LIMIT SHADOWING PROPERTY 1 Introduction

LIMIT SHADOWING PROPERTY Timo Eirola , Olavi Nevanlinna  , and Sergei Yu. Pilyugin # November 4, 1996  Institute of Mathematics, Helsinki University of Technology, Finland

# Faculty of Mathematics and Mechanics, University of St. Petersburg, Russia

Abstract. In this paper pseudoorbits of discrete dynamical systems are considered such

that the one{step errors of the orbits tend to zero with increasing indices. First it is shown that close to hyperbolic sets such orbits are shadowed by true trajectories of the system with shadowing errors also tending to zero. Then the rates of convergence are studied via considering pseudoorbits such that the error sequences belong to certain (weighted) lp { spaces and showing that the corresponding shadowing errors are there, too. Under certain conditions on the weights we establish weighted shadowing near nonhyperbolic sets. AMS subject classi cation: 58F15, 65L70 Key words: Hyperbolic sets, Shadowing property, Sacker{Sell spectrum

1 Introduction In simulation of dynamical systems improved accuracy usually has to be paid by increasing computation power or time. This suggests the natural simulation strategy to use low accuracy until "interesting" things are observed and reduce errors during computation in the hope that a true trajectory is approached and real dynamics of it is captured. This is an example of a situation where the one{step errors of pseudoorbits approximating trajectories of a dynamical system tend to zero. In this paper we look for trajectories that shadow such pseudoorbits and especially such for which the shadowing errors also tend to zero. Let  be a dynamical system on a metric space (X; d) . It is said that a sequence fxk gk2K ; K  Z ; is "{shadowed (or "{traced) by the trajectory through x 2 X if

d(k (x); xk ) < " for all k 2 K : The system has the pseudoorbit tracing property (POTP) if for given " > 0 1

there exists  > 0 such that any {trajectory1, i.e., a sequence fxk gk2Z with d(xk+1; (xk )) <  is "{shadowed (or "{traced) by a trajectory of  : A survey of shadowing results can be found in the book [8]. Often pseudotrajectories are obtained as results of numerical studies of dynamical systems. In this context POTP means that numerically found trajectories with uniformly small errors are close to real trajectories. In this paper we study a shadowing property close to the one described above. We say that  has the limit shadowing property if for any sequence fxk gk0 with d(xk+1; (xk )) k!1 ! 0 there is a point x such that

d(k (x); xk ) k!1 ! 0: 1 : And similarly for two{sided sequences fxk gk=?1 From the numerical point of view this property of a dynamical system  means the following: if we apply a numerical method that approximates  with "improving accuracy" so that one step errors tend to zero as time goes to in nity, then the numerically obtained trajectories tend to real ones. Such situations arise for example when we are not so interested in the initial (transient) behaviour of trajectories but want to get to areas where "interesting things" happen (e.g. attractors) and then improve accuracy. In section 2 we study limit shadowing of a dieomorphism  : IRn ! IRn : It is shown that in a neighbourhood of a hyperbolic set  has the limit shadowing property n k (Theorem o 2.1). Then in section 3 we study the rate of convergence of the sequence j (x) ? xk j k0 to zero. We prove that in a neighbourhood of a hyperbolic set  the following holds: if (1.1)

X

k2Z

jxk+1 ? (xk )jp

1=p

0 ;  2 (0; 1) and a family of linear subspaces Exs ; Exu ; x 2  ; of IRn such that (b1) Exs  Exu = IRn ; x 2  ; (b2) D(x)Ex = E(x) ; x 2  ;  = s; u ; (b3) (2.3) (2.4)

jDm(x)v j  Cm jvj jD?m(x)v j  Cm jvj

for v 2 Exs ; m  0 for v 2 Exu ; m  0 :

We can introduce the so{called Lyapunov (or adapted) metric d0 in a neighbourhood U0 of  (see: e.g. [11]). It has the following properties:

ajx ? yj  d0(x; y)  bjx ? yj

(2.5)

for some a; b > 0 ; and with respect to the norm generated by d0 we can take C = 1 in (2.3) and (2.4) with some 0 2 (; 1) instead of  : We work with d0; 0 below preserving the previous notation, that is jx ? yj stands for d0(x; y) and  for 0 : It follows from inequalities (2.5) that all results of this section are valid in the case of the standard Euclidean metric. The hyperbolic structure fExs ; Exu g can be extended to a neighbourhood of  : More exactly:

Lemma 2.1 (See: [2]) Fix  2 (; 1) : For any  > 0 there exists a neighbourhood U1 = U1 (;  ) of  and continuous families of linear subspaces Eexs ; Eexu ; x 2 U1 ; with the properties

a) Eex = Ex ; x 2  ;  = s; u ; b) Eexs  Eexu = IRn ; x 2 U1 ;

c) if Pxs is the projection of IRn onto Eexs along Eexu and Pxu = I ? Pxs ; then for x 2 U1 with (x) 2 U1 ; we have the following:

