Divergence free vector fields with orbital shadowing

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R E S E A R C H

Open Access

Divergence-free vector ﬁelds with orbital

shadowing

Manseob Lee

*

*_{Correspondence:}

[email protected] Department of Mathematics, Mokwon University, Daejeon, 302-729, Korea

Abstract

We show that a divergence-free vector field belongs to theC1_{-interior of the set of} divergence-free vector fields satisfying the orbital shadowing property when the vector field is Anosov.

MSC: 37C10; 37C50; 37D20

Keywords: divergence-free vector ﬁelds; star condition; shadowing; orbital shadowing; Anosov

1 Introduction

The shadowing theory is a very useful notion for the investigation of the stability condition. In fact, Robinson [] and Sakai [] proved that a diffeomorphism belongs to theC_-interior of the set of diffeomorphisms having the shadowing property coincides the structural sta-bility, that is, the diffeomorphism satisfies both Axiom A and the strong transversality condition. In general, if a diffeomorphism is-stable, that is, a diffeomorphism satisfies both Axiom A and a no-cycle condition, then there is a diffeomorphism which does not have the shadowing property (see, []). However, for another shadowing property, if a dif-feomorphism is-stable, then the diffeomorphism has another shadowing property.

In this article, we study another shadowing property which is called the orbital shadow-ing property. It is clear that if a diﬀeomorphism has the shadowshadow-ing property, then it has the orbital shadowing property. But the converse is not true. In fact, an irrational rotation map does not have the shadowing property, but it has the orbital shadowing property.

The orbital shadowing property was introduced by Pilyuginet al.[]. They showed that a diffeomorphism belongs to theC_{-interior of the set of diffeomorphisms having the orbital} shadowing property if and only if the diffeomorphism is structurally stable.

For a conservative diffeomorphism, Bessa [] proved that a conservative diffeomor-phism is in theC-interior of the set of all conservative diffeomorphisms satisfying the shadowing property if and only if it is Anosov. Lee and Lee [, ] proved that a conser-vative diffeomorphism is in theC_{-interior of the set of all conservative diffeomorphisms} satisfying the orbital shadowing property if and only if it is Anosov. Also, for a conservative vector field, that is, a divergence-free vector field, Ferreira [] proved that if a conservative vector field belongs to theC-interior of the set of all conservative vector fields satisfying the shadowing property, then it is Anosov. From the results, we study that if a conservative vector field belongs to theC_{-interior of the set of all conservative vector fields having the} orbital shadowing property, then it is Anosov. Our result is a generalization of the main theorem in [].

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2 Basic notions, deﬁnitions and results

LetMbe a closed, connected and smoothn(≥)-dimensional Riemannian manifold en-dowed with a volume form, which has a measureμ, called the Lebesgue measure. Given a Cr₍_r_≥_{), vector ﬁeld}_X_:_M_→_TM_{, the solution of the equation}_x₌_X₍_x_{) generates a}_Cr

flow,Xt_{; on the other hand, given a}_Cr_{flow, we can define a}_Cr–_{vector field by considering} X(x) =dX_dtt(x)|t=. We say thatXisdivergence-free(or aconservative vector field) if its diver-gence is equal to zero. Note that, by the Liouville formula, a flowXtis volume-preserving if and only if the corresponding vector fieldXis divergence-free. LetX_μ(M) denote the space ofCr_{divergence-free vector fields, and we consider the usual}_C_{Whitney topology} on this space. LetX∈X_μ(M). For anyδ>  andT> , we say that{(xi,ti) :ti≥T,i∈Z}is

Let⊆Mbe a compactXt_{-invariant set. We say that}_Xt_has_{the shadowing property}_on

if for any> , there isδ>  such that for any (δ, )-pseudo orbit{(xi,ti)}i∈Z⊂, let

whereB(A) is the neighborhood ofA. Note that the orbital shadowing property is a weak

version of the shadowing property: the diﬀerence is that we do not require a pointxiof a

pseudo-orbitξand the pointXti₍_y_{) of an exact orbit}O

X(y) to be close ‘at any time moment’;

instead, the sets of the points ofXandOX(y) are required to be close. Letbe a closed

Xt_{-invariant set. We say that}_is_hyperbolic_{if there are constants}_C_{>  and}_λ_{>  such}

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the exponential mapexp_p:TpM()→Mis well deﬁned for allp∈M, whereTpM() ={v∈

TpM:v ≤}. Givenx∈R, we consider its normal bundleNx=X(x)⊥⊂TxMand let

Nx(r) be ther-ball inNx. LetNx,r=expx(Nx(r)). For anyx∈Randt∈R, there arer> 

and aC _map_τ _:_N

x,r→Rwithτ(x) =tsuch thatXτ(y)(y)∈NXt₍_x_),for anyy∈N_x_,_r. We

sayτ(y) the ﬁrst return time ofy. Then we deﬁne thePoincarè map f by

f :Nx,r→NXt₍_x),,

y→f(y) =Xτ(y)(y).

LetN =_x∈RNxbe the normal bundle based onR. One can deﬁne the associatedlinear

Poincaré ﬂowbyPt

X(x) := Xt(x)◦DXt(x), where Xt₍_x₎:T_Xt₍_x₎M→N_Xt₍_x₎is the projection

along the direction ofX(Xt₍_x_)).

Denote byintOSμ(M) the set of divergence-free vector ﬁelds satisfying the orbital

shad-owing property.

