y}) + dim *W*^{s}(*Ï*_{Y}) < dim *M*. Since *Y* is Kupka-Smale, by Lemma 2.3 we know *W*^{u}(*Ï*_{Y}) â© *W*^{s}(*Ï*_{Y}) = â
. Since *X* has the *C*^{1} robustly barycenter property, this is a contradiction by Lemma 2.1. If *j* > *i* then dim *W*^{s}(*Ï*) + dim *W*^{u}(*Ï*) < dim *M* As in the case of *j* < *i*, we can take *Y* â đđą â©đ€_{1}(*X*). Then we have dim *W*^{s}(*Ï*_{Y}) + dim *W*^{u}(*Ï*_{Y}) < dim *M*. Since *Y* is Kupka-Smale, by Lemma 2.3 we know *W*^{s}(*Ï*_{Y}) â© *W*^{u}(*Ï*_{Y}) = â
. Since *X* has *C*^{1} robustly the barycenter property, this is a contradiction by Lemma 2.1. âĄ

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

#### Lemma 2.5

*Let* *X* â đ^{1}(*M*) *and* *Ï* â *Sing*(*X*). *Then for any* *C*^{1} *neighborhood* đ€(*X*) *of* *X* *there are* *ÎŽ*_{0} > 0 *and* *Î±* > 0 *such that if* đ(*ÎŽ*) : *T*_{Ï}M â *T*_{Ï}M *is a linear map with* â„đ(*ÎŽ*)-*D*_{Ï}Xâ„ < *ÎŽ* < *ÎŽ*_{0} *then there is* *Y*^{ÎŽ} â đ€(*X*) *satisfying*

$$\begin{array}{}{\displaystyle {Y}^{\mathrm{\xce\u017d}}(x)=\left\{\begin{array}{}({D}_{{\mathrm{e}\mathrm{x}\mathrm{p}}_{\mathrm{\xcf\x83}}^{\xe2\x88\x921}(x)}{\mathrm{e}\mathrm{x}\mathrm{p}}_{\mathrm{\xcf\x83}})\xe2\x88\x98\mathcal{O}(\mathrm{\xce\u017d})\xe2\x88\x98{\mathrm{e}\mathrm{x}\mathrm{p}}_{\mathrm{\xcf\x83}}^{\xe2\x88\x921}(x),& if\phantom{\rule{thinmathspace}{0ex}}x\xe2\x88\x88{B}_{\mathrm{\xce\pm}/4}(x)\\ X(x),& if\phantom{\rule{thinmathspace}{0ex}}x\xe2\x88\x89{B}_{\mathrm{\xce\pm}}(x).\end{array}\right.}\end{array}$$

*Furthermore*, *d*_{0}(*Y*^{ÎŽ}, *Y*^{0}) â0 *as* *ÎŽ* â 0. *Here* *Y*^{0} *is the vector field for* đ(0) = *D*_{Ï}X *and* *d*_{0} *is the* *C*^{0} *metric*.

By Lemma 2.5, *Y*^{0}|_{BÎ±/4}(*Ï*) is regarded as a linearization of *X*|_{BÎ±/4}(*Ï*) with respect to the exponential coordinates. If there is an interval *I* ââ and integral curve *Î¶*(*t*)(*t* â *I*) of the linear vector field đ(*ÎŽ*) in
$\begin{array}{}{\displaystyle {\mathrm{e}\mathrm{x}\mathrm{p}}_{\mathrm{\xcf\x83}}^{\xe2\x88\x921}}\end{array}$
(*B*_{Î±/4}(*Ï*)) â *T*_{Ï}M then the composition exp_{Ï} â *Î¶*:*I*(â â) â *M* is an integral curve of *Y*^{ÎŽ} in *B*_{Î±/4}(*Ï*) â *M* (see [17]).

#### Lemma 2.6

*Let* đ€(*X*) *be a* *C*^{1} *neighborhood of* *X*. *If a singular point* *Ï* *is not hyperbolic then there is* *Y* â đ€(*X*) *such that* *Y* *has two hyperbolic singular points with different indices*.

