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Divergence-free vector fields with inverse shadowing
Advances in Difference Equations volume 2013, Article number: 337 (2013)
Abstract
We show that if a divergence-free vector field has the -stably orbital inverse shadowing property with respect to the class of continuous methods , then the vector field is Anosov. The results extend the work of Bessa and Rocha (J. Differ. Equ. 250:3960-3966, 2011).
MSC:37C10, 37C27, 37C50.
1 Introduction
The notion of inverse shadowing property is a dual notion of the shadowing property. It was studied by [1–7]. In fact, Pilyugin [7] showed that every structurally stable diffeomorphism has the inverse shadowing property with respect to the class of continuous methods. In [3], Lee proved that if a diffeomorphism has the -stably inverse shadowing property with respect to the class of continuous methods , then the diffeomorphism is structurally stable. For vector fields, Lee and Lee [5] introduced the notion of inverse shadowing for flows and showed that every expansive flow with the shadowing property has the inverse shadowing property with respect to the class of continuous methods. Lee et al. [6] showed that the -interior of the set of vector fields with the orbital shadowing property with respect to the class coincides with the set of structurally stable vector fields. From the facts, Lee [4] showed that if a volume-preserving diffeomorphism has the -stably inverse shadowing property with respect to the class , then the diffeomorphism is Anosov. Moreover, if a volume-preserving diffeomorphism has the -stably orbital inverse shadowing property with respect to the class , then the diffeomorphism is Anosov. In this spirit, we study divergence-free vector fields with the inverse, orbital inverse shadowing property with respect to the class .
Let us be more detailed. Let M be a closed, connected, and Riemannian manifold endowed with a volume form μ, and let μ denote the Lebesgue measure associated to it. Given a () vector field , the solution of the equation gives rise to a flow, ; on the other hand, given a flow, we can define a vector field by considering . We say that X is divergence-free 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 a -pseudo orbit of if
for any , . A mapping is called a -method for X if for any , the map defined by
is a -pseudo orbit of X. Ψ is said to be complete if for any . Then a -method for X can be considered a family of -pseudo orbits of X. A method Ψ for X is said to be continuous if the map , given by for and , is continuous under the compact open topology on , where denotes the set of all functions from ℝ to M. The set of all complete -methods [resp. complete continuous -methods] for will be denoted by [resp. ]. We can see that if Y is another vector field which is sufficiently close to X in the topology, then it induces a complete continuous method for X. Let [resp. be the set of all complete continuous methods for X which are induced by vector fields Y with [resp. ], where is the metric on such that
and is the metric on such that
We say that a vector field X has the inverse shadowing property with respect to the class () if for any , there is such that for any -method and any point , there are and an increasing homeomorphism with such that
We denote by the set of divergence-free vector fields on M with the inverse shadowing property with respect to the class , where . Let be the -interior of the set of divergence-free vector fields on M with the inverse shadowing property with respect to the class , where .
We say that has the -stably inverse shadowing property with respect to the class if there is a -neighborhood of X such that for any , Y has the inverse shadowing property with respect to the class , where .
Remark 1.1 Let . Then , where .
We say that has the orbital inverse shadowing property with respect to the class if for any , there is such that for any -method and any point , there is such that
where . Note that if X has the inverse shadowing property with respect to the class , then X has the orbital shadowing property with respect to the class . But the converse is not true. Indeed, an irrational rotation map does not have the inverse shadowing property. And the map has the orbital shadowing property. Let be the -interior of the set of divergence-free vector fields on M with the orbital inverse shadowing property with respect to the class , where .
Let Λ be a closed -invariant set. We say that Λ is hyperbolic if there are constants , and a continuous splitting such that
for any and . If , then X is called Anosov.
Given a vector field X, we denote by the set of closed orbits of X and by the set of singularities of X, i.e., those points such that . Denote by . Let be the set of regular points. We assume 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 is the first return time of y. Then we define the Poincarè map f by
Let be the normal bundle based on R. We can define the associated linear Poincaré flow by
where is the projection along the direction of . We say that a vector field is topologically stable if for any , there is such that for any with , there is a semi-conjugacy from Y to X satisfying for all .
In [8], the authors proved that if a vector field is in the -interior of the set of topologically stable vector fields (not divergence-free vector fields), then X satisfies Axiom A and the strong transversality condition. Robinson [9] proved that if a vector field satisfies Axiom A and the strong transversality condition, then the vector field is structurally stable. For divergence-free vector fields, Bessa and Rocha [10] showed that if a divergence-free vector field is in the -interior of the set of topological stable vector fields , then X is Anosov.
