2.4 Feigenbaum renormalization.

We have already remarked that the rate of convergence to the limiting value

of the superstable points in the period doubling bifurcation, Feigenbaum™s con-

stant, is universal, i.e. not restricted to the logistic family. That is, if we let

δ = 4.6692....

denote Feigenbaum™s constant, then the superstable values sr in the period

doubling scenario satisfy

sr = s∞ ’ Bδ ’r + o(δ ’r )

where s∞ and B depend on the speci¬cs of the family, but δ applies to a large

class of such families.

There is another “universal” parameter in the story. Suppose that our family

fµ consists of maps with a single maximum, Xm , so that Xm must be one of

the points on any superstable periodic orbit. (In the case of the logistic family

1

Xm = 2 .) Let dr denote the di¬erence between Xm an the next nearest point

on the superstable 2r orbit; more precisely, de¬ne

r’1

2

(Xm ) ’ Xm .

d r = f sr

46 CHAPTER 2. BIFURCATIONS

Then

dr ∼ D(’±)r

where

.

± = 2.5029...

is again universal. This would appear to be a scale parameter (in x) associated

with the period doubling scenario. To understand this scale parameter, examine

the central portion of Fig 2.4 and observe that the graph of L—¦2 looks like an

µ

(inverted and) rescaled version of Lµ , especially if we allow a change in the

parameter µ. The rescaling is centered at the maximum, so in order to avoid

notational complexity, let us shift this maximum (for the logistic family) to the

1

origin by replacing x by y = x ’ 2 . In the new coordinates the logistic map is

given by

1 1 1 1

y ’ Lµ (y + ) ’ = µ( ’ y 2 ) ’ .

2 2 4 2

Let R denote the operator on functions given by

R(h)(y) := ’±h(h(y/±)). (2.12)

In other words, R sends a map h into its iterate h —¦ h followed by a rescaling.

We are going to not only apply the operator R, but also shift the parameter µ

in the maps

1 1

hµ (y) = µ( ’ y 2 ) ’

2 2

from one supercritical value to the next. So for each k = 0, 1, 2, . . . we set

gk0 := hsk

and then de¬ne

= Rgk0

gk,1

gk2 = Rgk1

gk3 = Rgk2

. .

. .

. .

It is observed (numerically) that for each k the functions gkr appear to be

approaching a limit, gk i.e.

gkr ’ gk .

So r

gk (y) = lim(’±)r gsk+r (y/(’±)r ).

2

Hence

Rgk = lim(’±)r+1 2r+1 gsk+r (y/(’±)r+1 ) = gk’1 .

It is also observed that these limit functions gk themselves are approaching a

limit:

gk ’ g.

47

2.4. FEIGENBAUM RENORMALIZATION.

Since Rgk = gk’1 we conclude that

Rg = g,

i.e. g is a ¬xed point for the Feigenbaum renormalization operator R. Notice

that rescaling commutes with R: If S denotes the operator (Sf )(y) = cf (y/c)

then

R(Sf )(y) = ’±(c(f (cf (y/(c±))/c) = S(R)f (y).

So if g is a ¬xed point, so is Sg. We may thus ¬x the scale in g by requiring

that

g(0) = 1.

The hope was then that there would be a unique function g (within an appro-

priate class of functions) satisfying

Rg = g, g(0) = 1,

or, spelling this out,

g(y) = ’±g —¦2 (’y/±), g(0) = 1. (2.13)

Notice that if we knew the function g, then setting y = 0 in (2.13) gives

1 = ’±g(1)

or

± = ’1/g(1).

In other words, assuming that we were able to establish all these facts and also

knew the function g, then the universal rescaling factor ± would be determined

by g itself. Feigenbaum assumed that g has a power series expansion in x2 took

the ¬rst seven terms in this expansion and substituted in (2.13). He obtained

a collection of algebraic equations which he solved and then derived ± close to

the observed “experimental” value. Indeed, if we truncate (2.13) we will get

a collection of algebraic equations. But these equations are not recursive, so

that at each stage of truncation modi¬cation is made in all the coe¬cients,

and also the nature of the solutions of these equations is not transparent. So

theoretically, if we could establish the existence of a unique solution to (2.13)

within a given class of functions the value of ± is determined. But the numerical

evaluation of ± is achieved by the renormalization property itself, rather than

from g(1) which is not known explicitly.

The other universal constant associated with the period doubling scenario,

the constant δ was also conjectured by Feigenbaum to be associated to the ¬xed

point g of the renormalization operator; this time with the linearized map J, i.e.

the derivative of the renormalization operator at its ¬xed point. Later on we will

see that in ¬nite dimensions, if the derivative J of a non-linear transformation

R at a ¬xed point has k eigenvalues > 1 in absolute value, and the rest < 1

in absolute value, then there exists a k-dimensional R invariant surface tangent

48 CHAPTER 2. BIFURCATIONS

at the ¬xed point to the subspace corresponding to the k eigenvalues whose

absolute value is > 1. On this invariant manifold, the map R is expanding.

Feigenbaum conjectured that for the operator R (acting on the appropriate

in¬nite dimensional space of functions) there is a one dimensional “expanding”

submanifold, and that δ is the single eigenvalue of J with absolute value greater

than 1.

In the course of the past twenty years, these conjectures of Feigenbaum

have been veri¬ed using high powered techniques from complex analysis, thanks

to the combined e¬ort of such mathematicians as Douady, Hubbard, Sullivan,

McMullen, and. . . .

2.5 Period 3 implies all periods

Throughout the following f will denote a continuous function on the reals whose

domain of de¬nition is assumed to include the given intervals in the various

statements.

Lemma 2.5.1 If I = [a, b] is a compact interval and I ‚ f (I) then f has a