#### Proof

Let đ€_{1}(*X*) â đ€(*X*) be a *C*^{1} neighborhood of *X*. Since a singular point *Ï* is not hyperbolic we have that *D*_{Ï}X has an eigenvalue Î» with *Re*(Î») = 0. By Lemma 2.5, there is *Y* â đ€_{1}(*X*) such that *Ï*_{Y} is a singular point of *Y* and *ÎŒ* is the only eigenvalue of *D*_{ÏY}*Y* with *Re*(*ÎŒ*) = 0. Then
$\begin{array}{}{\displaystyle {T}_{{\mathrm{\xcf\x83}}_{Y}}M={E}_{{\mathrm{\xcf\x83}}_{Y}}^{s}\xe2\x8a\x95{E}_{{\mathrm{\xcf\x83}}_{Y}}^{c}\xe2\x8a\x95{E}_{{\mathrm{\xcf\x83}}_{Y}}^{u}}\end{array}$
where,
$\begin{array}{}{\displaystyle {E}_{\mathrm{\xcf\x83}}^{s}}\end{array}$
is the eigenspace of *D*_{ÏY} *Y* associated with real part less than zero
$\begin{array}{}{\displaystyle {E}_{{\mathrm{\xcf\x83}}_{Y}}^{u}}\end{array}$
is the eigenspace of *D*_{ÏY}*Y* associated with real part greater than zero, and
$\begin{array}{}{\displaystyle {E}_{{\mathrm{\xcf\x83}}_{Y}}^{c}}\end{array}$
is the eigenspace of *D*_{ÏY}*Y* associated to *ÎŒ*.

Note that if dim
$\begin{array}{}{\displaystyle {E}_{{\mathrm{\xcf\x83}}_{Y}}^{c}}\end{array}$
= 2 then there are no singularities of *Y* nearby around *Ï*_{Y} in the neighborhood of *Ï*_{Y} (see [8, Theorem 6.2]).

Thus we consider dim
$\begin{array}{}{\displaystyle {E}_{{\mathrm{\xcf\x83}}_{Y}}^{c}}\end{array}$
= 1. Then there is *r* > 0 such that for all *v* â
$\begin{array}{}{\displaystyle {E}_{{\mathrm{\xcf\x83}}_{Y}}^{c}}\end{array}$
(*r*), *Y*(exp_{ÏY}) = 0, where
$\begin{array}{}{\displaystyle {E}_{{\mathrm{\xcf\x83}}_{Y}}^{c}}\end{array}$
(*r*) =
$\begin{array}{}{\displaystyle {E}_{{\mathrm{\xcf\x83}}_{Y}}^{c}}\end{array}$
â© *T*_{ÏY}*M*(*r*). We can take *Ï* â exp_{ÏY}(
$\begin{array}{}{\displaystyle {E}_{{\mathrm{\xcf\x83}}_{Y}}^{c}}\end{array}$
(*r*)) â {*Ï*_{y}} such that *Ï* is sufficiently close to *Ï*_{Y} and *Ï* is not a hyperbolic singularity for *Y*. We assume that index(*Ï*_{Y}) = index(*Ï*) = *j*. Then we can make a hyperbolic singular point which index is different from index(*Ï*_{Y}) = index(*Ï*) = *j*. By Lemma 2.5, take 0 < *Î±* < *d*(*Ï*_{Y}, *Ï*)/2, 0 < *ÎŽ* < *ÎŽ*_{0} and a linear map đ : *T*_{ÏY}*M* â *T*_{Ï}M such that đ(*v*) = â*ÎŽ* *v*, for all *v* â
$\begin{array}{}{\displaystyle {E}_{\mathrm{\xcf\x83}}^{c}}\end{array}$
, and đ(*v*) = *D*_{ÏY}*Y*(*v*), for all *v* â
$\begin{array}{}{\displaystyle {E}_{{\mathrm{\xcf\x83}}_{Y}}^{s}\xe2\x8a\x95{E}_{{\mathrm{\xcf\x83}}_{Y}}^{u}}\end{array}$
. By Lemma 2.5, there is *Z* â đ€_{1}(*X*) such that