Remark 1.2 We have the following implication: topological stability ⇒ inverse shadowing property with respect to the continuous method ⇒ orbital inverse shadowing property with respect to the continuous method .
From the above remark, we know that our result is a slight generalization of the main theorem in [10]. In this paper, we omit the phrase ‘with respect to the class ’ for simplicity. So, we say that X has the inverse, orbital inverse shadowing property means that X has the inverse, orbital inverse shadowing property with respect to the class . Note that if or , then it means that X has the -stably inverse, orbital inverse shadowing property with respect to the class , . The following is the main result of this paper.
Theorem 1.3 Let . Then
where is the set of divergence-free Anosov vector fields.
2 Proof of Theorem 1.3
Let M be as before, and let . The perturbations (Lemma 2.1) for volume-preserving vector fields allows to realize them 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 ,
-
, and
-
the family is on the parameter t.
The following result, proved in [[11], Lemma 3.2], is now stated for instead of because of the improved smooth pasting lemma proved in [[12], Lemma 5.2].
Lemma 2.1 Given and a vector field , there exists such that for all , for any periodic point p of period greater than 2, for any sufficiently small flowbox T of and for any one-parameter linear family such that for all , there exists satisfying the following properties:
-
(a)
Y is ϵ--close to X;
-
(b)
for all ;
-
(c)
, and
-
(d)
.
Remark 2.2 Let . By Zuppa’s theorem [13], we can find Y -close 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.
Theorem 2.3 [[14], Theorem 1.1]
If , then and X is Anosov.
Thus, to prove Theorem 1.3, it is enough to show that if X has the inverse shadowing property, or the orbital inverse shadowing property, then .
Proposition 2.4 Let . Then .
Proof Suppose that has the -stably inverse shadowing property. Then there is a -neighborhood of X such that for any , Y has the inverse shadowing property. Let with and be a small neighborhood of . We will derive a contradiction. Assume that there is an eigenvalue λ of such that . By Remark 2.2, there is such that , and has an eigenvalue λ with . Using Moser’s theorem (see [15]), there is a smooth conservative change of coordinates such that . Let be the Poincarè map associated to , and let V be a -neighborhood of f. Here, is the Poincarè section through p. By Lemma 2.1, we can find a small flowbox T of , , and there are , and such that
-
(a)
for all ,
-
(b)
,
-
(c)
,
-
(d)
for all , and
-
(e)
for all .
Then has an eigenvalue λ with . For , let be as in the definition of the inverse shadowing property of .
Take a linear map for all such that if , then , and is a hyperbolic linear Poincarè flow. Set . Then , and is a periodic point of . Since for any , there exist and an increasing homeomorphism with such that for all .
First, we assume that (the other case is similar). Then we can choose a vector v associated to λ such that . Since ,
Then we can take such that
where is the Poincarè map associated to . Since is a hyperbolic linear Poincarè flow, we can see that
for some . Thus
for some . This is a contradiction by the fact that Z has the inverse shadowing property.
Finally, we assume that λ is complex. By [[10], Lemma 3.2], there is such that is a rational rotation. Then there is such that is the identity. As in the previous argument, we get a contradiction. □
Proposition 2.5 Let . Then .
Proof Suppose that X has the -stably orbital inverse shadowing property. Then there is a -neighborhood of X such that for any , Y has the orbital inverse shadowing property. Let with and be a small neighborhood of . We will derive a contradiction. Assume that there is an eigenvalue λ of such that . As in the proof of Proposition 2.4, we get a hyperbolic linear Poincarè flow , and , and is a periodic point of . Since for any , there exist such that
for all . Take , and let .
If or , then as in the proof of Lemma 2.4, we get a contradiction. Indeed, since is a hyperbolic linear Poincarè flow, there is such that . Thus
for some . Since , this is a contradiction. □
End of the proof of Theorem 1.3 By Proposition 2.4 and Proposition 2.5, we have . Thus by Theorem 2.3, we get and X is Anosov. □
From the result of [10], we get the following corollary.
Corollary 2.6 Let . Then
where is the -interior of the set of divergence-free vector fields on M which are topologically stable.
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Acknowledgements
The first author is supported by the National Research Foundation (NRF) of Korea funded by the Korean Government (No. 2011-0015193). The second author 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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Lee, K., Lee, M. Divergence-free vector fields with inverse shadowing. Adv Differ Equ 2013, 337 (2013). https://doi.org/10.1186/1687-1847-2013-337
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DOI: https://doi.org/10.1186/1687-1847-2013-337