Divergence free vector fields with inverse shadowing

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R E S E A R C H

Open Access

Divergence-free vector ﬁelds with inverse

shadowing

Keonhee Lee

1

_{and Manseob Lee}

2*

*_{Correspondence:}

[email protected] 2_{Department of Mathematics,}

Mokwon University, Daejeon, 302-729, Korea

Full list of author information is available at the end of the article

Abstract

We show that if a divergence-free vector field has theC1-stably orbital inverse shadowing property with respect to the class of continuous methodsTd, 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

Keywords: topological stability; inverse shadowing; orbital inverse shadowing; continuous method; Anosov

1 Introduction

The notion of inverse shadowing property is a dual notion of the shadowing property. It was studied by [–]. In fact, Pilyugin [] showed that every structurally stable diffeo-morphism has the inverse shadowing property with respect to the class of continuous methods. In [], Lee proved that if a diffeomorphism has theC-stably inverse shadowing property with respect to the class of continuous methodsTd, then the diffeomorphism is structurally stable. For vector fields, Lee and Lee [] introduced the notion of inverse shad-owing for flows and showed that every expansive flow with the shadshad-owing property has the inverse shadowing property with respect to the class of continuous methods. Leeet al.[] showed that theC_{-interior of the set of vector fields with the orbital shadowing property}

with respect to the classTdcoincides with the set of structurally stable vector ﬁelds. From the facts, Lee [] showed that if a volume-preserving diﬀeomorphism has theC_-stably

in-verse shadowing property with respect to the classTd, then the diﬀeomorphism is Anosov. Moreover, if a volume-preserving diﬀeomorphism has theC_{-stably orbital inverse}

shad-owing property with respect to the classTd, 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 classTd.

Let us be more detailed. LetMbe a closed, connected, andC∞Riemannian manifold endowed with a volume formμ, and letμdenote the Lebesgue measure associated to it. Given aCr (r≥) vector field X: M→TM, the solution of the equationx=X(x) gives rise to aCr_flow,_Xt_{; on the other hand, given a}_Cr_{flow, we can define a}_Cr–_vector

ﬁeld by consideringX(x) =dX_dtt(x)|t=. We say thatXisdivergence-freeif its divergence is

equal to zero. Note that, by the Liouville formula, a flowXt _{is volume-preserving if and} only if the corresponding vector fieldXis divergence-free. LetXr_μ(M) denote the space ofCr _{divergence-free vector fields, and we consider the usual}_Cr_{Whitney topology on}

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this space. LetX∈X_μ(M) for anyδ>  andT> . We say that a (δ,T)-pseudo orbit of X∈X_μ(M) if

dXti₍_x i),xi+

<δ,

for anyti≥T,i∈Z. A mapping:R×M→Mis called a (δ,T)-methodforXif for any x∈M, the mapx:R→Mdeﬁned by

x(t) =(t,x), t∈R,

is a (δ,T)-pseudo orbit ofX.is said to becompleteif(,x) =xfor anyx∈M. Then a (δ,T)-method forXcan be considered a family of (δ,T)-pseudo orbits ofX. A method forXis said to becontinuousif the map:M→MR, given by(x)(t) =(t,x) forx∈M andt∈R, is continuous under the compact open topology onMR, whereMRdenotes the set of all functions fromRtoM. The set of all complete (δ, )-methods [resp. complete continuous (δ, )-methods] forX∈X_μ(M) will be denoted byTa(δ,X) [resp.Tc(δ,X)]. We can see that ifY is another vector ﬁeld which is suﬃciently close toXin theC_topology,

then it induces a complete continuous method forX. LetTh(δ,X) [resp.Td(δ,X] be the set of all complete continuous (δ, ) methods forXwhich are induced byC_{vector ﬁelds}_Y

withd(X,Y) <δ[resp.d(X,Y) <δ], wheredis theCmetric onXμ(M) such that

d(X,Y) =sup

x∈M

_X₍_x_{) –}_Y₍_x₎,

anddis theCmetric onXμ(M) such that

d(X,Y) =d(X,Y) +sup

x∈M

DX(x) –DY(x).

We say that a vector ﬁeldXhas theinverse shadowing propertywith respect to the class

Tα(α=a,c,h,d) if for any> , there isδ>  such that for any (δ,T)-method∈Tα(δ,X)

and any pointx∈M, there arey∈Mand an increasing homeomorphismh:R→Rwith h() =  such that

dXh(t)(x),(t,y)<, t∈R.

We denote byISμ,α(M) the set of divergence-free vector ﬁelds on Mwith the inverse

shadowing property with respect to the classTα, whereα=a,c,h,d. LetintISμ,α(M) be

theC_{-interior of the set of divergence-free vector ﬁelds on}_M_{with the inverse shadowing}

property with respect to the classTα, whereα=a,c,h,d.

