is bounded when starting from the point z₀=0. For example the above image is for m=4. A significant subset of this set consists of those values of c such that the iteration scheme approaches limits for which

z_n+1 = z_n.

Such a limit point z* satisfies the equation

z* = z*^m +c

For any c there is a limit point z*; i.e., such that if z₀=z* the iteration will remain at z* forever.

The crucial question is what are the limit points that are stable so that the iteration starting from z₀=0 will approach them.

Consider the deviations of the iteration values from the corresponding limit point; i.e.,

z_n+1 = z_n^m + c
z* = z*^m + c
Subtraction gives
z_n+1-z* = z_n^m - z*^m
= (z_n-z*)(Σz_n^m-1-jz*^j)
where j runs from 0 to m-1.

Thus

|z_n+1-z*| will be less than |z_n-z*| if |(Σz_n^m-1-jz*^j)|<1

For values of z_n close to z* this reduces to m|z*^m-1|<1. The boundary between the stable and unstable limit points is given by

|z*|=(1/m)^1/(m-1).

Such limit points are given by the equation

z* = (1/m)^1/(m-1)*e^iφ
for 0≤φ≤2π

It is noteworthy that the stability of points is independent of the value of the iteration parameter c. The values of c which give those limit points are simply

c = z* - z*^m
c = (1/m)^1/(m-1)cos(φ)−(1/m)^m/(m-1)cos(mφ)
+ i[(1/m)^1/(m-1)sin(φ)−(1/m)^m/(m-1)sin(mφ)]

This equation is a parametric equation for the set of c values. It shows how the points on the circle of radius (1/m)^1/(m-1) in the z* space map into the c space.

The plots below shows the full set of c values for m=2, 3, 4, 5, 6, 7 and 8.

The main structure of the Mandelbrot set for m has (m-1) lobes. It has the symmetry of the (m-1) cyclic group; i.e., a rotation through an angle of 2π/(m-1) leaves the figure unchanged. It also has a symmetry with respect to the real axis. Thus it has the symmetry of the (m-1) dihedral group. The complete Mandelbrot sets have this same symmetry.

Limit Two-Cycles

The iteration may approach a limit cycle rather than a limit point. For a two-period cycle of z₁* and z₂* the values would have to satisfy the equations

z₂* = z₁*^m + c
and
z₁* = z₂*^m + c

From these equations defining z₁* and z₂* it follows that

z₂*−z₁* = -(z₂*−z₁*)(z₂*^m-1+z₁*z₂*^m-2+z₁*²z₂^m-3+· · · + z₁*^m-1)
and with division by (z₁*−z₂*)
which is valid if z₂*≠z₁*
−1 = (z₂*^m-1+z₁*z₂*^m-2+z₁*²z₂*^m-3+· · · + z₁*^m-1)

This is a preliminary result which will be used later. For convenience let

C(x,y,k) = x^k + x^k-1y + x^k-2y² + · · · + y^k

Thus the above result can be expressed as

S(z₁*, z₂*, m-1) = −1

The Two-Cycle Values

The deviations from the two-cycle satisfy the equations

z_n+2-z₂* = (z_n+1-z₁*)S(z_n+1, z₁, m-1)
and
z_n+1-z₁* = (z_n-z₂*)S(z_n, z₂, m-1)
hence
z_n+2-z₂* = (z_n-z₂*)S(z_n+1, z₁, m-1)S(z_n, z₂, m-1)

Therefore if |z_n+2-z₂*| is to be less than |z_n-z₂*| it must be that

|S(z_n+1, z₁, m-1)S(z_n, z₂, m-1)| < 1

For values very close to a cycle pair this reduces to:

|(mz₁*^m-1)(mz₂*^m-1)| < 1
or, equivalently
|z₂*z₁*| < (1/m²)^1/(m-1)

For the boundary of the stable set equality prevails; i.e.,

|z₂*z₁*| = (1/m²)^1/(m-1)

