Hyperbolic Sets with the Strong Limit Shadowi ng Property*

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Hyperbolic Sets with the Strong Limit Shadowi ng Property*

KEON-HEELEE

Departmentof Mathematics,ChungnamNationalUniversity, Taejon, 305-764,Korea

(Received18June1999;Infinalform 20January2000)

Let b^bcaC dynamical system onacompact smooth manifoldM. Inthispaperwe introducethe notions ofweak limit shadowing property and strong limit shadowing property of subsets ofMwhich are not equivalent with that of shadowingproperty, and showthat for any hyperbolic submanifoldAofMthe restrictionb[^^isAnosovif and only ifAhas the strong limit shadowing property. Moreoverwefind hyperbolic sets which havethe stronglimitshadowing property.

Keywords: Anosovdynamical system;Birkhoffcenter; Hyperbolic manifolds; Shadow- ing property;Strongshadowing property; Weak shadowing property

AMS(MOS) SubjectClassifications: ^{58F, 54H}

LetM beacompact smoothmanifoldwith a metricrand

b"

^{M Ma}

C diffeomorphism (ordynamical system). Itis said that a sequence

{xk

:kE

7/}

^{in Mise shadowed}by apoint x*EMif

r(k(x*),Xk) <

for allk 77. We say thata subset YofM hasthe shadowing property orpseudoorbit tracingproperty (POTP) for

b

^{if for given e}

>

^{0 there}

exists 6

>

⁰ such that any 6-pseudotrajectory ⁱⁿ Y, i.e., a sequence (={xk ^{Y: k} 77} ^with

r((x), x+

^l)

^<

^{8, is e-shadowed by a point}

* This work wassupportedbyKoreaResearchFoundationGrant(KRF-99-0000).

Address for correspondence:Departmentof Mathematics, University ofQueensland, Brisbane,QLD 4072,Australia.e-mail:[email protected]

5O7

508 K.-H.LEE

x*EM.Inthiscas, the point x*iscalleda shadowing point of

.

^Ifthis

property holdswith Y=M, we say that

b

^{has thePOTP.}

Oftenpseudotrajectories are obtainedas resultsof numerical stud- ies of dynamical systems. In the context POTP means that numeri- cally found trajectories with uniformly small errors are close to real trajectories. In dynamical system theory, there ^{are various} types of shadowing property.

In 1997, Eirola,Nevanlinnaand Pilyuginintroducedthe concept of the limit shadowing propertyand studied theirproperties(see

[1]).

DEFINITION

A

subset Y of M ^{has the} limit shadowing property

(LmSP)

^for

b

^iffor any sequence {xk:k 7/} ⁱⁿ Ywith

r(ck(Xk),Xk+)

--*0 as

Ikl-*

^oo

thereis apoint x* Msuch that

r(rkk(x*),Xk)

--*0 as

Ik[-*

^c.

Ifthisproperty holdswith Y=M, ^{wesay that has the}LmSP.

From the numerical point of view, this property of a dynamical system means the following: if we apply a numericalmethod that approximates ^with "improving accuracy", ^so that one-step errors tend tozero as time go topositive and negative infinity then the ^nu- merically ^obtainedtrajectories tendto real ones.

A

closed invariant setA

c

Mis said to be hyperbolic for if

TAM

has a continuous splitting

TAM EA

^@

EuA

satisfying:

(1)

E andE^u are invariantunder thederivative map

TO;

(2) thereexist constantse

>

⁰and0

<

A

<

1 such that for anyn 7/+

max

(llr,"lll, IIT’-"III} < ^cA".

We say that

A

c Mis a hyperbolic

manifold

^for

b

^if

^A

^{is a C com-}

pact invariant submanifold of M with a hyperbolic structure as a subsetofM. IfMis hyperbolic for

b

^then

b

^{iscalledAnosov.}

Hirsch asksin [3], ifA

c

Misa hyperbolic manifold forb, doesit follow that

b

restricted to

A

isAnosov

(has

^a hyperbolicstructure)?

The answergiven by Franks and Robinson in [2]^wasnegative.

