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Volume preserving diffeomorphisms with orbital shadowing
Journal of Inequalities and Applications volume 2013, Article number: 18 (2013)
Let f be a volume-preserving diffeomorphism of a closed Riemannian manifold M. In this paper, we show that the following are equivalent:
(a) f belongs to the -interior of the set of volume-preserving diffeomorphisms with orbital shadowing,
(b) f is Anosov.
A fundamental problem in differentiable dynamical systems is to understand how a robust dynamic property on the underlying manifold would influence the behavior of the tangent map on the tangent bundle. For instance, in , Mañé proved that any structurally stable diffeomorphism is an Axiom A diffeomorphism. And in , Palis extended this result to Ω-stable diffeomorphisms. Let M be a closed Riemannian manifold endowed with a volume form ω. Let μ denote the Lebesgue measure associated to ω, and let d denote the metric induced on M by the Riemannian structure. Denote by the set of diffeomorphisms which preserves the Lebesgue measure μ endowed with the -topology. We know that every volume preserving diffeomorphism satisfying Axiom A is Anosov (for more details, see ).
For , a sequence of points () in M is called a δ-pseudo-orbit of f if for all . Let be a closed f-invariant set. We say that f has the shadowing property on Λ (or Λ is shadowable) if for every , there is such that for any δ-pseudo-orbit of f (), there is a point such that for all . It is easy to see that f has the shadowing property on Λ if and only if has the shadowing property on Λ for . The notion of pseudo-orbits often appears in several methods of the modern theory of dynamical systems. Moreover, the shadowing property plays an important role in the investigation of stability theory. In fact, Pilyugin  and Robinson  showed that if a diffeomorphism f is structurally stable, then f has the shadowing property. Moreover, Sakai  proved that if there is a -neighborhood of f such that for any , g has the shadowing property, then f is structurally stable. For each , let be the orbit of f through x; that is,
We say that f has the orbital shadowing property on Λ (or Λ is orbitally shadowable) if for any , there exists such that for any δ-pseudo-orbit , we can find a point such that
where denotes the ϵ-neighborhood of a set . f is said to have the weak shadowing property on Λ (or Λ is weakly shadowable) if for any , there exists such that for any δ-pseudo-orbit , there is a point such that . Note that if f has the shadowing property, then f has the orbital shadowing property, but the converse is not true (see ). It is easy to see that f has the orbital shadowing property on Λ if and only if has the orbital shadowing property on Λ for .
We say that Λ is hyperbolic if the tangent bundle has a Df-invariant splitting and there exist constants and such that
for all and .
We denote by the set of diffeomorphisms which have a -neighborhood such that for any , every periodic point of g is hyperbolic.
Very recently, Arbieto and Catalan  proved that every volume preserving diffeomorphism in is Anosov. To prove this, they used Mañé’s results in [, Proposition II.1] and showed that is hyperbolic if . Thus, we have the following theorem.
Theorem 1.1 [, Theorem 1.1]
Every diffeomorphism in is Anosov.
Let denote the -interior of the set of volume preserving diffeomorphisms in satisfying the orbital shadowing property. In , the authors proved that the -interior of the set of diffeomorphisms having the orbital shadowing property coincides with the set of structurally stable diffeomorphisms. Note that if a diffeomorphism satisfies structurally stable then it is not Anosov in general. But the converse is true. Finally, we prove the following theorem.
Theorem 1.2 The set of Anosov diffeomorphisms in coincides with the -interior of the set of diffeomorphisms in with orbital shadowing; that is, .
2 Proof of Theorem 1.2
Remark 2.1 Let . From Moser’s theorem (see ), we can find a smooth conservative change of coordinates such that , where is a small neighborhood of .
Recall that f has the orbital shadowing property on Λ (Λ is orbitally shadowable) if for any , there is such that for any δ-pseudo orbit of f, there is ∈M such that
Notice that in this definition, only δ-pseudo orbits of f are contained in Λ, but the shadowing point is not necessarily contained in Λ. To prove our result, we use Franks’ lemma which is proved in [, Proposition 7.4].
Lemma 2.2 Let , and be a -neighborhood of f in . Then there exist a -neighborhood of f and such that if , any finite f-invariant set , any neighborhood U of E, and any volume-preserving linear maps with for all , there is a conservative diffeomorphism coinciding with f on E and out of U, and for all .
