Divergence-free vector fields with orbital shadowing
© Lee; licensee Springer 2013
Received: 8 January 2013
Accepted: 17 April 2013
Published: 7 May 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
for any and . If , then X is called Anosov.
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
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 .
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].
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 .
for all , and ,
for all , and
for all .
, for ,
, for , and
, for .
Thus . This is a contradiction.
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.
where is the set of all divergence-free vector fields satisfying the shadowing property.
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).
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