Recently, Lee and Kim [6] showed that if

A

has the strong shadowing property then

[A: ^{A A}

^is

^Anosov,

and found hyperbolic setswhichhave the strong shadowing property.Itis awell-known fact indynamical systemtheorythatif

A

ishyperbolic for thenithasthe shadowing property, i.e., for given e

>

^{0 there exists 6}

>

^{0 such that}

any pseudotrajectory in

A

ise shadowed byapoint x* M.In general, the shadowing point x* neednotbelongto

A.

We say thatA has

the

strong shadowing property ifthe shadowing point x* belongs toA (for^{moredetails, see} [6]).

Inthispaper^weintroduce theconcept ofthe weaklimitshadowing property which is different from that of POTP. Moreover we will discuss that

[A

^is

^Anosov

^{if and only if}

^A

has the strong limit shadowing property, and find hyperbolic sets which have the strong limitshadowing property.

DEFINITION 2 A subset Y of M has the weak limit shadowing property (weak

LmSP)

^{for if there} exists a constant

do >

^{0 such}

that for any 0

<

^d

< do

and d-pseudotrajectory {Xk"^kE7/} ^{in Y} with

r(O(Xk),Xk+)

^{0 as}

Ikl

thereis apoint x*EMsuchthat

r(k(X*),Xk)

^{0 as}

[kl---,

^o.

Inthiscase, the point^{x* is}called a limit shadowing point of

.-If

^this

property holdswith Y=M, we say that has the weak LmSP.

It is easy to show that there exist systems which do not have the weakLmSP.

Example3 Consider the circle S with coordinate xE[0,1) ^and a diffeomorphism of S given by (x)=x. For any

do >

^{0 and}

0

<

^d

<

do, choose n satisfying (l/n)

<

^{d. Let} {Xk^:k 7/} ^{be a} sequence in S defined by

0 ifk=0

Xk-+(1/n+k) (modl)

^ifk>l

Xk+-(1/n--k) (modl)

^{ifk< -1}

Then is a d-pseudotrajectoryin S with

r(rb(Xk), Xk+l)

1

n

+

^{k *0 as}

Ikl

^o

Butwe can see that for any point x*ES

,

r(g)k(x*),Xk) -/*

^{0 as}

Ikl--*

^c.

Clearlywe knowthatthe LmSPimplies theweak LmSP. However the followingexampleshows that the weak LmSPis notequivalentto the LmSP. Moreover we can see that a dynamical systemwhich has the weakLmSP neednot have the POTP.

Example 4 Consider the circle S with coordinate xE[0, 1) ^{and a} diffeomorphism on S withthe following properties;

{ _}

b(x)=x

ifx

0,,

(1) {1) (1)

b(x)>x

^ifx

0,

^and

^b(x)<x

^ifxe

,1

Then we can see that

b

^{does not have the LmSP. In fact, let}

{Xk"

^k 7/} ^be asequence in S given by

0 if k>0,

Xk

ifk<0.

Then(is apseudotrajectorywith

r(b(xk), xt,+l)

^{0 as}

Ikl

^oo.

Butwe can seethat for any point x*ES

,

^wehave

r(g)k(x*),X)

0 as

Ik[--*

Moreoverit iseasy to showthat does nothave the POTP.

Toshowthat

b

^{hastheweakLmSP,weleta 0,b} (1/4),^c (1/2).

For anyn N,welet

V

denote the (1/n)-neighborhoodofthe points

s

{a,b,c}. Set

and

ForeachnEN, ^Choose

an >

^{0 such that}

{ 3}

an < inf r((x),x),r(-l(x),x)

^xE

UW’

_i=1

Forany o

<

^d

<

^{al0, let} {-{xg" k

_ ^7/}

^bea d-pseudotrajectoryin S with r(c(Xk),_Xk+1)--*0 as

Ikl-

^{oo. For} each integer n

_>

10, ^{we can} find

kn >

^{0 such that}

Ikl ^{>_ k}

^implies

^{r(p(Xk), Xk+l) <}

^On

Then we can consider fourpossiblecases.

Case 1 Suppose

{xk: I1 ^{> }}

^c

^(Va U ^vg U ^).