We introduce the notion of normally hyperbolic which was founded in . Let be an invariant submanifold of . We say that V is normally hyperbolic if there is a splitting such that
the splitting depends continuously on ,
() for all ,
there are constants and such that for every unit vector , and (), we have
for all .
Proposition 2.3 If , then every periodic point of f is hyperbolic.
Proof Take , and is a -neighborhood of . Let and be the number and -neighborhood of f corresponding to given by Lemma 2.2. To derive a contradiction, we assume that there exists a nonhyperbolic periodic point for some . To simplify the notation in the proof, we may assume that . Then there is at least one eigenvalue λ of such that .
By making use of Lemma 2.2, we linearize g at p using Moser’s theorem; that is, by choosing sufficiently small, we construct -nearby g such that
First, we may assume that with . Let v be the associated non-zero eigenvector such that . Then we can get a small arc
Take . Let be a number of the orbital shadowing property of corresponding to ϵ. Then by our construction of ,
Then it is clear that is normally hyperbolic for . Put . Given a constant , we construct a δ-pseudo orbit as follows. For fixed , choose distinct points in such that
Now, we define by for and . Since has the orbital shadowing property, must have the orbital shadowing property. Thus, we can find a point such that , and . For any , and
Then . Thus, for some . Now, we show that if is normally hyperbolic for , then the shadowing points belong to . Assume that there is a shadowing point . Then by the hyperbolicity, there are such that , where . This is a contradiction since has the orbital shadowing property. Thus, if is normally hyperbolic for , then the shadowing point belongs to . Since has the orbital shadowing property, from the above facts, we have . But and so is the identity map. Then does not have the orbital shadowing property. Thus, also does not have the orbital shadowing property.
Finally, if , then to avoid the notational complexity, we may assume that . As in the first case, by Lemma 2.2, there are and such that and
With a -small modification of the map , we may suppose that there is (the minimum number) such that for any . Then we can go on with the previous argument in order to reach the same contradiction. Thus, every periodic point of is hyperbolic. □
End of the proof of Theorem 1.2 Let . By Proposition 2.3, we see that . Thus, by Theorem 1.1, f is Anosov. □
Mãné R: An ergodic closing lemma. Ann. Math. 1982, 116: 503–540. 10.2307/2007021
Palis J:On the Ω-stability conjecture. Publ. Math. Inst. Hautes Études Sci. 1988, 66: 211–215.
Arbieto A, Catalan T: Hyperbolicity in the volume preserving scenario. Ergodic Theory Dyn. Syst. 2012. doi:10.1017/etds.2012.111
Pilyugin SY Lecture Notes in Math. 1706. In Shadowing in Dynamical Systems. Springer, Berlin; 1999.
Robinson C: Stability theorems and hyperbolicity in dynamical systems. Rocky Mt. J. Math. 1977, 7: 425–437. 10.1216/RMJ-1977-7-3-425
Sakai K: Pseudo orbit tracing property and strong transversality of diffeomorphisms on closed manifolds. Osaka J. Math. 1994, 31: 373–386.
Pilyugin SY, Rodionova AA, Sakai K: Orbital and weak shadowing properties. Discrete Contin. Dyn. Syst. 2003, 9: 287–308.
Moser J: On the volume elements on a manifold. Trans. Am. Math. Soc. 1965, 120: 286–294. 10.1090/S0002-9947-1965-0182927-5
Bonatti C, Díaz LJ, Pujals ER:A -generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks or sources. Ann. Math. 2003, 158: 355–418. 10.4007/annals.2003.158.355
Hirsh M, Pugh C, Shub M Lecture Notes in Math. In Invariant Manifolds. Springer, Berlin; 1977.
We wish to thank the referee for carefully reading of the manuscript and providing us many good suggestions. KL is supported by the National Research Foundation (NRF) of Korea funded by the Korean Government (No. 2011-0015193). ML 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 authors declare that they have no competing interests.
All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.
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Lee, K., Lee, M. Volume preserving diffeomorphisms with orbital shadowing. J Inequal Appl 2013, 18 (2013). https://doi.org/10.1186/1029-242X-2013-18
- normally hyperbolic
- orbital shadowing