(2.6) if w 2 Eexs ; then jPs(x) D(x)wj  jwj ; jPu(x) D(x)wj  2 jwj ;

(2.7) if w 2 Eexu ; then jPu(x) D(x)wj  1 jwj ; jPs(x) D(x)wj  2 jwj : 3

For a sequence of points  = fxk gk2Z  IRn and for a point x set d( ) = f(xk ) ? xk+1 gk2Z ; d+( ) = f(xk ) ? xk+1gk0 ;

n

n

o

o


(x;  ) = k (x) ? xk k2Z ;
+(x;  ) = k (x) ? xk k0 : First we recall some de nitions and preliminary results about shadowing in a neighbourhood of hyperbolic sets. We say that  is a {trajectory of  if (2.8) jd( )kj <  ; k 2 Z : Similarly, we call  = fxk gk0 a positive {trajectory, if the inequality of (2.8) is satis ed for k  0 :

Lemma 2.2 (A weak variant of Lemma A.1 of [8]). There exists a neighbourhood U2 of  and positive numbers L2 and 2 such that if   U2 is a {trajectory with  < 2 ; then there exists a point x such that j
(x;  )k j  L2  ; k 2 Z : And similarly for positive {trajectories. Remark. The proof in [8] is based on ideas of Bowen [3]. One can prove Lemma 2.2 also using techniques of Theorem 3.1 below. This approach was developed by Anosov [2]. It follows from Lemma 2.1 and from the Perron theorem that trajectories of  have stable and unstable manifolds while they remain in a neighbourhood of  (see [9] for details), so the following holds. Lemma 2.3 There exists a neighbourhood U3 of  and 3 > 0 such that if for two points x; y we have k (x); k (y) 2 U3 for k  0 and (2.9) jk (x) ? k (y)j  3 ; k  0 ; then  k (2.10) jk (x) ? k (y)j  2 1+2  jx ? yj ; k  0 : Remark. Inequality (2.9) with small 3 means that x and y belong to the same local stable manifold, hence (2.10) holds. The following theorem is a pure limit shadowing statement for positive pseudo{orbits. It says that if the one{step errors tend to zero, then there exists a true trajectory such that the shadowing errors tend to zero. For two{sided sequences see Prop. 3.2.

Theorem 2.1 There exists a neighbourhood U of  such that if a sequence  = fxk gk0 belongs to U and if (2.11) d+( )k k!1 ! 0; then there is a point x such that (2.12)
+(x;  )k k!1 ! 0:

4

Remark. For a xed point  this result was proved in [1] (Thm.13, chapter 9). One can prove our Thm. 2.1 by applying Proposition 11, chapter 10 in [1], but we prefer to give here a simple direct proof.

Proof. Take L2 ; 2; 3; U2 ; and U3 from Lemmas 2.2 and 2.3 and set Ue = U2 \ U3 : Take "0 > 0 and a neighbourhood U of  such that the open "0 {neighbourhood of U is in Ue : Let  = fxk gk0  U be such that (2.11) holds. Take " > 0 such that " < min(L22; 23 ; "0) :

(2.13)

For j = 1; 2; : : : nd (by (2.11)) an index kj such that d+( )k < L"2 j for k > kj : By Lemma 2.2 and (2.13) there exist points yj such that

jk (yj ) ? xk j < j" ; k > kj : It follows from (2.13) that k (yj ) 2 Ue for k > kj : We claim that (2.12) holds for x = y1 : (2.14)

Indeed, it follows from (2.14) and the choice of " that

jk (y1) ? k (yj )j < 2" < 3 ; k > kj : Hence by Lemma 2.3 (2.15)

  jk (y1) ? k (yj )j < 4" 1+2  k?kj ; k > kj :

For each j take lj > 0 such that (2.16) Then (2.14)-(2.16) imply which completes the proof.

3

lp

 l ?k 4 1+2  j j < 1j :

jxk ? k (y1)j < 2j" ; k  lj ;

{shadowing

The following states the lp {shadowing result. Here we consider two{sided sequences, i.e., indices running from ?1 to 1 : The theorem says that if the sequence of the one{step errors is in lp ; then the same holds also for the sequence of the shadowing errors. For p  1 denote by lp be the Banach space of sequences v = fvk gk2Z with the norm

kv k p =

X

k2Z

jvk jp

1=p

:

Also denote kvk1 = supk2Z jvk j : Note that in Theorems 3.1, 3.3, and in Proposition 3.2,  and  are as in section 2. 5

Theorem 3.1 There exists a neighbourhood U of  and numbers L ; 0 > 0 such that if for some p 2 [1; 1) and for a sequence  = fxk gk2Z  U we have kd( )kp < 0 ; then there exists a unique x such that

k
(x;  )kp  L kd( )kp :

