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Ck Invariant Manifolds for Maps of Banach Spaces
Journal of Mathematical Analysis and Applications 268, 1᎐24 Ž2002. doi:10.1006rjmaa.2001.7706, available online at http:rrwww.idealibrary.com on
C k Invariant Manifolds for Maps of Banach Spaces Mohamed Sami ElBialy Mathematics Department, Uni¨ ersity of Toledo, Toledo, Ohio 43606 E-mail: [email protected] Submitted by William A. Kirk Received October 2, 2000
In this work we give a unified proof for the existence, under appropriate gap conditions, of the standard invariant manifolds for a C k map of a Banach space near a fixed point where k G 1 is an integer. 䊚 2002 Elsevier Science ŽUSA . Key Words: invariant manifolds; linearization.
1. INTRODUCTION In w5x we proved a fixed-point theorem for a class of C k, ␦ , 0 - ␦ F 1, self-maps of a Banach space of functions. In this work we consider the case ␦ s 0. This theorem leads to a unified proof for the existence, under appropriate gap conditions, of the standard invariant manifolds Žpseudostable, pseudo-unstable, center-stable, center-unstable, weak-stable, weakunstable, strong-stable, strong-unstable, stable, and unstable . for C k, ␦ , 0 - ␦ F 1, maps of a Banach space. The case ␦ s 0 needs to be considered separately because the standard fiber contraction theorem w16x does not apply when ␦ s 0. We still have a fiber contraction, and we have a candidate for the kth derivative, but we cannot conclude that it is a unique attractive fixed point because, when ␦ s 0, the fiber contraction is not necessarily continuous in the base points, a condition needed for applying the fiber contraction theorem. As a result, if we use the fiber contraction theorem we can only show that these invariant manifolds are C ky 1, 1. The only places in the literature known to the author where the possible loss of differentiability of an invariant manifold when the map is C k , k g ⺞, is studied are w3x Žcenter-stable manifold., w6x Žsub-stable and weak-stable manifolds., w11x Žcenter manifold., and w22x Žpseudo-stable 1 0022-247Xr02 $35.00 䊚 2002 Elsevier Science ŽUSA . All rights reserved.
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MOHAMED SAMI ELBIALY
manifold.. It is obvious that these works do not cover all of the standard invariant manifolds when ␦ s 0. In this work we complete the case ␦ s 0 by giving a unified proof for the existence of all of the standard invariant manifolds, for the case ␦ s 0. Since in this article we use the results of w5x, we keep the style, notation, and definitions of w5x. For the sake of completeness we include all definitions and notation. Moreover, w5, Ž5.2.x will mean Eq. Ž5.2. in w5x. Also, w5, Lemma 3.5x will mean Lemma 3.5 of w5x. In Section 2 we give some definitions and state our main theorem. In Section 3 we give all of the definitions, notation, and setup used in w5x and indicate which conclusions in w5x are still valid when ␦ s 0. We also reduce the main theorem to Proposition 3.3, which we prove in Section 4. The idea of the proof is as follows: By applying the results in w5x we know that we have a C ky 1, 1 map y s w Ž x . which is an attractive fixed point for a transformation ⌫ defined on the appropriate function space. We also know that ⌬ky 1 w [ Ž w, Dw, . . . , D ky 1 w . is an attractive fixed point for the Ž k y 1.st lift of ⌫, which we denote by ⌫ky 1. Moreover, we know that the kth lift ⌫k is a fiber contraction which is not necessarily continuous in the base points. Thus, we can apply the standard contraction theorem to the invariant fiber above ⌬ky 1 w and obtain a fixed point, say . This is our candidate for D k w. We cannot use the fiber contraction theorem to conclude that D k w s because ⌫k is not necessarily continuous in the base points. To show that D k w s , it suffices to show that given x and h,
Ž ).
D ky 1 w Ž x q h . y D ky1 w Ž x . s
1
H0
Ž x q th . h dt.
Now let u o be any C k function Žsuch that ⌬k u o is in the function space defined below. and let u n s ⌫u ny1 , n G 1. Now, 5 ⌬ky 1 u n y ⌬ky1 w 5 ª 0 and
Ž 噛.
D ky 1 u n Ž x q h . y D ky1 u n Ž x . s
1
H0
D k u n Ž x q th . h dt.
The left-hand side of Ž噛. converges uniformly Ževerywhere. to the left-hand side of ŽU .. Let I s x q th ¬ 0 F h F 14 . To show that the right-hand side of Ž噛. converges uniformly on I to the right-hand side of ŽU . we restrict the fiber contraction ⌫ to a set W , which consists of the line segment I and its images under D i u n , 1 F i F k y 1, n G 0, and a few other sets given explicitly in Lemma 4.4, where we show that W is in fact compact. We all know that a continuous function on a compact set is uniformly continuous. We use this fact to show that the restriction of ⌫k to W is continuous in the restrictions of the base points. Then we can apply the fiber
C k INVARIANT MANIFOLDS
3
contraction theorem to the restricted ⌫k and conclude that D k u n < I ª < I as n ª ⬁. 1.1. In¨ ariant Manifolds for Maps. Consider a linear continuous map T : E ª E where E is a Banach space. The linear map is said to be -hyperbolic if its spectrum [ ŽT . lies off a circle of radius in the complex plane ⺓. If s 1, T is said to be hyperbolic. In either case we write the spectrum of T as s s j u , where s lies inside the circle of radius and u lies outside the same circle. Consider a C k, ␦ , 0 F ␦ F 1, map ⌽ s T q F, where F Ž0. s 0 and DF Ž0. s 0. When we apply our main Theorem 2.4 to a map of a Banach space, we obtain Theorem 5.2. In w5x we showed how to obtain Theorem 5.2 from our main Theorem 2.4. Following the referee’s advice, we include most of the details in Section 5 for completeness. We group the standard invariant manifolds of a -hyperbolic fixed point into two groups, PU and PS manifolds. PU manifolds are those invariant manifolds that correspond to u , the part of the spectrum that lies outside the circle of radius . The PU manifolds are the pseudo-unstable, centerunstable, unstable, and weak-stable manifolds. The last one is for contracting maps. PS manifolds are those invariant manifolds that correspond to s , the part of the spectrum that lies inside the circle of radius . The PS manifolds are the pseudo-stable, center-stable, stable, strong-stable, and weak-unstable manifolds. The last one is considered for expanding maps. If the map is invertible, PU manifolds of the map are the PS manifolds for the inverse of the map. For a noninvertible map, estimates needed for PU manifolds are more involved than those needed for PS manifolds. The reason for this is that for PU manifolds we need to use the Lipschitz inverse function theorem Žw29, p. 49x and w5, p. 236x.. We will elaborate more on this issue in Sections 5.3 and 5.5. Apart from that, PU and PS manifolds are very similar, as we can see from considering an invertible map. There is another grouping of invariant manifolds which is more serious than the PU-PS grouping, namely, what we call S-manifolds and W-manifolds. This grouping has much more profound roots, which we discuss in the following subsection. 1.2. S-manifolds, W-manifolds, Local In¨ ariant Manifolds, Uniqueness, and Smooth Cutoff Functions. There are no major distinctions between PU and PS manifolds when it comes to hypotheses, existence, and uniqueness. There is another grouping of invariant manifolds which reflects some fundamental differences regarding smallness assumptions Žin infinite-dimensional Banach spaces. and uniqueness. Recall that for maps of ⺢ n, local weak-stable manifolds are not unique because we need to use smooth cutoff functions to obtain the local
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MOHAMED SAMI ELBIALY
smallness assumptions in the form of global smallness assumptions. This is the case for weak-stable, weak-unstable, pseudo-stable, pseudo-unstable, center-stable, and center-unstable manifolds. We call these manifolds W-manifolds. We do not need to use smooth cutoff functions with what we call S-manifolds: stable, unstable, strong-stable, and strong-unstable manifolds. As a result, local S-manifolds are unique. In an infinite-dimensional Banach space, W-manifolds are totally out of luck because smooth cutoff functions are scarce. We elaborate on this issue in Section 5.6. The only places in the literature on invariant manifolds known to the author where the scarcity of smooth cutoff functions is addressed are w5᎐7, 20, 21x. It follows that for W-manifolds of maps of infinite-dimensional Banach spaces, the smallness assumptions are very restrictive because they truly have to be satisfied globally due to the scarcity of smooth cutoff functions. Thus, we can obtain only global W-manifolds when the global smallness assumptions are satisfied. These global W-manifolds are unique because we do not use cutoff functions.
2. MAIN RESULT We start by recalling some definitions from w5x and restating Theorem C of w5x, with ‘‘0 - ␦ F 1’’ replaced by ‘‘0 F ␦ F 1.’’ DEFINITION 2.1. 1. Let X and Y be Banach spaces. Let Z s X [ Y be endowed with the box norm. For a linear operator L, 5 L 5 will denote the operator norm of L. 2. Let X o ; X and q s 0, 1. We will consider two cases: ŽW.: X o s X and q s 0. ŽS.: X o s B1Ž0. and q s 1, where B1Ž0. ; X is the closed unit ball centered at the origin. 3. Let C 0, 1 Ž X o , Y . the space of uniformly Lipschitz continuous functions. Let 5 u 5# s sup x/0
uŽ x . < x
,
u g C 0, 1 Ž X o , Y . .
Notice that when q s 0, the norm 5 ⭈ 5# coincides with the standard sup-norm. 4. Let id X o: X o ª X o be given by id X oŽ x . s x. Consider a map Q: C 0, 1 Ž X o , Y . ª C 0, 1 Ž X o , X o . , u ¬ Qu .
5
C k INVARIANT MANIFOLDS
If Q u s id X o for all u g C 0, 1 Ž X o , Y . we say that Q is of type Ža. and write Q ' id X o. 5. Let T : X ª X be linear and continuous and let h g C 0, 1 Ž Z, X .. Suppose that for all u g C 0, 1 Ž X o , Y . we can solve the implicit relation
s Tx q h Ž , u Ž . . uniquely in in terms of x g X o . Thus, for all u g C 0, 1 Ž X o , Y ., we have a function Qu : Xo ª Xo , x ¬ s Qu Ž x . Q u s T q h( Ž id, u . ( Q u . Assume that for all u g C 0, 1 Ž X o , Y ., Q u g C 0, 1 Ž X o , X o .. Thus we have a map Q: C 0, 1 Ž X o , Y . ª C 0, 1 Ž X o , X o . , u ¬ Qu . In this case we say that Q is of type bŽT, h.. The following lemma is straightforward. LEMMA 2.2. Let Q: C 0, 1 Ž X o , Y . ª C 0, 1 Ž X o , X o . be of type bŽT, h.. Assume that LipŽ h. - 1 and hŽ0, 0. s 0. Let u g C 0, 1 Ž X o , Y . be such that uŽ0. s 0 and LipŽ u. F 1. Then LipŽŽ id, u.. s 1, Q uŽ0. s 0, and Lip Ž Q u . F
5T 5 1 y Lip Ž h .
.
DEFINITION 2.3. Let E1 and E2 be Banach spaces and let X o ; E1. For a function h g C k Ž X o , E2 ., define the following for 0 F j F k and 0 - ␥ F 1: 5 D j h 5 [ sup D j h Ž x . ¬ x g X o 4
 j, ␥ Ž h . [ ␥ Ž D h . [ sup j
½
D j h Ž x . y D j h Ž xX . < x y xX < ␥
x, xX g X o , x / xX ,
⌬ j h Ž x . [ Ž h Ž x . , Dh Ž x . , . . . , D j h Ž x . . . If max5 D⬘h 5, . . . , 5 D k h 5,  k, ␦ Ž h.4 - ⬁, we say that h g C k, ␦ Ž X o , E2 .. Notice that for 0 F j F k y 1,  j, 1Ž h. F 5 D jq1 h 5.
5
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MOHAMED SAMI ELBIALY
In this work we prove the following theorem: THEOREM 2.4. Let X, Y, Z, and X o be as in Definition 2.1. Let k s 1, 2, . . . and 0 F ␦ F 1. Assume the following: ŽH1.: Let M: Y ª Y; B1 , B2 , C: X ª X be continuous linear maps such that 5 M 5 5 C 5 - 1 and assume that either ŽW . or Ž S . holds: ŽW.: 5 M 5 F 1. ŽThus 5 M 5 5 C 5 q - 1 since q s 0.. Ž S .: 5 C 5 - 1. ŽThus 5 M 5 5 C 5 q - 1, since q s 1.. ŽC1.: Let F s Ž f, g, h1 , h 2 ., f, h1 , h 2 g C k, ␦ Ž Z, X ., g g C k, ␦ Ž Z, Y ., F Ž0, 0. s 0, and DF Ž0, 0. s 0. ŽC2.: 5 D j F 5 F ⑀ , j s 0, 1, . . . , k. ŽC3.:  j, 1Ž F . [ LipŽ D j F . - ⑀ , j s 0, 1, . . . , k y 1. If 0 - ␦ F 1, assume that  k, ␦ Ž D j F . - ⑀ . ŽH2.: Let J: C k, ␦ Ž X o , Y . ª C k, ␦ Ž X o , X o . be of type Ž a. or bŽ B1 , h1 .. Let H : C k , ␦ Ž Xo , Y . ª C k , ␦ Ž Xo , Xo . u ¬ Hu [ C q S F u SF u [ f ( Ž id, u . ( Ju . ŽH3.:
Let K : C k, ␦ Ž X o , Y . ª C k, ␦ Ž X o , X o . be of type Ž a. or bŽ B2 , h 2 ..
Let R F : C k , ␦ Ž Xo , Y . ª C k , ␦ Ž Xo , Xo . u ¬ R F u [ g ( Ž id, u . ( K u . For u g C k, ␦ Ž X o , Y . let
Ž 2.1.
⌫F u [ Mu( Hu q R F u.
Then, for sufficiently small ⑀ ) 0, ⌫F has a fixed point w g C k , ␦ Ž Xo , Y . , which is unique in the class of functions that satisfy w Ž 0. s 0 5 w 5# F 1 5 D w 5 F 1, j
 1 Ž D j w . F 1,
0FjFk 0 F j F k y 1.
C k INVARIANT MANIFOLDS
7
If 0 - ␦ F 1, w also satisfies
␦ Ž D k w . F 1. ŽH4.:
Suppose that g Ž x, y . satisfies g Ž x, y . - ⑀ Ž < x < kq ␦ q < y < . .