^{Let Xk}

V’

^for

some s E{a,b, c}. ^{Then, by} the choice of d, both _Xk-1 and _Xk+l cannotbelongto

V

^foru

^#

^{s.Andsowe have}

^{{x :lkl _> k}} Vff

^for

some fixeds E

{a,

b,

c}.

This means that_Xk Sas

Ikl

^c

Case 2 Suppose Xk

W

^{for some}

Ikl ^{>_ k.}

^Since r(flP(Xk),Xk)

> 2tn

andr(ck(Xk),Xk+ 1)<an, ^we have

r(Xk, Xk+l) >

OZn and

r(xk, Xk-1) > an

This means that

xi

- ^V

^{and x-i}

- ^V,

for some

> kn.

^{By the} choice of an, we can find

hn ^>

^{0, n}

^>

^{10, such}

thatifk

> hn

^then

XkVnc

^and

^X_kV.

This means that_Xk cand^x__k"- a, ask

Case3 SupposeXkE

W

^forsome

Ikl ^{> kn. As}

^{intheCase 2,we can}

show that either _Xk c and x__k b hold, or _Xk c and ^x_k a hold, ask o.

Case 4 Suppose Xk

W’

^{for some}

Ikl ^{>_ k..}

^{Then we can see that}

either _Xkb and X-ka hold, or _{Xk c} and _X-ka hold, as

At

any case,we can easilyfindx*eS such that

r(k(x*),Xk) -

^{0 as}

^Ikl

^c

This means that

b

^{hasthe weak LmSP.}

One of the main results about shadowing near a hyperbolic set ofa dynamical systemis the so-called the ShadowingLemma; which means thatif

A

isahyperbolicsetfor

b

^{thenithas a}neighborhood U whichhas the POTP (Shadowing property).

In [1], Eirola, Nevanlinna and Pilyugin obtained the similarresult withthe ShadowingLemma asfollows.

THEOREM5([1],^Theorem2.1)

If ^A

^ishyperbolic

for

^{qbthenthereexists}

aneighborhood U

of ^A

^{such that}

if

^{asequence{xk:k 72}belongsto U}

and^that

if ^r(ckk(x*),

^r(q(Xk),Xk)Xk +^--*)⁰

--

^{as0kas-,kcx.-,cx3 thenthereexists apointx* Msuch}

Similarly, we can consider the above theorem for two-sided se- quence. The proof of the following theorem is similar to that of Theorem 5, and we omit ithere.

THEOREM 6

/f A

ishyperbolic

for ck

^{then there exists a} neighborhood U

of ^A

^{whichhas theweak LmSP.}

Remarks 7 If

A c

Mishyperbolic ^for

b

^{thenithas a} neighborhood U which has the weak LmSP. However the limit shadowing point need notbelongto

A.

DEFINITION8

A

hyperbolic set

A c

M has thestronglimitshadowing property (strong LmSP) ^for

b

if there exist a neighborhood Uof

A

and a constant

do >

⁰such that for any 0

<

^d

< do

and any d-pseudo- trajectory {xg"kE7/} ⁱⁿ U with r(ck(Xk),_Xk+1) 0 as

[k]

^-,

,

there isx*E

A

satisfying

r(qbk(x*),

Xk)--0 as

]k -

^o.

STRONGSHADOWINGPROPERTY 513

THEOREM 9 Let

A

be a hyperbolic

manifoM for .

^Then

^{IA: A A}

isAnosov

if

^andonly

if ^A

^{has thestrongLmSP.}

Proof ^Suppose

^{A has the strong LmSP. Since}

^A

^is hyperbolic, $ is expansive on

A,

i.e., there exists an (expansive) constant c

>

⁰ such that if

r($k(x), qbk(y)) <

^{c for x E}

A,

yEM and all k ^7/ then x y.

n times

Wecansee that foreachn Z-

{0}, n

^{o... o is}also expansive on

A.