(3.1)

Proof. We begin with an elementary remark. Let B be a Banach space represented as (3.2) B = Bs  Bu : Assume that for a linear operator A : B ! B represented as  Ass Asu  A = Aus Auu

according to (3.2), we have

kAss k  0 ; k(Auu )?1k  0

for some 0 2 (0; 1) : Then there exist ; R > 0 such that if kAsu k   ; kAus k   ;

then A ? I is invertible and (A ? I )?1  R : Fix  2 (; 1) ; take 0 = 1+2  ; nd corresponding  ; R : Apply Lemma 2.1 to nd a neighbourhood U1 = U1(;  ) of  : Take a neighbourhood U of  with compact closure such that U [ (U )  U1 : Denote

n

o

Bp(r) = u 2 lp j kukp  r : Given a sequence  = fxk gk2Z let  = kd( )kp and  = fyk gk2Z ; where yk = xk + vk : Then  is a trajectory of  if and only if v = fvk gk2Z satis es (3.3) Obviously we have

vk+1 = (xk + vk ) ? xk+1 ;

k 2 Z:

(xk + vk ) ? xk+1 = [(xk ) ? xk+1] + D(xk )vk + k (vk ) ; where k (0) = 0 and @@vkk (0) = 0 : Since U is compact, there exist constants C1 ; 1 > 0 and a function (s) s! !0 0 such that kD(x)k  C1 ; x 2 U ; and j k (vk ) ? k (v 0k )j  (max(kvk1 ; kv 0k1 )) jvk ? v 0k j for kvk1 ; kv 0 k1  1 : Since kvk1  kvkp ; it follows from the inequalities j(xk + vk ) ? xk+1j  jd( )kj + (C1 + (1)) jvk j that, for  < 1 ; F (v) = ve ; where vek = (xk?1 + vk?1 ) ? xk ; k 2 Z (3.4)

6

de nes an operator from Bp(1) to lp : The set  = fxk + vk gk2Z is a trajectory of  if and only if v is a xed point of F : It follows from (3.4) that F is dierentiable at 0 with (DF (0)v)k depending only on vk?1 : Consider an extended hyperbolic structure fEexs ; Eexu g on U given by Lemma 2.1. Below we write Exs ; Exu instead of Eexs ; Eexu : We can represent

lp = lps  lpu ;

where

n

o

lp = v 2 lp j vk 2 Exk ; k 2 Z ;  = s; u : Take 2 (1; 2) such that  < (1 + )=2 = 0 : Take the corresponding  ; R : Suppose that U  U1 (0;  ) (see Lemma 2.1). By continuity of the families Exs ; Exu on the compact set U ; we can nd 2 > 0 with the property: if x; y 2 U ; jx ? yj < 2 ; then for v  2 Ex ;  = s; u ; the decomposition (3.5) v  = ve s + ve u ; ve  2 Ey ; satis es jve  j  jv  j : Assume that  < 2 : Let us show that for w = ws + wu ; where w = Pxk w ; the decomposition of we = D(xk )w : we = wbss + wbsu + wbus + wbuu ; where wb = Pxk+1 D(xk )w ; ;  2 fs; ug satis es jwbssj  0jwsj ; jwbsuj   jwu j : (3.6) jwbusj   jwsj ; jwbuu j  10 jwu j First, by Lemma 2.1 the decomposition of we at (xk ) :

we = wess + wesu + weus + weuu ; where we = P(xk ) D(xk )w ; ;  2 fs; ug satis es

jwessj  jwsj ; jwesuj  2 jwu j : jweusj  2 jws j ; jweuu j  1 jwu j Then (3.6) follows from (3.5), since jxk+1 ? (xk )j   < 2 :

Now since (DF (0)v)k+1 = D(xk )vk we get, according to the decomposition of lp, that  ss su  DF (0) = AAus AAuu ; where kAss k  0 ; k(Auu )?1k  0 ; kAsu k ; kAus k   :

Hence DF (0) ? I is invertible and (I ? DF (0))?1  R : Now we solve the equation F (v) = v ; which is equivalent to (3.3). Set

G(v) = (I ? DF (0))?1(F (v) ? DF (0)v) : 7

Then F (v) = v is clearly equivalent to v = G(v) : De ne H (v) = F (v) ? DF (0)v ; i.e., H (v)k = [(xk?1) ? xk ] + k?1 (vk?1) : Then

kH (0)kp =

(3.7)

It follows from the properties of

X

k2Z

jd( )kj p

1=p

:

that, for v1; v2 2 Bp (1) ;

kH (v1 ) ? H (v2)kp  (max(kv1 k1 ; kv2 k1 )) kv1 ? v2 kp : It follows from (3.7) and (3.8) that kG(0)kp  R and kG(v1 ) ? G(v2 )kp  R (max kvi k1 )kv1 ? v2kp : i=1;2 (3.8)

