Generic diffeomorphisms with weak limit shadowing

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R E S E A R C H

Open Access

Generic diffeomorphisms with weak limit

shadowing

Gang Lu

1

_{, Keonhee Lee}

2

_{and Manseob Lee}

3*

*_{Correspondence:}

[email protected]

3_{Department of Mathematics,}

Mokwon University, Daejeon, 302-729, Korea

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

Abstract

In this paper, we show that if aC1-generic diffeomorphism has the weak limit shadowing property on the chain recurrent set, then the diffemorphism satisfies Axiom A and the no-cycle condition.

MSC: 37C50; 37D30

Keywords: shadowing; weak limit shadowing;

-stable; generic

1 Introduction

LetMbe a closedC∞manifold, and letDiff(M) be the space of diﬀeomorphisms ofM en-dowed with theC_{-topology. Denote by}_d_{the distance on}_M_{induced from a Riemannian}

metric · on the tangent bundleTM. Letf∈Diff(M) andbe a closedf-invariant set. Forδ> , a sequence of points{xi}bi=a(–∞ ≤a<b≤ ∞) inMis called aδ-pseudo orbitof

f ifd(f(xi),xi+) <δfor alla≤i≤b– .

We say thatf has theshadowing propertyonif for every> , there isδ>  such that for anyδ-pseudo orbit{xi}bi=a⊂off (–∞ ≤a<b≤ ∞), there is a pointy∈Msuch that

d(fi₍_y_),_x

i) <for alla≤i≤b– . In the dynamical systems, the shadowing theory is a very

useful notion. In fact, it deals with the stability theorem (see []). For instance, Robinson [] proved that if a diﬀeomorphismfis structurally stable, then it has the shadowing property. In [] Sakai showed thatf belongs to theC-interior of the shadowing property if and only iff is structurally stable. In this paper, we deal with another shadowing property, that is, the weak limit shadowing property which was studied by [].

We say thatfhas theweak limit shadowing propertyon(orisweak limit shadowable

forf) if there exists aδ>  with the following property: if a sequence{xi}i∈Z⊂is a

δ-pseudo orbit off, for which relationsd(f(xi),xi+)→ asi→+∞andd(f–(xi+),xi)→

asi→–∞hold, then there is a pointy∈Msuch thatd(fi₍_y_),_x

i)→ asi→ ±∞. Note

that iff has the limit shadowing property, thenf has the weak limit shadowing property. But the converse is not true (see [, Example ]). Denote byP(f) the set of periodic points off. ThenP(f)⊂(f)⊂R(f), where(f) is the set of non-wandering points off, and R(f) is the set of chain recurrent points off. Note that iff satisﬁes Axiom A and the no-cycle condition, then(f) =R(f). We say thatf has thes-limit shadowing propertyonif for any> , there is aδ>  such that for anyδ-limit pseudo orbitξ={xi}i∈Z⊂, there is

a pointy∈Msuch thatd(fi₍_y_),_x

i) <for alli∈Z, andd(fi(y),xi)→ asi→ ±∞. Clearly,

the weak limit shadowing property is a weak notion of thes-limit shadowing property. We say thatishyperbolicif the tangent bundleTMhas aDf-invariant splittingEs⊕Eu

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and there exist constantsC>  and  <λ<  such that

Dxfn|Exs≤Cλ

n _and _D

xf–n|Eux≤Cλ

n

for allx∈andn≥. If=M, thenf is Anosov. Very recently, Sakai [] showed that if a

C-generic diﬀeomorphismf has thes-limit shadowing property onR(f), thenf satisﬁes Axiom A and the no-cycle condition. The result is motivation for this study. The main theorem of the paper is as follows.

Theorem . For C_{-generic f}_,_{if f has the weak limit shadowing property on}_R₍_f_),_{then f} satisﬁes AxiomAand the no-cycle condition.

2 Proof of Theorem 1.1

Let M be as before andf ∈Diff(M). Letp∈P(f) be a hyperbolic saddle with period π(p) > . The stable manifoldWs(p) and the unstable manifoldWu(p) are deﬁned as fol-lows. It is well known that ifpis a hyperbolic periodic point off with a periodk, then the sets

Ws(p) =x∈M:fkn(x)→pasn→ ∞ and

Wu(p) =x∈M:f–kn(x)→pasn→ ∞

areC-injectively immersed submanifolds ofM. Letp,q∈P(f) be saddles. LetP(f) be the set of periodic points off. Denote byOf(p) the periodicf-orbit ofp∈P(f). We denote

p∼qif the intersectionsWs(Of(p))Wu(Of(q))=∅andWu(Of(p))Ws(Of(q))=∅.

Then we know that ifp∼q, thenindex(p) =index(q). Hereindex(p) is the dimension of the stable manifold ofp, that is,dimWs₍_p_).

Proposition . There is a residual setG⊂Diff(M)such that for any f ∈G,if f|R(f)has the weak limit shadowing property,then for any saddles p,q∈P(f),index(p) =index(q).