Ž 2.2.
Then, for sufficiently small ⑀ ) 0,
Ž 2.3.
D j w Ž x . F < x < kq ␦yj ,
0 F j F k.
In w5x we proved Theorem 2.4 for 0 - ␦ F 1. Therefore we know that Theorem 2.4 is true for C ky 1, 1 maps. In this work we consider the case ␦ s 0. In what follows we need the fiber contraction theorem, which is proved in w16, p. 136x. 2.5. THE FIBER CONTRACTION THEOREM. Let ᑜ be a metric space and let : ᑜ ª ᑜ be a map ha¨ ing an attracti¨ e fixed point p Ž i.e., ᭙ u g ᑜ: n Ž u. ª p as n ª ⬁.. Let ᑳ be a metric space and let u ¬ u g ᑜ 4 be a family of maps u : ᑳ ª ᑳ such that the formula ⌫ Ž u, . s Ž Ž u., u Ž .. defines a continuous map ⌫: ᑜ = ᑳ ª ᑜ = ᑳ. Let q g ᑳ be a fixed point for p. Then Ž p, q . is an attracti¨ e fixed point for ⌫ pro¨ ided that
Ž 2.4.
lim sup Lip Ž
n
Ž u.
. - 1,
᭙u g ᑜ.
nª⬁
Notice that we do not assume that the spaces ᑜ and ᑳ are complete because we assume that the map has an attractive fixed point p g ᑜ and the map p has a fixed point q g ᑳ.
3. THE SETUP In this section we give all the definitions, notation, and setup used in w5x which we need for the case ␦ s 0. We also indicate which conclusions in w5x are still valid when ␦ s 0. DEFINITION 3.1. 1. Let 5 u 5# be as in Definition 2.1 and define the following spaces: ᑲ o s u g C 0 Ž X o , Y . ¬ 5 u 5# - ⬁ 4 , ᑳ o s u g ᑲ o Ž X o , Y . ¬ u Ž 0 . s 0, 5 u 5# F 1, Lip Ž u . F 1 4 . Let ⌫o [ ⌫F : ᑳ o ª ᑳ o be given by Ž2.1..
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MOHAMED SAMI ELBIALY
2. For 1 F j F k, let L Ž X j, Y . be the Banach space of symmetric continuous j-multilinear maps from X j to Y with the standard norm 5 A 5 [ sup
½
< A < < <
5
0/gX .
For g C 0 Ž X o , L Ž X j, Y .. and 0 - ␥ F 1, let 5 5 [ sup Ž x . , xgX o
␥ Ž . [ sup
Ž x . y Ž xX . < x y xX < ␥
½
x / xX , x, xX g X o .
5
Notice that  1Ž . s LipŽ .. 3. For 1 F j F k, let ᑲ j s j g C 0 Ž X o , L Ž X j , Y . . 5 j 5 - ⬁ ,
½
5
ᑳ j [ j g ᑲ j ¬ 5 j 5 F 1,  1 Ž j . F 1 4 ,
1FjFky1
ᑳ k [ k g ᑲ k ¬ 5 k 5 F 14 , ⌺ o [ Ž u . s o , ⌺ j [ Ž u, 1 , . . . , j . ,
1 F j F k y 1,
⌺ [ ⌺ ky 1 , 5 ⌺ j 5# [ max 5 u 5#, 5 1 5 , . . . , 5 j 5 4 ,
1 F j F k y 1.
It is obvious that each Ž ᑲ j, 5 ⭈ 5. is a Banach space and that ᑳ j is a closed subset. 4. Let z [ HuŽ x ., [ JuŽ x ., and [ K uŽ x .. 5. Let k s 1 and 1 g ᑳ 1. Let J⌺1 1Ž x . [ Ju1 1Ž x ., where
¡I,
J⌺1 1 Ž x . [
~
¢
J is of type Ž a . ,
I y Dh1 Ž , u Ž . . Ž I, 1 Ž . .
y1
B1 ,
J is of type b Ž B1 , h1 . .
Let K⌺1 1Ž x . [ Ku1 1Ž x ., where
¡I,
K⌺1 1 Ž x . [
~
¢
K is of type Ž a . ,
I y Dh 2 Ž , u Ž . . Ž I, 1 Ž . . K is of type b Ž B2 , h 2 . .
y1
B2 ,
9
C k INVARIANT MANIFOLDS
Let H ⌺1 1 Ž x . [ Hu1 1 Ž x . [ C q Su1 1 Ž x . [ C q Df Ž , u Ž . . Ž I, 1 Ž . . Ju1 1 Ž x . . Let ⌫1 : ᑳ o = ᑳ 1 ª ᑳ o = ᑳ 1 ,
Ž 3.1.
⌫1 Ž u, 1 . [ ⌫⌬ F Ž u, 1 . [ Ž ⌫o u, ⌫1u 1 . , ⌫1u 1 Ž x . [ M 1 Ž z . Hu1 1 Ž x . q Dg Ž , u Ž . . Ž I, 1 Ž . . Ku1 1 Ž x . . 6. Let k G 2 and let k g ᑳ k . Let
¡I,
J⌺k k
J is of type Ž a . ,
Ž x . [~ I y Dh1 Ž , u Ž . . Ž I, 1 Ž . . J is of type b Ž B1 , h1 .
¢
A Ž x . [ Dh1 Ž , u Ž . . Ž 0, k Ž . . J⌺1 1 Ž x .
¡I,
K⌺k k
y1
Ž x . [~
¢
k
AŽ x .
q Q kh1 ; J Ž ⌺, x .
K is of type Ž b . , y1
I y Dh 2 Ž , u Ž . . Ž I, 1 Ž . .
BŽ x .
K is of type b Ž B2 , h 2 .
B Ž x . [ Dh 2 Ž , u Ž . . Ž 0, k Ž . . K⌺1 1 Ž x .
k
q Q kh 2 ; K Ž ⌺, x .
H ⌺k k Ž x . [ Df Ž , u Ž . . Ž 0, k Ž . . J⌺1 Ž x . 1 Ž x .
k
q Df Ž , u Ž . . Ž I, 1 Ž . . J⌺k k Ž x . q Q kf ; J Ž ⌺, x . , where Pk Ž ⌺, x . [ Pk Ž ⌬k f Ž , u Ž . . , ⌺ Ž z . , ⌺ Ž . , H ⌺Ž ky1. Ž x . . , Q kh ; L Ž ⌺, x . [ Q kh ; L Ž ⌬k h Ž , u Ž . . , ⌺ Ž . , N ⌺Ž ky1. Ž x . . ,
[ Lu Ž x . ,
h s f , g , h1 , h 2 ,
L s J, K ,
N ⌺Ž ky1. Ž x . [ Ž N ⌺1 1 Ž x . , . . . , N ⌺ky 1 ky1 Ž x . . , Pk Ž 0, ⭈ , ⭈ , ⭈ . ' 0,
Pk Ž ⭈, 0, ⭈ , ⭈ . ' 0,
Q kh ; L Ž 0, ⭈ , ⭈ . ' 0,
N s H , J, K,
Pk Ž ⭈, ⭈ , ⭈ , 0 . ' 0,
Q kh ; L Ž ⭈, ⭈ , 0 . ' 0.
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MOHAMED SAMI ELBIALY
Moreover, Pk Ž t 1 , t 2 , t 3 , t4 . and Q kh; L Ž t 1 , t 2 , t 3 . are polynomials in the components of their arguments and linear in the components of t 1. 7. For k G 1, let s k and ky1
Bky 1 [
Ł ᑳi
is0
⌫k : ᑜ ky1 = ᑳ k ª ᑜ ky1 = ᑳ k , ⌫k Ž ⌺, . [ ⌫⌬k F Ž ⌺, . [ Ž ⌫ky 1 Ž ⌺ . , ⌫k⌺ . , ⌫k⌺ Ž x . [ M Ž z . H ⌺1 1 Ž x .
k
q M 1 Ž z . H ⌺k Ž x .
q Pk Ž ⌺, x . q Q kg ; K Ž ⌺, x . q Dg Ž , u Ž . . Ž 0, Ž . . K⌺1 1 Ž x .
k
q Dg Ž , u Ž . . Ž I, 1 Ž . . K⌺k Ž x . . 8. Notice that if w is a C k fixed point for ⌫o , then ⌫k Ž ⌬k w . s ⌬k w. 3.2. Information about ⌫k Which Can be Obtained from w5x. Most of the estimates in w5x are valid for the case ␦ s 0. The only estimates that are not valid are the ones that lead to the conclusion that ⌫k⌺ is continuous in ⌺. These are the estimates in w5, Lemma 4.7x. More precisely, we know the following for sufficiently small ⑀ ) 0: 1. Theorem 2.4 is proven in w5x for the case 0 - ␦ F 1. If we apply it, with smoothness C ky 1, 1, to ⌫ky 1: ᑜ ky1 ª ᑜ ky1 , ᑜ ky1 s ᑜ ky2 = ᑳ ky 1 , we conclude that ⌫ky 1 is well defined Ži.e., ⌫ky1 maps ᑜ ky1 to itself. and has a unique attractive fixed point ⌺# s Ž w, 1 , . . . , ky1 . g ᑜ ky1 and j s D j w for j s 1, . . . , k y 1. 2. For all ⌺ g ᑜ ky 1 , the map ⌫k⌺ maps ᑳ k to itself, and hence ⌫k maps ᑜ ky 1 = ᑳ k to itself. This is shown in w5, Lemma 4.4x, where the proof is valid for ␦ s 0. 3. For sufficiently small ⑀ ) 0, there is 0 - s w 5 M 5 5 C 5 k q O Ž ⑀ .x - 1 such that for all ⌺ g ᑜ ky 1 , for all , g ᑳ k ,
Ž 3.2.
5 ⌫k⌺ y ⌫k⌺ 5 F 5 y 5 .
This is shown in w5, Proposition 4.6x, where the proof is valid for ␦ s 0.
C k INVARIANT MANIFOLDS
11
4. It follows that ⌫k : ᑜ ky1 = ᑳ k ª ᑜ ky1 = ᑳ k is well defined and is a fiber contraction which is not necessarily continuous in ⌺. Since ⌺# g ᑜ ky 1 is fixed under ⌫ky1 , the fiber ᑠ# [ ⌺#4 = ᑳ k is invariant under ⌫# [ ⌫ky 1 = ⌫k⌺# . By Ž3.2. and the standard contraction mapping theorem, ⌫# has a unique attractive fixed point Ž ⌺#, . g ᑠ#. 5. However, we do not know that ⌫k⌺ is continuous in ⌺, which is one of the hypotheses of the Fiber Contraction Theorem 2.5. Therefore we cannot conclude that Ž ⌺#, . is a unique attractive fixed point for ⌫k : ᑜ ky 1 = ᑳ k ª ᑜ ky1 = ᑳ k , which is essential in showing that s D k w. 6. In spite of this temporary setback, we have a candidate for D k w, namely, the fixed point of ⌫#. We will show that s D k w directly using Proposition 3.3 below. 7. Before we proceed to show that in fact s D k w, let us see why we can prove that ⌫k⌺ is continuous in ⌺ only when 0 - ␦ F 1, but not when ␦ s 0: When 0 - ␦ F 1, we include the requirement ␦ Ž k . F 1 in the definition of ᑳ k . Then, the map D k F and g ᑳ k are uniformly ␦-Holder continuous. This is precisely what we used in the proof of ¨ Lemma 4.7 of w5x to show that ⌫k⌺ is also uniformly ␦-Holder continuous ¨ in ⌺ when 0 - ␦ F 1. PROPOSITION 3.3. X o , then
Ž 3.3.
If x and h are in X such that both x and x q h are in
D ky 1 w Ž x q h . y D ky1 w Ž x . s
1
H0
Ž x q th . h dt.
It follows that D ky 1 w is continuously differentiable and D k w s .
4. PROOF OF PROPOSITION 3.3 DEFINITION 4.1. Let x and h be as in Proposition 3.3. Notice that x q th g X, 0 F t F 1. Define the following: 1. Let u o g ᑳ o be such that ⌬k u o g ᑜ ky1 = ᑳ k . Let u n [ ⌫o u ny1 , n G 1. Thus, for n G 0, ⌬k u n g ᑜ ky1 = ᑳ k . Let u⬁ [ w. 2. Let Un [ ⌬ky 1 u n and U⬁ [ W [ ⌬ky 1 w. Thus, for n G 0, Un g ᑜ ky 1 , Un s Ž ⌫ky1 . n ŽUo ., and 5 Un y W 5# ª 0 as n ª ⬁. 3. Let : w0, 1x ª X o be given by Ž t . [ x q th. Let I [ Žw0, 1x.. 4. Let n [ Un < I , for n G 0, and ⬁ [ W < I s U⬁ < I .
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MOHAMED SAMI ELBIALY
5. Let ⺦ [ n ¬ 0 F n F ⬁ 4 , ⺩ [ ˆ s < I ¬ g ᑳ k 4 , 5 n 5 o [ 5 Un < I 5#, 5 ˆ 5 1 [ 5 < I 5 s sup gI
ˆ s < I g ⺩.
Ž . ,
6. Let ⌳: ⺦ = ⺩ ª ⺦ = ⺩ be the obvious restriction of ⌫k to ⺦ = ⺩ given by
Ž 4.1.
⌳ Ž j , ˆ . [ ⌳ o j , ⌳ j ˆ [ ⌫k Ž Uj , .
ž
/
I
.
where ⌳ o and ⌳ j are defined below. Let z j s Hu jŽ ., j s Ju jŽ ., and j s K u jŽ .. Thus for g I, ⌳ o j Ž . [ ⌳ o Ž Uj < I . Ž . [ ⌫ky1Uj
I
Ž .,
⌳ j ˆ Ž . [ ⌫kUj < I Ž . s M Ž z j . HU1j Du j Ž .
Ž 4.2.
k
q MDu j Ž z j . HUkj Ž .
q Pk Ž Uj , . q Q kg ; K Ž Uj , . q Dg Ž j , u j Ž j . .Ž 0, Ž j . . KU1j Du j Ž .
k
q Dg Ž j , u Ž j . .Ž I, Du j Ž j . . KUkj Ž . . Remarks 4.2. 1. It is obvious that Ž⺦, 5 ⭈ 5 o . is a complete metric space since it consists of one sequence and its limit point. Also Ž⺩, 5 ⭈ 5 1 . is a metric space. Recall that the fiber contraction theorem is stated for metric spaces. 2. Recall that
Ž 4.3.
