borhood of a, but it may also include points far away, and may be a very

complicated set as we saw in the example of Newton™s method applied to a

cubic.

A ¬xed point, r, is called a repeller or a repelling or an unstable ¬xed point

if

|F (r)| > 1. (1.33)

Points near a repelling ¬xed point (as in the case of our renormalization group

example, in the next section) are pushed away upon iteration.

An attractive ¬xed point s with

F (s) = 0 (1.34)

is called superattractive or superstable. Near a superstable ¬xed point, s, (as in

the case of Newton™s method) the iterates converge exponentially to s.

The notation F —¦n will mean the n-fold composition,

F —¦n = F —¦ F —¦ · · · —¦ F (ntimes).

A ¬xed point of F —¦n is called a periodic point of period n . If p is a periodic

point of period n, then so are each of the points

p, F (p), F —¦2 (p), . . . , F —¦(n’1) (p)

and the chain rule says that at each of these points the derivative of F —¦n is the

same and is given by

(F —¦n ) (p) = F (p)F (F (p)) · · · F (F —¦(n’1) (p)).

If any one of these points is an attractive ¬xed point for F n then so are all the

others. We speak of an attractive periodic orbit. Similarly for repelling.

A periodic point will be superattractive for F —¦n if and only if at least one of

the points p, F (p), . . . F —¦(n’1) (p) satis¬es F (q) = 0.

1.5 Renormalization group

We illustrate these notions in an example: consider a hexagonal lattice in the

plane. This means that each lattice point has six nearest neighbors. Let each

site be occupied or not independently of the others with a common probability

0 ¤ p ¤ 1 for occupation. In percolation theory the problem is to determine

whether or not there is a positive probability for an in¬nitely large cluster of

occupied sites. (By a cluster we mean a connected set of occupied sites.) We

plot some ¬gures with p = .2, .5, and .8 respectively. For problems such as this

there is a critical probability pc : for p < pc the probability of of an in¬nite cluster

is zero, while it is positive for for p > pc . One of the problems in percolation

theory is to determine pc for a given lattice.

19

1.5. RENORMALIZATION GROUP

Figure 1.1: p=.2

20 CHAPTER 1. ITERATIONS AND FIXED POINTS

Figure 1.2: p=.5

21

1.5. RENORMALIZATION GROUP

Figure 1.3: p=.8

22 CHAPTER 1. ITERATIONS AND FIXED POINTS

1

For the case of the hexagonal lattice in the plane, it turns out that pc = 2 .

We won™t prove that here, but arrive at the value 1 as the solution to a problem

2

which seems to be related to the critical probability problem in many cases. The

idea of the renormalization group method is that many systems exhibit a similar

behavior at di¬erent scales, a property known as self similarity. Understanding

the transformation properties of this self similarity yields important information

about the system. This is the goal of the renormalization group method.

Rather than attempt a general de¬nition, we use the hexagonal lattice as a

¬rst and elementary illustration. Replace the original hexagonal lattice by a

coarser hexagonal lattice as follows: pick three adjacent vertices on the original

hexagonal lattice which form an equilateral triangle. This then organizes the

lattice into a union of disjoint equilateral triangles, all pointing in the same

direction, where, alternately, two adjacent lattice points on a row form a base of

a triangle and the third lattice point is a vertex of a triangle from an adjacent

row . The center of these triangles form a new (coarser) hexagonal lattice, in

fact one where the distance between sites has been increased by a factor of three.

See the ¬gures.

Each point on our new hexagonal lattice is associated with exactly three

points on our original lattice. Now assign a probability, p to each point of our

new lattice by the principle of majority rule: a new lattice point will be declared

occupied if a majority of the associated points of the old lattice are occupied.

Since our triangles are disjoint, these probabilities are independent. We can

achieve a majority if all three sites are occupied (which occurs with probability

p3 ) or if two out of the three are occupied (which occurs with probability p2 (1’p)

with three choices as to which two sites are occupied). Thus

p = p3 + 3p2 (1 ’ p). (1.35)

This has three ¬xed points: 0, 1, 1 . The derivative at 1 is 2 > 1, so it is repelling.

3

2 2

The points 0 and 1 are superattracting. So starting with any p > 1 , iteration

2

leads rapidly towards the state where all sites are occupied, while starting with

p < 1 leads rapidly under iteration towards the totally empty state. The point

2

1

is an unstable ¬xed point for the renormalization transformation.

2

23

1.5. RENORMALIZATION GROUP

Figure 1.4: The original hexagonal lattice organized into groups of three adja-

cent vertices.

24 CHAPTER 1. ITERATIONS AND FIXED POINTS

Figure 1.5: The new hexagonal lattice with edges emanating from each vertex,

indicating the input for calculating p from p.

Chapter 2

Bifurcations

2.1 The logistic family.

In population biology one considers iteration of the “logistic function”

Lµ (x) = µx(1 ’ x). (2.1)

1

Here 0 < µ is a real parameter. The ¬xed points of Lµ are 0 and 1 ’ µ . Since

Lµ (x) = µ ’ 2µx,

1

Lµ (0) = µ, Lµ (1 ’ ) = 2 ’ µ.

µ

As x represents a proportion of a population, we are mainly interested only in

1

0 ¤ x ¤ 1. The maximum of Lµ is always achieved at x = 2 , and the maximum

value is µ . So for 0 < µ ¤ 4, Lµ maps [0, 1] into itself.

4

For µ > 4, portions of [0, 1] are mapped into the range x > 1. A second

operation of Lµ maps these points to the range x < 0 and then are swept o¬ to

’∞ under successive applications of Lµ .

We now examine the behavior of Lµ more closely for varying ranges of µ.

0<µ¤1

2.1.1