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Theory and Modern Applications

Divergence-free vector fields with inverse shadowing

Abstract

We show that if a divergence-free vector field has the C 1 -stably orbital inverse shadowing property with respect to the class of continuous methods T d , 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 [17]. 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 C 1 -stably inverse shadowing property with respect to the class of continuous methods T d , 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 C 1 -interior of the set of vector fields with the orbital shadowing property with respect to the class T d coincides with the set of structurally stable vector fields. From the facts, Lee [4] showed that if a volume-preserving diffeomorphism has the C 1 -stably inverse shadowing property with respect to the class T d , then the diffeomorphism is Anosov. Moreover, if a volume-preserving diffeomorphism has the C 1 -stably orbital inverse shadowing property with respect to the class T d , 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 T d .

Let us be more detailed. Let M be a closed, connected, and C Riemannian manifold endowed with a volume form μ, and let μ denote the Lebesgue measure associated to it. Given a C r (r1) vector field X:MTM, the solution of the equation x =X(x) gives rise to a C r flow, X t ; on the other hand, given a C r flow, we can define a C r 1 vector field by considering X(x)= d X t ( x ) d t | t = 0 . We say that X is divergence-free if its divergence is equal to zero. Note that, by the Liouville formula, a flow X t is volume-preserving if and only if the corresponding vector field X is divergence-free. Let X μ r (M) denote the space of C r divergence-free vector fields, and we consider the usual C r Whitney topology on this space. Let X X μ 1 (M) for any δ>0 and T>0. We say that a (δ,T)-pseudo orbit of X X μ 1 (M) if

d ( X t i ( x i ) , x i + 1 ) <δ,

for any t i T, iZ. A mapping Ψ:R×MM is called a (δ,T)-method for X if for any xM, the map Ψ x :RM defined by

Ψ x (t)=Ψ(t,x),tR,

is a (δ,T)-pseudo orbit of X. Ψ is said to be complete if Ψ(0,x)=x for any xM. Then a (δ,T)-method for X can be considered a family of (δ,T)-pseudo orbits of X. A method Ψ for X is said to be continuous if the map Ψ :M M R , given by Ψ (x)(t)=Ψ(t,x) for xM and tR, is continuous under the compact open topology on M R , where M R denotes the set of all functions from to M. The set of all complete (δ,1)-methods [resp. complete continuous (δ,1)-methods] for X X μ 1 (M) will be denoted by T a (δ,X) [resp. T c (δ,X)]. We can see that if Y is another vector field which is sufficiently close to X in the C 0 topology, then it induces a complete continuous method for X. Let T h (δ,X) [resp. T d (δ,X] be the set of all complete continuous (δ,1) methods for X which are induced by C 1 vector fields Y with d 0 (X,Y)<δ [resp. d 1 (X,Y)<δ], where d 0 is the C 0 metric on X μ 1 (M) such that

d 0 (X,Y)= sup x M { X ( x ) Y ( x ) } ,

and d 1 is the C 1 metric on X μ 1 (M) such that

d 1 (X,Y)= d 0 (X,Y)+ sup x M D X ( x ) D Y ( x ) .

We say that a vector field X has the inverse shadowing property with respect to the class T α (α=a,c,h,d) if for any ϵ>0, there is δ>0 such that for any (δ,T)-method Ψ T α (δ,X) and any point xM, there are yM and an increasing homeomorphism h:RR with h(0)=0 such that

d ( X h ( t ) ( x ) , Ψ ( t , y ) ) <ϵ,tR.

We denote by IS μ , α (M) the set of divergence-free vector fields on M with the inverse shadowing property with respect to the class T α , where α=a,c,h,d. Let int IS μ , α (M) be the C 1 -interior of the set of divergence-free vector fields on M with the inverse shadowing property with respect to the class T α , where α=a,c,h,d.

We say that X X μ 1 (M) has the C 1 -stably inverse shadowing property with respect to the class T α if there is a C 1 -neighborhood U(X) X μ 1 (M) of X such that for any YU(X), Y has the inverse shadowing property with respect to the class T d , where α=a,c,h,d.

Remark 1.1 Let X X μ 1 (M). Then IS μ , a (M) IS μ , c (M) IS μ , h (M) IS μ , d (M), where α=a,c,h,d.

