0.4

0.3

0.2

0.1

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

√

Figure 2.12: Graphical iteration of P = L—¦3 with µ = 1 + 8 ’ .002 and initial

µ

point .5034. The solid lines are iteration steps of size less than .07 representing

bottleneck steps. The dotted lines are the longer steps.

57

2.6. INTERMITTENCY.

0.6

0.55

0.5

0.45

0.4

0.4 0.45 0.5 0.55 0.6

Figure 2.13: Zooming in on the central portion of the preceding ¬gure.

58 CHAPTER 2. BIFURCATIONS

Chapter 3

Conjugacy

3.1 A¬ne equivalence

An a¬ne transformation of the real line is a transformation of the form

x ’ h(x) = Ax + B

where A and B are real constants with A = 0. So an a¬ne transformation

consists of a change of scale (and possibly direction if A < 0) given by the

factor A, followed by a shift of the origin given by B. In the study of linear

phenomena, we expect that the essentials of an object be invariant under a

change of scale and a shift of the origin of our coordinate system.

For example, consider the logistic transformation, Lµ (x) = µx(1 ’ x) and

the a¬ne transformation

µ

hµ (x) = ’µx + .

2

We claim that

hµ —¦ Lµ —¦ h’1 = Qc (3.1)

µ

where

Qc (x) = x2 + c (3.2)

and where c is related to µ by the equation

µ2 µ

c=’ +. (3.3)

4 2

In other words, we are claiming that if c and µ are related by (3.3) then we have

hµ (Lµ (x)) = Qc (hµ (x)).

To check this, the left hand side expands out to be

µ µ

= µ 2 x2 ’ µ 2 x + ,

’µ[µx(1 ’ x)] +

2 2

59

60 CHAPTER 3. CONJUGACY

while the right hand side expands out as

µ 2 µ2 µ µ

+ = µ 2 x2 ’ µ 2 x +

(’µx + ) ’

2 4 2 2

giving the same result as before, proving (3.1).

2

We say that the transformations Lµ and Qc , c = ’ µ + µ are conjugate by

4 2

the a¬ne transformation, hµ .

More generally, let f : X ’ X and g : Y ’ Y be maps of the sets X and Y

to themselves, and let h : X ’ Y be a one to one map of X onto Y . We say

that h conjugates f into g if

h —¦ f —¦ h’1 = g,

or, what amounts to the same thing, if

h —¦ f = g —¦ h.

We shall frequently write this equation in the form of a commutative diagram

f

X ’’’ X

’’

¦ ¦

¦ ¦

h h

Y ’’’ Y

’’

g

The statement that the diagram is commutative means that going along the

upper right hand path (so applying h —¦ f ) is equal to traversing the left lower

path (which is g —¦ f ).

Notice that if h —¦ f —¦ h’1 = g, then

g circn = h —¦ f —¦n —¦ h’1 .

So the problem of studying the iterates of g is the same (up to the transformation

h) as that of f , providing that the properties we are interested in studying are

not destroyed by h.

Certainly a¬ne transformations will always be allowed. Let us generalize

the preceding computation by showing that any quadratic transformation (with

non-vanishing leading term) is conjugate (by an a¬ne transformation) to a

transformation of the form Qc for suitable c. More precisely:

Proposition 3.1.1 Let f = ax2 + bx + d then f is conjugate to Qc by the a¬ne

map h(x) = Ax + B where

b b2

b

A = a, B = , and c = ad + ’ .