Take 3 2 (0; 1) such that R (3 ) < 12 : Since kvk1  kvkp we have

kG(v1) ? G(v2)kp  21 kv1 ? v2 kp

for v1 ; v2 2 Bp(3) : That is G contracts on the ball Bp (3) : Set 0 = min(2; 2R3 ) ; L = 2R :

Then if  < 0 ; we have for v 2 Bp(L) :

kG(v)kp  kG(0)kp + kG(v) ? G(0)kp  R + 12 2R = L :

Hence G is a contraction of Bp (L) and G has a unique xed point v = fvk gk2Z with kv kp  L : It follows from our construction that fyk gk2Z = fxk + vk gk2Z is a trajectory, i.e., yk = k (y0) and

X

k2Z

  X  p1=p jxk ? k (y0)jp 1=p = jvk j  L : k2Z

This completes the proof. Remark. If  = fxk gk0 is a sequence such that kd+( )kp   with  < 0=2 ; then assuming also that d(x0; ) <  we can take xe0 2  such that jxe0 ? x0j <  and extend  to e = fxk gk2Z by de ning xk = k (xe0) for k < 0 : Then e satis es kd(e)kp  2 ; and applying Theorem 3.1 we can nd  = fyk gk2Z { a trajectory of  { such that k ? ekp  2L ; which implies k
+(y0;  )kp  2L : Remark. In [4] Easton studied the following property of a dynamical system  on a metric space (X; d) : given " > 0 there exists a  > 0 such that if a sequence of points fxk gk2Z satis es P d(x ; (x )) <  k k k+1 then there exists a point x such that

P d(x ; k (x)) < " : k k 8

This was called the strong shadowing property. In case p = 1 Theorem 3.1 shows that a dieomorphism has the strong shadowing property in a neighbourhood of a hyperbolic set. If we take p = 1 the shadowing can be asked in two ways: either considering bounded sequences or sequences with zero limits, both with the standard supnorm topology. The rst case is the basic shadowing result (Lemma 2.2). For the latter we get

Proposition 3.2 There exists a neighbourhood U of  and L; 0 > 0 such that if for a sequence  = fxk gk2Z  U we have kd( )k1 < 0 and limjkj!1 d( )k = 0 ; then there exists a unique x such that k
(x;  )k1  Lkd( )k1 and limjkj!1 jxk ? k (x)j = 0 : Proof. In the proof of previous theorem take, in the place of lp ; the Banach space c0 of sequences fuk gk2Z  IRn such that limjkj!1 juk j = 0 with kuk = maxk juk j : Otherwise the proof is similar. Next we will consider weighted analogues of inequalities (1.1) and (1.2) (3.9)

X

(3.10) with some  > 1 :

k2Z

jkj jxk+1 ? (xk )j p

X k2Z

jkj jk (x) ? xk jp

1=p

1=p

;

 L

Theorem 3.3 Assume that

(3.11) 1=p < 1 : Then there exists a neighbourhood U of  and numbers L ; 0 > 0 such that if a sequence satis es (3.9) with  < 0 ; then there exists a unique x such that (3.10) holds.

Proof. Take a neighbourhood U0 of  and an adapted metric such that (3.11) still holds with the corresponding slightly increased  : Take  >  and 2 (1; 2) such that

0 = 1=p  < 1 : For this 0 take the corresponding ; R : (See the beginning of the proof of Theorem 3.1.) Now the proof goes parallel to that of Theorem 3.1. The only dierence is in obtaining a bound for kI ? DF (0)k ; where F is de ned as before, but now in the Banach space of sequences with norm  X jkj p 1=p kvk;p =  jvj : k2Z

9

Instead of inequalities (3.6) we now take small enough 2 > 0 so that for we = D(xk )w we get jwbssj  jwsj ; jwbsuj  1=p jwu j (3.12) jwbusj  1=p jwsj ; jwbuu j  1 jwu j

o

n

Then for a sequence ve = D(xk?1)v sk?1 k2Z ; where vek = ve sk + ve uk with v sk ; ve sk 2 Exsk ; ve uk 2 Exuk we get X X X jkj s p X jkj  jve k j   ( jv sk?1j)p = ( )p jk+1j jv sk jp  ( )p jkj jv sk jp ; i.e.,

k2Z kve sk

;p  

k2Z

k2Z

k2Z

1=p  kv s k

;p

: Hence for

 Ass Asu  DF (0) = Aus Auu

we get kAss k  1=p  = 0 : Inequalities k(Auu )?1k  0 ; kAsu k ; kAus k   are obtained similarly. The rest of the proof is the same as in Theorem 3.1. Clearly a result for one{sided sequences similar to that of the remark after Theorem 3.1 is valid also here. Now we show that under certain conditions on the weights we get limit weighted shadowing without assuming the invariant set to be hyperbolic. First we study one-sided sequences and "strong enough" weights.. For an increasing sequence r = frk gk0 of positive weights and p  1 denote lr;p the Banach space of sequences with norm

kvkr;p =

X

k0

rk jvk jp

1=p

:

Theorem 3.4 Let  be a C 1 {dieomorphism of IRn and U  IRn bounded. Assume the weight sequence satis es rk+1  rk for k  k0 ; where (3.13) 1=p > M = max ( 1 ; sup kD(x)?1k ) : x2U

Then there exist L; 0 > 0 such that if  = fxk gk0  U satis es kd+( )kr;p < 0 ; then there exists a unique x such that

(3.14)

+ 


(x ;  ) r;p  L d+( ) r;p :

Proof. First note that taking c = 0min ?j rj > 0 we have rk  ck : As in the proof j k0 of previous theorems we look for a sequence fvk gk0 2 lr;p satifying (xk + vk ) = xk+1 + vk+1 ; k  0 ; 10

i.e., F (v) = Sv ; where F (v)k = (xk + vk ) ? xk+1 and S is the left shift:

S (v0 ; v1 ; : : :) = (v1; v2; : : :) : Let us rst show that S ? DF (0) has a bounded right inverse in lr;p : For this we will need the following: if faj gj 0 is a nonnegative sequence, and p;  > 1 ; then (3.15)

X j 0

aj  C (; p)

X j 0

1=p j apj ;

Indeed, by Holder's inequality:

X

j 0

aj =

X

j 0

j=paj ?j=p 

where C (; p) =

X j 0

j apj

1=p X j 0

!1? p1

1

 p?1 1

 p?1 ? 1

(?j=p) p?1 p

:

 p?p 1

from which (3.15) follows. For p = 1 the inequality (3.15) holds trivially with C (; 1) = 1 Equation Sv ? DF (0)v = w is equivalent to

vk+1 = D(xk )vk + wk ; k  0 :

(3.16)

Set Dk;k = I and Dj +1;k = D(xj )D(xj ?1) : : : D(xk ) : We claim that the sequence fvk gk0 de ned by 1 X (3.17) vk = ? Dj?+11 ;k wj ; k  0 j =k

satis es (3.16). To show the convergence of (3.17) take e = (M p + )=2 and  = e=M p : Then by (3.13) and (3.15) 1 1 X X jDj?+11 ;k wj j  M j +1?kjwj j = M M j jwk+j j j =0 j =k j =k 1 p 1=p  X  e j  j M jwk+j j  MC (; p) Mp j =0 1 1=p X ej ?k jwj jp : = MC (; p)

jvk j 

(3.18)

1 X

j =k

Since e <  and j ?k  rcj ; the last line of (3.18) is at most Mc C (; p) kwkr;p : Hence the sums of (3.17) converge. Further,

vk+1 = ?

1 X j =k+1

Dj?+11 ;k+1wj = ?Dk+1;k

= ?D(xk )

1 X j =k

1 X

(Dj +1;k+1Dk+1;k )?1wj

j =k+1

Dj?+11 ;k wj + wk = D(xk )vk + wk ;

11

i.e., v satis es (3.16). A bound for the norm of v comes using (3.18): 1 X k=0

rk jvk jp  M pC (; p)p = M pC (; p)p

1 1 X X k=0

rk

j =k

ej ?k jwj jp

k ej ?k r jw jp j j j =0 k=0 rj j r 1 X X

:

Here we have: j r kX 0 ?1 r X k k ej ?k 0 j ? k j ? k e e    + r r r k=0 j k=0 j k=k0 j j ej ?k kX 0 ?1 ej ?k X ek0 +  = C :  +  0 e ? 1  ? e j ?k0 j ?k j r X k

k=0

k=k0

Hence Sv ? DF (0)v = w has a solution v depending linearly on w such that kvkr;p  MC (; p)C01=p kwkr;p ; i.e., S ? DF (0) has a bounded right inverse. It follows that S ? DF (0) is invertible, provided it is one{to{one. So, assume v is such that Sv ?DF (0)v = 0 and vm 6= 0 for some m : Then for k > m we have

vk = D(xk?1)vk?1 ; and so that v is not in is equivalent to

1 X

jvk j  M ?1 jvk?1j  M m?k jvm j ;

rk jvk jp  c jvm jp

k=m lr;p : Hence

M mp

1 X k=m

(=M p)k = 1

S ? DF (0) has a bounded inverse in lr;p : Now F (v) = Sv

v = (S ? DF (0))?1(F (v) ? DF (0)v) : Solving this goes as in the proof of Theorem 3.1.