We say that X X μ 1 (M) has the orbital inverse shadowing property with respect to the class T d if for any ϵ>0, there is δ>0 such that for any (δ,1)-method Ψ T d (δ,X) and any point xM, there is yM such that

d H ( Orb X ( x ) ¯ , Orb ( y , Ψ ) ¯ ) <ϵ,

where Orb(y,Ψ)={Ψ(t,y):tR}. Note that if X has the inverse shadowing property with respect to the class T d , then X has the orbital shadowing property with respect to the class T d . 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 int OIS μ , α (M) be the C 1 -interior of the set of divergence-free vector fields on M with the orbital inverse shadowing property with respect to the class T α , where α=a,c,h,d.

Let Λ be a closed X t -invariant set. We say that Λ is hyperbolic if there are constants C>0, λ>0 and a continuous splitting T Λ M= E s X(x) E u such that

D X t | E s ( x ) C e λ t and D X t | E u ( x ) C e λ t

for any xΛ and t>0. If Λ=M, then X is called Anosov.

Given a vector field X, we denote by PO(X) the set of closed orbits of X and by Sing(X) the set of singularities of X, i.e., those points xM such that X(x)= 0 . Denote by Crit(X)=Sing(X)PO(X). Let R:=MSing(X) be the set of regular points. We assume that the exponential map exp p : T p M(1)M is well defined for all pM, where T p M(1)={v T p M:v1}. Given xR, we consider its normal bundle N x = X ( x ) T x M, and let N x (r) be the r-ball in N x . Let N x , r = exp x ( N x (r)). For any xR and tR, there are r>0 and a C 1 map τ: N x , r R with τ(x)=t such that X τ ( y ) (y) N X t ( x ) , 1 for any y N x , r . We say τ(y) is the first return time of y. Then we define the Poincarè map f by

f : N x , r N X t ( x ) , 1 , y f ( y ) = X τ ( y ) ( y ) .

Let N= x R N x be the normal bundle based on R. We can define the associated linear Poincaré flow by

P X t (x):= Π X t ( x ) D X t (x),

where Π X t ( x ) : T X t ( x ) M N X t ( x ) is the projection along the direction of X( X t (x)). We say that a vector field X X μ 1 (M) is topologically stable if for any ϵ>0, there is δ>0 such that for any Y X μ 1 (M) with d 0 (X,Y)<δ, there is a semi-conjugacy (h,τ) from Y to X satisfying d(h(x),x)<ϵ for all xM.

In [8], the authors proved that if a vector field is in the C 1 -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 C 1 -interior of the set of topological stable vector fields X X μ 1 (M), then X is Anosov.

Remark 1.2 We have the following implication: topological stability inverse shadowing property with respect to the continuous method T d orbital inverse shadowing property with respect to the continuous method T d .

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 T d ’ 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 T d . Note that if Xint IS μ , α 1 (M) or Xint OIS μ , α 1 (M), then it means that X has the C 1 -stably inverse, orbital inverse shadowing property with respect to the class T α , α=a,c,h,d. The following is the main result of this paper.

Theorem 1.3 Let X X μ 1 (M). Then

int IS μ , d 1 (M)=int OIS μ 1 (M)= A μ 1 (M),

where A μ 1 (M) is the set of divergence-free Anosov vector fields.

2 Proof of Theorem 1.3

Let M be as before, and let X X μ 1 (M). The perturbations (Lemma 2.1) for volume-preserving vector fields allows to realize them as perturbations of a fixed volume-preserving flow. Fix X X μ 1 (M) and τ>0. A one-parameter area-preserving linear family { A t } t R associated to { X t (p);t[0,τ]} is defined as follows:

  • A t : N p N p is a linear map for all tR,

  • A t =id for all t0 and A t = A τ for all tτ,

  • A t SL(n,R), and

  • the family A t is C on the parameter t.

The following result, proved in [[11], Lemma 3.2], is now stated for X X μ 1 (M) instead of X X μ 4 (M) because of the improved smooth C 1 pasting lemma proved in [[12], Lemma 5.2].

Lemma 2.1 Given ϵ>0 and a vector field X X μ 1 (M), there exists ξ 0 = ξ 0 (ϵ,X) such that for all τ[1,2], for any periodic point p of period greater than 2, for any sufficiently small flowbox T of { X t (p);t[0,τ]} and for any one-parameter linear family { A t } t [ 0 , τ ] such that A t A t 1 < ξ 0 for all t[0,τ], there exists Y X μ 1 (M) satisfying the following properties:

  1. (a)

    Y is ϵ- C 1 -close to X;

  2. (b)

    Y t (p)= X t (p) for all tR;

  3. (c)

    P Y τ (p)= P X τ (p) A τ , and

  4. (d)

    Y | T c X | T c .

