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Divergence-free vector fields with orbital shadowing
Advances in Difference Equations volume 2013, Article number: 132 (2013)
We show that a divergence-free vector field belongs to the -interior of the set of divergence-free vector fields satisfying the orbital shadowing property when the vector field is Anosov.
MSC:37C10, 37C50, 37D20.
The shadowing theory is a very useful notion for the investigation of the stability condition. In fact, Robinson  and Sakai  proved that a diffeomorphism belongs to the -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 Ω-stable, that is, a diffeomorphism satisfies both Axiom A and a no-cycle condition, then there is a diffeomorphism which does not have the shadowing property (see, ). However, for another shadowing property, if a diffeomorphism is Ω-stable, 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. . They showed that a diffeomorphism belongs to the -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  proved that a conservative diffeomorphism is in the -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 -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 divergence-free vector field, Ferreira  proved that if a conservative vector field belongs to the -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 -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 .
2 Basic notions, definitions and results
Let M be a closed, connected and smooth -dimensional Riemannian manifold endowed with a volume form, which has a measure μ, called the Lebesgue measure. Given a (), vector field , the solution of the equation generates a flow, ; on the other hand, given a flow, we can define a vector field by considering . We say that X is divergence-free (or a conservative vector field) if its divergence is equal to zero. Note that, by the Liouville formula, a flow is volume-preserving if and only if the corresponding vector field X is divergence-free. Let denote the space of divergence-free vector fields, and we consider the usual Whitney topology on this space. Let . For any and , we say that is a -pseudo orbit of if
for any , . Define Rep as the set of increasing homeomorphisms such that . Fix and define as follows:
Let be a compact -invariant set. We say that has the shadowing property on Λ if for any , there is such that for any -pseudo orbit , let for any , and for any , there exist a point and an increasing homeomorphism with such that
for any . If , then has the shadowing property. Now, we introduce the notion of the orbital shadowing property. For , we denote to be the orbit of X through x; that is, . We say that has the orbital shadowing property if for any there is such that for any -pseudo orbit there is a point such that
where is the neighborhood of A. Note that the orbital shadowing property is a weak version of the shadowing property: the difference is that we do not require a point of a pseudo-orbit ξ and the point of an exact orbit to be close ‘at any time moment’; instead, the sets of the points of X and are required to be close. Let Λ be a closed -invariant set. We say that Λ is hyperbolic if there are constants and such that a continuous splitting satisfying
for any and . If , then X is called Anosov.
Given a vector field X, we denote by the set of singularities of X, i.e., those points such that . Let be the set of regular points. We know that the exponential map is well defined for all , where . Given , we consider its normal bundle and let be the r-ball in . Let . For any and , there are and a map with such that for any . We say the first return time of y. Then we define the Poincarè map f by
Let be the normal bundle based on R. One can define the associated linear Poincaré flow by , where is the projection along the direction of .
Denote by the set of divergence-free vector fields satisfying the orbital shadowing property.
Theorem 2.1 Let . If , then X has no singularity and X is Anosov.
3 Proof of Theorem 2.1
Let be a compact, -invariant and regular set. We say that Λ is hyperbolic for if admits a -invariant splitting such that there is satisfying
for all . The following is well known and one can find a proof in .
Theorem 3.1 Λ is a hyperbolic set of if and only if the linear Poincaré flow restricted on Λ has a hyperbolic splitting .
Consider a splitting over Λ, for , such that all the subbundles have constant dimensions. This splitting is dominated if it is -invariant, and there is such that for every , we have
for any .
The following was proved in .
Theorem 3.2 [, Proposition 4.1]
If admits a linear hyperbolic singularity of a saddle type, then does not admit any dominated splitting over .
From the Theorem 3.2, we know that if a vector field X admits a dominated splitting, then .
Franks’ lemma for divergence-free vector fields allows to realize the perturbations as perturbations of a fixed volume-preserving flow. Fix and . A one-parameter area-preserving linear family associated to is defined as follows:
is a linear map for all ,
, for all and for all ,
the family is on the parameter t.
The following result is proved in [, Lemma 3.2].
Lemma 3.3 Given and a vector field , there exists such that for all , for any periodic point p of period greater than 2, for any sufficient small flowbox of and for any one-parameter linear family such that for all , there exists satisfying the following properties:
Y is ϵ--close to X;
for all ;
Remark 3.4 Let . By Zuppa’s theorem , we can find Y -closed to X such that , and has an eigenvalue λ with .
A divergence-free vector field X is a divergence-free star vector field if there exists a -neighborhood of X in such that if , then every point in is hyperbolic. The set of divergence-free star vector fields is denoted by . Then we get the following.
Theorem 3.5 [, Theorem 1] If , then and X is Anosov.
Thus, to prove Theorem 3.7, it is enough to show that if X is in the , then .
Lemma 3.6 If , then .
Proof Let . Then there is a -neighborhood of X such that for any , Y has the orbital shadowing property. Let with and be a small neighborhood of p. We will derive a contradiction. Assume that there is an eigenvalue λ of such that . By Remark 3.4, there is such that , and has an eigenvalue λ with . We define the map with the map being the Poincarè map associated to . Here is a smooth conservative map with (see, ). Let be a -neighborhood of f. Here is the Poincarè section through p. By Lemma 3.3, we can find a small flowbox of , and there are , and such that
for all , and ,
for all , and
for all .
By the notion of Lemma 3.3, we can assume that has an eigenvalue . Firstly, we assume that (the other case is similar). Then we can choose a vector v associated to λ such that , and we set . Since ,
For , let be as in the definition of the orbital shadowing property of . Set . There is such that , and for , where for . We construct a pseudo-orbit of belonging to as follows:
, for ,
, for , and
, for .
Then is a -pseudo orbit of and it is contained in . By the orbital shadowing property, we can take a point such that
If , then we know that there is such that , and . Then
Thus . This is a contradiction.
If , there is such that . Then for some , we have
which is a contradiction.
Finally, we assume that λ is complex. By [, Lemma 3.2], there is such that is a rational rotation. Then there is such that 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, . Thus by Theorem 3.5, and X is Anosov. □
By  and our main result, we have the following.
Corollary 3.7 Let . Then
where is the set of all divergence-free vector fields satisfying the shadowing property.
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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. 2011-0007649).
The author declares that they have no competing interests.
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Cite this article
Lee, M. Divergence-free vector fields with orbital shadowing. Adv Differ Equ 2013, 132 (2013). https://doi.org/10.1186/1687-1847-2013-132
- divergence-free vector fields
- star condition
- orbital shadowing