Ž ⌫ky 1Uj , ⌫kU D k u j . s ⌫k Ž Uj , D k u j . s ⌫k Ž ⌬k u j . s ⌬k Ž ⌫o u j . s ⌬k u jq1 s Ž Ujq1 , D k u jq1 . . j
Restricting Ž4.3. to I, we obtain ⌳ o j s ⌫ky1Uj < I s Ujq1 < I s jq1 ,
Ž 4.4.
⌳ j Ž D k u j < I . s ⌫ Uj D k u j < I s D k u jq1 < I .
13
C k INVARIANT MANIFOLDS
PROPOSITION 4.3. The following are true for sufficiently small ⑀ ) 0: 1. ⌳ maps ⺦ = ⺩ to itself. 2. The fiber ⺩⬁ [ ⬁4 = ⺩ is in¨ ariant under ⌳. 3. ⌳: ⺦ = ⺩ ª ⺦ = ⺩ is a fiber contraction Ž not necessarily continuous in Uj yet .. 4. ⌳⬁ : ⺩⬁ ª ⺩⬁ has a unique attracti¨ e fixed point . 5. In fact, s < I , where is the unique attracti¨ e fixed point of ⌫kW : W 4 = ᑳ k ª W 4 = ᑳ k . Proof. Assertions Ž1. ᎐ Ž4. follow from Section 3.2. As for Ž5., it is obvious that < I g ⺩ is a fixed point for ⌳⬁ s ⌫ w < I . But is the unique fixed point of ⌳⬁ . Thus s < I . To be able to apply the Fiber Contraction Theorem we need to show that ⌳Ž j , ˆ. is continuous in j . We need the following elementary but important lemma. LEMMA 4.4. Let 0 F i F k y 1, 0 F j F ⬁, and L s H, J, K. The following sets are compact subsets of Y: Uj s u j Ž I . ,
Zj L s L u jŽ I . ,
js⬁
Us
D Uj ,
V j L , i s D i u j Ž L u jŽ I . . ,
js⬁
ZL s
js0
js⬁
D Zj L ,
V
js0
L, i
s
D V j L, i, js0
ky1
V
L
s
DV
L, i
,
is0
W s U j ZH j Z J j ZK j V H j V J j V K. Proof. Since a finite union of compact sets is compact, and since a continuous image of the interval w0, 1x is compact, we need to consider only U , Z L and V L, i. We will prove that V L, i is compact. The other two cases are easier. Recall that X and Y are Banach. Since V⬁ L, i is compact, it has a finite ␣-net, for all ␣ ) 0. That is, there is a finite set of points x 1 , . . . , x n␣ 4 ; V⬁ L, i, such that V⬁
L, i
; OŽ ␣. [
n␣
D O Ž xi , ␣ . ,
O Ž x, ␣ . [ xX g X o ¬ < xX y x < - ␣ 4 .
is1
Fix g I and let j [ L u jŽ ., 0 F j F ⬁. By assertions Ž1. and Ž2. of w5, Lemma 2.3x,
Ž 4.5.
<j y ⬁ < s L u Ž . y L w Ž . F O Ž ⑀ . 5 u j y w 5# ª 0, j
14
MOHAMED SAMI ELBIALY
as j ª ⬁, uniformly in g X o . Moreover, for 0 F i F k y 1,
Ž 4.6.
D i u j Ž j . y D i w Ž ⬁ . F D i u j Ž j . y D i w Ž j . q D i w Ž j . y D i w Ž ⬁ . F 5 D i u j y D i w 5 q  1 Ž w . <j y ⬁ < F 2 5 Uj y W 5# ª 0,
as j ª ⬁, uniformly in g X o . It follows that for all ␣ ) 0, there is j o such that if j G j0 , then V j L, i ; O Ž ␣ .. The finite union of compact sets V 0L, i j V 1L, i j ⭈⭈⭈ j V j oL,y1i is compact. Thus, for all ␣ ) 0, V L, i has a finite ␣-net. Since Y is also a metric space, it follows that V L, i is compact. LEMMA 4.5. As abo¨ e, let L s H, J, K ; N s H , I , K; h s f, g, h1 , h 2 ; A s C, B1 , B2 ; and j s L u jŽ ., g I, 0 F j F ⬁, where the correspondence between L, N , h, and A should be ob¨ ious from hypotheses ŽH1. and ŽH2.. The following are true for sufficiently small ⑀ ) 0:
Ž 4.7.
<j y n < F O Ž ⑀ . Ž u j y u n . < I # F Ž u j y u n . < I #
Ž 4.8.
u j Ž j . y u n Ž n . F 5 A 5 q q O Ž ⑀ . F 5 A5 q q O Ž ⑀ .
Ž u j y un . < I # j y n
o
.
For all ␣ ) 0, there is c ) 0 such that if 5 j y n 5 o - c, then for all g I, the following are true for 1 F j F ⬁, 1 F n F ⬁:
Ž 4.9.
D k h Ž j , u j Ž j . . y D k h Ž n , u n Ž n . . - ␣ ,
Ž 4.10.
Ž j . y Ž n . - ␣
Ž 4.11.
LU1j Ž . y LU1n Ž . - ␣
Ž 4.12.
Pk Ž j , . y Pk Ž n , . - ␣ ,
Ž 4.13.
Q kh ; L Ž j , . y Q kh ; L Ž n , . - ␣ ,
Ž 4.14.
LUkj Ž . y LUkn Ž . - ␣ ,
kG1
kG2 kG2 k G 2.
Proof. Notice that Lemma 2.3 of w5x is applicable for ␦ s 0, k G 1, since it is proved at the C 0, 1 level. Now, estimate Ž4.7. follows from w5, Lemma 2.3Ž1,2.x, and estimate Ž4.8. follows from w5, Lemma 2.3Ž3.x.
C k INVARIANT MANIFOLDS
15
To prove estimate Ž4.9. let G s D k h. Since G is continuous and W is compact, it follows that G < W is uniformly continuous. Thus, for all ␣ ) 0, there is d ) 0, such that if , X , , X g W , < y X < - d, and < y X < - d, then < G Ž , . y G Ž X , X .< - ␣ . Let c s min d, d w 5 A 5 q q O Ž ⑀ .xy1 4 . Assume that 5 j y n 5 o - c. Then, by Ž4.7., <j y n < - c - d, and by Ž4.8., u j Ž j . y u n Ž n . - 5 A 5 q q O Ž ⑀ . c - d. Thus, < G Žj , u j Žj .. y G Žn , u nŽn ..< - ␣ . The proof of estimate Ž4.10. is similar since is continuous and W is compact. In fact it is simpler since we need only estimate Ž4.7.. To prove estimate Ž4.11. notice that all of the functions involved in LU1i Ž . are continuous and hence uniformly continuous on W . Now the estimate follows from estimates Ž4.7. ᎐ Ž4.10., by an argument similar to the one used to prove Ž4.9.. Notice that in the case k s 1, ⌫u1 does not include P and Q terms. As for estimates Ž4.12. and Ž4.13., notice that for k G 2, all of the terms in Pk Ž i , . and Q kh; L Ž i , ., except Ž ⌬k h.(Ž id, u i ., are at least C 1 and have uniformly bounded first derivatives, which implies uniform continuity on all of X o . The term Ž ⌬k h.(Ž id, u i . is treated in Ž4.9.. Thus, Pk Ž i , . and Q kh; L Ž i , . are uniformly continuous on W . In view of estimates Ž4.7. ᎐ Ž4.10., an argument similar to that used in the proof of estimate Ž4.9. shows that estimates Ž4.12. and Ž4.13. are true. Now estimate Ž4.14. follows from estimates Ž4.7. ᎐ Ž4.13. by an argument similar to that used to prove estimate Ž4.9.. PROPOSITION 4.6. continuous in .
For sufficiently small ⑀ ) 0, ⌳Ž , ˆ. s Ž ⌳ o , ⌳ˆ. is
Proof. We know that ⌳ o is continuous in . We need to show that ⌳ˆ is continuous in . Considering Ž4.2., we can see that all terms are at least uniformly Lipschitz continuous, except the terms Ž z ., Ž ., HUk Ž ., Pk ŽU, ., Q kg, K ŽU, ., and KUk Ž .. However, all of these terms are continuous and hence uniformly continuous on the compact set W . In view of Lemma 4.5, for all ␣ ) 0, there is c ) 0, such that if 5 y 5 o - c, then 5 ⌳ˆ y ⌳ˆ 5 1 - ␣ . This shows that ⌳ˆ is continuous in . PROPOSITION 4.7. For sufficiently small ⑀ ) 0, the fiber contraction ⌳: ⺦ = ⺩ ª ⺦ = ⺩ has a unique attracti¨ e fixed point gi¨ en by Ž ⬁ , < I .. Proof. In Proposition 4.3 we showed that ⌳: ⺦ = ⺩ ª ⺦ = ⺩ is a fiber contraction and that on the invariant fiber ⺩⬁ , the map ⌳⬁ : ⺩⬁ ª ⺩⬁ has a unique attractive fixed point s < I . By Proposition 4.6, ⌳ˆ is continuous in . By the fiber contraction theorem, Ž ⬁ , < I . is a unique attractive fixed point for ⌳: ⺦ = ⺩ ª ⺦ = ⺩.
16
MOHAMED SAMI ELBIALY
Proof of Proposition 3.3. Let u n ¬ n s 0, 1, 2, . . . 4 be as in Definition 4.1. Each u n is C k Therefore Ln Ž x, h . [ D ky 1 u n Ž x q h . y D ky 1 u n Ž x . s
1
H0
D k u n Ž x q th . h dt \ Rn Ž x, h . .
We also know that 5 D ky 1 u n y D ky1 w 5 ª 0 as n ª ⬁ on X o . Thus LnŽ x, h. converges uniformly to the left-hand side of Ž3.3. on X o . It remains to show that RnŽ x, h. converges to the right-hand side of Ž3.3.. Recall that Ž ⬁ , < I . is the unique attractive fixed point of ⌳. Thus, by Ž4.4.,
< I y D k u n < I ª 0, as n ª 0. It follows that 1
H0
D k u n Ž x q th . h y Ž x q th . h dt F < h < D k u n < J y < J ª 0
as n ª 0. This finishes the proof of Proposition 3.3 and Theorem 2.4.
5. INVARIANT MANIFOLDS FOR MAPS 5.1. The Setup. Consider a linear continuous map T : E ª E where E is a Banach space. The linear map is said to be -hyperbolic if its spectrum [ ŽT . lies off a circle of radius in the complex plane ⺓. If s 1, T is said to be hyperbolic. In either case we write the spectrum of T as s s j u , where s lies inside the circle of radius and u lies outside the same circle. It follows from the spectral decomposition theorem that if T is -hyperbolic, then there is a unique T-invariant splitting E s Es [ Eu , such that the spectrum of Tp [ T ¬ E p is p , p s s, u. It is obvious that Tu : Eu ª Eu is an automorphism, but Ts : Es ª Es might not be. For any 0 - a - - b, there are adapted norms on Es and Eu such that < Ts x s < - a < x s < ,
Ž 5.1.
< Tuy1 x u < -
1 b
< xu <,
0 / x s g Es 0 / x u g Eu .
Let 0 F ␦ F 1 when k G 1; and let ␦ s 1 when k s 0. Without loss of generality, we consider a C k, ␦ map F s T q G, where G Ž x . s oŽ x . near
C k INVARIANT MANIFOLDS
17
x s 0, which takes the form
Ž 5.2.
xUs s Ts x s q g s Ž x s , x u . xUu s Tu x u q g u Ž x s , x u . .
Recall the definitions of 5 w 5# and  j, ␦ given in Definitions 2.1 and 2.3, respectively. THEOREM 5.2. Consider the map Ž5.2. in the setup of Section 5.1. Let ⑀ ) 0. D0: Suppose that G is uniformly Lipschitz continuous, LipŽ G . - ⑀ , and 5 G 5 - ⑀ . Then, for sufficiently small ⑀ ) 0, the following hold: PU0: The map F has a PU in¨ ariant manifold of the form
Ž 5.3.
xs s w Ž xu . ,
which is unique in the class of functions
w g C 0 , 1 Ž Eu , Es . ¬ w Ž 0. s 0, Lip Ž w . F 1, 5 w 5# F 1 4 . PS0: The map F has a PS in¨ ariant manifold of the form
Ž 5.4.
xu s ¨ Ž xs . ,
which is unique in the class of functions
w g C 0 , 1 Ž Es , Eu . ¬ ¨ Ž 0. s 0, Lip Ž ¨ . F 1, 5 ¨ 5# F 1 4 . In the following, if ␦ s 0, ignore the statements about  k, ␦ , k G 1. D1: Let k s 1 and 0 F ␦ F 1. Suppose that G g C 1, ␦ Ž E, E ., 5 G 5 ⑀ , 5 DG 5 - ⑀ , and  1, ␦ Ž G . - ⑀ . Then, for sufficiently small ⑀ ) 0, the following hold: PU1: If a - b1q ␦ , then w is unique in the class of functions
w g C 1, ␦ Ž Eu , Es . ¬ w Ž 0. s 0, 5 w 5# F 1, 1, ␦ Ž w . F 1, 5 Dw 5 F 1 4 . Moreo¨ er, Dw Ž0. s 0, and, hence, the in¨ ariant manifold is tangent to Eu . PS1: If a1q ␦ - b, then ¨ is unique in the class of functions
¨ g C 1, ␦ Ž Es , Eu . ¬ ¨ Ž 0. s 0, 5 ¨ 5# F 1, 1, ␦ Ž ¨ . F 1, 5 D¨ 5 F 1 4 . Moreo¨ er, D¨ Ž0. s 0, and, hence, the in¨ ariant manifold is tangent to Es . Dk: Let k ) 1 and 0 F ␦ F 1. Suppose that G g C k, ␦ Ž E, E ., 5 G 5 ⑀ , 5 D j G 5 - ⑀ , and  j, ␦ Ž G . - ⑀ , for 1 F j F k. Then, for sufficiently small
18
MOHAMED SAMI ELBIALY
⑀ ) 0, the following hold: PUk: If a - b kq ␦ , then w is unique in the class of functions
w g C k , ␦ Ž Eu , Es . ¬ w Ž 0. s 0, 5 w 5# F 1,  j, ␦ Ž w . F 1, 5 D j w 5 F 1, 1 F j F k4 . PSk:
If a kq ␦ - b, then ¨ is unique in the class of functions
¨ g C k , ␦ Ž Es , Eu . ¬ ¨ Ž 0. s 0, 5 ¨ 5# F 1,  j, ␦ Ž ¨ . F 1, 5 D j ¨ 5 F 1, 1 F j F k4 . DvU:
Suppose that for some Q ) 0, g s Ž x s , x u . F Q < x s < q < x u < kq ␦ .
