- Open Access
Vector fields with stably limit shadowing
Advances in Difference Equations volume 2013, Article number: 255 (2013)
Let X be a vector field on a closed smooth manifold M. In this paper, we show that if X belongs to the -interior of the set of all vector fields having the limit shadowing property, then it is transitive Anosov.
Discrete case dynamical system results can be extended to the case of continuous, but not always, in particular results which involve the hyperbolic structure. For instance, it is well known that if a diffeomorphism has a -neighborhood such that every periodic point of is hyperbolic, then the non-wandering set is hyperbolic. However, the result is not true for the case of vector fields (see ). The notion of the limit shadowing property was introduced and studied by Eirola, Nevanlinna, and Pilyugin [2, 3]. It is different from the shadowing property (see ). Various shadowing properties are used in the investigation of the orbit structure. For instance, Sakai  and Robinson  proved that a diffeomorphism is structurally stable if and only if it belongs to the set of all diffeomorphisms having the shadowing property. For vector fields, Lee and Sakai  proved that if a vector field does not admit singularities, then the -interior of the set of all vector fields having the shadowing property coincides with the set of structurally stable vector fields. Recently, in , Pilyugin showed that a diffeomorphism belongs to the set of all diffeomorphisms having the limit shadowing property if and only if it is Ω-stable, that is, Axiom A and the no-cycle condition. From this, Carvalho  proved that the -interior of the limit shadowing property is equal to the set of transitive Anosov diffeomorphisms. Very recently, Ribeiro  proved that for -generic vector fields, if a vector field has the limit shadowing property in a closed isolated set, then it is a transitive hyperbolic set. Moreover, if the closed set is the whole space, then it is a transitive Anosov flow. In this result, we study the -interior of the set of all vector fields having the limit shadowing property, which extends the result of .
Let M be a closed -dimensional smooth Riemmanian manifold, and let d be the distance on M induced from a Riemannian metric on the tangent bundle TM, and denote by the set of -vector fields on M endowed with the -topology. Then every generates a -flow ; that is, a -map such that is a diffeomorphism satisfying and for all and .
For any , a sequence is a δ-pseudo orbit of X (or δ-chain of X) if for any .
For the sequence , we denote
We say that X has the shadowing property if for any , there is satisfying the following property: given any δ-pseudo orbit for all , there is a point and an increasing homeomorphism such that
for any and for any .
A sequence is a limit pseudo orbit of X if for , ,
The following definition introduced by Yujin et al.  is different from the notion of Ribeiro . We say that X has the limit shadowing property if for any , there is and an increasing homeomorphism with such that for ,
Denote by the set of all diffeomorphisms having the limit shadowing property. We say that X has the -stably limit shadowing property if there is a -neighborhood of X such that for any , Y has the limit shadowing property.
Let Λ be an -invariant compact set. The Λ is called hyperbolic for if there are constants , and a splitting such that the tangent flow leaves invariant the continuous splitting and
for and . If , then X is Anosov.
A point is called non-wandering of if for any neighborhood U of x, there is such that . The set of non-wandering points of X is denoted by . Then we know that . Here is the set of singularities of X, and is the set of periodic orbits of X. Define . Here, for any , σ is called the critical element of X. We say that is transitive if there is a point such that , where is the omega limit set of x. The following is the main result of this paper.
Theorem 1.1 Let . If X belongs to the -interior of , then X is transitive Anosov.
2 Proof of Theorem 1.1
Let M be as before, and let . We say that a vector field is robustly transitive if there exists a -neighborhood of X such that for all , Y is transitive. For robustly transitive vector fields, Doering  showed that if a compact manifold M is three-dimensional, then every robustly transitive vector field is Anosov. In , Vivier proved that robustly transitive flows on the whole n-dimensional compact without boundary manifold M must have a dominated splitting for the linear Poincaré flow and have no singularities.
Theorem 2.1 [, Theorem 1]
If X is a robustly transitive vector field, then X admits no singular points.
We say that a vector field is homogeneous on if there is a -neighborhood of X such that
for all , there are no sinks nor sources in U,
every critical element of Y in is hyperbolic, and
the index of the continuation on of every critical point does not change.
