ρ(t) = 0

1

∀ t¤

ρ(t) = 1

2

|ρ (t)| < K ∀t

where K is some number,

K > 2.

For a ¬xed let r be su¬ciently small so that the on the ball, Br (0) we have

the estimate

dφx < ,

2K

which is possible since dφ0 = 0 and dφ is continuous. Now de¬ne

x

ψ(x) = ρ( )φ(x),

r

and continuously extend to

x ≥ r.

ψ(x) = 0,

Notice that

r

x¤

ψ(x) = φ(x), .

2

Let us now check the Lipschitz constant of ψ. There are three alternatives: If

x1 and x2 both belong to Br (0) we have

x1 x2

ψ(x1 ) ’ ψ(x2 ) )φ(x1 ) ’ ρ(

= ρ( )φ(x2 )

r r

x1 x2 x2

¤ |ρ( ) ’ ρ( ) φ(x1 ) ’ φ(x2 )

) φ(x1 ) + ρ(

r r r

¤ (K x1 ’ x2 /r) — x1 — ( /2K) + ( /2K) — x1 ’ x2

¤ x1 ’ x2 |.

If x1 ∈ Br (0), x2 ∈ Br (0), then the second term in the expression on the second

line above vanishes and the ¬rst term is at most ( /2) x1 ’ x2 . If neither x1

nor x2 belong to Br (0) then ψ(x1 ) ’ ψ(x2 ) = 0 ’ 0 = 0. We have veri¬ed that

Lip[ψ] < and so have proved the theorem.

128 CHAPTER 7. HYPERBOLICITY.

7.2 invariant manifolds

Let p be a hyperbolic ¬xed point of a di¬eomorphism, f . The stable manifold

of f at p is de¬ned as the set

W s (p) = W s (p, f ) = {x| lim f n (x) = p}. (7.9)

n’∞

Similarly, the unstable manifold of f at p is de¬ned as

W u (p) = W u (p, f ) = {x| lim f ’n (x) = p}. (7.10)

n’∞

We have de¬ned W s and W u as sets. We shall see later on in this section that

in fact they are submanifolds, of the same degree of smoothness as f . The

terminology, while standard, is unfortunate. A point which is not exactly on

W s (p) is swept away under iterates of f from any small neighborhood of p. This

is the content of our ¬rst proposition below. So it is a very unstable property

to lie on W s . Better terminology would be “contracting” and “expanding”

submanifolds. But the usage is standard, and we will abide by it. In any event,

the sets W s (p) and W u (p) are, by their very de¬nition, invariant under f .

In the case that f = A is a hyperbolic linear transformation on a Banach

space E = S • U , then W s (0) = S and W u (0) = U as follows immediately

from the de¬nitions. The main result of this section will be to prove that in the

general case, the stable manifold of f at p will be a submanifold whose tangent

at p is the stable subspace of the linear transformation dfp .

Notice that for a hyperbolic ¬xed point, replacing f by f ’1 interchanges the

roles of W s and W u . So in much of what follows we will formulate and prove

theorems for either W s or for W u . The corresponding results for W u or for W s

then follow automatically.

Let A be a hyperbolic linear transformation on a Banach space E = S • U ,

and consider any ball, Br = Br (0) of radius r about the origin. If x ∈ Br does

not lie on S © Br , this means that if we write x = xs • xu with xs ∈ S and

xu ∈ U then xu = 0. Then

An x A n xs + A n xu

=

A n xu

≥

≥ c n xu .

If we choose n large enough, we will have cn xu > r. So eventually, An x ∈ Br .

Put contrapositively,

S © Br = {x ∈ Br |An x ∈ Br ∀n ≥ 0}.

Now consider the case of a hyperbolic ¬xed point, p, of a di¬eomorphism, f . We

may introduce coordinates so that p = 0, and let us take A = df0 . By the C 0

conjugacy theorem, we can ¬nd a neighborhood, V of 0 and homeomorphism

h : Br ’ V

129

7.2. INVARIANT MANIFOLDS

with

h —¦ f = A —¦ h.

Then

f n (x) = h’1 —¦ An —¦ h (x)

will lie in U for all n ≥ 0 if and only if h(x) ∈ S(A) if and only if An h(x) ’ 0.

This last condition implies that f n (x) ’ p. We have thus proved

Proposition 7.2.1 Let p be a hyperbolic ¬xed point of a di¬eomorphism, f .

For any ball, Br (p) of radius r about p, let

Br (p) = {x ∈ Br (p)|f n (x) ∈ Br (p)∀n ≥ 0}.

s s

(7.11)

Then for su¬ciently small r, we have

Br (p) ‚ W s (p).

s

s

Furthermore, our proof shows that for su¬ciently small r the set Br (p) is a

s

topological submanifold in the sense that every point of Br (p) has a neighbor-

s

hood (in Br (p)) which is the image of a neighborhood, V in a Banach space

under a homeomorphism, H. Indeed, the restriction of h to S gives the desired

homeomorphism.

Remark. In the general case we can not say that Br (p) = Br (p)©W s (p) because

s

a point may escape from Br (p), wander around for a while, and then be drawn

towards p.

But the proposition does assert that Br (p) ‚ W s (p) and hence, since W s is

s

invariant under f ’1 , we have

f ’n [Br (p)] ‚ W s (p)

s

for all n, and hence

f ’n [Br (p)] ‚ W s (p).

s

n≥0

On the other hand, if x ∈ W s (p), which means that f n (x) ’ p, eventually

f n (x) arrives and stays in any neighborhood of p. Hence p ∈ f ’n [Br (p)] for