c’

The estimate on the Lipschitz constant comes from

Lip[Gf (v)] ¤ Lip[fs —¦ (v, id)]Lip[(fu —¦ (v, id)]

1

¤ Lip[fs ]Lip[v]Lip · ’1

c’

1

¤ (Lip[As ] + Lip[ps —¦ (f ’ A)]) ·

c’1 ’

c+

¤ .

c’1’

133

7.2. INVARIANT MANIFOLDS

In going from the ¬rst line to the second we have used the preceding lemma.

In particular, if

2 < c’1 ’ c (7.18)

then

Lip[Gf (v)] ¤ 1.

Let us now obtain a condition on δ which will guarantee that

Gf (v)(U (r) ‚ S(r).

Since

fu —¦ (v, id)U (r) ⊃ U (r),

we have

[fu —¦ (v, id)]’1 U (r) ‚ U (r).

Hence, from the de¬nition of Gf (v), it is enough to arrange that

fs —¦ (v, id)[U (r)] ‚ S(r).

For x ∈ U (r) we have

¤ ps —¦ (f ’ A)(v(x), x) + As v(x)

fs (v(x), x)

¤ |(f ’ A)(v(x), x)| + c v(x)

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

¤ |(v(x), x)| + δ + cr

¤ r + δ + cr.

So we would like to have

( + c)r + δ < r

or

δ ¤ r(1 ’ c ’ ). (7.19)

If this holds, then Gf maps X into X.

We now want conditions that guarantee that Gf is a contraction on X,

where we take the sup norm. Let (w, x) be a point in S(r) • U (r) such that

fu (w, x) ∈ U (r). Let v ∈ X, and consider

|(w, x) ’ (v(x), x)| = w ’ v(x) ,

which we think of as the distance along S from the point (w, x) to graph(v).

Suppose we apply f . So we replace (w, x) by f (w, x) = (fs (w, x), fu (w, x)) and

graph(v) by f (graph(v)) = graph(Gf (v)). The corresponding distance along S

is fs (w, x) ’ Gf (v)(fu (w, x) . We claim that

fs (w, x) ’ Gf (v)(fu (w, x)) ¤ (c + 2 ) w ’ v(x) . (7.20)

Indeed,

fs (v(x), x) = Gf (v)(fu (v(x), x)

134 CHAPTER 7. HYPERBOLICITY.

by the de¬nition of Gf , so we have

fs (w, x) ’ Gf (v)(fu (w, x)) ¤ fs (w, x) ’ fs (v(x), x) +

+ Gf (v)(fu ((v(x), x) ’ Gf (v)(fu (w, x))

¤ Lip[fs ]|(w, x) ’ (v(x), x)| +

+Lip[fu ]|(v(x), x) ’ (w, x)|

¤ Lip[fs ’ ps A + ps A] w ’ v(x) +

+Lip[fu ’ pu A] w ’ v(x)

¤ ( + c + ) w ’ v(x)

which is what was to be proved.

Consider two elements, v1 and v2 of X. Let z be any point of U (r), and

apply (7.20) to the point

(w, x) = (v1 ([fu —¦ (v1 , id)]’1 ](z)), [fu —¦ (v1 , id)]’1 ](z))

which lies on graph(v1 ), and where we take v = v2 in (7.20). The image of

(w, x) is the point (Gf (v1 )(z), z) which lies on graph(Gf (v1 )), and, in particular,

fu (w, x) = z. So (7.20) gives

Gf (v1 )(z)’Gf (v2 )(z) ¤ (c+2 ) v1 ([fu —¦(v1 , id)]’1 ](z))’v2 ([fu —¦(v1 , id)]’1 ](z) .

Taking the sup over z gives

Gf (v1 ) ’ Gf (v2 ) ¤ (c + 2 ) v1 ’ v2 sup . (7.21)

sup

Intuitively, what (7.20) is saying is that Gf multiplies the S distance between

two graphs by a factor of at most (c + 2 ). So Gf will be a contraction in the

sup norm if

2 < 1’c (7.22)

which implies (7.18). To summarize: we have proved that Gf is a contraction

in the sup norm on X if (7.17), (7.19) and (7.22) hold, i.e.

δ < r min(c’1 ’ 1 ’ 2 , 1 ’ c ’ ).

2 < 1 ’ c,

Notice that since c < 1, we have c’1 ’ 1 > 1 ’ c so both expressions occurring

in the min for the estimate on δ are positive.

Now the uniform limit of continuous functions which all have Lip[v] ¤ 1 has

Lipschitz constant ¤ 1. In other words, X is closed in the sup norm as a subset

of the space of continuous maps of U (r) into S(r), and so we can apply the

contraction ¬xed point theorem to conclude that there is a unique ¬xed point,

g ∈ X of Gf . Since g ∈ X, condition (i) of the theorem is satis¬ed. As for (ii),

let (g(x), x) be a point on graph(g) which is the image of the point (g(y), y)

under f , so

(g(x), x) = f (g(y), y)

135

7.2. INVARIANT MANIFOLDS

which implies that

x = [fu —¦ (g, id)](y).

We can write this equation as

pu —¦ f|graph(g) = [fu —¦ (g, id)] —¦ (pu )|graph(g) .

In other words, the projection pu conjugates the restriction of f to graph(g)

into [fu —¦ (g, id)]. Hence the restriction off ’1 to graph(g) is conjugated by pu

into [fu —¦ (g, id)]’1 . But, by (7.16), the map [fu —¦ (g, id)]’1 is a contraction since

c’1 ’ 1 > 1 ’ c > 2

so

c’1 ’ > 1 + > 1.

The fact that Lip[g] ¤ 1 implies that

|(g(x), x) ’ (g(y), y)| = x ’ y

since we are using the box norm. So the restriction of pu to graph(g) is an

isometry between the (restriction of) the box norm on graph(g)and the norm

on U . So we have proved statement (ii), that the restriction of f ’1 to graph(g)

is a contraction.

We now turn to statement (iii) of the theorem. Suppose that (w, x) is a

point in S(r) • U (r) with f (w, x) ∈ S(r) • U (r). By (7.20) we have

fs (w, x) ’ g(fu (w, x) ¤ (c + 2 ) w ’ g(x)

since Gf (g) = g. So if the ¬rst n iterates of f applied to (w, x) all lie in

S(r) • U (r), and if we write

f n (w, x) = (z, y),

we have

z ’ g(y) ¤ (c + 2 )n w ’ g(x) ¤ (c + 2 )r.

f n (S(r) • U (r)) we must have z = g(y), in other

So if the point (z, y) is in

words

f n (S(r) • U (r)) ‚ graph(g).

But

graph(g) = f [graph(g)] © [S(r) • U (r)]

so

f n (S(r) • U (r)),

graph(g) ‚

proving that