Re ned result for positive pseudo{orbits

We saw that weighted lp {shadowing of pseudo{orbits can be obtained for slowly increasing weights in the hyperbolic situation (Thm 3.3) or generally for strong enought weights for positive pseudo{orbits. Then the question arises: which (geometric) speeds of decreasing one{step errors can be seen in the shadowing errors? Here we need the concept of Sacker{Sell spectrum. Let  be a compact invariant set of a dieomorphism  : IRn ! IRn of class C 1 . Sacker and Sell [10] studied the so-called spectrum (; ) and the resolvent set R(; ) = IR n (; ) of the pair ;  . Let us formulate one of their main results. We say that the set  is invariantly connected if it cannot be represented as the union of two disjoint nonempty compact invariant sets.

Proposition 3.5 ([10]). Assume that  is invariantly connected. Then 12

a) the spectrum (; ) is the union of k  n compact intervals: (; ) = [a1 ; b1] [ : : : [ [ak ; bk ] b) for any  2 R(; ) there exist constants C > 0 ;  2 (0; 1) and a continuous family of linear subspaces Ex? ; Ex+  IRn ; x 2  , such that (b.1) Ex?  Ex+ = IRn (b.2) D(x)Ex = E(x) k Dk (x)v  Ck jvj ; v 2 Ex?; k  0 (b.3) ?k D?k (x)v  Ck jvj ; v 2 Ex+ ; k  0

Remark 3.1 In [10] the spectrum is de ned in such a way that analogues of (b.3) hold k for exp(?k) D (x)v , hence (; ) and the original Sacker { Sell spectrum are related by the transformation  7! exp(?). Of course, this does not change the geometry of the spectra.

Remark 3.2 Relations between (; ) and the Mather spectrum [7] were studied in [6]. For   1 ; p  1 we work again with the Banach space l;p of positive sequences with the norm  X k p 1=p kvk;p =  jvk j : k0

Theorem 3.6 Assume that  is invariantly connected and that for some ; p  1 we have e := 1=p 2 R(; ) : Then there exists a neighbourhood W of  and numbers 0 ; L > 0 such that if a sequence of points  = fxk gk0  W satis es

d+( )

  ; 0 ;p

then there is a point x such that


+(x;  )

 L

d+( )

: (3.19) ;p ;p

In the proof we use the following two lemmas:

Lemma 3.1 Assume that e 2 R(; ) ;  2 (0; 1) ; and C are as in Prop. 3.5. Then, for given " > 0; 0 2 (; 1) there exists, in a neighbourhood W of  ; a Lyapunov norm j
jx and a (not necessarily D {invariant) extension of families Ex? ; Ex+ to W ; and positive numbers M ; 1 such that for x; x0 2 W ; d(x0; (x)) < 1 ; we have (3.20) e jPx?0 D(x)vjx0  (1 + ")0jvjx ; e jPx+0 D(x)vjx0  "jvjx ; v 2 Ex? (3.21) e jPx+0 D(x)vjx0   (11+ ") jvjx ; e jPx?0 D(x)vjx0  "jvjx ; v 2 Ex+ ; 0 where Px? is the projection onto Ex? along Ex+ and Px+ = I ? Px? ; x 2 W : Further, n 1 (3.22) M jvjx  jvj  M jvjx ; x 2 W ; v 2 IR : 13

Proof. We construct the Lyapunov norm in the usual way as follows. For x 2  ; v = v? + v+ 2 Ex?  Ex+ set  1=2 jvjx = jv? j2x + jv+ j2x ;

where

jv? jx =

N X j =0

ej ?0 j jDj (x)v?j ; jv+ jx =

N X j =0

e?j ?0 j jD?j (x)v+j ;

and N is such that C (=0 )N +1 < 1 : Then for v? 2 Ex? we get:

N X ej +1?0 j jDj ((x))D(x)v?j j =0 1 0N X = 0 @ ej ?0 j jDj (x)v?j + eN +1?0 N ?1 jDN +1(x)v?jA 0jN=1 1 X  0 @ ej ?0 j jDj (x)v?j + ?0 N ?1 C N +1jv? jA  0jv?jx :

e jD(x)v?j(x) =

j =1

Similarly, for v+ 2 Ex+ we get:

X e jD(x) v+ j(x) = e?j +1?0 j jD?j ((x))D(x)v+j N

1 0N ?j1=0 X 1 =  @ e?j ?0 j jD?j (x)v?j + e0 jD(x)v+jA 0 j =0 1 0N ?1 X e 1     @ e?j ?0 j jD?j (x)v?j + CN +10eN +1 jD?N (x)v+jA  1 jv+ jx : 0 j =0 0 These estimates allow us to to extend j
jx to a neighbourhood of  so that (3.20) and (3.21) hold. For details see: [5] and the proof of Thm. 3.1.