From the definition, we know the following theorem.
Theorem 2.2 [, Theorem 4.1]
Let be a robustly transitive homogeneous flow on an -manifold M. Then X is Anosov.
Let Λ be a closed -invariant set. We say that Λ is attracting if for some neighborhood U of Λ satisfying for all . An attractor of X is a transitive attracting set of X and a repeller is an attractor for −X. We say that Λ is a proper attractor or proper repeller if .
Lemma 2.3 [, Proposition 3]
A vector field X is chain transitive in an isolated set Λ if and only if Λ has no proper attractor for X.
By Lemma 2.3, if , then we omit the isolated condition. Then we have the following.
Lemma 2.4 If X has the limit shadowing property, then X has no proper attractor.
Proof To derive a contradiction, we may assume that X has a proper attractor Λ. By definition, and . Since Λ is an attractor, there is such that , where is the η-neighborhood of Λ. Since , choose and . Then we can construct a limit pseudo orbit as follows: for all , (i) , , and (ii) , , . By the limit shadowing property, there are and an increasing homeomorphism with such that for , as . Then we can find such that
for . Thus, for all , for all . Since is an attracting neighborhood of Λ, for ,
Since Λ is an -invariant set, we know that . Thus we have that
This is a contradiction. □
We introduce the notion of chain transitive set, which is a weaker notion of transitive set. We say that a set Λ is chain transitive if for any and , there is a δ-pseudo orbit with for any and . The following is a version of the vector fields of the result of Gu .
Lemma 2.5 [, Theorem 4.1]
Let . If X has the limit shadowing property, then X is transitive if and only if X is chain transitive.
By Lemma 2.5, X is transitive, X does not contain sinks or sources. Thus, for , we just consider saddle-type periodic orbits. Let γ be a hyperbolic closed orbit of a vector field . We define the stable and unstable manifolds of γ by
Denote by the dimension of the stable manifold of γ.
Lemma 2.6 Let be hyperbolic orbits. If X has the limit shadowing property, then .
Proof Let be hyperbolic orbits. Suppose that X has the limit shadowing property. Take and such that and , where is the period of a. For , put , , , and , , . Then
is a limit pseudo orbit of X. By the limit shadowing property, there are and an increasing homeomorphism with such that for , as . Then we can find such that for ,
Then, for , we have . If , then
Also, we can find such that for ,
Then, for , we have . If , then
Thus , and so , where is the orbits of y. □
A vector field is Kupka-Smale if its critical orbits are all hyperbolic and their stable and unstable manifolds intersect transversely. It is well known that the Kupka-Smale vector field is a residual subset in . Denote by the set of all vector fields satisfying the Kupka-Smale.
Lemma 2.7 [, Lemma 3.4]
Let be a Kupka-Smale vector field, and let be hyperbolic orbits. If , then .
Lemma 2.8 Let X have the -stably limit shadowing property, and let be as in the definition. Then, for any and for any hyperbolic , .
Proof Suppose that X has the -stably limit shadowing property. Then there is a -neighborhood of X such that for any , Y has the limit shadowing property. Let be hyperbolic orbits. To derive a contradiction, we may assume that . Then we know that or . Assume that . Since X has the -stably limit shadowing property, we can choose such that , where and are the continuations of η and γ, respectively. Since , by Lemma 2.7, . Since Z has the limit shadowing property, by Lemma 2.6 this is a contradiction. □
Lemma 2.9 [, Theorem 4.3]
Let be a -neighborhood of X. If is not hyperbolic, then there is such that Y has two hyperbolic orbits with different indices.
Proof of Theorem 1.1 Let X have the -stably limit shadowing property. Then there is a -neighborhood of X such that for any , Y has the limit shadowing property. To get the conclusion, it is enough to show that every is hyperbolic by Lemma 2.8. By contradiction, we may assume that is not hyperbolic. Then, by Lemma 2.9, there is such that Z has two hyperbolic periodic orbits with different indices. Since Z has the limit shadowing property, this is a contradiction by Lemma 2.8. □
The author carried out the proof of the theorem and approved the final manuscript.
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We thank the referees very much for their helpful comments and suggestions. The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2011-0007649).
The author declares that they have no competing interests.