We say thatX∈X_μ(M) has theC_{-stably inverse shadowing property with respect to}

the classTαif there is aC-neighborhoodU(X)⊂Xμ(M) ofXsuch that for anyY∈U(X),

Y has the inverse shadowing property with respect to the classTd, whereα=a,c,h,d.

Remark . LetX∈X_μ(M). ThenISμ,a(M)⊂ISμ,c(M)⊂ISμ,h(M)⊂ISμ,d(M), where

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We say thatX∈X_μ(M) has the orbital inverse shadowing property with respect to the classTdif for any> , there isδ>  such that for any (δ, )-method∈Td(δ,X) and any pointx∈M, there isy∈Msuch that

dH

OrbX(x),Orb(y,)<,

whereOrb(y,) ={(t,y) :t∈R}. Note that ifXhas the inverse shadowing property with respect to the class Td, then Xhas the orbital shadowing property with respect to the classTd. 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 intOISμ,α(M) be theC-interior of the set of divergence-free vector ﬁelds onMwith the

orbital inverse shadowing property with respect to the classTα, whereα=a,c,h,d.

Letbe a closedXt-invariant set. We say thatishyperbolicif there are constants C> ,λ>  and a continuous splittingTM=Es⊕ X(x) ⊕Eusuch that

DXt|Es₍_x₎≤Ce–λt and DX–t|_Eu₍_x₎≤Ce–λt

for anyx∈andt> . If=M, thenXis calledAnosov.

Given a vector fieldX, we denote byPO(X) the set of closed orbits ofXand bySing(X) the set ofsingularitiesofX,i.e., those pointsx∈Msuch thatX(x) =. Denote by Crit(X) = Sing(X)∪PO(X). LetR:=M\Sing(X) be the set ofregularpoints. We assume that the exponential mapexp_p:TpM()→Mis well defined for allp∈M, whereTpM() ={v∈ TpM:v ≤}. Givenx∈R, we consider its normal bundleNx= X(x)⊥⊂TxM, and let Nx(r) be ther-ball inNx. LetNx,r=expx(Nx(r)). For anyx∈Randt∈R, there arer>  and aC mapτ :Nx,r→Rwithτ(x) =tsuch thatXτ(y)(y)∈NXt₍_x_),for anyy∈N_x_,r. We sayτ(y) is thefirst return timeofy. 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. We can deﬁne the associatedlinear Poincaré ﬂowby

Pt_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)). We say that a vector ﬁeldX∈X_μ(M) istopologically stableif for any> , there isδ>  such that for any Y∈X_μ(M) withd(X,Y) <δ, there is a semi-conjugacy (h,τ) fromY toX

satisfyingd(h(x),x) <for allx∈M.

In [], the authors proved that if a vector ﬁeld is in theC_{-interior of the set of}

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Remark . We have the following implication: topological stability⇒inverse shadowing property with respect to the continuous methodTd⇒orbital inverse shadowing property with respect to the continuous methodTd.

From the above remark, we know that our result is a slight generalization of the main theorem in []. In this paper, we omit the phrase ‘with respect to the classTd’ for sim-plicity. So, we say thatXhas the inverse, orbital inverse shadowing property means thatX has the inverse, orbital inverse shadowing property with respect to the classTd. Note that ifX∈intIS_μ_,_α(M) orX∈intOIS_μ_,_α(M), then it means thatXhas theC_{-stably inverse,}

orbital inverse shadowing property with respect to the classTα,α=a,c,h,d. The following

is the main result of this paper.

Theorem . Let X∈X_μ(M).Then

intIS_μ_,_d(M) =intOIS_μ(M) =Aμ(M),

whereA

μ(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(M). The perturbations (Lemma .) for preserving vector fields allows to realize them as perturbations of a fixed volume-preserving flow. FixX∈X_μ(M) andτ > . Aone-parameter area-preserving linear family

{At}t∈Rassociated to{Xt(p);t∈[,τ]}is deﬁned as follows: • At:Np→Npis a linear map for allt∈R,

• At=idfor allt≤andAt=Aτ for allt≥τ,

• At∈SL(n,R), and

• the familyAtisC∞on the parametert.

The following result, proved in [, Lemma .], is now stated forX∈X_μ(M) instead of X∈X_μ(M) because of the improvedsmooth Cpasting lemmaproved 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 suﬃciently small ﬂowboxT of{Xt(p);t∈[,τ]}and for any one-parameter linear family{At}t∈[,τ]such that A_tA–

t <ξfor all t∈[,τ],there exists Y∈Xμ(M)satisfying the following properties:

(a) Yis-C_{-close to}_X_;

(b) Yt₍_p_{) =}_Xt₍_p₎_{for all}_t_∈_R_; (c) Pτ

Y(p) =PτX(p)◦Aτ,and

(d) Y|_Tc≡X|_Tc.