Lemma 3.2 Let A be a linear map between Banach spaces V0 and V1 ; which has a right inverse Ay : V1 ! V0 (i.e., AAy = I on V1 ). If B : V0 ! V1 is linear such that kA ? B k kAy k < 1 ; then B is right invertible with y kB yk  1 ? kAk?A Bkk kAy k : Proof. Since k(A ? B )Ayk < 1 ; the series  2 Be = Ay + Ay(A ? B )Ay + Ay (A ? B )Ay + : : : converges and   B Be = (A + (B ? A)) Ay + Ay(A ? B )Ay + Ay(A ? B )Ay(A ? B )Ay + : : : = I + (A ? B )Ay + (A ? B )Ay(A ? B )Ay + : : : +(B ? A)Ay + (B ? A)Ay(A ? B )Ay + (B ? A)Ay(A ? B )Ay(A ? B )Ay + : : : = I ; 14

i.e., B y = Be is a right inverse of B :

Proof. (of Thm. 3.6). Take " > 0 such that  := (1 + ")0 < 1 and 1 ? (1"+ ") < 1 : 0 As before, given a pseudotrajectory   W with kd( )k;p  1 ; we can write the equation for a shadowing trajectory  + v as vk+1 = (xk + vk ) ? xk+1 ; k = 0; 1; 2; : : : ; i.e., Sv = F (v) ; where S; F : l;p ! l;p (Sv)k = vk+1 ; F (v)k = (xk + vk ) ? xk+1 : First we show that S ? DF (0) has a right inverse, which is uniformly bounded for   W satisfying kd( )k;p  1 :

Consider S and F as maps E ! Ee ; where E and Ee are the Banach spaces of sequences fvk gk0 with norms

k v kp =

1  1    X X k jPx?k vk jpxk + jPx+k vk jpxk ; kvkp = k+1 jPx?k+1 vk jpxk+1 + jPx+k+1 vk jpxk+1 ;

k=0

k=0

respectively. By (3.22) these norms are equivalent to that of l;p ; i.e., there exists C > 0 independent of  such that 1 (3.23) C kvk  kvk;p  C kvk : Denote E = E ?  E + ; Ee = Ee?  Ee+ ; where

n o n o E  = v 2 E j vk 2 Exk ; k  0 ; Ee = v 2 Ee j vk 2 Exk+1 ; k  0 ;

Split S and DF (0) accordingly:  0  ; DF (0) =  A?? A?+  : S = S?? 0 S++ A+? A++

y : Ee? ! E ? S++ has norm 1 and S?? has a right inverse S??  0 for k = 0 y (S??u)k = u ; k?1 for k  1 which also has norm 1 : By (3.20) we get for v 2 E ?

kA?? vkp =

1 1 X X k+1 jPx?k+1 D(xk )vk jpxk+1  (1 + ")pp0 k jvk jpxk = (1 + ")pp0 kvkp ; k=0

k=0

i.e., kA?? k   ; and

kA+? vkp =

1 1 X X k+1 jPx+k+1 D(xk )vk jpxk+1  "p k jvk jpxk = "p kvkp ; k=0

k=0

15

y k = 1 ; we get by Lemma 3.2 that implying kA+? k  " : Since kA?? k < 1 ; and kS?? S?? ? A?? has a right inverse Ee? ! E ? with bound y k kS??  1 ?1  : k(S?? ? A?? )yk  (3.24) y 1 ? kA?? k kS?? k

Similarly, for v 2 E + we get

kA++ k  1= and kA?+ k  " : It is easy to see that A++ maps E + onto Ee+ so that A++ is invertible with kA?++1 k   : Again by Lemma 3.2 S++ ? A++ has a right inverse Ee+ ! E + with bound (3.25) k(S++ ? A++ )yk  1 ?  :   By (3.24) and (3.25) S ? A0?? A0 has right inverse which has norm bound 1=(1 ? ) : ++  0 A  ? + Further, A 0 has norm at most " : So, by Lemma 3.2, S ? DF (0) has a right +? inverse Ee ! E with norm (1 ? ) = 1 k(S ? DF (0))yk  1 ?1="= (3.26) (1 ? ) 1 ?  ? " :

Hence by (3.23) it is right invertible from l;p to l;p with norm 2 k(S ? DF (0))yk;p  R = 1 ?C ? " : This bound does not depend on  ; provided kd( )k;p  1 : Consider equation v = G(v) ; where G(v) = (S ? DF (0))y(F (v) ? DF (0)v) : By multiplication with S ? DF (0) we see that a solution of this also satis es Sv = F (v) : As in the proof of Thm 3.1 we can take 0 2 (0; 1) such that G has Lipschitz constant < 12 in the ball kvk;p < 2R0 uniformly in  satisfying kd( )k;p  1 : If now  = kd( )k;p  0 ; then G contracts the ball kvk;p < 2R ; i.e., equation v = G(v) ; has a solution there. Remark. Note that, in contrast to Thm. 3.4, the point x is not necessarily unique. It is not dicult to show that x is unique if (; ) \ [e; 1) = ; (in this case Ex+ = Rn for x 2 ).

