for a full understanding, we will try to describe some of the issues involved in

the study of equations (4.2) and (4.1) from a more naive viewpoint. Consider,

for example, F = Lµ , 1 < µ < 3. If we start with any initial seed other than

x0 = 0, it is clear that the limiting probability is

p(Ik ) = 1,

1

if the ¬xed point,1 ’ ∈ Ik and

µ

p(Ik ) = 0

√

otherwise. Similarly, if 3 < µ < 1 + 6, and we start with any x0 other than 0

1

or the ¬xed point, 1 ’ µ then clearly the limiting probability will be p(I) = 1

if both points of period two belong to I, p(I) = 1 if I contains exactly one of

2

the two period two points, and p(I) = 0 otherwise. These are all examples of

discrete measures in the sense that there is a ¬nite (or countable) set of points,

{zk }, each assigned a positive number, m(zk ) and

µ(I) = m(zk ).

zk ∈I

We are making the implicit assumption that this series converges for every

bounded interval. The integral of a function, φ, with respect to the discrete

measure, µ, denoted by φ, µ or by φµ is de¬ned as

φµ = φ(xk )m(xk ).

This de¬nition makes sense under the assumption that the series on the right

hand side is absolutely convergent. The rule for computing the push forward,

87

4.1. HISTOGRAMS AND INVARIANT DENSITIES

F— µ (when de¬ned) is very simple. Indeed, let {yl } be the set of points of the

form yl = F (xk ) for some k, and set

n(yl ) = m(xk ).

F (xk )=yl

Notice that there is some problem with this de¬nition if there are in¬nitely

many points xk which map to the same yl . Once again we must make some

convergence assumption. For example, if the map F is everywhere ¬nite-to-one,

there will be no problem. Thus the push forward of a discrete measure is a

discrete measure given by the above formula.

At the other extreme, a measure is called absolutely continuous (with

respect to Lebesgue measure) if there is an integrable function, ρ, called the

density so that

µ(I) = ρ(x)dx.

I

For any continuous function, φ we de¬ne the integral of φ with respect to µ as

φ, µ = φµ = φ(x)ρ(x)dx

if the integral is absolutely convergent. Suppose that the map F is piecewise

di¬erentiable and in fact satis¬es |F (x)| = 0 except at a ¬nite number of points.

These points are called critical points for the map F and their images are called

critical values. Suppose that A is an interval containing no critical values, and

to ¬x the ideas, suppose that F ’1 (A) is the union of ¬nitely many intervals, Jl

each of which is mapped monotonically (either strictly increasing or decreasing)

onto A. The change of variables formula from ordinary calculus says that for

any function g = g(y) we have

g(y)dy = g(F (x))|F (x)|dx,

A Jk

where y = F (x). So if we set g(y) = ρ(x)|1/F (x)| we get

1

ρ(x) dy = ρ(x)dx = µ(Jk ).

|F (x)| Jk

Summing over k and using the de¬nition (4.2) we see that F— µ has the density

ρ(x)

σ(y) = . (4.3)

|F (x)|

F (xk )=y

Equation (4.3) is sometimes known as the Perron Frobenius equation, and the

transformation ρ ’ σ as the Perron Frobenius operator.

Getting back to our histogram, if we expect the limit measure to be of the

absolutely continuous type, so

1

p(Ik ) ≈ ρ(x) — , x ∈ Ik

N

88 CHAPTER 4. SPACE AND TIME AVERAGES

then we expect that

nk N

ρ(x) ≈ lim , x ∈ Ik

m’∞ m + 1

as the formula for the limiting density.

the histogram of L4

4.2

We wish to prove the following assertions:

(i) The measure, µ, with density

1

σ(x) = (4.4)

x(1 ’ x)

π

is invariant under L4 . In other words it satis¬es

L4— µ = µ.

(ii) Up to a multiplicative constant, (4.4) is the only continuous density invariant

under L4

(iii) If we pick the initial seed generically, then the normalized histogram con-

verges to (4.4).

We give two proofs of (i). The ¬rst is a direct veri¬cation of the Perron

Frobenius formula (4.3) with y = F (x) = 4x(1 ’ x) so |F (x)| = |F (1 ’ x)| =

4|1 ’ 2x|. Notice that the σ given by (4.4) satis¬es σ(x) = σ(1 ’ x) so (4.3)

becomes

1 2

= .

4x(1 ’ x)(1 ’ 4x(1 ’ x)) π4|1 ’ 2x| x(1 ’ x)

π

But this follows immediately from the identity

1 ’ 4x(1 ’ x) = (2x ’ 1)2 .

For our second proof, consider the tent transformation, T . For any interval, I

contained in [0, 1], T ’1 (I) consists of the union of two intervals, each of half the

length of I. In other words the ordinary Lebesgue measure is preserved by the

tent transformation: T— ν = ν where ν has density ρ(x) ≡ 1. Put another way,

the function ρ(x) ≡ 1 is the solution of the Perron Frobenius equation

ρ(x) ρ(1 ’ x)

ρ(T x) = + . (4.5)

2 2

It follows immediately from the de¬nitions, that

(F —¦ G)— µ = F— (G— µ),

where F and G are two transformations , and µ is a measure. In particular,

since h —¦ T = L4 —¦ h where

πx

h(x) = sin2 ,

2

4.2. THE HISTOGRAM OF L4 89

600