To prove Proposition ., we need the following lemma.

Lemma . Let p,q∈P(f)be saddles.If f has the weak limit shadowing property onR(f),

then Ws_(O

f(p))∩Wu(Of(q))=∅.

Proof Suppose thatf has the weak limit shadowing property onR(f). For any saddles

p,q∈P(f), we show thatWu(Of(p))∩Ws(Of(q))=∅. For the sake of simplicity, we may

assume thatf(p) =pandf(q) =q. Letδ>  be the number of the weak limit shadowing property off such thatd(p,q) <δ. We constructδ-limit pseudo orbitξ={xi}i∈Z⊂R(f) as

follows. (i)x=p, (ii)x–i=f–i(p) for alli> , (iii)xi=fi(q) for alli≥. Then theδ-limit

pseudo orbit

ξ=. . . ,x–,p= (x),q(=x),x, . . .

={. . . ,p,p,q,q, . . .},

and it is clear thatξ ⊂R(f). Sincef has the weak shadowing property onR(f), there is a pointy∈Msuch thatd(fi(y),xi)→ asi→ ±∞. Thenfi(y)→pasi→–∞and

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The following is called the Kupka-Smale theorem.

Lemma . There is a residual setG⊂Diff(M)such that for any f ∈G,every periodic point is hyperbolic,and the stable manifolds and the unstable manifolds of periodic points are all transverse.

Proof of Proposition. Letf ∈G, and letp,q∈P(f) be saddles. Suppose thatf has the

weak limit shadowing property onR(f). Letδ>  be the number of the weak limit shad-owing property of f such thatd(p,q) <δ. Then we will drive a contradiction, we may assume thatindex(p)=index(q). Then we know thatdimWs₍_p_{) +}_dim_Wu₍_q_{) <}_dim_M_or

dimWu₍_p_{) +}_dim_Ws₍_q_{) <}_dim_M_{. In this proof, we consider that}_dim_Ws₍_p_{) +}_dim_Wu₍_q_{) <}

dimM(the other case is similar). Sincef ∈G,Ws(p)∩Wu(q) =∅. Sincef has the weak

limit shadowing property onR(f), by Lemma .,Ws₍_p₎_∩_Wu₍_q₎₌_∅_{. This is a}

contradic-tion.

Letp∈P(f) be a hyperbolic saddle with a periodπ(p) > . Then there are the local stable manifoldWs(p) and the unstable manifoldWu(p) ofpfor some=(p) > . It is easily seen that ifd(fn₍_x_),_fn₍_p₎₎_≤_{for all}_n_≥_{, then}_x_∈_Ws

(p), and ifd(fn(x),fn(p))≤for alln≤, thenx∈Wu(p). The following lemma shows that iff has thes-limit shadowing property onR(f), then the numbers of sinks and sources are ﬁnite (see [, Lemma ]). From the above facts, we show that iff has the weak limit shadowing property onR(f), then the numbers of sinks and sources are ﬁnite.

Lemma . Let f have the weak limit shadowing property onR(f),and letδ> be the

number of the weak limit shadowing property of f.For any saddle q∈P(f),if p∈P(f)is a sink or a source,then d(p,q)≥δ.

Proof We will derive a contradiction. Suppose thatq∈P(f) is a saddle andp∈P(f) is a sink withd(p,q) <δ. For the sake of simplicity, we may assume thatf(p) =p,f(q) =q. Since

qis a saddle, there is(q) >  such that if for anyx∈M,d(fi₍_x_),_fi₍_q₎₎_≤₍_q_{) as}_i_{→ ∞}_{, then}

x∈Ws₍_q₎(q), and ifx∈M,d(fi₍_x_),_fi₍_q₎₎_≤₍_q_{) as}_i_→_–_∞_{, then}_x_∈_Wu

(q)(q). Then we may

assume thatd(p,q) >(q). Then we construct aδ-limit pseudo orbitξ={xi}i∈Z⊂R(f) as

follows. Putx–i=f–i(p) fori≥ andxi=fi(q) fori≥. Thenξ={xi}i∈Zis clearly aδ-limit

pseudo orbit off, andξ={xi}i∈Z⊂R(f). Sincef has the weak limit shadowing property

onR(f), there is a pointy∈Msuch thatd(fi₍_y_),_x

i)→ asi→ ±∞. Sincepis a sink,

d(f–i₍_y_),_x

–i) =d(f–i(y),p)→ asi→ ∞. Theny=p. Sinced(fi(y),xi) =d(fi(y),q)→

asi→ ∞, there isk>  such thatd(fk+i₍_y_),_fk+i₍_q_{)) =}_d₍_fk+i₍_y_),_q₎_≤₍_q_{) for}_i_≥_{. Then}

fk₍_y₎_∈_Ws

(q)(q). Sincey=p, we know thatd(p,q)≤(q). This is a contradiction.