Theorem . Let X ∈X_μ(M).If X ∈intOSμ(M), then X has no singularity and X is

Anosov.

3 Proof of Theorem 2.1

Let⊂Mbe a compact,Xt-invariant and regular set. We say thatishyperbolicforPt_X ifNadmits aPt_X-invariant splittingN=s⊕usuch that there isl>  satisfying

P_Xls₍_x₎≤



 and P –l

Xu₍_x₎≤

 

for allx∈. The following is well known and one can ﬁnd a proof in [].

Theorem . is a hyperbolic set of Xt_{if and only if the linear Poincaré ﬂow P}t

Xrestricted

onhas a hyperbolic splitting N=s⊕u.

Consider a splittingN=N_{⊕ · · · ⊕}_Nk_over_{, for }_≤_k_≤_n_{– , such that all the}

sub-bundles have constant dimensions. This splitting isdominatedif it isPt

X-invariant, and

there isl>  such that for every ≤i<j≤k, we have

P_Xl_Ni₍_x₎·PX–lNj₍_Xl₍_x₎₎≤

 

for anyx∈.

The following was proved in [].

Theorem .[, Proposition .] If X∈X(M)admits a linear hyperbolic singularity of a saddle type,then P_Xt does not admit any dominated splitting over M\Sing(X).

From the Theorem ., we know that if a vector ﬁeldXadmits a dominated splitting, thenSing(X) =∅.

Franks’ lemma for divergence-free vector fields allows to realize the perturbations as perturbations of a fixed volume-preserving flow. Fix X∈X_μ(M) and τ > . A one-parameter area-preserving linear family{At}t∈Rassociated to{Xt(p);t∈[,τ]}is defined

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• At:Np→Npis a linear map for allt∈R, • At=id, for allt≤andAt=Aτ for allt≥τ, • At∈SL(n,R)and

• the familyAtisC∞on the parametert.

The following result is proved in [, Lemma .].

Lemma . Given> and a vector field X∈X_μ(M),there existsξ=ξ(,X)such that for allτ ∈[, ],for any periodic point p of period greater than,for any sufficient small flowboxT of{Xt(p);t∈[,τ]}and for any one-parameter linear family{At}t∈[,τ]such that

A divergence-free vector fieldXis adivergence-free star vector fieldif there exists aC -neighborhoodU(X) ofXinX_μ(M) such that ifY ∈U(X), then every point inCrit(Y) is hyperbolic. The set of divergence-free star vector fields is denoted byG

μ(M). Then we get

Up be a small neighborhood of p. We will derive a contradiction. Assume that there is

an eigenvalueλofPπ

we assume thatλ=  (the other case is similar). Then we can choose a vectorvassociated toλsuch thatv=α/, and we setIv={t·v: ≤t≤}. Sinceϕp–(v)∈ϕ–p (Np)\ {p},

gϕ_p–(v)=ϕ_p–◦Pπ_Y(p)◦ϕp

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For  <<α/, let  <δ<be as in the deﬁnition of the orbital shadowing property of Zt_{. Set}_J

p=ϕp–(Iv). There is k∈Nsuch thatxk=ϕp–(v),v=pand|vi–vi+|<δ for ≤i≤k– , wherevi=ti·vfor ≤i≤k– . We construct a (δ, ) pseudo-orbit ofZt

belonging toJpas follows: (a) xi=ϕp–(v),ti=πfori< ,

(b) xi=g(ϕp–(vk)),ti=πfor≤i≤k– , and (c) xi=gi–k(ϕp–(vi)),ti=πfori≥k.

Thenξ={(xi,ti) :i∈Z}is a (δ, )-pseudo orbit ofZtand it is contained inJp. By the orbital

shadowing property, we can take a pointz∈Msuch that

OZ(z)⊂B(ξ) and ξ⊂B

OZ(z)

.

If z ∈Jp, then we know that there is T >  such that ZT(z)∈ B(x)∩Jp, and

d(ZT₍_z_),_x

k) =α/. Then

dx,ZT(z)

=dϕ_p–(v),ϕp–

ZT₍_z₎₌_d_ϕ–

p (), ϕp–

ZT₍_z₎

=dp,ϕ_p–ZT₍_z₎₌α

 >.

ThusOZ(z)⊂B(ξ). This is a contradiction.

Ifz∈M\Jp, there isT>  such thatZT(z)∈B(x). Then for somej=nπ, we have

<d(x,xk)≤d

x,ZT₍_z₎₊_d_ZT+j₍_z_),_x k

< ,

which is a contradiction.

Finally, we assume thatλis complex. By [, Lemma .], there isZ∈U(X) such that Pπ

Z(p) is a rational rotation. Then there isl>  such thatPlZ+π(p) is the identity. Then, as in

the previous argument, we get a contradiction.

End of the proof of Theorem .. By Lemma ., X∈G

μ(M). Thus by Theorem ., Sing(X) =∅andXis Anosov.

By [] and our main result, we have the following.

Corollary . Let X∈Xμ(M).Then

intSμ(M) =intOSμ(M) =Aμ(M),

whereintSμ(M)is the set of all divergence-free vector ﬁelds satisfying the shadowing

prop-erty.

Competing interests

The author declares that they have no competing interests.

Acknowledgements

We wish to thank the referee for carefully reading of the manuscript and providing us with many good suggestions. This work 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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