Ž 5.5.
PUv: Then for sufficiently small Q ) 0, D j w Ž x u . F < x u < kq ␦yj ,
Ž 5.6. DvS:
Ž 5.7.
0 F j F k.
Suppose that for some Q ) 0, g u Ž x s , x u . F Q < x s < kq ␦ q < x u < .
PSv: Then for sufficiently small Q ) 0,
Ž 5.8.
D j ¨ Ž x s . F < x s < kq ␦yj ,
0 F j F k.
Now we show how to obtain Theorem 5.2 from the main Theorem 2.4. 5.3. The PU Manifolds. Given a circle of radius in the complex plane ⺓, a PU manifold is an invariant manifold that corresponds to the part of the spectrum that lies outside that circle. For the pseudo-unstable and center-unstable manifolds we have - 1. For the unstable manifold we have s 1. For the strong unstable manifold we have ) 1. For a contraction, i.e., when 5 T 5 - 1, a weak-stable manifold corresponds to - 1. The graph of a map of the form x s s w Ž x u . is invariant under the map Ž5.2. iff w Ž xUu . s w Ž x u .U . If we write x s xUu and z s x u , this identity can be rewritten as
Ž 5.9.
w Ž x . s Ts w Ž z . q g s Ž w Ž z . , z . x s Tu z q g u Ž w Ž z . , z . .
19
C k INVARIANT MANIFOLDS
We take advantage of the fact that Tu is invertible and invert the second identity in Ž5.9., using the Lipschitz inverse function theorem Žw29, p. 49x and w5, p. 236x.. We state the theorem in our notation and setup. LEMMA 5.4 ŽLipschitz Inverse Function Theorem.. If - 1, let X o s X Ž case W .. If G 1, let X o s B1Ž0. Ž case S .. Let w g C 0, 1 Ž X o , Y . be such that LipŽ w . F 1. Assume that g u g C 0, 1 Ž Z, Y . ,
g u Ž 0, 0 . s 0,
5 Tuy1 5Lip Ž g u . - 1.
Let Gw : X ª X be gi¨ en by Gw Ž z . [ Tu z q g u Ž w Ž z . , z .
z g X.
Then, Gw is a uniformly Lipschitz continuous homeomorphism onto X. Moreo¨ er, Jw [ Gwy1 is uniformly Lipschitz continuous and satisfies Jw s Tuy1 y Tuy1 g u ( Ž w, id . ( Jw Lip Ž Jw . F
5 Tuy1 5 1 y 5 Tuy1 5 Lip Ž g u .
Jw Ž X . s X . If - 1 Ž case ŽW .., then Jw Ž X o . s Jw Ž X . s X s X o . If G 1 Ž case Ž S .., then for sufficiently small LipŽ g u ., we ha¨ e Jw Ž X o . 9 X o . Thus, Gw Ž X o . < X o . Let X [ Eu and Y [ Es . The map Ž5.2. satisfies LipŽ g u . - ⑀ . Thus for sufficiently small ⑀ ) 0, 5 Tuy1 5 LipŽ g u . - 1. In view of Lemma 5.4, we can rewrite Ž5.9. in the form of the standard graph transform, ⌫F w s Ts w q g s ( Ž w, id . ( Tu q g u ( Ž w, id .
Ž 5.10.
y1
s Ts w ( Hw q g s ( Ž w, id . ( Hw Hw s Tu q g u ( Ž w, id .
y1
.
If G 1, we have 5 Tuy1 5 - 1. Thus, we let X 0 s B1Ž0. Žcase ŽS... If - 1, let X 0 s X Žcase ŽW ... Let M s Ts C s Tuy1 f Ž x, y . s h Ž x, y . s yCg u Ž y, x . g Ž x, y . s g s Ž y, x . .
20
MOHAMED SAMI ELBIALY
Notice that we reversed the order of Ž x, y . in the definitions of Ž f, g, h.. In view of Lemma 5.4, we define y1
Hw s K w s Jw s Tu q g u ( Ž w, id . s C q f ( Ž id, w . ( Hw
y1
R w s g s ( Ž w, id . ( Tu q g u ( Ž w, id . s g s ( Ž w, id . ( Hw s g s ( Ž w, id . ( K w .
Notice that in case ŽW., 5 M 5 s 5 Ts 5 - 1, as required by our main Theorem 2.4. In case ŽS., 5 C 5 s 5 Tuy1 5 - 1, and by Lemma 5.4, Hw maps X 0 onto itself. Now we can see that the graph transform Ž5.10. takes the form Ž2.1. that we study in Theorem 2.4. Moreover, if a - b kq ␦ , then
Ž 5.11.
5 M 5 5 C 5 kq ␦ s 5 Ts 5 5 Tuy1 5 kq ␦ F
a b
kq ␦
- 1.
Thus, assertions PU0, PU1, PUk, and PUv of Theorem 5.2 follow from Theorem 2.4. Notice that Ž5.11. is trivially satisfied for the ŽS.-PU manifolds, the unstable and strong stable manifolds since 1 F - b - b kq ␦ and a - . 5.5. The PS Manifolds. Given a circle of radius in the complex plane ⺓, a PS manifold is an invariant manifold that corresponds to the part of the spectrum that lies inside that circle. For the pseudo-stable and centerstable manifolds, ) 1. For the stable manifold, s 1. For the strong-stable manifold, - 1. For an expanding map, i.e., 5 T 5 ) 1, a weak-unstable manifold corresponds to ) 1. The graph of a function x u s ¨ Ž x s . is invariant under the map Ž5.2. iff Ž ¨ xUs . s ¨ Ž x s .U , that is,
Ž 5.12.
¨ Ž xUs . s Tu¨ Ž x s . q g u Ž x s , ¨ Ž x s . .
xUs s Ts x s q g s Ž x s , ¨ Ž x s . . .
The second equation in Ž5.12. is not necessarily invertible since Ts might not be invertible. However, Tu is invertible, which allows us to rewrite Ž5.12. in the following form with x s x s and z s xUs :
Ž 5.13.
¨ Ž x . s Tuy1 ¨ Ž z . y Tuy1 g u Ž x, ¨ Ž x . .
z s Ts x q g s Ž x, ¨ Ž x . . .
21
C k INVARIANT MANIFOLDS
Thus, we can define ⌫F by
Ž 5.14.
⌫F ¨ s Tuy1 ¨ ( Ts q g s ( Ž id, ¨ . y Tuy1 g u ( Ž id, ¨ . .
The transformation Ž5.14. is not the standard graph transform. However, we can write it in the form Ž2.1. of Theorem 2.4, without using Lemma 5.4, as follows: Let X s Es and Y s Eu . If F 1, we have 5 Ts 5 - 1. Thus we let X o s B1Ž0. Žcase ŽS... If ) 1, let X o s X Žcase ŽW... Let M s Tuy1 , C s Ts , g s yTuy1 g u , f s g s , J ' K ' id X o. Let Hw s Ts q g s ( Ž id, w . s C q f ( Ž id, w . ( Jw s C q S F w
Ž 5.15. R F w s g u ( Ž id, w . s g ( Ž id, w . ( K w ⌫F w s Mw( Hw q R F w. In case ŽS., 5 C 5 s 5 Ts 5 - 1. Thus, if < x < - 1, Hw Ž x . F Ž5 C 5 q ⑀ .< x < - 1, for sufficiently small ⑀ ) 0. It follows that Hw maps X o s B1Ž0. into itself. In case ŽW., 5 M 5 s 5 Tuy1 5 - 1. It follows that Ž5.15. is in the form Ž2.1. of Theorem 2.4. If a kq ␦ - b, we have
Ž 5.16.
5 M 5 5 C 5 kq ␦ s 5 Tuy1 5 5 Ts 5 kq ␦ F
a kq ␦ b
- 1.
Thus, assertions PS0, PS1, PSk, and PSv of Theorem 5.2 follow from Theorem 2.4. Notice that Ž5.16. is trivially satisfied for ŽS.-PS manifolds Žstable and strong stable., since a kq ␦ - a - F 1 and - b. 5.6. S-manifolds, W-manifolds, Local In¨ ariant Manifolds, Uniqueness, and Smooth Cutoff Functions. There is no major distinction between PU and PS manifolds when it comes to hypotheses, existence, and uniqueness. There is another grouping of the standard invariant manifolds which reflects some fundamental differences regarding smallness assumptions Žin infinite dimensional Banach spaces. and uniqueness. Recall that for maps of ⺢ n, local weak-stable manifolds are not unique because we need to use smooth cutoff functions to obtain the local smallness assumptions in the form of the global smallness assumptions D0, D1, Dk, DvU, and DvS. This is the case for weak-stable, weak-unstable, pseudo-stable, pseudo-unstable, center-stable, and center-unstable manifolds. We call these manifolds W-manifolds. We do not need to use smooth cutoff functions with what we call S-manifolds: stable, unstable, strong-stable, and strong-unstable manifolds. As a result, local S-manifolds are unique. We will show later how to obtain the local smallness assumptions for an S-manifold.
22
MOHAMED SAMI ELBIALY
In an infinite-dimensional Banach space, W-manifolds are totally out of luck because smooth cutoff functions are scarce. In w19x, it is shown that the space of continuous functions on an interval does not admit a C 1 cutoff function. In w25x, it is shown that a separable Banach space admits a Frechet differentiable cutoff function iff its dual space is separable. The ´ only places in the literature on invariant manifolds known to the author where this problem is addressed are w5᎐7, 20x. It follows that for W-manifolds of maps of infinite-dimensional Banach spaces, the smallness assumptions are very restrictive because they truly have to be satisfied globally due to the scarcity of smooth cutoff functions. Thus, we can obtain only global W-manifolds when the global smallness assumptions are satisfied. These global W-manifolds are unique because we do not use cutoff functions. 1. Recall that for S-manifolds Žstrong-stable, strong-unstable, stable, and unstable manifolds. we have 5 C 5 - 1, and we can, as we will show soon, re-formulate the problem so that X o s B1Ž0.. We cannot do that for W-manifolds Žpseudo-unstable, pseudo-stable, center-unstable, center-stable, weak-stable, and weak-unstable manifolds. because it either is a weak manifold or corresponds to a part of the spectrum that straddles or intersects the unit circle Žthe first 4.. Thus 5 C 5 l 1. They satisfy condition ŽW. of Theorem 2.4. Notice that we can think of the weak-stable Žunstable . manifold as the stable Žunstable . manifold in a pseudo-unstable Žstable. manifold. This way we can see that it inherits its problems from the pseudo-unstable Žstable. manifold which corresponds to part of the spectrum that straddles the unit circle. Now, it is obvious that the problem that W-manifolds are facing result from a situation where part of the spectrum straddles the unit circle or intersects it. 2. The smallness assumptions for S-manifolds can be obtained by replacing the original map, say ⌽, by ⌽ Ž x . s
1
⌽Ž x. .
Notice that the linear part of the map does not change. Moreover, the graph of a map y s w Ž x . ,
< x < F 1,
is invariant under ⌽ iff the graph of the map y s w Ž x . [ w Ž xr . ,
< x < F ,
C k INVARIANT MANIFOLDS
23
is invariant under ⌽. Now, since 5 C 5 - 1, for sufficiently small ⑀ ) 0, the map Hw maps B1Ž0. to itself, and we can take X o s B1Ž0.. Thus, we do not need any cutoff functions. 3. On the other hand, the graph transform for a W-manifold has to be studied on all of the Banach space X because 5 C 5 l 1. In this case one would like to obtain the smallness assumptions on the nonlinear part of the map by means of a smooth cutoff function. And this is where the trouble starts. 4. Now we can see that for W-manifolds of maps of infinite-dimensional Banach spaces, the smallness assumptions are very restrictive because they truly have to be satisfied globally due to the scarcity of smooth cutoff functions.
REFERENCES 1. V. Afraimovich, On smooth changes in variables, Selecta Math. So¨ iet 9, No. 3 Ž1990.. 2. P. W. Bates, K. Lu, and C. Zeng, Existence and persistence of invariant manifolds for semi-flows in Banach spaces, preprint. 3. S.-N. Chow and K. Lu, C k centre unstable manifolds, Proc. Roy. Soc. Edinburgh Sect. A 108 Ž1988., 303᎐320. 4. S.-N. Chow and K. Lu, Invariant manifolds for flows in Banach spaces, J. Differential Equations 74 Ž1988., 285᎐317. 5. M. S. ElBialy, On pseudo-stable and pseudo-unstable manifolds for maps, J. Math. Anal. Appl. 232 Ž1999., 229᎐258. 6. M. S. ElBialy, Sub-stable and weak-stable manifolds associated with finitely non-resonant spectral subspaces, Mathematische Zeitschrift, accepted. 7. M. S. ElBialy, Local contractions of Banach spaces and spectral gap conditions, J. Funct. Anal. 182 Ž2001., 108᎐150. 8. M. S. ElBialy, Embedding lemmas for functions of a Banach space that vanish up to higher order terms, in preparation. 9. N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Uni¨ . Math. J. 21 Ž1973., 193᎐226. 10. N. Fenichel, Asymptotic stability with rate conditions, Indiana Uni¨ . Math. J. 23 Ž1973., 1109᎐1137. 11. S. A. van Gils and A. Vanderbauwhede, Center manifolds and contractions on scales of Banach spaces, J. Funct. Anal. 72 Ž1987., 209᎐224. 12. J. Hadamard, Sur l’itreration et les solutions asymptotiques des ´ equations differentielles, ´ Bull. Soc. Math. France 29 Ž1901., 224᎐228. 13. M. P. Heble, Approximation of differentiable functions on Hibert space, Contemp. Math. 54 Ž1986., 17᎐33. 14. M. P. Heble, ‘‘Approximation Problems in Analysis and Probability,’’ North-Holland, Amsterdam, 1989. 15. D. Henry, ‘‘Geometric Theory of Parabolic Equations,’’ Lecture Notes in Mathematics, Vol. 840, Springer-Verlag, New York, 1981. 16. M. W. Hirsh, C. C. Pugh, Stable manifolds and hyperbolic sets, in ‘‘Proceedings of the Symp. in Pure Mathematics,’’ Vol. 14, pp. 133᎐164, Am. Math. Soc., Providence, 1970.
