Skip to main content

Volume preserving diffeomorphisms with orbital shadowing

Abstract

Let f be a volume-preserving diffeomorphism of a closed C Riemannian manifold M. In this paper, we show that the following are equivalent:

(a) f belongs to the C 1 -interior of the set of volume-preserving diffeomorphisms with orbital shadowing,

(b) f is Anosov.

MSC:37C50, 34D10.

1 Introduction

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 [1], Mañé proved that any C 1 structurally stable diffeomorphism is an Axiom A diffeomorphism. And in [2], Palis extended this result to Ω-stable diffeomorphisms. Let M be a closed C 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 Diff μ (M) the set of diffeomorphisms which preserves the Lebesgue measure μ endowed with the C 1 -topology. We know that every volume preserving diffeomorphism satisfying Axiom A is Anosov (for more details, see [3]).

For δ>0, a sequence of points { x i } i = a b (a<b) in M is called a δ-pseudo-orbit of f if d(f( x i ), x i + 1 )<δ for all aib1. Let ΛM be a closed f-invariant set. We say that f has the shadowing property on Λ (or Λ is shadowable) if for every ϵ>0, there is δ>0 such that for any δ-pseudo-orbit { x i } i = a b Λ of f (a<b), there is a point yM such that d( f i (y), x i )<ϵ for all aib1. It is easy to see that f has the shadowing property on Λ if and only if f n has the shadowing property on Λ for nZ{0}. 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 [4] and Robinson [5] showed that if a diffeomorphism f is structurally stable, then f has the shadowing property. Moreover, Sakai [6] proved that if there is a C 1 -neighborhood U(f) of f such that for any gU(f), g has the shadowing property, then f is structurally stable. For each xM, let O f (x) be the orbit of f through x; that is,

O f (x)= { f n ( x ) : n Z } .

We say that f has the orbital shadowing property on Λ (or Λ is orbitally shadowable) if for any ϵ>0, there exists δ>0 such that for any δ-pseudo-orbit ξ= { x i } i Z Λ, we can find a point yM such that

O f (y) B ϵ (ξ)andξ B ϵ ( O f ( y ) ) ,

where B ϵ (A) denotes the ϵ-neighborhood of a set AM. f is said to have the weak shadowing property on Λ (or Λ is weakly shadowable) if for any ϵ>0, there exists δ>0 such that for any δ-pseudo-orbit ξ= { x i } i Z Λ, there is a point yM such that ξ B ϵ ( O f (y)). Note that if f has the shadowing property, then f has the orbital shadowing property, but the converse is not true (see [7]). It is easy to see that f has the orbital shadowing property on Λ if and only if f n has the orbital shadowing property on Λ for nZ{0}.

We say that Λ is hyperbolic if the tangent bundle T Λ M has a Df-invariant splitting E s E u and there exist constants C>0 and 0<λ<1 such that

D x f n | E x s C λ n and D x f n | E x u C λ n

for all xΛ and n0.

We denote by F μ (M) the set of diffeomorphisms f Diff μ (M) which have a C 1 -neighborhood U(f) Diff μ (M) such that for any gU(f), every periodic point of g is hyperbolic.

Very recently, Arbieto and Catalan [3] proved that every volume preserving diffeomorphism in F μ (M) is Anosov. To prove this, they used Mañé’s results in [[1], Proposition II.1] and showed that P ( f ) ¯ is hyperbolic if f F μ (M). Thus, we have the following theorem.

Theorem 1.1 [[3], Theorem 1.1]

Every diffeomorphism in F μ (M) is Anosov.

Let int OS μ (M) denote the C 1 -interior of the set of volume preserving diffeomorphisms in Diff μ (M) satisfying the orbital shadowing property. In [7], the authors proved that the C 1 -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 AN μ (M) of Anosov diffeomorphisms in Diff μ (M) coincides with the C 1 -interior of the set of diffeomorphisms in Diff μ (M) with orbital shadowing; that is, AN μ (M)=int OS μ (M).