Remark 2.2 Let X X μ 1 (M). By Zuppa’s theorem [13], we can find Y C 1 -close to X such that Y X μ (M), Y π (p)=p and P Y π (p) has an eigenvalue λ with |λ|=1.

A divergence-free vector field X is a divergence-free star vector field if there exists a C 1 -neighborhood U(X) of X in X μ 1 (M) such that if YU(X), then every point in Crit(Y) is hyperbolic. The set of divergence-free star vector fields is denoted by G μ 1 (M). Then we get the following theorem.

Theorem 2.3 [[14], Theorem 1.1]

If X G μ 1 (M), then Sing(X)= 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 X G μ 1 (M).

Proposition 2.4 Let Xint IS μ , d 1 (M). Then X G μ 1 (M).

Proof Suppose that X X μ 1 (M) has the C 1 -stably inverse shadowing property. Then there is a C 1 -neighborhood U(X) of X such that for any YU(X), Y has the inverse shadowing property. Let pγPO(X) with X π (p)=p and U p be a small neighborhood of pM. We will derive a contradiction. Assume that there is an eigenvalue λ of P X π (p) such that |λ|=1. By Remark 2.2, there is YU(X) such that Y X μ (M), Y Y π (p)=p and P Y π (p) has an eigenvalue λ with |λ|=1. Using Moser’s theorem (see [15]), there is a smooth conservative change of coordinates φ p : U p T p M such that φ p (p)= 0 . Let f: φ p 1 ( N p ) N p be the Poincarè map associated to Y t , and let V be a C 1 -neighborhood of f. Here, N p is the Poincarè section through p. By Lemma 2.1, we can find a small flowbox T of Y [ 0 , t 0 ] , 0< t 0 <π, and there are Z U 0 (Y)U(X), gV and α>0 such that

  1. (a)

    Z t = Y t for all tR,

  2. (b)

    P Z t 0 (p)= P Y t 0 (p),

  3. (c)

    Z | T c =Y | T c ,

  4. (d)

    g(x)= φ p 1 P Y π (p) φ p (x) for all x B α (p) φ p 1 ( N p ), and

  5. (e)

    g(x)=f(x) for all x B 4 α (p) φ p 1 ( N p ).

Then P Z π (p) has an eigenvalue λ with |λ|=1. For 0<ϵ<α/8, let 0<δ<ϵ be as in the definition of the inverse shadowing property of Z t .

Take a linear map A t : N p N p for all tR such that if P Z π (p) A t 0 P Z π (p)< δ 0 , then d 1 (Z,W)<δ, and P Z π (p) A t 0 is a hyperbolic linear Poincarè flow. Set P W t (p)= P Z π (p) A t 0 . Then W t T d (Z), and pγ is a periodic point of W t . Since ZU(X) for any xM, there exist yM and an increasing homeomorphism h:RR with h(0)=0 such that d( Z h ( t ) (x), W t (y))<ϵ for all tR.

First, we assume that λ=1 (the other case is similar). Then we can choose a vector v associated to λ such that v<α/4. Since φ p 1 (v) φ p 1 ( N p ){p},

g ( φ p 1 ( v ) ) = φ p 1 P Y π (p) φ p ( φ p 1 ( p ) ) = φ p 1 P Y π (p)(v)= φ p 1 (v).

Then we can take z φ p 1 ( N p ) such that

p B ϵ { g 1 i ( z ) : i Z } B α / 2 (p) φ p 1 ( N p ),

where g 1 is the Poincarè map associated to W t . Since P W t (p) is a hyperbolic linear Poincarè flow, we can see that

d ( g i ( p ) , g 1 i ( z ) ) >ϵ

for some iZ. Thus

d ( Z h ( t ) ( p ) , W t ( z ) ) >ϵ

for some tR. 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 ZU(X) such that P Z π (p) is a rational rotation. Then there is l>0 such that P Z l + π (p) is the identity. As in the previous argument, we get a contradiction. □

Proposition 2.5 Let Xint OIS μ , d 1 (M). Then X G μ 1 (M).

Proof Suppose that X has the C 1 -stably orbital inverse shadowing property. Then there is a C 1 -neighborhood U(X) of X such that for any YU(X), Y has the orbital inverse shadowing property. Let pγPO(X) with X π (p)=p and U p be a small neighborhood of pM. We will derive a contradiction. Assume that there is an eigenvalue λ of P X π (p) such that |λ|=1. As in the proof of Proposition 2.4, we get a hyperbolic linear Poincarè flow P W t (p), and W t T d (Z), and pγ is a periodic point of W t . Since ZU(X) for any xM, there exist yM such that

d H ( O ( x , Z t ) ¯ , O ( y , W t ) ¯ ) <ϵ

for all tR. Take t =min{|t|: W t (y) φ p 1 ( N p )}, and let w= W t (y) φ p 1 ( N p ).

