= sin2 (π ’ πx)

= sin2 πx

= sin2 2θ

= 4 sin2 θ(1 ’ sin2 θ)

= L4 (h(x))

where we have used the fact that sin(π ’ ±) = sin ± to pass from the second line

to the third. So we have veri¬ed our claim in all cases.

Many interesting properties of a transformation are preserved under conju-

gation by a homeomorphism. (A homeomorphism is a bijective continuous map

with continuous inverse.) For example, if p is a periodic point of period n of f ,

so that f —¦n (p) = p, then

g —¦n (h(p)) = h —¦ f —¦n (p) = h(p)

if h —¦ f = g —¦ h. So periodic points are carried into periodic points of the same

period under a conjugacy. We will consider several other important properties

of a transformation as we go along, and will prove that they are invariant under

conjugacy. So what our result means is that if we prove these properties for

T , we conclude that they are true for Lµ . Since we have veri¬ed that L4 is

conjugate to Q’2 , we conclude that they hold for Q’2 as well.

Here is another example of a conjugacy, this time an a¬ne conjugacy. Con-

sider

V (x) = 2|x| ’ 2.

V is a map of the interval [’2, 2] into itself. Consider

h2 (x) = 2 ’ 4x.

So h2 (0) = 2, h2 (1) = ’2. In other words, h2 maps the interval [0, 1] in a one

to one fashion onto the interval [’2, 2]. We claim that

V —¦ h2 = h2 —¦ T.

Indeed,

V (h2 (x)) = 2|2 ’ 4x| ’ 2.

1

For 0 ¤ x ¤ 2 this equals 2(2 ’ 4x) ’ 2 = 2 ’ 8x = 2 ’ 4(2x) = h2 (T x). For

1

2 ¤ x ¤ 1 we have V (h2 (x)) = 8x ’ 6 = 2 ’ 4(2 ’ 2x) = h2 (T x). So we have

veri¬ed the required equation in all cases. The e¬ect of the a¬ne transformation,

h2 is to enlarge the graph of T , shift it, and turn it upside down. But as far as

iterations are concerned, these changes do not e¬ect the essential behavior.

3.2. CONJUGACY OF T AND L4 65

0.5

0

-0.5

V(x)

-1

-1.5

-2

-2.5

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

x

Figure 3.3: V (x) = 2|x| ’ 2.

66 CHAPTER 3. CONJUGACY

3.3 Chaos

A transformation F is called (topologically) transitive if for any two open (non

empty) intervals, I and J, one can ¬nd initial values in I which, when iterated,

will eventually take values in J. In other words, we can ¬nd an x ∈ I and an

integer n so that F n (x) ∈ J.

For example, consider the tent transformation, T . Notice that T maps the

interval [0, 1 ] onto the entire interval [0, 1], and also maps the interval [ 1 , 1] onto

2 2

the entire interval, [0, 1]. So T —¦2 maps each of the intervals [0, 1 ], [ 1 , 1 ], [ 1 , 3 ]

4 42 24

and [ 3 ] onto the entire interval [0, 1]. More generally, T —¦n maps each of the 2n

4

intervals [ 2k , k+1 ], 0 ¤ k ¤ 2n ’ 1 onto the entire interval [0, 1]. But any open

n 2n

interval I contains some interval of the form [ 2k , k+1 ] if we choose n su¬ciently

n 2n

large. For example it is enough to choose n so large that 23 is less than the

n

—¦n

length of I. So for this value on n, T maps I onto the entire interval [0, 1],

and so, in particular, there will be points, x, in I with F (x) ∈ J.

Proposition 3.3.1 Suppose that g —¦ h = h —¦ f where h is continuous and sur-

jective, and suppose that f is transitive. Then g is transitive.

Proof. We are given non-empty open I and J and wish to ¬nd an n and an

x ∈ I so that g —¦n (x) ∈ J. To say h is continuous means that h’1 (J) is a

union of open intervals. To say that h is surjective implies that h’1 (J) is not

empty. Let L be one of the intervals constituting h’1 (J). Similarly, h’1 (I) is

a union of open intervals. Let K be one of them. By the transitivity of f we

can ¬nd an n and a y ∈ K with f —¦n (y) ∈ L. Let x = h(y). Then x ∈ I and

g —¦n (x) = g —¦n (h(y)) = h(f —¦n (y)) ∈ h(L) ‚ J. QED.

As a corollary we conclude that if f is conjugate to g, then f is transitive

if and only if g is transitive. (Just apply the proposition twice, once with the

roles of f and g interchanged.) But in the proposition we did not make the

hypothesis that h was bijective or that it had a continuous inverse. We will

make use of this more general assertion.

A set S of points is called dense if every non-empty open interval, I, contains

a point of S. The behavior of density under continuous surjective maps is also

very simple:

Proposition 3.3.2 If h : X ’ Y is a continuous surjective map, and if D is

a dense subset of X then h(D) is a dense subset of Y .

Proof. Let I ‚ Y be a non-empty open interval. Then h’1 (I) is a union of

open intervals. Pick one of them, K and then a point y ∈ D © K which exists

since D is dense. But then f (y) ∈ f (D) © I. QED

We de¬ne PER(f ) to be the set of periodic points of the map f . If h—¦f = g—¦h,

then f —¦n (p) = p implies that g —¦n (h(p)) = h(f —¦n (p)) = h(p) so

h[PER(f )] ‚ PER(g).

67

3.4. THE SAW-TOOTH TRANSFORMATION AND THE SHIFT

In particular, if h is continuous and surjective, and if PER(f ) is dense, then so

is PER(g).

Following Devaney and recent work (1992) by J. Banks et.al. Amer. Math.

Monthly 99 (1992) 332-334, let us call f chaotic if f is transitive and PER(f )

is dense. It follows from the above discussion that

Proposition 3.3.3 If h : X ’ Y is surjective and continuous, if f : X ’ X

is chaotic, and if h —¦ f = g —¦ h, then g is chaotic.

We have already veri¬ed that the tent transformation, T , is transitive. We

claim that PER(T ) is dense on [0, 1] and hence that T is chaotic. To see this,