Remark . LetX∈X_μ(M). By Zuppa’s theorem [], we can ﬁndY C-close toXsuch thatY∈X∞_μ(M),Yπ₍_p_{) =}_p_and_Pπ

Y(p) has an eigenvalueλwith|λ|= .

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

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Theorem .[, Theorem .] If X∈G

μ(M),then Sing(X) =∅and X is Anosov.

Thus, to prove Theorem ., it is enough to show that ifXhas the inverse shadowing property, or the orbital inverse shadowing property, thenX∈G

μ(M).

Proposition . Let X∈intIS_μ_,_d(M).Then X∈G μ(M).

Proof Suppose thatX∈X_μ(M) has theC_{-stably inverse shadowing property. Then there}

is aC_{-neighborhood}_U₍_X_{) of}_X_{such that for any}_Y_∈_U₍_X_),_Y_{has the inverse shadowing}

property. Letp∈γ ∈PO(X) withXπ₍_p_{) =}_p_and_U

p be a small neighborhood ofp∈M. We will derive a contradiction. Assume that there is an eigenvalueλofPπ

X(p) such that

|λ|= . By Remark ., there isY∈U(X) such thatY∈X∞_μ(M),Yπ

Y(p) =pandPπY(p) has an eigenvalueλwith|λ|= . Using Moser’s theorem (see []), there is a smooth conser-vative change of coordinatesϕp:Up→TpMsuch thatϕp(p) =→– . Letf :ϕ–p (Np)→Np deﬁnition of the inverse shadowing property ofZt_.

Take a linear mapAt:Np→Npfor allt∈Rsuch that ifPπZ(p)◦At–PZπ(p)<δ, then

d(Z,W) <δ, andPπZ(p)◦Atis a hyperbolic linear Poincarè ﬂow. SetPtW(p) =PπZ(p)◦At. Then Wt _∈_T_d(_Z_{), and}_p_∈_γ _{is a periodic point of}_Wt_{. Since}_Z_∈_U₍_X_{) for any}_x_∈_M_, there existy∈Mand an increasing homeomorphismh:R→Rwithh() =  such that d(Zh(t)₍_x_),_Wt₍_y_{)) <}_{for all}_t_∈_R_.

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for somet∈R. This is a contradiction by the fact thatZhas the inverse shadowing prop-erty.

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

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

previous argument, we get a contradiction.

Proposition . Let X∈intOIS_μ_,_d(M).Then X∈G μ(M).

Proof Suppose thatXhas theC_{-stably orbital inverse shadowing property. Then there}

is a C_{-neighborhood}_U₍_X_{) of}_X _{such that for any}_Y_∈_U₍_X_),_Y _{has the orbital inverse}

shadowing property. Letp∈γ ∈PO(X) withXπ₍_p_{) =}_p_and_U

pbe a small neighborhood ofp∈M. We will derive a contradiction. Assume that there is an eigenvalueλofPπ

X(p) such that|λ|= . As in the proof of Proposition ., we get a hyperbolic linear Poincarè ﬂowPt_W(p), andWt∈Td(Z), andp∈γ is a periodic point ofWt. SinceZ∈U(X) for any x∈M, there existy∈Msuch that

dH

Ox,Zt_,O_y_,_Wt_<

for allt∈R. Taket=min{|t|:Wt₍_y₎_∈_ϕ–

p (Np)}, and letw=Wt

(y)∈ϕ_p–(Np).

Ifλ∈Rorλ∈C, then as in the proof of Lemma ., we get a contradiction. Indeed, since Pt

W(p) is a hyperbolic linear Poincarè ﬂow, there isj>  such thatg j

(w) /∈Bα/(p)∩ϕp–(Np). Thus

dH

Ox,Zt_,_O_w_,_Wt_>

for somet∈R. SinceZ∈U(X), this is a contradiction.

End of the proof of Theorem. By Proposition . and Proposition ., we have X∈

G

μ(M). Thus by Theorem ., we getSing(X) =∅andXis Anosov.

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

Corollary . Let X∈X_μ(M).Then

intT S_μ(M) =intIS_μ_,_d(M) =intOIS_μ(M) =intA_μ(M),

whereintT S_μ(M)is the C_{-interior of the set of divergence-free vector ﬁelds on M which}

are topologically stable.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, Chungnam National University, Daejeon, 305-764, Korea.}2_{Department of Mathematics,}

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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).

Received: 4 March 2013 Accepted: 23 October 2013 Published:21 Nov 2013

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