$$\begin{array}{}{\displaystyle Z(x)=({D}_{\mathrm{e}\mathrm{x}{\mathrm{p}}_{{\text{\u0106\x93}}_{\mathrm{Y}}}^{\xe2\x88\x921}(\mathrm{x})}{\mathrm{e}\mathrm{x}\mathrm{p}}_{{\mathrm{\xcf\x83}}_{Y}})\xe2\x88\x98\mathcal{O}\xe2\x88\x98{\mathrm{e}\mathrm{x}\mathrm{p}}_{{\mathrm{\xcf\x83}}_{Y}}^{\xe2\x88\x921}(x),\phantom{\rule{thinmathspace}{0ex}}\text{if}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}x\xe2\x88\x88{B}_{\mathrm{\xce\pm}/4}({\mathrm{\xcf\x83}}_{Y}).}\end{array}$$

Then there is the singular point *Ï*_{Z} such that *Ï*_{Z} is hyperbolic and index(*Ï*_{Z}) = *j* + 1. Since *Z*(*x*) = *Y*(*x*) for all *x* â *B*_{Î±}(*Ï*_{Y}), *Ï* 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* â đ€_{1}(*X*)) and a linear map *L* : *T*_{Ï}M â *T*_{Ï}M such that for some 0 < *Î±*_{1} â€ *Î±*, *W*(*x*) =
$\begin{array}{}{\displaystyle ({D}_{\mathrm{e}\mathrm{x}{\mathrm{p}}_{\mathrm{\xe2\x88\x85}}^{\xe2\x88\x921}(\mathrm{x})}{\mathrm{e}\mathrm{x}\mathrm{p}}_{\mathrm{\xcf\x84}})\xe2\x88\x98L\xe2\x88\x98{\mathrm{e}\mathrm{x}\mathrm{p}}_{\mathrm{\xcf\x84}}^{\xe2\x88\x921}(x),}\end{array}$
if *x* â *B*_{Î±1/4}(*Ï*), and *W*(*x*) = *Z*(*x*) if *x* â â *B*_{Î±1}(*Ï*). Then *Ï* is hyperbolic singularity for *W* which index is *j*. Thus the vector filed *W* has two hyperbolic singular points *Ï*_{Z} with index(*Ï*_{Z}) = *j* + 1 and *Ï* with index(*Ï*) = *j*.âĄ

#### Proposition 2.7

*If a vector field* *X* â đ^{1}(*M*) *has the* *C*^{1} *robustly barycenter property then every singular points is hyperbolic*.

#### Proof

Suppose, by contradiction, that there is a *Ï* â *Sig*(*X*) such that *Ï* is not hyperbolic. By lemma 2.6, there is *Y* *C*^{1} close to *X* such that *Y* has two hyperbolic singular points *Ï*_{Y} and *Ï* 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 âĄ

#### Lemma 2.8

*Let* *Îł* *and* *Î·* *be hyperbolic periodic orbits and let* đ€(*X*) *be a* *C*^{1} *neighborhood of* *X*. *If a vector field* *X* *has the* *C*^{1} *robustly barycenter property*, *then for any* *Y* â đ€(*X*), *we have* index(*Îł*_{Y}) = index(*Î·*_{Y}), *where* *Îł*_{Y} *and* *Î·*_{Y} *are the continuations of* *Îł* *and* *Î·*, *respectively*.

#### Proof

Let *Îł* and *Î·* be hyperbolic closed orbits, and let đ€(*X*) be a *C*^{1} neighborhood of *X*. Then there is *Y* â đ€_{1}(*X*) âđ€(*X*) such that *Îł*_{Y} and *Î·*_{Y} are the continuations of *Îł* and *Î·*, respectively.