If λR or λC, then as in the proof of Lemma 2.4, we get a contradiction. Indeed, since P W t (p) is a hyperbolic linear Poincarè flow, there is j>0 such that g 1 j (w) B α / 4 (p) φ p 1 ( N p ). Thus

d H ( O ( x , Z t ) ¯ , O ( w , W t ) ¯ ) >ϵ

for some tR. Since ZU(X), this is a contradiction. □

End of the proof of Theorem 1.3 By Proposition 2.4 and Proposition 2.5, we have X G μ 1 (M). Thus by Theorem 2.3, we get Sing(X)= and X is Anosov. □

From the result of [10], we get the following corollary.

Corollary 2.6 Let X X μ 1 (M). Then

int TS μ 1 (M)=int IS μ , d 1 (M)=int OIS μ 1 (M)=int A μ 1 (M),

where int TS μ 1 (M) is the C 1 -interior of the set of divergence-free vector fields on M which are topologically stable.

References

  1. Corless R, Pilugin SY: Approximate and real trajectories for generic dynamical systems. J. Math. Anal. Appl. 1995, 189: 409-423. 10.1006/jmaa.1995.1027

    Article  MathSciNet  MATH  Google Scholar 

  2. Han Y, Lee K: Inverse shadowing for structurally stable for flows. Dyn. Syst. 2004, 19: 371-388. 10.1080/1468936042000269569

    Article  MathSciNet  MATH  Google Scholar 

  3. Lee K: Continuous inverse shadowing and hyperbolicity. Bull. Aust. Math. Soc. 2003, 67: 15-26. 10.1017/S0004972700033487

    Article  MathSciNet  MATH  Google Scholar 

  4. Lee M: Volume-preserving diffeomorphisms with inverse shadowing. J. Inequal. Appl. 2012, 2012(275):1-9.

    MathSciNet  MATH  Google Scholar 

  5. Lee K, Lee Z: Inverse shadowing for expansive flows. Bull. Korean Math. Soc. 2003, 40: 703-713.

    Article  MathSciNet  MATH  Google Scholar 

  6. Lee K, Lee Z, Zhang Y: Structural stability of vector fields with orbital inverse shadowing. J. Korean Math. Soc. 2008, 45: 1505-1521. 10.4134/JKMS.2008.45.6.1505

    Article  MathSciNet  MATH  Google Scholar 

  7. Pilygin SY: Inverse shadowing by continuous methods. Discrete Contin. Dyn. Syst. 2002, 8: 29-38.

    Article  MathSciNet  Google Scholar 

  8. Moriyasu K, Sakai K, Sumi N: Vector fields with topological stability. Trans. Am. Math. Soc. 2001, 353: 3391-3408. 10.1090/S0002-9947-01-02748-9

    Article  MathSciNet  MATH  Google Scholar 

  9. Robinson C: Structural stability of vector fields. Ann. Math. 1974, 99: 154-175. 10.2307/1971016

    Article  MathSciNet  MATH  Google Scholar 

  10. Besa M, Rocha J: Topological stability for conservative systems. J. Differ. Equ. 2011, 250: 3960-3966. 10.1016/j.jde.2011.01.009

    Article  MathSciNet  MATH  Google Scholar 

  11. Bessa M, Rocha J:On C 1 -robust transitivity of volume-preserving flows. J. Differ. Equ. 2008, 245(11):3127-3143. 10.1016/j.jde.2008.02.045

    Article  MathSciNet  MATH  Google Scholar 

  12. Bessa, M, Rocha, J: Contributions to the geometric and ergodic theory of conservative flows. Ergod. Theory Dyn. Syst. (2012, at press)

  13. Zuppa C:Regularisation C des champs vectoriels qui préservent lélément de volume. Bol. Soc. Bras. Mat. 1979, 10: 51-56. 10.1007/BF02584629

    Article  MathSciNet  MATH  Google Scholar 

  14. Ferreira C: Stability properties of divergence-free vector fields. Dyn. Syst. 2012, 27: 223-238. 10.1080/14689367.2012.655710

    Article  MathSciNet  MATH  Google Scholar 

  15. 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  MathSciNet  MATH  Google Scholar 

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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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Keywords

  • topological stability
  • inverse shadowing
  • orbital inverse shadowing
  • continuous method
  • Anosov