In these notes I will use mathematical notation similar to the syntax of Mathematica.

11. Counting in the Mandelbrot set: an excursion into number theory.

I this lecture I will discuss some coutning problems
associated with periodic points in the Mandelbrot set. We saw in Lecture
7 that a point ** c** of period n must be a root of the
polynomial

`G[0, c_, z_] := z; G[n_, c_, z_] := G[n-1,
c, z]^2 + c.`

This polynomial is easily seen to degree 2^(n-1). This raises the follwoing questions.

- How do we know that
has`G[n, c, 0]`distinct roots?`2^(n-1)` - How many of these roots actually have period
, rather than some lower period?`n` - How many of the period n points are inside cardioids and how many are insdie buds?
- How many of them are real?

Each of these questions requires some elementry number theory to answer.

**Why are there 2^(n-1)**

Here are the first few values of ** G[n, c,
0]**, given by the code

`Do[Print["G[",n,", c, 0]
= ",InputForm[Expand[G[n,c,0]]]], {n, 2, 4}] `

The output is

`G[2, c, 0] = c + c^2 `

`G[3, c, 0] = c + c^2 + 2*c^3 + c^4 `

`G[4, c, 0] = c + c^2 + 2*c^3 + 5*c^4 +
6*c^5 + 6*c^6 + 4*c^7 + c^8 `

It is clear from th definition of ** G**
that the degree of this polynomial doubles each time we increase

The *Fundamental Theorem of Algebra* says
that any polynomial of degree d of the form

`p[c] = a[0] + a[1]*c + ... a[d-1]*c^(d-1)
+ c^d`

an be factored as

`p[c] = (c - r[1])(c - r[2])...(c - r[d]),`

and the complex numbers r[1] through r[d] are
its roots. In genral these do not have to be distinct. For example we could
have ** p[c] = (c - r)^d**, in which ase the roots would all
be the same.

There is a way to use the derivative of ** p**
at a root

`p[c] = (c - r)^2*q[c]`

where ** q[c]** is a polynomial of
degree

`p'[c] = 2(c-r)*q[c] + (c - r)^2*q[c]`

` = (c-r)*(2*q[c]
+ (c-r)*q[c]).`

Since this derivative is divisible by ** (c-r)**,
it must vanish when

*If the derivative of p does
not vanish at a root r, then that root only occurs once. *(11.1)

We will use this fact to prove that ** G[n,
c, 0]** has no repeated roots by showing that its derivative is
nonzero at each of its roots. By definition,

We want to show that this is nonzero whenever
** c** is a root

Now suppose that the root ** r** is
a whole number. (This is almost never the case, but nevermind.) The polynomials
appearing in (11.2) all have integer coefficients, so the expression on
the right is an odd number and hence nonzero. This means that

*An integer root of *`G[n,
c, 0]`* cannot be a repeated root*. (11.3)

This statement as it stands is not worth very
much, because the only integers that are ever roots of a ** G[n, c,
0] **are 0 and -1, but there is a way to leverage this idea into
something more useful. The key concept is the following.

*Definition 11.4*.
*A complex number z is an algebraic
integer if it is the root of a polynomial
of the form*

`p[z] = a[0] + a[1]*z + ... a[d-1]*z^(d-1)
+ z^d`

*here the coefficients *`a[0]`*
though *`a[d-1]`* are all integers.*

Algebraic integers are discussed in many textbooks on number theory. They have many of the same formal properties as ordinary integers. The sum, difference and product of any two algebraic integers is again an algebraic integer, and one can distinguish between those that are even (i.e. two times another algebraic integer) and those which are odd.

Algebraic integers are relevant here because each
root of ** G[n, c, 0] **is one by definition, since

Why does this argument not apply to preperiodic points?

**How many
period n points are there?**

We now know that ** G[n, c, 0] **has
exactly

` 1 = Period[1]`

` 2 = Period[2] + Period[1]`

` 4 = Period[3] + Period[1]`

` 8 = Period[4] + Period[2] + Period[1]`

`16 = Period[5] + Period[1]`

`32 = Period[6] + Period[3] + Period[2]
+ Period[1]`

` etc. `

It is possible to solve these equations for ** Period[n]**
in terms the values of

** 2^(n-1) == Sum[Period[Divisors[n][[i]]],
{i, 1, Length[Divisors[n]]}] **(11.4)

This notation is cumbersome so we shorten it by defining

`DivisorSum[f_, n_] := `

` Sum[f[Divisors[n][[i]]],{i,1,Length[Divisors[n]]}]`

so that (11.4) can be rewritten as

`2^(n-1) = DivisorSum[Period, n]`

There is a method for solving equations of this
type called the *Moebius inversion formula*. The Möbius
function ** MoebiusMu[n]** is defined to be

`f[n] = DivisorSum[g, n],`

in other words ** f[n]** is the sum
over all divisors

`MoebiusDivisorSum[f_, n_] := `

` Sum[MoebiusMu[n/Divisors[n][[i]]]*f[Divisors[n][[i]]],{i,1,Length[Divisors[n]]}]`

Then the inversion formula says that

in other words** g[n]** is a similar
sum of the quantities

** Period[n_] := MoebiusDivisorSum[h,
n], **(11.6)

in other words,` Period[n]`* *is
the sum over all divisors ** d** of

For example for a prime number p, (11.6) gives

`Period[p] = 2^(p-1) - 1`

`Period[p^2] = 2^(p^2-1) - 2^(p-1),`

and if q is another prime number we get

`Period[p*q] = 2^(p*q-1) - 2^(p-1) - 2^(q-1)
+ 1.`

**How many period n points are
in buds?**

Now that we know the number ** Period[n]**
of points of period

For this we need the Euler
totient function ** EulerPhi[n]**, which is defined to be
the number of integers between 0 and

Let ** Buds[n]** be the number of period

Now suppose we want to determine ** Buds[10]**,
the numebr of period 10 buds attached directly or indirectly to the main
cardioid. There are

primary buds of period 10,`EulerPhi[10] = 4`secondary bud of period 10 attached to each of the Buds[5] = 4 buds of period 5, and`EulerPhi[2] = 1`secondary buds of period 10 attached to the one bud of period 2.`EulerPhi[5] = 4`

It is known that the sum over all divisors ** d**
of

**How many period n points are
real?**

*This page was last revised on February 26, 1998.*