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17. M. W. Hirsh, C. C. Pugh and M. Shub, ‘‘Invariant Manifolds,’’ Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, New York, 1977. 18. M. C. Irwin, A new proof of the pseudo-stable manifold theorem, J. London Math. Soc. 21 Ž1980., 557᎐566. 19. N. Kurzweil, On approximation in real Banach spaces, Studia Math. 14 Ž1954., 213᎐231, 1954. 20. R. de la Llave, Invariant manifolds associated to non-resonant spectral subspaces, J. Statist. Phys. 87, Nos. 1᎐2 Ž1997., 211᎐249. 21. R. de la Llave, Regularity of the composition operator in spaces of Holder functions, ¨ Discrete and Continuous Dynamical Systems 5 Ž1999., 157᎐184. 22. R. de la Llave and C. E. Wayne, On Irwin’s proof of the pseudo-stable manifold theorem, Math. Z. 219 Ž1995., 301᎐321. 23. R. de la Llave and C. E. Wayne, Invariant manifolds associated to non-resonant invariant subspaces for some non-invertible maps and semi-flows, preprint. 24. J. G. Llavona, ‘‘Approximation of Continuously Differentiable Functions,’’ North-Holland, Amsterdam, 1986. 25. E. B. Leach and J. H. M. Whitfiels, Differentiable functions and rough norms on Banach spaces, Proc. Amer. Math. Soc. 33 Ž1979., 120᎐126. 26. N. Moulis, Approximation de fonctions differentiables sur certains espaces de Banach, Ann. Inst. Fourier 21, No. 4 Ž1971., 293᎐345. ¨ 27. O. Perron, Uber Stabilitat ¨ und asymptotisches Verhalten der Integrale von Differentialgleichungssystemen, Math. Z. 29 Ž1928., 129᎐190. 28. Ya. B. Pesin, On the existence of invariant fiberings for diffeomorphism of a smooth manifold, USSR Sborn. 20, No. 2 Ž1973., 213᎐222. 29. M. Shub, ‘‘Global Stability of Dynamical Systems,’’ Springer-Verlag, New York, 1987.
C k Invariant Manifolds for Maps of Banach Spaces Mohamed Sami ElBialy Mathematics Department, Uni¨ ersity of Toledo, Toledo, Ohio 43606 E-mail: [email protected] Submitted by William A. Kirk Received October 2, 2000
In this work we give a unified proof for the existence, under appropriate gap conditions, of the standard invariant manifolds for a C k map of a Banach space near a fixed point where k G 1 is an integer. 䊚 2002 Elsevier Science ŽUSA . Key Words: invariant manifolds; linearization.
1. INTRODUCTION In w5x we proved a fixed-point theorem for a class of C k, ␦ , 0 - ␦ F 1, self-maps of a Banach space of functions. In this work we consider the case ␦ s 0. This theorem leads to a unified proof for the existence, under appropriate gap conditions, of the standard invariant manifolds Žpseudostable, pseudo-unstable, center-stable, center-unstable, weak-stable, weakunstable, strong-stable, strong-unstable, stable, and unstable . for C k, ␦ , 0 - ␦ F 1, maps of a Banach space. The case ␦ s 0 needs to be considered separately because the standard fiber contraction theorem w16x does not apply when ␦ s 0. We still have a fiber contraction, and we have a candidate for the kth derivative, but we cannot conclude that it is a unique attractive fixed point because, when ␦ s 0, the fiber contraction is not necessarily continuous in the base points, a condition needed for applying the fiber contraction theorem. As a result, if we use the fiber contraction theorem we can only show that these invariant manifolds are C ky 1, 1. The only places in the literature known to the author where the possible loss of differentiability of an invariant manifold when the map is C k , k g ⺞, is studied are w3x Žcenter-stable manifold., w6x Žsub-stable and weak-stable manifolds., w11x Žcenter manifold., and w22x Žpseudo-stable 1 0022-247Xr02 $35.00 䊚 2002 Elsevier Science ŽUSA . All rights reserved.
2
MOHAMED SAMI ELBIALY
manifold.. It is obvious that these works do not cover all of the standard invariant manifolds when ␦ s 0. In this work we complete the case ␦ s 0 by giving a unified proof for the existence of all of the standard invariant manifolds, for the case ␦ s 0. Since in this article we use the results of w5x, we keep the style, notation, and definitions of w5x. For the sake of completeness we include all definitions and notation. Moreover, w5, Ž5.2.x will mean Eq. Ž5.2. in w5x. Also, w5, Lemma 3.5x will mean Lemma 3.5 of w5x. In Section 2 we give some definitions and state our main theorem. In Section 3 we give all of the definitions, notation, and setup used in w5x and indicate which conclusions in w5x are still valid when ␦ s 0. We also reduce the main theorem to Proposition 3.3, which we prove in Section 4. The idea of the proof is as follows: By applying the results in w5x we know that we have a C ky 1, 1 map y s w Ž x . which is an attractive fixed point for a transformation ⌫ defined on the appropriate function space. We also know that ⌬ky 1 w [ Ž w, Dw, . . . , D ky 1 w . is an attractive fixed point for the Ž k y 1.st lift of ⌫, which we denote by ⌫ky 1. Moreover, we know that the kth lift ⌫k is a fiber contraction which is not necessarily continuous in the base points. Thus, we can apply the standard contraction theorem to the invariant fiber above ⌬ky 1 w and obtain a fixed point, say . This is our candidate for D k w. We cannot use the fiber contraction theorem to conclude that D k w s because ⌫k is not necessarily continuous in the base points. To show that D k w s , it suffices to show that given x and h,
Ž ).
D ky 1 w Ž x q h . y D ky1 w Ž x . s
1
H0
Ž x q th . h dt.
Now let u o be any C k function Žsuch that ⌬k u o is in the function space defined below. and let u n s ⌫u ny1 , n G 1. Now, 5 ⌬ky 1 u n y ⌬ky1 w 5 ª 0 and
Ž 噛.
D ky 1 u n Ž x q h . y D ky1 u n Ž x . s
1
H0
D k u n Ž x q th . h dt.
The left-hand side of Ž噛. converges uniformly Ževerywhere. to the left-hand side of ŽU .. Let I s x q th ¬ 0 F h F 14 . To show that the right-hand side of Ž噛. converges uniformly on I to the right-hand side of ŽU . we restrict the fiber contraction ⌫ to a set W , which consists of the line segment I and its images under D i u n , 1 F i F k y 1, n G 0, and a few other sets given explicitly in Lemma 4.4, where we show that W is in fact compact. We all know that a continuous function on a compact set is uniformly continuous. We use this fact to show that the restriction of ⌫k to W is continuous in the restrictions of the base points. Then we can apply the fiber
C k INVARIANT MANIFOLDS
3
contraction theorem to the restricted ⌫k and conclude that D k u n < I ª < I as n ª ⬁. 1.1. In¨ ariant Manifolds for Maps. Consider a linear continuous map T : E ª E where E is a Banach space. The linear map is said to be -hyperbolic if its spectrum [ ŽT . lies off a circle of radius in the complex plane ⺓. If s 1, T is said to be hyperbolic. In either case we write the spectrum of T as s s j u , where s lies inside the circle of radius and u lies outside the same circle. Consider a C k, ␦ , 0 F ␦ F 1, map ⌽ s T q F, where F Ž0. s 0 and DF Ž0. s 0. When we apply our main Theorem 2.4 to a map of a Banach space, we obtain Theorem 5.2. In w5x we showed how to obtain Theorem 5.2 from our main Theorem 2.4. Following the referee’s advice, we include most of the details in Section 5 for completeness. We group the standard invariant manifolds of a -hyperbolic fixed point into two groups, PU and PS manifolds. PU manifolds are those invariant manifolds that correspond to u , the part of the spectrum that lies outside the circle of radius . The PU manifolds are the pseudo-unstable, centerunstable, unstable, and weak-stable manifolds. The last one is for contracting maps. PS manifolds are those invariant manifolds that correspond to s , the part of the spectrum that lies inside the circle of radius . The PS manifolds are the pseudo-stable, center-stable, stable, strong-stable, and weak-unstable manifolds. The last one is considered for expanding maps. If the map is invertible, PU manifolds of the map are the PS manifolds for the inverse of the map. For a noninvertible map, estimates needed for PU manifolds are more involved than those needed for PS manifolds. The reason for this is that for PU manifolds we need to use the Lipschitz inverse function theorem Žw29, p. 49x and w5, p. 236x.. We will elaborate more on this issue in Sections 5.3 and 5.5. Apart from that, PU and PS manifolds are very similar, as we can see from considering an invertible map. There is another grouping of invariant manifolds which is more serious than the PU-PS grouping, namely, what we call S-manifolds and W-manifolds. This grouping has much more profound roots, which we discuss in the following subsection. 1.2. S-manifolds, W-manifolds, Local In¨ ariant Manifolds, Uniqueness, and Smooth Cutoff Functions. There are no major distinctions between PU and PS manifolds when it comes to hypotheses, existence, and uniqueness. There is another grouping of invariant manifolds which reflects some fundamental differences regarding smallness assumptions Žin infinite-dimensional Banach spaces. and uniqueness. Recall that for maps of ⺢ n, local weak-stable manifolds are not unique because we need to use smooth cutoff functions to obtain the local
4
MOHAMED SAMI ELBIALY
smallness assumptions in the form of global smallness assumptions. This is the case for weak-stable, weak-unstable, pseudo-stable, pseudo-unstable, center-stable, and center-unstable manifolds. We call these manifolds W-manifolds. We do not need to use smooth cutoff functions with what we call S-manifolds: stable, unstable, strong-stable, and strong-unstable manifolds. As a result, local S-manifolds are unique. In an infinite-dimensional Banach space, W-manifolds are totally out of luck because smooth cutoff functions are scarce. We elaborate on this issue in Section 5.6. The only places in the literature on invariant manifolds known to the author where the scarcity of smooth cutoff functions is addressed are w5᎐7, 20, 21x. It follows that for W-manifolds of maps of infinite-dimensional Banach spaces, the smallness assumptions are very restrictive because they truly have to be satisfied globally due to the scarcity of smooth cutoff functions. Thus, we can obtain only global W-manifolds when the global smallness assumptions are satisfied. These global W-manifolds are unique because we do not use cutoff functions.
2. MAIN RESULT We start by recalling some definitions from w5x and restating Theorem C of w5x, with ‘‘0 - ␦ F 1’’ replaced by ‘‘0 F ␦ F 1.’’ DEFINITION 2.1. 1. Let X and Y be Banach spaces. Let Z s X [ Y be endowed with the box norm. For a linear operator L, 5 L 5 will denote the operator norm of L. 2. Let X o ; X and q s 0, 1. We will consider two cases: ŽW.: X o s X and q s 0. ŽS.: X o s B1Ž0. and q s 1, where B1Ž0. ; X is the closed unit ball centered at the origin. 3. Let C 0, 1 Ž X o , Y . the space of uniformly Lipschitz continuous functions. Let 5 u 5# s sup x/0
uŽ x . < x
,
u g C 0, 1 Ž X o , Y . .
Notice that when q s 0, the norm 5 ⭈ 5# coincides with the standard sup-norm. 4. Let id X o: X o ª X o be given by id X oŽ x . s x. Consider a map Q: C 0, 1 Ž X o , Y . ª C 0, 1 Ž X o , X o . , u ¬ Qu .
5
C k INVARIANT MANIFOLDS
If Q u s id X o for all u g C 0, 1 Ž X o , Y . we say that Q is of type Ža. and write Q ' id X o. 5. Let T : X ª X be linear and continuous and let h g C 0, 1 Ž Z, X .. Suppose that for all u g C 0, 1 Ž X o , Y . we can solve the implicit relation
s Tx q h Ž , u Ž . . uniquely in in terms of x g X o . Thus, for all u g C 0, 1 Ž X o , Y ., we have a function Qu : Xo ª Xo , x ¬ s Qu Ž x . Q u s T q h( Ž id, u . ( Q u . Assume that for all u g C 0, 1 Ž X o , Y ., Q u g C 0, 1 Ž X o , X o .. Thus we have a map Q: C 0, 1 Ž X o , Y . ª C 0, 1 Ž X o , X o . , u ¬ Qu . In this case we say that Q is of type bŽT, h.. The following lemma is straightforward. LEMMA 2.2. Let Q: C 0, 1 Ž X o , Y . ª C 0, 1 Ž X o , X o . be of type bŽT, h.. Assume that LipŽ h. - 1 and hŽ0, 0. s 0. Let u g C 0, 1 Ž X o , Y . be such that uŽ0. s 0 and LipŽ u. F 1. Then LipŽŽ id, u.. s 1, Q uŽ0. s 0, and Lip Ž Q u . F
5T 5 1 y Lip Ž h .
.
DEFINITION 2.3. Let E1 and E2 be Banach spaces and let X o ; E1. For a function h g C k Ž X o , E2 ., define the following for 0 F j F k and 0 - ␥ F 1: 5 D j h 5 [ sup D j h Ž x . ¬ x g X o 4
 j, ␥ Ž h . [ ␥ Ž D h . [ sup j
½
D j h Ž x . y D j h Ž xX . < x y xX < ␥
x, xX g X o , x / xX ,
⌬ j h Ž x . [ Ž h Ž x . , Dh Ž x . , . . . , D j h Ž x . . . If max5 D⬘h 5, . . . , 5 D k h 5,  k, ␦ Ž h.4 - ⬁, we say that h g C k, ␦ Ž X o , E2 .. Notice that for 0 F j F k y 1,  j, 1Ž h. F 5 D jq1 h 5.
