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 Open Access
Divergencefree vector fields with orbital shadowing
 Manseob Lee^{1}Email author
https://doi.org/10.1186/168718472013132
© Lee; licensee Springer 2013
Received: 8 January 2013
Accepted: 17 April 2013
Published: 7 May 2013
Abstract
We show that a divergencefree vector field belongs to the ${C}^{1}$interior of the set of divergencefree vector fields satisfying the orbital shadowing property when the vector field is Anosov.
MSC:37C10, 37C50, 37D20.
Keywords
 divergencefree vector fields
 star condition
 shadowing
 orbital shadowing
 Anosov
1 Introduction
The shadowing theory is a very useful notion for the investigation of the stability condition. In fact, Robinson [1] and Sakai [2] proved that a diffeomorphism belongs to the ${C}^{1}$interior of the set of diffeomorphisms having the shadowing property coincides the structural stability, that is, the diffeomorphism satisfies both Axiom A and the strong transversality condition. In general, if a diffeomorphism is Ωstable, that is, a diffeomorphism satisfies both Axiom A and a nocycle condition, then there is a diffeomorphism which does not have the shadowing property (see, [3]). However, for another shadowing property, if a diffeomorphism is Ωstable, then the diffeomorphism has another shadowing property.
In this article, we study another shadowing property which is called the orbital shadowing property. It is clear that if a diffeomorphism has the shadowing property, then it has the orbital shadowing property. But the converse is not true. In fact, an irrational rotation map does not have the shadowing property, but it has the orbital shadowing property.
The orbital shadowing property was introduced by Pilyugin et al. [3]. They showed that a diffeomorphism belongs to the ${C}^{1}$interior of the set of diffeomorphisms having the orbital shadowing property if and only if the diffeomorphism is structurally stable.
For a conservative diffeomorphism, Bessa [4] proved that a conservative diffeomorphism is in the ${C}^{1}$interior of the set of all conservative diffeomorphisms satisfying the shadowing property if and only if it is Anosov. Lee and Lee [5, 6] proved that a conservative diffeomorphism is in the ${C}^{1}$interior of the set of all conservative diffeomorphisms satisfying the orbital shadowing property if and only if it is Anosov. Also, for a conservative vector field, that is, a divergencefree vector field, Ferreira [7] proved that if a conservative vector field belongs to the ${C}^{1}$interior of the set of all conservative vector fields satisfying the shadowing property, then it is Anosov. From the results, we study that if a conservative vector field belongs to the ${C}^{1}$interior of the set of all conservative vector fields having the orbital shadowing property, then it is Anosov. Our result is a generalization of the main theorem in [7].
2 Basic notions, definitions and results
for any $x\in \mathrm{\Lambda}$ and $t>0$. If $\mathrm{\Lambda}=M$, then X is called Anosov.
Let $\mathcal{N}={\bigcup}_{x\in R}{\mathcal{N}}_{x}$ be the normal bundle based on R. One can define the associated linear Poincaré flow by ${P}_{X}^{t}(x):={\mathrm{\Pi}}_{{X}_{t}(x)}\circ D{X}^{t}(x)$, where ${\mathrm{\Pi}}_{{X}^{t}(x)}:{T}_{{X}^{t}(x)}M\to {N}_{{X}^{t}(x)}$ is the projection along the direction of $X({X}^{t}(x))$.
Denote by $int{\mathcal{OS}}_{\mu}(M)$ the set of divergencefree vector fields satisfying the orbital shadowing property.
Theorem 2.1 Let $X\in {\mathfrak{X}}_{\mu}^{1}(M)$. If $X\in int{\mathcal{OS}}_{\mu}(M)$, then X has no singularity and X is Anosov.
3 Proof of Theorem 2.1
for all $x\in \mathrm{\Lambda}$. The following is well known and one can find a proof in [8].
Theorem 3.1 Λ is a hyperbolic set of ${X}^{t}$ if and only if the linear Poincaré flow ${P}_{X}^{t}$ restricted on Λ has a hyperbolic splitting ${N}_{\mathrm{\Lambda}}={\mathrm{\Delta}}^{s}\oplus {\mathrm{\Delta}}^{u}$.
for any $x\in \mathrm{\Lambda}$.
The following was proved in [9].
Theorem 3.2 [[9], Proposition 4.1]
If $X\in {\mathfrak{X}}^{1}(M)$ admits a linear hyperbolic singularity of a saddle type, then ${P}_{X}^{t}$ does not admit any dominated splitting over $M\setminus Sing(X)$.
From the Theorem 3.2, we know that if a vector field X admits a dominated splitting, then $Sing(X)=\mathrm{\varnothing}$.
Franks’ lemma for divergencefree vector fields allows to realize the perturbations as perturbations of a fixed volumepreserving flow. Fix $X\in {\mathfrak{X}}_{\mu}^{1}(M)$ and $\tau >0$. A oneparameter areapreserving linear family ${\{{A}_{t}\}}_{t\in \mathbb{R}}$ associated to $\{{X}_{t}(p);t\in [0,\tau ]\}$ is defined as follows:

${A}_{t}:{N}_{p}\to {N}_{p}$ is a linear map for all $t\in \mathbb{R}$,

${A}_{t}=\mathit{id}$, for all $t\le 0$ and ${A}_{t}={A}_{\tau}$ for all $t\ge \tau $,

${A}_{t}\in SL(n,\mathbb{R})$ and

the family ${A}_{t}$ is ${C}^{\mathrm{\infty}}$ on the parameter t.
The following result is proved in [[10], Lemma 3.2].
 (a)
Y is ϵ${C}^{1}$close to X;
 (b)
${Y}^{t}(p)={X}^{t}(p)$ for all $t\in \mathbb{R}$;
 (c)
${P}_{Y}^{\tau}(p)={P}_{X}^{\tau}(p)\circ {A}_{\tau}$, and
 (d)
$Y{}_{{\mathcal{T}}^{c}}\equiv X{}_{{\mathcal{T}}^{c}}$.
Remark 3.4 Let $X\in {\mathfrak{X}}_{\mu}^{1}(M)$. By Zuppa’s theorem [11], we can find Y ${C}^{1}$closed to X such that $Y\in {\mathfrak{X}}_{\mu}^{\mathrm{\infty}}(M)$, ${Y}^{\pi}(p)=p$ and ${P}_{Y}^{\pi}(p)$ has an eigenvalue λ with $\lambda =1$.
A divergencefree vector field X is a divergencefree star vector field if there exists a ${C}^{1}$neighborhood $\mathcal{U}(X)$ of X in ${\mathfrak{X}}_{\mu}^{1}(M)$ such that if $Y\in \mathcal{U}(X)$, then every point in $Crit(Y)$ is hyperbolic. The set of divergencefree star vector fields is denoted by ${\mathcal{G}}_{\mu}^{1}(M)$. Then we get the following.
Theorem 3.5 [[12], Theorem 1] If $X\in {\mathcal{G}}_{\mu}^{1}(M)$, then $Sing(X)=\mathrm{\varnothing}$ and X is Anosov.
Thus, to prove Theorem 3.7, it is enough to show that if X is in the $int{\mathcal{OS}}_{\mu}(M)$, then $X\in {\mathcal{G}}_{\mu}^{1}(M)$.
Lemma 3.6 If $X\in int{\mathcal{OS}}_{\mu}^{1}(M)$, then $X\in {\mathcal{G}}_{\mu}^{1}(M)$.
 (a)
${Z}^{t}(p)={Y}^{t}(p)$ for all $t\in \mathbb{R}$, ${P}_{Z}^{{t}_{0}}(p)={P}_{Y}^{{t}_{0}}(p)$ and $Z{}_{{\mathcal{T}}^{c}}=Y{}_{{\mathcal{T}}^{c}}$,
 (b)
$g(x)={\phi}_{p}^{1}\circ {P}_{Y}^{\pi}(p)\circ {\phi}_{p}(x)$ for all $x\in {B}_{\alpha}(p)\cap {\phi}_{p}^{1}({N}_{p})$, and
 (c)
$g(x)=f(x)$ for all $x\notin {B}_{4\alpha}(p)\cap {\phi}_{p}^{1}({N}_{p})$.
 (a)
${x}_{i}={\phi}_{p}^{1}({v}_{0})$, ${t}_{i}=\pi $ for $i<0$,
 (b)
${x}_{i}=g({\phi}_{p}^{1}({v}_{k}))$, ${t}_{i}=\pi $ for $0\le i\le k1$, and
 (c)
${x}_{i}={g}^{ik}({\phi}_{p}^{1}({v}_{i}))$, ${t}_{i}=\pi $ for $i\ge k$.
Thus ${\mathcal{O}}_{Z}(z)\not\subset {B}_{\u03f5}(\xi )$. This is a contradiction.
which is a contradiction.
Finally, we assume that λ is complex. By [[10], Lemma 3.2], there is $Z\in \mathcal{U}(X)$ such that ${P}_{Z}^{\pi}(p)$ is a rational rotation. Then there is $l>0$ such that ${P}_{Z}^{l+\pi}(p)$ is the identity. Then, as in the previous argument, we get a contradiction. □
End of the proof of Theorem 3.7. By Lemma 3.6, $X\in {\mathcal{G}}_{\mu}^{1}(M)$. Thus by Theorem 3.5, $Sing(X)=\mathrm{\varnothing}$ and X is Anosov. □
By [7] and our main result, we have the following.
where $int{\mathcal{S}}_{\mu}(M)$ is the set of all divergencefree vector fields satisfying the shadowing property.
Declarations
Acknowledgements
We wish to thank the referee for carefully reading of the manuscript and providing us with many good suggestions. This work is supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology, Korea (No. 20110007649).
Authors’ Affiliations
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