Let

e(df) sup{e >

^{0 e is an} expansiveconstant of

b

^{n w.r.t.}

A},

and let

inf(e(’),

^{n 7/-}

{0}}

--

Then we have c

>

^{0. Since}

^A

^has the strong limit shadowing prop- erty, we can find a neighborhood

U0

^of

A

and a constant

do >

⁰

such that forany0

<

^d

< do

and any d-pseudotrajectory

in

U0

^with r($(Xk),Xk+l)--*O as

Ikl-,oo,

there is a point x*

A

satisfying

r(qbk(x*),Xk) ---

^{0 as}

^Ikl

Choosed

>

0 suchthat

d

< min(c, do)

^and

B(A, d) c U0.

Put U

B(A,

d).

Firstwc showthat

ke;dpk(u)= A.

Itisclear that

A

c

kez,k(u)

since

A

is invariant. To show

that f"k.d?k(u)Ch,

^{we let}

$k(u).

Thenwehave

ba(y)

Uforall k ^7/. Let_Xk

dpk(y)

for k ^7/.

The {Xk"^k

7/}

is a d-pseudotrajcctoryin U. Hence thereexists a point x*E

A

such that

r(dpk(x*),Xk)

^--,0 as

[k[

^o.

Choosen

>

^{0 such that if}

[k[ ^_>

^{n then}

r(qbk(x*),Xk) r(qbk(x*), d?k(y)) <

^c.

Put

t2n(x*)

a E

A

and

t2n(y)

b. Thenwehave

r(2k(a), 2k(b)) <

^{c for all k}

^e

^7/.

This meansthata b and soy x*

A.

Nextweshow that

[A

^isstructurally stable. Let

b Diff’(A)

beC near to

[A.

^Thenwe can find

e

^Diff

^(M)

^{such that}

isC nearto

b

^and

Ia ^b.

If we apply [4, ^Theorem 7.3] ^which says that the maximal hyper- bolic sets enjoy a type of structural stability, then we can find a

k -k

homeomorphism h"

f"l

^e

^{b (U)} Ok

e

b (U)

^{such that} (1)

oh ho.b

^on

["kbg(U)=A,

and

(2) ^{h is} C^onear tothe identity map on

A.

Since

(A) ^A,

^wehave

^{A c} ke ’k(U)

^Put

^g=h-lA.

^Since

^A

^isa

compact manifold and g is C^o near to the identity map on

A,

g issurjective. Hence we geth(A)=h(g(A))=

A.

This means that

blA

^is

structurallystable.

Ifwe apply [8, ^Theorem 5], ^{we can} see that

blA: ^{A A}

^is

^Anosov.

The converse is obviousby Theorem 6, and^socompletes the proofof the theorem.

Hyperbolicmanifold whichdonothavethe stronglimitshadowing propertycanbe foundintheexample given by Franks andRobinson [2].

Now we wish to find hyperbolic sets which have the strong limit shadowing property. Put

{x

^x

where w(x) ^and a(x) ^denote the positive and negative limit set of x.

Then C(b)^isanonemptyclosed invariantsubsetofM.Wesay thata point xEM is called nonwandering if for any neighborhood U ofx andaninteger

no >

^{0thereexists an}integern

> no

^with

^b(U)

^U

# 0.

Apointx Mis saidtobe chainrecurrentif foranye

>

⁰there exists an e pseudotrajectory for $ from x to x. The set of nonwandering points and the set of chain recurrent pointsof $ will be denoted by fl($) and CR($), respectively. Thenwe have the followinginclusions

Per

(b) ^c C(b) ^c f(b) ^c CR(ck).

THEOREM 10

If

^{theset C(qb) ishyperbolic}

for

^{qb then ithas thestrong}

limit shadowing property.

Proof

^{If thesetC($)is}hyperbolic then$ isexpansive^onC($).Then

b

^nisalso expansive on C(b)^foreach nonzero integern. Put

e(b) sup{e >

^{O"eis an} expansiveconstantof $n w.r.t.

C($)},

and

e

inf{e(n)’n

^7/-

^{0}}.