5
6
MOHAMED SAMI ELBIALY
In this work we prove the following theorem: THEOREM 2.4. Let X, Y, Z, and X o be as in Definition 2.1. Let k s 1, 2, . . . and 0 F ␦ F 1. Assume the following: ŽH1.: Let M: Y ª Y; B1 , B2 , C: X ª X be continuous linear maps such that 5 M 5 5 C 5 - 1 and assume that either ŽW . or Ž S . holds: ŽW.: 5 M 5 F 1. ŽThus 5 M 5 5 C 5 q - 1 since q s 0.. Ž S .: 5 C 5 - 1. ŽThus 5 M 5 5 C 5 q - 1, since q s 1.. ŽC1.: Let F s Ž f, g, h1 , h 2 ., f, h1 , h 2 g C k, ␦ Ž Z, X ., g g C k, ␦ Ž Z, Y ., F Ž0, 0. s 0, and DF Ž0, 0. s 0. ŽC2.: 5 D j F 5 F ⑀ , j s 0, 1, . . . , k. ŽC3.:  j, 1Ž F . [ LipŽ D j F . - ⑀ , j s 0, 1, . . . , k y 1. If 0 - ␦ F 1, assume that  k, ␦ Ž D j F . - ⑀ . ŽH2.: Let J: C k, ␦ Ž X o , Y . ª C k, ␦ Ž X o , X o . be of type Ž a. or bŽ B1 , h1 .. Let H : C k , ␦ Ž Xo , Y . ª C k , ␦ Ž Xo , Xo . u ¬ Hu [ C q S F u SF u [ f ( Ž id, u . ( Ju . ŽH3.:
Let K : C k, ␦ Ž X o , Y . ª C k, ␦ Ž X o , X o . be of type Ž a. or bŽ B2 , h 2 ..
Let R F : C k , ␦ Ž Xo , Y . ª C k , ␦ Ž Xo , Xo . u ¬ R F u [ g ( Ž id, u . ( K u . For u g C k, ␦ Ž X o , Y . let
Ž 2.1.
⌫F u [ Mu( Hu q R F u.
Then, for sufficiently small ⑀ ) 0, ⌫F has a fixed point w g C k , ␦ Ž Xo , Y . , which is unique in the class of functions that satisfy w Ž 0. s 0 5 w 5# F 1 5 D w 5 F 1, j
 1 Ž D j w . F 1,
0FjFk 0 F j F k y 1.
C k INVARIANT MANIFOLDS
7
If 0 - ␦ F 1, w also satisfies
␦ Ž D k w . F 1. ŽH4.:
Suppose that g Ž x, y . satisfies g Ž x, y . - ⑀ Ž < x < kq ␦ q < y < . .
Ž 2.2.
Then, for sufficiently small ⑀ ) 0,
Ž 2.3.
D j w Ž x . F < x < kq ␦yj ,
0 F j F k.
In w5x we proved Theorem 2.4 for 0 - ␦ F 1. Therefore we know that Theorem 2.4 is true for C ky 1, 1 maps. In this work we consider the case ␦ s 0. In what follows we need the fiber contraction theorem, which is proved in w16, p. 136x. 2.5. THE FIBER CONTRACTION THEOREM. Let ᑜ be a metric space and let : ᑜ ª ᑜ be a map ha¨ ing an attracti¨ e fixed point p Ž i.e., ᭙ u g ᑜ: n Ž u. ª p as n ª ⬁.. Let ᑳ be a metric space and let u ¬ u g ᑜ 4 be a family of maps u : ᑳ ª ᑳ such that the formula ⌫ Ž u, . s Ž Ž u., u Ž .. defines a continuous map ⌫: ᑜ = ᑳ ª ᑜ = ᑳ. Let q g ᑳ be a fixed point for p. Then Ž p, q . is an attracti¨ e fixed point for ⌫ pro¨ ided that
Ž 2.4.
lim sup Lip Ž
n
Ž u.
. - 1,
᭙u g ᑜ.
nª⬁
Notice that we do not assume that the spaces ᑜ and ᑳ are complete because we assume that the map has an attractive fixed point p g ᑜ and the map p has a fixed point q g ᑳ.
3. THE SETUP In this section we give all the definitions, notation, and setup used in w5x which we need for the case ␦ s 0. We also indicate which conclusions in w5x are still valid when ␦ s 0. DEFINITION 3.1. 1. Let 5 u 5# be as in Definition 2.1 and define the following spaces: ᑲ o s u g C 0 Ž X o , Y . ¬ 5 u 5# - ⬁ 4 , ᑳ o s u g ᑲ o Ž X o , Y . ¬ u Ž 0 . s 0, 5 u 5# F 1, Lip Ž u . F 1 4 . Let ⌫o [ ⌫F : ᑳ o ª ᑳ o be given by Ž2.1..
8
MOHAMED SAMI ELBIALY
2. For 1 F j F k, let L Ž X j, Y . be the Banach space of symmetric continuous j-multilinear maps from X j to Y with the standard norm 5 A 5 [ sup
½
< A < < <
5
0/gX .
For g C 0 Ž X o , L Ž X j, Y .. and 0 - ␥ F 1, let 5 5 [ sup Ž x . , xgX o
␥ Ž . [ sup
Ž x . y Ž xX . < x y xX < ␥
½
x / xX , x, xX g X o .
5
Notice that  1Ž . s LipŽ .. 3. For 1 F j F k, let ᑲ j s j g C 0 Ž X o , L Ž X j , Y . . 5 j 5 - ⬁ ,
½
5
ᑳ j [ j g ᑲ j ¬ 5 j 5 F 1,  1 Ž j . F 1 4 ,
1FjFky1
ᑳ k [ k g ᑲ k ¬ 5 k 5 F 14 , ⌺ o [ Ž u . s o , ⌺ j [ Ž u, 1 , . . . , j . ,
1 F j F k y 1,
⌺ [ ⌺ ky 1 , 5 ⌺ j 5# [ max 5 u 5#, 5 1 5 , . . . , 5 j 5 4 ,
1 F j F k y 1.
It is obvious that each Ž ᑲ j, 5 ⭈ 5. is a Banach space and that ᑳ j is a closed subset. 4. Let z [ HuŽ x ., [ JuŽ x ., and [ K uŽ x .. 5. Let k s 1 and 1 g ᑳ 1. Let J⌺1 1Ž x . [ Ju1 1Ž x ., where
¡I,
J⌺1 1 Ž x . [
~
¢
J is of type Ž a . ,
I y Dh1 Ž , u Ž . . Ž I, 1 Ž . .
y1
B1 ,
J is of type b Ž B1 , h1 . .
Let K⌺1 1Ž x . [ Ku1 1Ž x ., where
¡I,
K⌺1 1 Ž x . [
~
¢
K is of type Ž a . ,
I y Dh 2 Ž , u Ž . . Ž I, 1 Ž . . K is of type b Ž B2 , h 2 . .
y1
B2 ,
9
C k INVARIANT MANIFOLDS
Let H ⌺1 1 Ž x . [ Hu1 1 Ž x . [ C q Su1 1 Ž x . [ C q Df Ž , u Ž . . Ž I, 1 Ž . . Ju1 1 Ž x . . Let ⌫1 : ᑳ o = ᑳ 1 ª ᑳ o = ᑳ 1 ,
Ž 3.1.
⌫1 Ž u, 1 . [ ⌫⌬ F Ž u, 1 . [ Ž ⌫o u, ⌫1u 1 . , ⌫1u 1 Ž x . [ M 1 Ž z . Hu1 1 Ž x . q Dg Ž , u Ž . . Ž I, 1 Ž . . Ku1 1 Ž x . . 6. Let k G 2 and let k g ᑳ k . Let
¡I,
J⌺k k
J is of type Ž a . ,
Ž x . [~ I y Dh1 Ž , u Ž . . Ž I, 1 Ž . . J is of type b Ž B1 , h1 .
¢
A Ž x . [ Dh1 Ž , u Ž . . Ž 0, k Ž . . J⌺1 1 Ž x .
¡I,
K⌺k k
y1
Ž x . [~
¢
k
AŽ x .
q Q kh1 ; J Ž ⌺, x .
K is of type Ž b . , y1
I y Dh 2 Ž , u Ž . . Ž I, 1 Ž . .
BŽ x .
K is of type b Ž B2 , h 2 .
B Ž x . [ Dh 2 Ž , u Ž . . Ž 0, k Ž . . K⌺1 1 Ž x .
k
q Q kh 2 ; K Ž ⌺, x .
H ⌺k k Ž x . [ Df Ž , u Ž . . Ž 0, k Ž . . J⌺1 Ž x . 1 Ž x .
k
q Df Ž , u Ž . . Ž I, 1 Ž . . J⌺k k Ž x . q Q kf ; J Ž ⌺, x . , where Pk Ž ⌺, x . [ Pk Ž ⌬k f Ž , u Ž . . , ⌺ Ž z . , ⌺ Ž . , H ⌺Ž ky1. Ž x . . , Q kh ; L Ž ⌺, x . [ Q kh ; L Ž ⌬k h Ž , u Ž . . , ⌺ Ž . , N ⌺Ž ky1. Ž x . . ,
[ Lu Ž x . ,
h s f , g , h1 , h 2 ,
L s J, K ,
N ⌺Ž ky1. Ž x . [ Ž N ⌺1 1 Ž x . , . . . , N ⌺ky 1 ky1 Ž x . . , Pk Ž 0, ⭈ , ⭈ , ⭈ . ' 0,
Pk Ž ⭈, 0, ⭈ , ⭈ . ' 0,
Q kh ; L Ž 0, ⭈ , ⭈ . ' 0,
N s H , J, K,
Pk Ž ⭈, ⭈ , ⭈ , 0 . ' 0,
Q kh ; L Ž ⭈, ⭈ , 0 . ' 0.
10
MOHAMED SAMI ELBIALY
Moreover, Pk Ž t 1 , t 2 , t 3 , t4 . and Q kh; L Ž t 1 , t 2 , t 3 . are polynomials in the components of their arguments and linear in the components of t 1. 7. For k G 1, let s k and ky1
Bky 1 [
Ł ᑳi
is0
⌫k : ᑜ ky1 = ᑳ k ª ᑜ ky1 = ᑳ k , ⌫k Ž ⌺, . [ ⌫⌬k F Ž ⌺, . [ Ž ⌫ky 1 Ž ⌺ . , ⌫k⌺ . , ⌫k⌺ Ž x . [ M Ž z . H ⌺1 1 Ž x .
k
q M 1 Ž z . H ⌺k Ž x .
q Pk Ž ⌺, x . q Q kg ; K Ž ⌺, x . q Dg Ž , u Ž . . Ž 0, Ž . . K⌺1 1 Ž x .
k
q Dg Ž , u Ž . . Ž I, 1 Ž . . K⌺k Ž x . . 8. Notice that if w is a C k fixed point for ⌫o , then ⌫k Ž ⌬k w . s ⌬k w. 3.2. Information about ⌫k Which Can be Obtained from w5x. Most of the estimates in w5x are valid for the case ␦ s 0. The only estimates that are not valid are the ones that lead to the conclusion that ⌫k⌺ is continuous in ⌺. These are the estimates in w5, Lemma 4.7x. More precisely, we know the following for sufficiently small ⑀ ) 0: 1. Theorem 2.4 is proven in w5x for the case 0 - ␦ F 1. If we apply it, with smoothness C ky 1, 1, to ⌫ky 1: ᑜ ky1 ª ᑜ ky1 , ᑜ ky1 s ᑜ ky2 = ᑳ ky 1 , we conclude that ⌫ky 1 is well defined Ži.e., ⌫ky1 maps ᑜ ky1 to itself. and has a unique attractive fixed point ⌺# s Ž w, 1 , . . . , ky1 . g ᑜ ky1 and j s D j w for j s 1, . . . , k y 1. 2. For all ⌺ g ᑜ ky 1 , the map ⌫k⌺ maps ᑳ k to itself, and hence ⌫k maps ᑜ ky 1 = ᑳ k to itself. This is shown in w5, Lemma 4.4x, where the proof is valid for ␦ s 0. 3. For sufficiently small ⑀ ) 0, there is 0 - s w 5 M 5 5 C 5 k q O Ž ⑀ .x - 1 such that for all ⌺ g ᑜ ky 1 , for all , g ᑳ k ,
Ž 3.2.
5 ⌫k⌺ y ⌫k⌺ 5 F 5 y 5 .
This is shown in w5, Proposition 4.6x, where the proof is valid for ␦ s 0.
C k INVARIANT MANIFOLDS
11
4. It follows that ⌫k : ᑜ ky1 = ᑳ k ª ᑜ ky1 = ᑳ k is well defined and is a fiber contraction which is not necessarily continuous in ⌺. Since ⌺# g ᑜ ky 1 is fixed under ⌫ky1 , the fiber ᑠ# [ ⌺#4 = ᑳ k is invariant under ⌫# [ ⌫ky 1 = ⌫k⌺# . By Ž3.2. and the standard contraction mapping theorem, ⌫# has a unique attractive fixed point Ž ⌺#, . g ᑠ#. 5. However, we do not know that ⌫k⌺ is continuous in ⌺, which is one of the hypotheses of the Fiber Contraction Theorem 2.5. Therefore we cannot conclude that Ž ⌺#, . is a unique attractive fixed point for ⌫k : ᑜ ky 1 = ᑳ k ª ᑜ ky1 = ᑳ k , which is essential in showing that s D k w. 6. In spite of this temporary setback, we have a candidate for D k w, namely, the fixed point of ⌫#. We will show that s D k w directly using Proposition 3.3 below. 7. Before we proceed to show that in fact s D k w, let us see why we can prove that ⌫k⌺ is continuous in ⌺ only when 0 - ␦ F 1, but not when ␦ s 0: When 0 - ␦ F 1, we include the requirement ␦ Ž k . F 1 in the definition of ᑳ k . Then, the map D k F and g ᑳ k are uniformly ␦-Holder continuous. This is precisely what we used in the proof of ¨ Lemma 4.7 of w5x to show that ⌫k⌺ is also uniformly ␦-Holder continuous ¨ in ⌺ when 0 - ␦ F 1. PROPOSITION 3.3. X o , then
Ž 3.3.
If x and h are in X such that both x and x q h are in
D ky 1 w Ž x q h . y D ky1 w Ž x . s
1
H0
Ž x q th . h dt.
It follows that D ky 1 w is continuously differentiable and D k w s .