Suppose that *Îł* has index *i* and *Î·* has index *j* with *i* â *j*. If *j* < *i* then dim *W*^{u}(*Îł*) + dim *W*^{s}(*Î·*) â€ dim *M*. Take *Y* â đđą â© đ€_{1}(*X*). Then we have dim *W*^{u}(*Îł*_{Y}) + dim *W*^{s}(*Î·*_{Y}) â€ dim *M*. Since *Y* is Kupka-Smale, by Lemma 2.3 we know *W*^{u}(*Îł*_{Y}) â© *W*^{s}(*Î·*_{Y}) = â
. Since *X* has the *C*^{1} robustly barycenter property, this is a contradiction by Lemma 2.1. Other case is similar. âĄ

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* đ€(*X*) *be a* *C*^{1} *neighborhood of* *X* *and let* *Îł* *be a periodic orbit of* *X*. *If a periodic point* *p* â *Îł* *is not hyperbolic then there is* *Y* â đ€(*X*) *such that* *Y* *has two hyperbolic periodic orbits with different indices*.

#### Proposition 2.10

*Let* đ€(*X*) *be a* *C*^{1} *neighborhood of* *X*. *Suppose that* *X* *has the* *C*^{1} *robustly barycenter property*. *Then for any* *Y* â đ€(*X*), *every periodic orbits of* *Y* *is hyperbolic*.

#### Proof

Let đ€(*X*) be a *C*^{1} neighborhood of *X*. To derive a contradiction, we may assume that there is *Y* â đ€(*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. âĄ

#### Theorem 2.11

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

#### Proof

Let đ€(*X*) be a *C*^{1} neighborhood of *X*. Suppose that *Sing*(*X*) â â
. Then there are a hyperbolic *Ï* â *Sing*(*X*) with index *i* and a hyperbolic periodic orbit *Îł* with index *j*. Then there is a *C*^{1} neighborhood đ€_{1}(*X*) â đ€(*X*) of *X* such that for any *Y* â đ€_{1}(*X*), there are the continuations *Ï*_{Y} and *Îł*_{Y} of *Ï* and *Îł*, respectively. Thus we know that dim *W*^{s}(*Ï*) = dim *W*^{s}(*Ï*_{Y}), dim *W*^{u}(*Ï*) = dim *W*^{u}(*Ï*_{Y}), dim *W*^{s}(*Îł*) = dim *W*^{s}(*Îł*_{Y}) and dim *W*^{u}(*Îł*) = dim *W*^{u}(*Îł*_{Y}).

If *j* < *i* then dim *W*^{u}(*Ï*) + dim *W*^{s}(*Îł*) â€ dim *M*. Take a vector field *Z* â đđą â© đ€_{1}(*X*) such that dim *W*^{u}(*Ï*_{Z}) + dim *W*^{s}(*Îł*_{Z}) â€ dim *M*. By Lemma 2.3, *W*^{u}(*Ï*_{Z}) â© *W*^{s}(*Îł*_{Z}) = â
. This is a contradiction by Lemma 2.1.

If *j* â„ *i* then dim *W*^{s}(*Ï*)+dim *W*^{u}(*Îł*) â€ dim *M*. As in the case *j* < *i*, we can take a vector field *Y* â đđą â© đ€_{1}(*X*) such that dim *W*^{s}(*Ï*_{Y})+dim *W*^{u}(*Îł*_{Y}) â€ dim *M*. By Lemma 2.3, *W*^{s}(*Ï*_{Y}) â© *W*^{u}(*Îł*_{Y}) = â
. This is a contradiction. Thus if a vector field *X* has the *C*^{1} robustly barycenter property then *X* has no singularities. âĄ

#### Proof of Theorem A

Suppose that *X* has the *C*^{1} robustly barycenter property. Then by Theorem 2.11, *Sing*(*X*) = â
. 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. âĄ

If a vector field *X* is transitive, then it is clear that *Î©*(*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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