Letpbe a periodic point off, and let  <δ< . We sayphas aδ-weak eigenvalueprovided

Dpfπ(p)has an eigenvalueλsuch that ( –δ)π(p)<|λ|< ( +δ)π(p). We say that the periodic

point has a real spectrum if all of its eigenvalues are real and a simple spectrum if all of its eigenvalues have multiplicity one. The following lemma will play a crucial role in our proof.

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(a) for anyη> ,if for anyC_{-neighborhood}_U₍_f₎_of_f_,_{there exist}_g_∈_U₍_f₎_and

pg,qg∈P(g)with the same period such thatd(pg,qg) <η,then there existp,q∈P(f)

with the same period such thatd(p,q) <η;

(b) for anyη> ,if for anyC-neighborhoodU(f)off,there existg∈U(f)andpg∈P(g)

with anη-weak eigenvalue,then there existp∈P(f)with aη-weak eigenvalue; (c) for anyη> ,ifq∈P(f)with anη-weak eigenvalue and a real spectrum,then there

existsp∈P(f)with anη-weak eigenvalue with a simple real spectrum.

Lemma .[, Lemma .] There is a residual setG⊂Diff(M)such that for any f ∈G, for anyη> ,if for any C_{-neighborhood}_U₍_f₎_{of f}_,_{there exist g}_∈_U₍_f₎_{and p}

g,qg∈P(g)with

the same period such that d(pg,qg) <ηwith diﬀerent indices,then there exist p,q∈P(f)with

the same period such that d(p,q) <ηwith diﬀerent indices.

The following so-called Franks lemma will play an essential role in our proof.

Lemma . LetU(f)be any given C_{-neighborhood of f}_._{Then there exists}_{> } _{and a} C_{-neighborhood}_U

(f)⊂U(f)of f such that for given g∈U(f),a ﬁnite set{x,x, . . . ,xN},

a neighborhood U of{x,x, . . . ,xN}and linear maps Li:TxiM→Tg(xi)M satisfyingLi–

Dxig ≤for all≤i≤N,there exists g∈U(f)such that g(x) =g(x)if x∈ {x,x, . . . ,xN}∪ (M\U)and Dxig

₌_L

ifor all≤i≤N.

If p∈P(f) is hyperbolic, then for anyg∈U(f), there is a unique hyperbolic periodic pointpg∈P(g) nearbypsuch thatπ(pg) =π(p) andindex(pg) =index(p), whereindex=

dimWs₍_p_{). Such a}_p

gis called thecontinuationofp.

Lemma . There is a residual setG⊂Diff(M)such that for any f ∈G,if f has the weak limit shadowing property onR(f),then there isη> such that f has noη-weak eigenvalue.

Proof Letf ∈G=G∩G. To derive a contradiction, we may assume that for anyη> ,

there is a hyperbolic periodic pointqgofg(C-nearbyf) such thatqghas anη-weak

eigen-value and a simple real spectrum. Letδ>  be the number of the weak limit shadowing property off such that  <η<δ/. For the sake of simplicity, we assume thatqgis a ﬁxed

point. By Lemma ., there ish C_{-close to}_g_and_{h C}_-nearby_f _{such that}_q

hhas  as an

eigenvalue. By Lemma . and as in the proof of [, Lemma .], we can construct anhl

(l> )-invariant small arcIqhofh

l_containing_q

hsuch thatd(ph,rh) <η, whereph,rhare

the end points ofIqh,h

l₍_p

h) =ph,hl(rh) =rhandph,rhare hyperbolic saddles and

diﬀer-ent indices. Sincef ∈G, there existp,r∈P(f) with the same period such thatd(p,r) <η

with diﬀerent indices. Since f has the weak limit shadowing property onR(f), and by Lemma ., p,r∈P(f) are saddles. Since d(p,r) <δ, by Proposition ., we know that

index(p) =index(r). But, sinceindex(p)=index(r), this is a contradiction. Denote by F(M) the C_{-interior of the set of diﬀeomorphisms of}_M_{whose periodic}

points are all hyperbolic. In [], Hayashi showed that iff ∈F(M), thenf satisﬁes Axiom A and the no-cycle condition. To prove Theorem ., it is enough to showf ∈F(M).

End of proof of Theorem. Letf ∈G, and letf have the weak limit shadowing property

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pointpg such that the pointpg has anη/ weak eigenvalue. Sincef ∈G, there isp∈P(f)

such thatphas anη-weak eigenvalue. By Lemma ., this is a contradiction.

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.

Author details

1_{Department of Mathematics, School of Science, Shenyang University of Technology, Shenyang, 110178, P.R. China.} 2_{Department of Mathematics, Chungnam National University, Daejeon, 305-764, Korea.}3_{Department of Mathematics,}

Mokwon University, Daejeon, 302-729, Korea.

Acknowledgements

We wish to thank the referee for carefully reading of the manuscript and for providing us with many good suggestions. GL is supported by the Doctoral Science Foundation of Liaoning Province by Hall of Liaoning province science and technology (No. 2012-1055). 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).

Received: 1 November 2012 Accepted: 11 January 2013 Published: 4 February 2013 References

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doi:10.1186/1687-1847-2013-27