ńņš. 22 |

functions deļ¬ned on [0, 1] and satisfying

f (0) = 0

g(0) = 0

f (1) = 1

g(1) = 1

f (x) < x āx = 0, 1

g(x) < x āx = 0, 1.

Then there exists a continuous, monotone increasing function h deļ¬ned on [0, 1]

with

h(0) = 0, h(1) = 1,

and

h ā—¦ f = g ā—¦ h.

Proof. Choose any point (x0 , y0 ) in the open square

0 < x < 1, 0 < y < 1.

If (x0 , y0 ) is to be a point on the curve y = h(x), then the equation h ā—¦ f = g ā—¦ h

implies that the point (x1 , y1 ) also lies on this curve, where

x1 = f (x0 ), y1 = g(y0 ).

By induction so will the points (xn , yn ) where

xn = f n (x0 ), yn = g n (y0 ).

By hypothesis

x0 > x1 > x2 > ...,

and since there is no solution to f (x) = x for 0 < x < 1 the limit of the

xn , n ā’ ā must be zero. Also for the yn . So the sequence of points (xn , yn )

approaches (0, 0) as n ā’ +ā. Similarly, as n ā’ ā’ā the points (xn , yn )

approach (1, 1). Now choose any continuous, strictly monotone function

y = h(x),

deļ¬ned on

x1 ā¤ x ā¤ x 0

with

h(x1 ) = y1 , h(x0 ) = y0 .

Extend its deļ¬nition to the interval x2 ā¤ x ā¤ x1 by setting

h(x) = g(h(f ā’1 (x))), x2 ā¤ x ā¤ x1 .

74 CHAPTER 3. CONJUGACY

Notice that at x1 we have

g(h(f ā’1 (x1 ))) = g(h(x0 )) = g(y0 ) = y1 ,

so the deļ¬nitions of h at the point x1 are consistent. Since f and g are monotone

and continuous, and since h was chosen to be monotone on x1 ā¤ x ā¤ x0 ,

we conclude that h is monotone on x2 ā¤ x ā¤ x1 and hence continuous and

monotone on all of x2 ā¤ x ā¤ x0 . Continuing in this way, we deļ¬ne h on the

interval xn+1 ā¤ x ā¤ xn , n ā„ 0 by

h = g n ā—¦ h ā—¦ f ā’n .

Setting h(0) = 0, we get a continuous and monotone increasing function deļ¬ned

on 0 ā¤ x ā¤ x0 . Similarly, we extend the deļ¬nition of h to the right of x0 up

to x = 1. By its very construction, the map h conjugates f into g, proving the

proposition.

Notice that as a corollary of the method of proof, we can conclude

Proposition 3.6.2 Let f and g be two monotone increasing functions deļ¬ned

in some neighborhood of the origin and satisfying

f (0) = g(0) = 0, |f (x)| < |x|, |g(x)| < |x|, āx = 0.

Then there exists a homeomorphism, h deļ¬ned in some neighborhood of the

origin with h(0) = 0 and

h ā—¦ f = g ā—¦ h.

Indeed, just apply the method (for n ā„ 0) to construct h to the right of the

origin, and do an analogous procedure to construct h to the left of the origin.

As a special case we obtain

Proposition 3.6.3 Let f and g be diļ¬erentiable functions with f (0) = g(0) = 0

and

0 < f (0) < 1, 0 < g (0) < 1. (3.6)

Then there exists a homeomorphism h deļ¬ned in some neighborhood of the origin

with h(0) = 0 and which conjugates f into g.

The mean value theorem guarantees that the hypotheses of the preceding propo-

sition are satisļ¬ed.

Also, it is clear that we can replace (3.6) by any of the conditions

1 < f (0), 1 < g (0)

0 > f (0) > ā’1, 0 > g (0) > ā’1

ā’1 > f (0), ā’1 > g (0),

and the conclusion of the proposition still holds.

It is important to observe that if f (0) = g (0), then the homeomorphism,

h, can not be a diļ¬eomorphism. That is, h can not be diļ¬erentiable with

75

3.7. SEQUENCE SPACE AND SYMBOLIC DYNAMICS.

a diļ¬erentiable inverse. In fact, h can not have a non-zero derivative at the

origin. Indeed, diļ¬erentiating the equation g ā—¦ h = h ā—¦ f at the origin gives

g (0)h (0) = h (0)f (0),

and if h (0) = 0 we can cancel it form both sides of the equation so as to obtain

f (0) = g (0). (3.7)

What is true is that if (3.7) holds, and if

|f (0)| = 1, (3.8)

then we can ļ¬nd a diļ¬erentiable h with a diļ¬erentiable inverse which conjugates

f into g.

We postpone the proof of this result until we have developed enough ma-

chinery to deal with the n-dimensional result. These theorems are among my

earliest mathematical theorems. A complete characterization of transformations

of R near a ļ¬xed point together with the conjugacy by smooth maps if (3.7)

and (3.8) hold, were obtained and submitted for publication in 1955 and pub-

lished in the Duke Mathematical Journal. The discussion of equivalence under

homeomorphism or diļ¬eomorphism in n-dimensions was treated for the case of

contractions in 1957 and in the general case in 1958, both papers appearing

in the American Journal of Mathematics. We will return to these matters in

Chapter ??.

3.7 Sequence space and symbolic dynamics.

In this section we will illustrate a powerful method for studying dynamical

systems by examining the quadratic transformation

Qc : x ā’ x2 + c

for values of c < ā’2.

For any value of c, the two possible ļ¬xed points of Qc are

ā ā

1 1

(1 ā’ 1 ā’ 4c), (1 + 1 ā’ 4c)

pā’ (c) = p+ (c) =

2 2

by the quadratic formula. These roots are real with pā’ (c) < p+ (c) for c < 1/4.

The graph of Qc lies above the diagonal for x > p+ (c), hence the iterates of any

x > p+ (c) tend to +ā. If x0 < ā’p+ (c), then x1 = Qc (x0 ) > p+ (c), and so the

further iterates also tend to +ā. Hence all the interesting action takes place in

the interval [ā’p+ , p+ ]. The function Qc takes its minimum value, c, at x = 0,

and

ā

1

c = ā’p+ (c) = ā’ (1 + 1 ā’ 4c)

2

76 CHAPTER 3. CONJUGACY

when c = ā’2. For ā’2 ā¤ c ā¤ 1/4, the iterate of any point in [ā’p+ , p+ ] remains

ńņš. 22 |