some n. We have thus proved that for su¬ciently small r we have

W s (p) = f ’n [Br (p)].

s

(7.12)

n≥0

s

We will prove that Br (p) is a submanifold. It will then follow from (7.12) that

W s (p) is a submanifold. The global disposition of W s (p), and in particular

its relation to the stable and unstable manifolds of other ¬xed points, is a key

ingredient in the study of the long term behavior of dynamical systems. In this

s

section our focus is purely local, to prove the smooth character of the set B r (p).

We follow the treatment in [?].

We will begin with the hypothesis that f is merely Lipschitz, and give a proof

(independent of the C 0 linearization theorem) of the existence and Lipschitz

130 CHAPTER 7. HYPERBOLICITY.

character of the W u . We will work in the following situation: A is a hyperbolic

linear isomorphism of a Banach space E = S • U with

A’1 x ¤ a x , x ∈ U.

Ax ¤ a x , x ∈ S,

We let S(r) denote the ball of radius s about the origin in S, and U (r) the ball

of radius r in U . We will assume that

f : S(r) — U (r) ’ E

is a Lipschitz map with

f (0) ¤ δ (7.13)

and

Lip[f ’ A] ¤ . (7.14)

We wish to prove the following

Theorem 7.2.1 Let c < 1. There exists an = (a) and a δ = δ(a, , r) so that

if f satis¬es (7.13) and (7.14) then there is a map

g : Eu (r) ’ Es (r)

with the following properties:

(i) g is Lipschitz with Lip[g] ¤ 1.

(ii) The restriction of f ’1 to graph(g) is contracting and hence has a ¬xed point,

p, on graph(g).

(iii) We have

f n (S(r) • U (r)) = W u (p) © [S(r) • U (p)].

graph(g) =

The idea of the proof is to apply the contraction ¬xed point theorem to the

space of maps of U (r) to S(r). We want to identify such a map, v, with its

graph:

graph(v) = {(v(x), x), x ∈ U (r)}.

Now

f [graph(v)] = {f (v(x), x)} = {(fs (v(x), x), fu (v(x), x))},

where we have introduced the notation

fs = ps —¦ f, fu = pu —¦ f,

where ps denotes projection onto S and pu denotes projection onto U .

Suppose that the projection of f [graph(v)] onto U is injective and its image

contains U (r). This means that for any y ∈ U (r) there is a unique x ∈ U (r)

with

fu (v(x), x) = y.

So we write

x = [fu —¦ (v, id)]’1 (y)

131

7.2. INVARIANT MANIFOLDS

where we think of (v, id) as a map of U (r) ’ E and hence of

fu —¦ (v, id)

as a map of U (r) ’ U . Then we can write

f [graph(v)] = {(fs (v([fu —¦ (v, id)]’1 (y), y)} = graphG[ f (v)]

where

Gf (v) = fs —¦ (v, id) —¦ [fu —¦ (v, id)]’1 . (7.15)

The map v ’ Gf (v) is called the graph transform (when it is de¬ned). We are

going to take

X = Lip1 (U (r), S(r))

to consist of all Lipschitz maps from U (r) to S(r) with Lipschitz constant ¤ 1.

The purpose of the next few lemmas is to show that if and δ are su¬ciently

small then the graph transform, Gf is de¬ned and is a contraction on X. The

contraction ¬xed point theorem will then imply that there is a unique g ∈ X

which is ¬xed under Gf , and hence that graph(g) is invariant under f . We will

then ¬nd that g has all the properties stated in the theorem.

In dealing with the graph transform it is convenient to use the box metric,

| |, on S • U where

|xs • xu | = max{ xs , xu }

i.e.

|x| = max{ ps (x) , pu (x) }.

We begin with

Lemma 7.2.1 If v ∈ X then

Lip[fu —¦ (v, id) ’ Au ] ¤ Lip[f ’ A].

Proof. Notice that

pu —¦ A(v(x), x) = pu (As (v(x)), Au x) = Au x

so

fu —¦ (v, id) ’ Au = pu —¦ [f ’ A] —¦ (v, id).

We have Lip[pu ] ¤ 1 since pu is a projection, and

Lip(v, id) ¤ max{Lip[v], Lip[id]} = 1

since we are using the box metric. Thus the lemma follows.

Lemma 7.2.2 Suppose that 0 < < c’1 and

Lip[f ’ A] < .

Then for any v ∈ X the map fu —¦ (v, id) : Eu (r) ’ Eu is a homeomorphism

whose inverse is a Lipschitz map with

1

Lip [fu —¦ (v, id)]’1 ¤ . (7.16)

c’1 ’

132 CHAPTER 7. HYPERBOLICITY.

Proof.Using the preceding lemma, we have

’1

Lip[fu ’ Au ] < < c’1 < A’1 ’1

= (Lip[Au ]) .

u

By the Lipschitz implicit function theorem we conclude that fu —¦ (v, id) is a

homeomorphism with

1 1

Lip [fu —¦ (v, id)]’1 ¤ ¤ ’1

A’1 c’

’1 ’ Lip[f —¦ (v, id) ’ A ]

u u u

by another application of the preceding lemma. QED. We now wish to show

that the image of fu —¦ (v, id) contains U (r) if and δ are su¬ciently small:

By the proposition in section 5.2 concerning the image of a Lipschitz map, we

know that the image of U (r) under fu —¦ (v, id) contains a ball of radius r/»

about [fu —¦ (v, id)](0) where » is the Lipschitz constant of [fu —¦ (v, id)]’1 . By

the preceding lemma, r/» = r(c’1 ’ ). Hence fu —¦ (v, id)(U (r)) contains the

ball of radius

r(c’1 ’ ) ’ fu (v(0), 0)

about the origin. But

¤ fu (0, 0) + fu (v(0), 0) ’ fu (0, 0)

fu (v(0), 0)

¤ fu (0, 0) + (fu ’ pu A)(v(0), 0) ’ (fu ’ pu A)(0, 0)

¤ |f (0)| + |(f ’ A)(v(0), 0) ’ (f ’ A)(0, 0)|

¤ |f (0)| + r.

The passage from the second line to the third is because pu A(x, y) = Au y = 0

if y = 0. The passage from the third line to the fourth is because we are using

the box norm. So

r(c’1 ’ ) ’ fu (v(0), 0) ≥ r(c’1 ’ 2 ) ’ δ

if (7.13) holds. We would like this expression to be ≥ r, which will happen if

δ ¤ r(c’1 ’ 1 ’ 2 ). (7.17)

We have thus proved

Proposition 7.2.2 Let f be a Lipschitz map satisfying (7.13) and (7.14) where

2 < c’1 ’1 and (7.17) holds. Then for every v ∈ X, the graph transform, G f (v)

is de¬ned and

c+