4 Example: limit shadowing of periodic ows Consider a system of dierential equations (4.1) x_ (t) = f (t; x(t)) ; 16

x(t) 2 IRn :

Assume that f and @x f are continuous and bounded and that f is 1{periodic in t : Denote by x(t; t0; x0) the solution of (4.1) with initial value x(t0) = x0 and let  be the Poincare map of (4.1), that is,

(x0) = x(1; 0; x0) : Our assumptions on f imply that  is a dieomorphism of class C 1 : Assume that  has a hyperbolic attractor  : We consider two types of approximations of  : using Picard-Lindelof {iterations and one{step discretizations.

Approximation with Picard-Lindelof {iterations Denote by T the Picard operator on C [0; 1] :

Tx(t) = x0 +

Zt 0

f (; x( )) d

and set j (x0) = (T j xe0)(1) ; where xe0(t) = x0 ; t 2 [0; 1] : The following is a standard result: j +1 j j (x0) ? (x0)j  C (M (4.2) j + 1)! ; where M = supt;x jfx (t; x)j and C = eMM supt;x jf (t; x)j : Let xk+1 = jk (xk ) ; k  0 with x0 close to  ; j0 big enough, fjk gk0 increasing, and limk!1 jk = 1 : Then by Theorem 2.1 there exists x such that limk!1 k (x) ? xk = 0 : In order to get lp {shadowing from Theorem 3.1 , one needs

!

M jk p < 1 ; k=0 (jk + 1)! for which one easily obtains from the Stirling formula that growth jk  j0 + p1 ln(k) is enough. For weights rk = k (see Theorem 3.3) we can set jk  j0 + k ln(p ) : On the other hand, if jk 's increase for example like jk = j0 + k ; then using Theorem 3.4 with weights  (j + 1)! p k rk = k1+ "M jk +1 1 X

we obtain that there exists a trajectory, which is approached with essentially the speed of decrease of the approximation errors.

Approximation with one{step discretizations

Assuming f is smooth enough, any standard one{step discretization (e.g. a Runge{Kutta method) gives a map h (t0; x0) = ( t0 + h ; x(t0 + h; t0; x0) + O(hq+1) ) ; where q  1 is the order of the method. 17

Take an increasing sequence of natural numbers fnk gk0 ; x0 close to  ; set hk = 1=nk ; and de ne a sequence fxk gk0 by nhkk (0; xk ) = (1; xk+1) :

Then j(xk ) ? xk+1j  Cn?k q : Again, if limk!1 n k = 1 and n0 is big enough, then by Theorem 2.1 there exists x such that limk!1 k (x) ? xk = 0 : For lp {shadowing from Theorem 3.1 , it suces to have nk  n0 + k=pq with some  > 1 : For weights rk = k (see Theorem 3.3) we can set nk  n0 + k=pq k=pq :

References [1] E. Akin. The General Topology of Dynamical Systems. AMS, Grad. Stud. in Math., vol.1, 1993. [2] D. V. Anosov. Ob odnom klasse invariantnyh mnoestv gladkih dinamiqeskih sistem (On a class of invariant sets of smooth dynamical systems). Proc. 5th Int. Conf. on Nonl. Oscill., Kiev, 2:39{45, 1970. [3] R. Bowen. Equilibrium states and the ergodic theory of Anosov dieomorphisms. Springer{Verlag, Berlin{Heidelberg, 1975. LN in Math. 470. [4] R. Easton. Chain transitivity and the domain of in uence of an invariant set. In: The structure of attractors in dynamical systems, North Dakota State Univ, June 1977, Springer LN in Math. 668, pages 95{102, 1978. [5] A. Katok. Local properties of hyperbolic sets. Appendix to the Russian translation of: Z. Nitecki, Dierentiable Dynamics, 1971. [6] G. Kronin. Perturbations of Mather and Sacker{Sell spectra (to appear). [7] Z. Nitecki. Dierentiable Dynamics. MIT Press, Boston, 1971. [8] S. Yu. Pilyugin. The space of dynamical systems with the C 0 {topology. Springer{ Verlag, Berlin{Heidelberg, 1994. LN in Math. 1571. [9] V. A. Pliss. Integral~nye mnoestva periodiqeskih sistem differencial~nyh uravneni$i (Integral sets of periodic systems of dierential equations). Nauka, Moscow, 1977. [10] R. J. Sacker and G. R. Sell. A spectral theory for linear dierential systems. J. Di. Equations, 27:320{358, 1978. [11] M. Shub. Global stability of dynamical systems. Springer-Verlag, New York, 1987.

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