4. PROOF OF PROPOSITION 3.3 DEFINITION 4.1. Let x and h be as in Proposition 3.3. Notice that x q th g X, 0 F t F 1. Define the following: 1. Let u o g ᑳ o be such that ⌬k u o g ᑜ ky1 = ᑳ k . Let u n [ ⌫o u ny1 , n G 1. Thus, for n G 0, ⌬k u n g ᑜ ky1 = ᑳ k . Let u⬁ [ w. 2. Let Un [ ⌬ky 1 u n and U⬁ [ W [ ⌬ky 1 w. Thus, for n G 0, Un g ᑜ ky 1 , Un s Ž ⌫ky1 . n ŽUo ., and 5 Un y W 5# ª 0 as n ª ⬁. 3. Let : w0, 1x ª X o be given by Ž t . [ x q th. Let I [ Žw0, 1x.. 4. Let n [ Un < I , for n G 0, and ⬁ [ W < I s U⬁ < I .
12
MOHAMED SAMI ELBIALY
5. Let ⺦ [ n ¬ 0 F n F ⬁ 4 , ⺩ [ ˆ s < I ¬ g ᑳ k 4 , 5 n 5 o [ 5 Un < I 5#, 5 ˆ 5 1 [ 5 < I 5 s sup gI
ˆ s < I g ⺩.
Ž . ,
6. Let ⌳: ⺦ = ⺩ ª ⺦ = ⺩ be the obvious restriction of ⌫k to ⺦ = ⺩ given by
Ž 4.1.
⌳ Ž j , ˆ . [ ⌳ o j , ⌳ j ˆ [ ⌫k Ž Uj , .
ž
/
I
.
where ⌳ o and ⌳ j are defined below. Let z j s Hu jŽ ., j s Ju jŽ ., and j s K u jŽ .. Thus for g I, ⌳ o j Ž . [ ⌳ o Ž Uj < I . Ž . [ ⌫ky1Uj
I
Ž .,
⌳ j ˆ Ž . [ ⌫kUj < I Ž . s M Ž z j . HU1j Du j Ž .
Ž 4.2.
k
q MDu j Ž z j . HUkj Ž .
q Pk Ž Uj , . q Q kg ; K Ž Uj , . q Dg Ž j , u j Ž j . .Ž 0, Ž j . . KU1j Du j Ž .
k
q Dg Ž j , u Ž j . .Ž I, Du j Ž j . . KUkj Ž . . Remarks 4.2. 1. It is obvious that Ž⺦, 5 ⭈ 5 o . is a complete metric space since it consists of one sequence and its limit point. Also Ž⺩, 5 ⭈ 5 1 . is a metric space. Recall that the fiber contraction theorem is stated for metric spaces. 2. Recall that
Ž 4.3.
Ž ⌫ky 1Uj , ⌫kU D k u j . s ⌫k Ž Uj , D k u j . s ⌫k Ž ⌬k u j . s ⌬k Ž ⌫o u j . s ⌬k u jq1 s Ž Ujq1 , D k u jq1 . . j
Restricting Ž4.3. to I, we obtain ⌳ o j s ⌫ky1Uj < I s Ujq1 < I s jq1 ,
Ž 4.4.
⌳ j Ž D k u j < I . s ⌫ Uj D k u j < I s D k u jq1 < I .
13
C k INVARIANT MANIFOLDS
PROPOSITION 4.3. The following are true for sufficiently small ⑀ ) 0: 1. ⌳ maps ⺦ = ⺩ to itself. 2. The fiber ⺩⬁ [ ⬁4 = ⺩ is in¨ ariant under ⌳. 3. ⌳: ⺦ = ⺩ ª ⺦ = ⺩ is a fiber contraction Ž not necessarily continuous in Uj yet .. 4. ⌳⬁ : ⺩⬁ ª ⺩⬁ has a unique attracti¨ e fixed point . 5. In fact, s < I , where is the unique attracti¨ e fixed point of ⌫kW : W 4 = ᑳ k ª W 4 = ᑳ k . Proof. Assertions Ž1. ᎐ Ž4. follow from Section 3.2. As for Ž5., it is obvious that < I g ⺩ is a fixed point for ⌳⬁ s ⌫ w < I . But is the unique fixed point of ⌳⬁ . Thus s < I . To be able to apply the Fiber Contraction Theorem we need to show that ⌳Ž j , ˆ. is continuous in j . We need the following elementary but important lemma. LEMMA 4.4. Let 0 F i F k y 1, 0 F j F ⬁, and L s H, J, K. The following sets are compact subsets of Y: Uj s u j Ž I . ,
Zj L s L u jŽ I . ,
js⬁
Us
D Uj ,
V j L , i s D i u j Ž L u jŽ I . . ,
js⬁
ZL s
js0
js⬁
D Zj L ,
V
js0
L, i
s
D V j L, i, js0
ky1
V
L
s
DV
L, i
,
is0
W s U j ZH j Z J j ZK j V H j V J j V K. Proof. Since a finite union of compact sets is compact, and since a continuous image of the interval w0, 1x is compact, we need to consider only U , Z L and V L, i. We will prove that V L, i is compact. The other two cases are easier. Recall that X and Y are Banach. Since V⬁ L, i is compact, it has a finite ␣-net, for all ␣ ) 0. That is, there is a finite set of points x 1 , . . . , x n␣ 4 ; V⬁ L, i, such that V⬁
L, i
; OŽ ␣. [
n␣
D O Ž xi , ␣ . ,
O Ž x, ␣ . [ xX g X o ¬ < xX y x < - ␣ 4 .
is1
Fix g I and let j [ L u jŽ ., 0 F j F ⬁. By assertions Ž1. and Ž2. of w5, Lemma 2.3x,
Ž 4.5.
<j y ⬁ < s L u Ž . y L w Ž . F O Ž ⑀ . 5 u j y w 5# ª 0, j
14
MOHAMED SAMI ELBIALY
as j ª ⬁, uniformly in g X o . Moreover, for 0 F i F k y 1,
Ž 4.6.
D i u j Ž j . y D i w Ž ⬁ . F D i u j Ž j . y D i w Ž j . q D i w Ž j . y D i w Ž ⬁ . F 5 D i u j y D i w 5 q  1 Ž w . <j y ⬁ < F 2 5 Uj y W 5# ª 0,
as j ª ⬁, uniformly in g X o . It follows that for all ␣ ) 0, there is j o such that if j G j0 , then V j L, i ; O Ž ␣ .. The finite union of compact sets V 0L, i j V 1L, i j ⭈⭈⭈ j V j oL,y1i is compact. Thus, for all ␣ ) 0, V L, i has a finite ␣-net. Since Y is also a metric space, it follows that V L, i is compact. LEMMA 4.5. As abo¨ e, let L s H, J, K ; N s H , I , K; h s f, g, h1 , h 2 ; A s C, B1 , B2 ; and j s L u jŽ ., g I, 0 F j F ⬁, where the correspondence between L, N , h, and A should be ob¨ ious from hypotheses ŽH1. and ŽH2.. The following are true for sufficiently small ⑀ ) 0:
Ž 4.7.
<j y n < F O Ž ⑀ . Ž u j y u n . < I # F Ž u j y u n . < I #
Ž 4.8.
u j Ž j . y u n Ž n . F 5 A 5 q q O Ž ⑀ . F 5 A5 q q O Ž ⑀ .
Ž u j y un . < I # j y n
o
.
For all ␣ ) 0, there is c ) 0 such that if 5 j y n 5 o - c, then for all g I, the following are true for 1 F j F ⬁, 1 F n F ⬁:
Ž 4.9.
D k h Ž j , u j Ž j . . y D k h Ž n , u n Ž n . . - ␣ ,
Ž 4.10.
Ž j . y Ž n . - ␣
Ž 4.11.
LU1j Ž . y LU1n Ž . - ␣
Ž 4.12.
Pk Ž j , . y Pk Ž n , . - ␣ ,
Ž 4.13.
Q kh ; L Ž j , . y Q kh ; L Ž n , . - ␣ ,
Ž 4.14.
LUkj Ž . y LUkn Ž . - ␣ ,
kG1
kG2 kG2 k G 2.
Proof. Notice that Lemma 2.3 of w5x is applicable for ␦ s 0, k G 1, since it is proved at the C 0, 1 level. Now, estimate Ž4.7. follows from w5, Lemma 2.3Ž1,2.x, and estimate Ž4.8. follows from w5, Lemma 2.3Ž3.x.
C k INVARIANT MANIFOLDS
15
To prove estimate Ž4.9. let G s D k h. Since G is continuous and W is compact, it follows that G < W is uniformly continuous. Thus, for all ␣ ) 0, there is d ) 0, such that if , X , , X g W , < y X < - d, and < y X < - d, then < G Ž , . y G Ž X , X .< - ␣ . Let c s min d, d w 5 A 5 q q O Ž ⑀ .xy1 4 . Assume that 5 j y n 5 o - c. Then, by Ž4.7., <j y n < - c - d, and by Ž4.8., u j Ž j . y u n Ž n . - 5 A 5 q q O Ž ⑀ . c - d. Thus, < G Žj , u j Žj .. y G Žn , u nŽn ..< - ␣ . The proof of estimate Ž4.10. is similar since is continuous and W is compact. In fact it is simpler since we need only estimate Ž4.7.. To prove estimate Ž4.11. notice that all of the functions involved in LU1i Ž . are continuous and hence uniformly continuous on W . Now the estimate follows from estimates Ž4.7. ᎐ Ž4.10., by an argument similar to the one used to prove Ž4.9.. Notice that in the case k s 1, ⌫u1 does not include P and Q terms. As for estimates Ž4.12. and Ž4.13., notice that for k G 2, all of the terms in Pk Ž i , . and Q kh; L Ž i , ., except Ž ⌬k h.(Ž id, u i ., are at least C 1 and have uniformly bounded first derivatives, which implies uniform continuity on all of X o . The term Ž ⌬k h.(Ž id, u i . is treated in Ž4.9.. Thus, Pk Ž i , . and Q kh; L Ž i , . are uniformly continuous on W . In view of estimates Ž4.7. ᎐ Ž4.10., an argument similar to that used in the proof of estimate Ž4.9. shows that estimates Ž4.12. and Ž4.13. are true. Now estimate Ž4.14. follows from estimates Ž4.7. ᎐ Ž4.13. by an argument similar to that used to prove estimate Ž4.9.. PROPOSITION 4.6. continuous in .
For sufficiently small ⑀ ) 0, ⌳Ž , ˆ. s Ž ⌳ o , ⌳ˆ. is
Proof. We know that ⌳ o is continuous in . We need to show that ⌳ˆ is continuous in . Considering Ž4.2., we can see that all terms are at least uniformly Lipschitz continuous, except the terms Ž z ., Ž ., HUk Ž ., Pk ŽU, ., Q kg, K ŽU, ., and KUk Ž .. However, all of these terms are continuous and hence uniformly continuous on the compact set W . In view of Lemma 4.5, for all ␣ ) 0, there is c ) 0, such that if 5 y 5 o - c, then 5 ⌳ˆ y ⌳ˆ 5 1 - ␣ . This shows that ⌳ˆ is continuous in . PROPOSITION 4.7. For sufficiently small ⑀ ) 0, the fiber contraction ⌳: ⺦ = ⺩ ª ⺦ = ⺩ has a unique attracti¨ e fixed point gi¨ en by Ž ⬁ , < I .. Proof. In Proposition 4.3 we showed that ⌳: ⺦ = ⺩ ª ⺦ = ⺩ is a fiber contraction and that on the invariant fiber ⺩⬁ , the map ⌳⬁ : ⺩⬁ ª ⺩⬁ has a unique attractive fixed point s < I . By Proposition 4.6, ⌳ˆ is continuous in . By the fiber contraction theorem, Ž ⬁ , < I . is a unique attractive fixed point for ⌳: ⺦ = ⺩ ª ⺦ = ⺩.
16
MOHAMED SAMI ELBIALY
Proof of Proposition 3.3. Let u n ¬ n s 0, 1, 2, . . . 4 be as in Definition 4.1. Each u n is C k Therefore Ln Ž x, h . [ D ky 1 u n Ž x q h . y D ky 1 u n Ž x . s
1
H0
D k u n Ž x q th . h dt \ Rn Ž x, h . .
We also know that 5 D ky 1 u n y D ky1 w 5 ª 0 as n ª ⬁ on X o . Thus LnŽ x, h. converges uniformly to the left-hand side of Ž3.3. on X o . It remains to show that RnŽ x, h. converges to the right-hand side of Ž3.3.. Recall that Ž ⬁ , < I . is the unique attractive fixed point of ⌳. Thus, by Ž4.4.,
< I y D k u n < I ª 0, as n ª 0. It follows that 1
H0
D k u n Ž x q th . h y Ž x q th . h dt F < h < D k u n < J y < J ª 0
as n ª 0. This finishes the proof of Proposition 3.3 and Theorem 2.4.
5. INVARIANT MANIFOLDS FOR MAPS 5.1. The Setup. Consider a linear continuous map T : E ª E where E is a Banach space. The linear map is said to be -hyperbolic if its spectrum [ ŽT . lies off a circle of radius in the complex plane ⺓. If s 1, T is said to be hyperbolic. In either case we write the spectrum of T as s s j u , where s lies inside the circle of radius and u lies outside the same circle. It follows from the spectral decomposition theorem that if T is -hyperbolic, then there is a unique T-invariant splitting E s Es [ Eu , such that the spectrum of Tp [ T ¬ E p is p , p s s, u. It is obvious that Tu : Eu ª Eu is an automorphism, but Ts : Es ª Es might not be. For any 0 - a - - b, there are adapted norms on Es and Eu such that < Ts x s < - a < x s < ,
Ž 5.1.
< Tuy1 x u < -
1 b
< xu <,
0 / x s g Es 0 / x u g Eu .
Let 0 F ␦ F 1 when k G 1; and let ␦ s 1 when k s 0. Without loss of generality, we consider a C k, ␦ map F s T q G, where G Ž x . s oŽ x . near
C k INVARIANT MANIFOLDS
17
x s 0, which takes the form
Ž 5.2.
xUs s Ts x s q g s Ž x s , x u . xUu s Tu x u q g u Ž x s , x u . .