2 Proof of Theorem 1.2

Remark 2.1 Let f Diff μ 1 (M). From Moser’s theorem (see [8]), we can find a smooth conservative change of coordinates φ x :U(x) T x M such that φ x (x)=0, where U(x) is a small neighborhood of xM.

Recall that f has the orbital shadowing property on Λ (Λ is orbitally shadowable) if for any ϵ>0, there is δ>0 such that for any δ-pseudo orbit ξ= { x i } i Z Λ of f, there is M such that

O f (y) B ϵ (ξ)andξ B ϵ ( O f ( y ) ) .

Notice that in this definition, only δ-pseudo orbits of f are contained in Λ, but the shadowing point yM is not necessarily contained in Λ. To prove our result, we use Franks’ lemma which is proved in [[9], Proposition 7.4].

Lemma 2.2 Let f Diff μ 1 (M), and U be a C 1 -neighborhood of f in Diff μ 1 (M). Then there exist a C 1 -neighborhood U 0 U of f and ϵ>0 such that if g U 0 , any finite f-invariant set E={ x 1 ,, x m }, any neighborhood U of E, and any volume-preserving linear maps L j : T x j M T g ( x j ) M with L j D x j gϵ for all j=1,,m, there is a conservative diffeomorphism g 1 U coinciding with f on E and out of U, and D x j g 1 = L j for all j=1,,m.

We introduce the notion of normally hyperbolic which was founded in [10]. Let VM be an invariant submanifold of f Diff μ (M). We say that V is normally hyperbolic if there is a splitting T V M=TV N s N u such that

  • the splitting depends continuously on xV,

  • D x f( N x σ )= N f ( x ) σ (σ=s,u) for all xV,

  • there are constants C>0 and 0<λ<1 such that for every unit vector x T x V, v s N x s and v u N x u (xV), we have

    D x f n ( v s ) C λ n D x f n ( v ) and D x f n ( v u ) C 1 λ 1 D x f n ( v )

for all n0.

Proposition 2.3 If fint OS μ (M), then every periodic point of f is hyperbolic.

Proof Take fint OS μ (M), and U(f) is a C 1 -neighborhood of f Diff μ (M). Let ϵ>0 and V(f) U 0 (f) be the number and C 1 -neighborhood of f corresponding to U(f) given by Lemma 2.2. To derive a contradiction, we assume that there exists a nonhyperbolic periodic point pP(g) for some gV(f). To simplify the notation in the proof, we may assume that g(p)=p. Then there is at least one eigenvalue λ of D p g such that |λ|=1.

By making use of Lemma 2.2, we linearize g at p using Moser’s theorem; that is, by choosing α>0 sufficiently small, we construct g 1 C 1 -nearby g such that

g 1 (x)={ φ p 1 D p g φ p ( x ) if  x B α ( p ) , g ( x ) if  x B 4 α ( p ) .

Then g 1 (p)=g(p)=p.

First, we may assume that λR with λ=1. Let v be the associated non-zero eigenvector such that v=α/4. Then we can get a small arc

I v ={tv:1t1} φ p ( B α ( p ) ) .

Take ϵ=α/8. Let 0<δ<ϵ be a number of the orbital shadowing property of g 1 corresponding to ϵ. Then by our construction of g 1 ,

φ p 1 ( I v ) B α (p).

Then it is clear that φ p 1 ( I v ) is normally hyperbolic for g 1 . Put J p = φ p 1 ( I v ). Given a constant δ>0, we construct a δ-pseudo orbit ξ= { x i } i Z J p as follows. For fixed kZ, choose distinct points x 0 =p, x 1 , x 2 ,, x k in J p such that

  1. (a)

    d( x i , x i + 1 )<δ for i=0,1,,k1,

  2. (b)

    d( x i 1 , x i )<δ for i=0,,k1,

  3. (c)

    x 0 =x and d( x k , x k )>2ϵ.