Then we have e

>

^{0. Since} C(b) ^{has the} strong shadowing property for

b

^{[6, Theorem 7], we can find 0}

^<

⁶

^{< (e/2)}

such that any 6- pseudotrajectoryin C(b) ^is

(e/2)

shadowed by ^apointin C(b). By Theorem 6, there exist a neighborhood U of C(b) ^and a constant d’

>

^{0 such that for any 0}

<

^d

<

d’ and any d-pseudotrajectory {Xk ^kE7/} ⁱⁿ Uwith r(qb(Xk),_Xk+1) 0 and

Ikl -

o, there exists a point x*EM such that

r(qbk(x*),Xk)

^{--,0 as}

Ikl-

^o.

Choose

do >

^{0 such that}

do<min d’,-

^{and B}

A,do

^cU.

If we let

Uo

B(A, (1/2)d0) ^then

Uo

^and (1/3)d0 ^{are required. For} any 0

<

^d

<

(1/3)d0, ^let

{Xk"

k 7/} ^{be a} d-pseudotrajectory in

U0

with r($(Xk),Xk+l)^{0 as}

Ikl- .

^{Thenwe can find a point x* M}

satisfying

r(qk(x*),Xk)

^---+0 as

Ikl

^c

⁽¹⁾

For each k7/, choose yk.A with r(xk, Yk)< (1/2)do. ^Then

’=

{Yk"^k 7/} ^{isa c} pseudotrajectoryin

A.

Infact, wehave

r(dP(yk),

_Yk+l

<_ r(d/)(yk), dP(Xk)) + ^r(dp(Xk),

Xk+l

+ ^r(Xk+l

Yk+l

< - ⁺

^d

^{+- do < a}

Since C() has the strong shadowing property, there exists a point y* C() such that

r(k(y*),yk) < e

1

⁽²⁾

By the fact (1), we can choose N>0 such that if

[k[ >

N then

r(k(x*),

Xk)

< (1/2)d0.

^{Thenwe have}

r(Ok(x*), ^Ok(y*)) < r(O(x*),Xk) + ^r(xl,yk) + ^{r(yk, Ok(y*))}

< do ⁺ do ^{+-e <}

^e

for each

Ikl ^>_

^{N. Put}

^2N(x*)

^{a and}

^2g(y,)

^{bEC(). Then we}

have

r(

^2N

(a), 2v (b)) <

^e

for all k

EZ.

Since

2N

^{is expansive on C() with an expansive}

constant e, wehavea b and sox* y*EC().ThismeansthatC() has the strong limitshadowing property.

If the chain recurrent set CR() ^is hyperbolic then we have CR() C(). Hence ^we get the following corollary.

COROLLARY 11

If

^{the set CR() is hyperbolic then it has the strong}

limitshadowingproperty.

References

[1] Eirola, T., Nevanlinna, O. and Pilyugin, S. (1997). Limit shadowing property, Numer.Funct.Anal. Optim., 18, 75-92.

[2] Franks, J.and Robinson,C.(1976). Aquasi-Anosov diffeomorphism that isnot

Anosov, Trans., Amer.Math.Soc.,223, 276-278.

[3] Hirsch, M.(1970).^Oninvariantsubsets ofhyperbolic sets; In:"Essaysintopology and related topics",pp. 126-146.

[4] Hirsch, M. and Pugh, C. (1970). Stable manifolds and hyperbolic sets, Proc.

Sympos. Pure Math., 14, 125-163.

[5] Kato, K. (1988).Stability and the pseudo-orbit tracing property for diffeomorph- isms,Mem. Fac.Sci.Kfchi. Univ.Set.A Math.,9, 37-58.

[6] Lee, K. and Kim,J. (1999). Hyperbolic manifolds with the stronglyshadowing property, Bull. Austral. Math.Soc.,60, 37-43.

[7] Mane, R. (1978). ^Invariantsets ofAnosovdiffeomorphisms, lnvent. Math., 46, 147-152.

[8] Ombach, J. (1996). Shadowing, expansiveness and hyperbolic homeomorphisms, J.Austral. Math.Soc., 61,57-72.

[9] Palis, J. and Pugh, C. (1975). Fifty problems on dynamical systems, Springer LectureNotesonMathematics,468, 345-353.

[10] Zeghib, A. (1995). Subsystems of Anosov systems, Amer. J. Math., 117, 1431-1448.