Recall the definitions of 5 w 5# and  j, ␦ given in Definitions 2.1 and 2.3, respectively. THEOREM 5.2. Consider the map Ž5.2. in the setup of Section 5.1. Let ⑀ ) 0. D0: Suppose that G is uniformly Lipschitz continuous, LipŽ G . - ⑀ , and 5 G 5 - ⑀ . Then, for sufficiently small ⑀ ) 0, the following hold: PU0: The map F has a PU in¨ ariant manifold of the form
Ž 5.3.
xs s w Ž xu . ,
which is unique in the class of functions
w g C 0 , 1 Ž Eu , Es . ¬ w Ž 0. s 0, Lip Ž w . F 1, 5 w 5# F 1 4 . PS0: The map F has a PS in¨ ariant manifold of the form
Ž 5.4.
xu s ¨ Ž xs . ,
which is unique in the class of functions
w g C 0 , 1 Ž Es , Eu . ¬ ¨ Ž 0. s 0, Lip Ž ¨ . F 1, 5 ¨ 5# F 1 4 . In the following, if ␦ s 0, ignore the statements about  k, ␦ , k G 1. D1: Let k s 1 and 0 F ␦ F 1. Suppose that G g C 1, ␦ Ž E, E ., 5 G 5 ⑀ , 5 DG 5 - ⑀ , and  1, ␦ Ž G . - ⑀ . Then, for sufficiently small ⑀ ) 0, the following hold: PU1: If a - b1q ␦ , then w is unique in the class of functions
w g C 1, ␦ Ž Eu , Es . ¬ w Ž 0. s 0, 5 w 5# F 1, 1, ␦ Ž w . F 1, 5 Dw 5 F 1 4 . Moreo¨ er, Dw Ž0. s 0, and, hence, the in¨ ariant manifold is tangent to Eu . PS1: If a1q ␦ - b, then ¨ is unique in the class of functions
¨ g C 1, ␦ Ž Es , Eu . ¬ ¨ Ž 0. s 0, 5 ¨ 5# F 1, 1, ␦ Ž ¨ . F 1, 5 D¨ 5 F 1 4 . Moreo¨ er, D¨ Ž0. s 0, and, hence, the in¨ ariant manifold is tangent to Es . Dk: Let k ) 1 and 0 F ␦ F 1. Suppose that G g C k, ␦ Ž E, E ., 5 G 5 ⑀ , 5 D j G 5 - ⑀ , and  j, ␦ Ž G . - ⑀ , for 1 F j F k. Then, for sufficiently small
18
MOHAMED SAMI ELBIALY
⑀ ) 0, the following hold: PUk: If a - b kq ␦ , then w is unique in the class of functions
w g C k , ␦ Ž Eu , Es . ¬ w Ž 0. s 0, 5 w 5# F 1,  j, ␦ Ž w . F 1, 5 D j w 5 F 1, 1 F j F k4 . PSk:
If a kq ␦ - b, then ¨ is unique in the class of functions
¨ g C k , ␦ Ž Es , Eu . ¬ ¨ Ž 0. s 0, 5 ¨ 5# F 1,  j, ␦ Ž ¨ . F 1, 5 D j ¨ 5 F 1, 1 F j F k4 . DvU:
Suppose that for some Q ) 0, g s Ž x s , x u . F Q < x s < q < x u < kq ␦ .
Ž 5.5.
PUv: Then for sufficiently small Q ) 0, D j w Ž x u . F < x u < kq ␦yj ,
Ž 5.6. DvS:
Ž 5.7.
0 F j F k.
Suppose that for some Q ) 0, g u Ž x s , x u . F Q < x s < kq ␦ q < x u < .
PSv: Then for sufficiently small Q ) 0,
Ž 5.8.
D j ¨ Ž x s . F < x s < kq ␦yj ,
0 F j F k.
Now we show how to obtain Theorem 5.2 from the main Theorem 2.4. 5.3. The PU Manifolds. Given a circle of radius in the complex plane ⺓, a PU manifold is an invariant manifold that corresponds to the part of the spectrum that lies outside that circle. For the pseudo-unstable and center-unstable manifolds we have - 1. For the unstable manifold we have s 1. For the strong unstable manifold we have ) 1. For a contraction, i.e., when 5 T 5 - 1, a weak-stable manifold corresponds to - 1. The graph of a map of the form x s s w Ž x u . is invariant under the map Ž5.2. iff w Ž xUu . s w Ž x u .U . If we write x s xUu and z s x u , this identity can be rewritten as
Ž 5.9.
w Ž x . s Ts w Ž z . q g s Ž w Ž z . , z . x s Tu z q g u Ž w Ž z . , z . .
19
C k INVARIANT MANIFOLDS
We take advantage of the fact that Tu is invertible and invert the second identity in Ž5.9., using the Lipschitz inverse function theorem Žw29, p. 49x and w5, p. 236x.. We state the theorem in our notation and setup. LEMMA 5.4 ŽLipschitz Inverse Function Theorem.. If - 1, let X o s X Ž case W .. If G 1, let X o s B1Ž0. Ž case S .. Let w g C 0, 1 Ž X o , Y . be such that LipŽ w . F 1. Assume that g u g C 0, 1 Ž Z, Y . ,
g u Ž 0, 0 . s 0,
5 Tuy1 5Lip Ž g u . - 1.
Let Gw : X ª X be gi¨ en by Gw Ž z . [ Tu z q g u Ž w Ž z . , z .
z g X.
Then, Gw is a uniformly Lipschitz continuous homeomorphism onto X. Moreo¨ er, Jw [ Gwy1 is uniformly Lipschitz continuous and satisfies Jw s Tuy1 y Tuy1 g u ( Ž w, id . ( Jw Lip Ž Jw . F
5 Tuy1 5 1 y 5 Tuy1 5 Lip Ž g u .
Jw Ž X . s X . If - 1 Ž case ŽW .., then Jw Ž X o . s Jw Ž X . s X s X o . If G 1 Ž case Ž S .., then for sufficiently small LipŽ g u ., we ha¨ e Jw Ž X o . 9 X o . Thus, Gw Ž X o . < X o . Let X [ Eu and Y [ Es . The map Ž5.2. satisfies LipŽ g u . - ⑀ . Thus for sufficiently small ⑀ ) 0, 5 Tuy1 5 LipŽ g u . - 1. In view of Lemma 5.4, we can rewrite Ž5.9. in the form of the standard graph transform, ⌫F w s Ts w q g s ( Ž w, id . ( Tu q g u ( Ž w, id .
Ž 5.10.
y1
s Ts w ( Hw q g s ( Ž w, id . ( Hw Hw s Tu q g u ( Ž w, id .
y1
.
If G 1, we have 5 Tuy1 5 - 1. Thus, we let X 0 s B1Ž0. Žcase ŽS... If - 1, let X 0 s X Žcase ŽW ... Let M s Ts C s Tuy1 f Ž x, y . s h Ž x, y . s yCg u Ž y, x . g Ž x, y . s g s Ž y, x . .
20
MOHAMED SAMI ELBIALY
Notice that we reversed the order of Ž x, y . in the definitions of Ž f, g, h.. In view of Lemma 5.4, we define y1
Hw s K w s Jw s Tu q g u ( Ž w, id . s C q f ( Ž id, w . ( Hw
y1
R w s g s ( Ž w, id . ( Tu q g u ( Ž w, id . s g s ( Ž w, id . ( Hw s g s ( Ž w, id . ( K w .
Notice that in case ŽW., 5 M 5 s 5 Ts 5 - 1, as required by our main Theorem 2.4. In case ŽS., 5 C 5 s 5 Tuy1 5 - 1, and by Lemma 5.4, Hw maps X 0 onto itself. Now we can see that the graph transform Ž5.10. takes the form Ž2.1. that we study in Theorem 2.4. Moreover, if a - b kq ␦ , then
Ž 5.11.
5 M 5 5 C 5 kq ␦ s 5 Ts 5 5 Tuy1 5 kq ␦ F
a b
kq ␦
- 1.
Thus, assertions PU0, PU1, PUk, and PUv of Theorem 5.2 follow from Theorem 2.4. Notice that Ž5.11. is trivially satisfied for the ŽS.-PU manifolds, the unstable and strong stable manifolds since 1 F - b - b kq ␦ and a - . 5.5. The PS Manifolds. Given a circle of radius in the complex plane ⺓, a PS manifold is an invariant manifold that corresponds to the part of the spectrum that lies inside that circle. For the pseudo-stable and centerstable manifolds, ) 1. For the stable manifold, s 1. For the strong-stable manifold, - 1. For an expanding map, i.e., 5 T 5 ) 1, a weak-unstable manifold corresponds to ) 1. The graph of a function x u s ¨ Ž x s . is invariant under the map Ž5.2. iff Ž ¨ xUs . s ¨ Ž x s .U , that is,
Ž 5.12.
¨ Ž xUs . s Tu¨ Ž x s . q g u Ž x s , ¨ Ž x s . .
xUs s Ts x s q g s Ž x s , ¨ Ž x s . . .
The second equation in Ž5.12. is not necessarily invertible since Ts might not be invertible. However, Tu is invertible, which allows us to rewrite Ž5.12. in the following form with x s x s and z s xUs :
Ž 5.13.
¨ Ž x . s Tuy1 ¨ Ž z . y Tuy1 g u Ž x, ¨ Ž x . .
z s Ts x q g s Ž x, ¨ Ž x . . .
21
C k INVARIANT MANIFOLDS
Thus, we can define ⌫F by
Ž 5.14.
⌫F ¨ s Tuy1 ¨ ( Ts q g s ( Ž id, ¨ . y Tuy1 g u ( Ž id, ¨ . .
The transformation Ž5.14. is not the standard graph transform. However, we can write it in the form Ž2.1. of Theorem 2.4, without using Lemma 5.4, as follows: Let X s Es and Y s Eu . If F 1, we have 5 Ts 5 - 1. Thus we let X o s B1Ž0. Žcase ŽS... If ) 1, let X o s X Žcase ŽW... Let M s Tuy1 , C s Ts , g s yTuy1 g u , f s g s , J ' K ' id X o. Let Hw s Ts q g s ( Ž id, w . s C q f ( Ž id, w . ( Jw s C q S F w
Ž 5.15. R F w s g u ( Ž id, w . s g ( Ž id, w . ( K w ⌫F w s Mw( Hw q R F w. In case ŽS., 5 C 5 s 5 Ts 5 - 1. Thus, if < x < - 1, Hw Ž x . F Ž5 C 5 q ⑀ .< x < - 1, for sufficiently small ⑀ ) 0. It follows that Hw maps X o s B1Ž0. into itself. In case ŽW., 5 M 5 s 5 Tuy1 5 - 1. It follows that Ž5.15. is in the form Ž2.1. of Theorem 2.4. If a kq ␦ - b, we have
Ž 5.16.
5 M 5 5 C 5 kq ␦ s 5 Tuy1 5 5 Ts 5 kq ␦ F
a kq ␦ b
- 1.
Thus, assertions PS0, PS1, PSk, and PSv of Theorem 5.2 follow from Theorem 2.4. Notice that Ž5.16. is trivially satisfied for ŽS.-PS manifolds Žstable and strong stable., since a kq ␦ - a - F 1 and - b. 5.6. S-manifolds, W-manifolds, Local In¨ ariant Manifolds, Uniqueness, and Smooth Cutoff Functions. There is no major distinction between PU and PS manifolds when it comes to hypotheses, existence, and uniqueness. There is another grouping of the standard invariant manifolds which reflects some fundamental differences regarding smallness assumptions Žin infinite dimensional Banach spaces. and uniqueness. Recall that for maps of ⺢ n, local weak-stable manifolds are not unique because we need to use smooth cutoff functions to obtain the local smallness assumptions in the form of the global smallness assumptions D0, D1, Dk, DvU, and DvS. This is the case for weak-stable, weak-unstable, pseudo-stable, pseudo-unstable, center-stable, and center-unstable manifolds. We call these manifolds W-manifolds. We do not need to use smooth cutoff functions with what we call S-manifolds: stable, unstable, strong-stable, and strong-unstable manifolds. As a result, local S-manifolds are unique. We will show later how to obtain the local smallness assumptions for an S-manifold.
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MOHAMED SAMI ELBIALY
In an infinite-dimensional Banach space, W-manifolds are totally out of luck because smooth cutoff functions are scarce. In w19x, it is shown that the space of continuous functions on an interval does not admit a C 1 cutoff function. In w25x, it is shown that a separable Banach space admits a Frechet differentiable cutoff function iff its dual space is separable. The ´ only places in the literature on invariant manifolds known to the author where this problem is addressed are w5᎐7, 20x. It follows that for W-manifolds of maps of infinite-dimensional Banach spaces, the smallness assumptions are very restrictive because they truly have to be satisfied globally due to the scarcity of smooth cutoff functions. Thus, we can obtain only global W-manifolds when the global smallness assumptions are satisfied. These global W-manifolds are unique because we do not use cutoff functions. 1. Recall that for S-manifolds Žstrong-stable, strong-unstable, stable, and unstable manifolds. we have 5 C 5 - 1, and we can, as we will show soon, re-formulate the problem so that X o s B1Ž0.. We cannot do that for W-manifolds Žpseudo-unstable, pseudo-stable, center-unstable, center-stable, weak-stable, and weak-unstable manifolds. because it either is a weak manifold or corresponds to a part of the spectrum that straddles or intersects the unit circle Žthe first 4.. Thus 5 C 5 l 1. They satisfy condition ŽW. of Theorem 2.4. Notice that we can think of the weak-stable Žunstable . manifold as the stable Žunstable . manifold in a pseudo-unstable Žstable. manifold. This way we can see that it inherits its problems from the pseudo-unstable Žstable. manifold which corresponds to part of the spectrum that straddles the unit circle. Now, it is obvious that the problem that W-manifolds are facing result from a situation where part of the spectrum straddles the unit circle or intersects it. 2. The smallness assumptions for S-manifolds can be obtained by replacing the original map, say ⌽, by ⌽ Ž x . s
1
⌽Ž x. .
Notice that the linear part of the map does not change. Moreover, the graph of a map y s w Ž x . ,
< x < F 1,
is invariant under ⌽ iff the graph of the map y s w Ž x . [ w Ž xr . ,
< x < F ,
C k INVARIANT MANIFOLDS
23
is invariant under ⌽. Now, since 5 C 5 - 1, for sufficiently small ⑀ ) 0, the map Hw maps B1Ž0. to itself, and we can take X o s B1Ž0.. Thus, we do not need any cutoff functions. 3. On the other hand, the graph transform for a W-manifold has to be studied on all of the Banach space X because 5 C 5 l 1. In this case one would like to obtain the smallness assumptions on the nonlinear part of the map by means of a smooth cutoff function. And this is where the trouble starts. 4. Now we can see that for W-manifolds of maps of infinite-dimensional Banach spaces, the smallness assumptions are very restrictive because they truly have to be satisfied globally due to the scarcity of smooth cutoff functions.
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