Now, we define ξ= { x i } i Z by x k i + j = x j for iZ and j=k1,k2,,1,0,1,,k1. Since g 1 has the orbital shadowing property, g 1 | J p must have the orbital shadowing property. Thus, we can find a point yM such that ξ B ϵ ( O g 1 (y)), and O g 1 (y) B ϵ (ξ). For any v I v , φ p 1 (v) J p B α (p) and

g 1 ( φ p 1 ( v ) ) = φ p 1 D p g φ p ( φ p 1 ( v ) ) .

Then g 1 ( φ p 1 (v))= φ p 1 (v). Thus, g 1 l ( J p )= J p for some l>0. Now, we show that if J p is normally hyperbolic for g 1 , then the shadowing points belong to J p . Assume that there is a shadowing point yM J p . Then by the hyperbolicity, there are l,kZ such that d( g 1 l (y), x k )>ϵ, where x k ξ= { x i } i Z . This is a contradiction since g 1 | J p has the orbital shadowing property. Thus, if J p is normally hyperbolic for g 1 , then the shadowing point belongs to J p . Since g 1 | J p has the orbital shadowing property, from the above facts, we have y J p . But g 1 l ( J p )= J p and so g 1 l | J p is the identity map. Then g 1 l | J p does not have the orbital shadowing property. Thus, g 1 | J p also does not have the orbital shadowing property.

Finally, if λC, then to avoid the notational complexity, we may assume that g(p)=p. As in the first case, by Lemma 2.2, there are α>0 and g 1 V(f) such that g 1 (p)=g(p)=p and

g 1 (x)={ φ p 1 D p g φ p ( x ) if  x B α ( p ) , g ( x ) if  x B 4 α ( p ) .

With a C 1 -small modification of the map D p g, we may suppose that there is l>0 (the minimum number) such that D p g l (v)=v for any v φ p ( B α (p)) T p M. Then we can go on with the previous argument in order to reach the same contradiction. Thus, every periodic point of fint OS μ (M) is hyperbolic. □

End of the proof of Theorem 1.2 Let fint OS μ (M). By Proposition 2.3, we see that f F μ (M). Thus, by Theorem 1.1, f is Anosov. □

References

  1. Mãné R: An ergodic closing lemma. Ann. Math. 1982, 116: 503–540. 10.2307/2007021

    Article  MATH  Google Scholar 

  2. Palis J:On the C 1 Ω-stability conjecture. Publ. Math. Inst. Hautes Études Sci. 1988, 66: 211–215.

    Article  MATH  MathSciNet  Google Scholar 

  3. Arbieto A, Catalan T: Hyperbolicity in the volume preserving scenario. Ergodic Theory Dyn. Syst. 2012. doi:10.1017/etds.2012.111

    Google Scholar 

  4. Pilyugin SY Lecture Notes in Math. 1706. In Shadowing in Dynamical Systems. Springer, Berlin; 1999.

    Google Scholar 

  5. Robinson C: Stability theorems and hyperbolicity in dynamical systems. Rocky Mt. J. Math. 1977, 7: 425–437. 10.1216/RMJ-1977-7-3-425

    Article  MATH  Google Scholar 

  6. Sakai K: Pseudo orbit tracing property and strong transversality of diffeomorphisms on closed manifolds. Osaka J. Math. 1994, 31: 373–386.

    MATH  MathSciNet  Google Scholar 

  7. Pilyugin SY, Rodionova AA, Sakai K: Orbital and weak shadowing properties. Discrete Contin. Dyn. Syst. 2003, 9: 287–308.

    Article  MATH  MathSciNet  Google Scholar 

  8. Moser J: On the volume elements on a manifold. Trans. Am. Math. Soc. 1965, 120: 286–294. 10.1090/S0002-9947-1965-0182927-5

    Article  MATH  Google Scholar 

  9. Bonatti C, Díaz LJ, Pujals ER:A C 1 -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

    Article  MATH  Google Scholar 

  10. Hirsh M, Pugh C, Shub M Lecture Notes in Math. In Invariant Manifolds. Springer, Berlin; 1977.

    Google Scholar 

Download references

Acknowledgements

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).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Manseob Lee.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1029-242X-2013-18

Keywords

  • normally hyperbolic
  • hyperbolic
  • orbital shadowing
  • shadowing
  • Anosov