This means that z₂*z₁* is on a circle in the complex plane of radius (1/m²)^1/(m-1) . This means that

z₂*z₁* = (1/m²)^1/(m-1)e^iφ
= (1/m)^2/(m-1)e^iφ
= m^−2/(m-1)e^iφ

Now consider the equation which determines the values for a two-cycle. It was previously established that

−1 = (z₂*^m-1+z₁*z₂*^m-2+z₁*²z₂*^m-3+· · · + z₁*^m-1)

Multiplying this equation by z₁*^m-1 and grouping together terms of the form (z₁*z₂*)^k gives

−z₁*^m-1 = (re^iφ)^m-1 + (re^iφ)^m-2z₁*² + (re^iφ)^m-3z₁*⁴ + · · · + z₁*^2(m-1)

where re^iφ=z₁*z₂*, r being equal to (1/m²)^1/(m-1). This is the polynomial equation whose solutions give z₁. Because of symmetry of the previous equation with respect to z₁ and z₂ the equation for z₂ is exactly the same. The order of the polynomial is 2(m-1) so the solutions are grouped into (m-1) pairs.

The Bifurcation Points Between the Points
Associated with Limit Points and Limit Two-Cycles

There are roots of a polynomial which are of multiplicity two. These correspond to the case in which an infinitesimal change in coefficients of the polynomial would result in two nearby roots rather than one root. The condition for a root of multiplicity two is it makes both the polynomial and its derivative equal to zero.

Before examining the case of the boundary between sequences converging to a fixed point (if effect one-cycles) and sequences converging to a two cycle, it worthwhile to look at the case of two of the fixed point solutions converging. This not the same as the elements for a two-cycle converging into a one-cycle.

A complex number z' is of multiplicity two if

z'^m − z' + c = 0
and
mz'^m-1 − 1 =0

Therefore, from the second equation

z' = (1/m)^1/(m-1)e^{ik(2π/(m-1))} for k=1, 2, ..., (m-1)

In order for these values to be roots of the polynomial it is necessary that

c = (1/m)^1/(m-1)e^{ik(2π/(m-1))} − (1/m)^m/(m-1)e^{ik(2πm/(m-1))}

These values of c are just discrete points on the boundary curve of the main body of the Mandelbrot set. They are separated by an angle of 2π/(m-1). The diagram below shows these points in green for the case of m=4.

Now let us consider the bifurcation points. Since the condition for stability is

|z₂*z₁*| = (1/m²)^1/(m-1)

if z₂* → z₁* then the condition to be satisfied by z₁* is

|z₁*²| = (1/m²)^1/(m-1)
and hence
z₁*² = (1/m²)^1/(m-1)e^ikθ
where θ=(2π)/(m-1) and k runs from 1 to (m-1)

The condition on z₁*² means that

z₁* = ±(1/m)^1/(m-1)e^ikθ/2 for k=1,...,(m-1)
or, equivalently
z₁* = (1/m)^1/(m-1)e^ikθ/2 for k=1,...,2(m-1)

The plot of the values of c corresponding to these values of z₁* are shown below.

The points shown in blue are the ones which correspond to the bifurcation points. Those in green at the cusps of the curve correspond to multiplicity-two fixed point solutions.

It is shown elsewhere that the entire Mandelbrot set for exponent m has this type of symmetry for rotations through an angle of 2π/(m-1).

The Boundary Curve Between
Stable and Unstable Two-Cycles

If w is defined as z₁/(re^iφ)^1/2 the polynomial can be put into the form

1 + w² + w⁴ + · · · + w^2(m-1) = −w^m-1/(re^iφ)^(m-1)/2
or, equivalently
(w^2m-1)/(w²-1) = = −w^m-1/(re^iφ)^(m-1)/2

In principle for a given value of m the polynomial can be solved for each value of φ and the value of c determined from

c = z₁* − z₂*^m

where z₂* = re^iφ/z₁*.

Multiplying the above equation through by (w²-1) introduces an additional two roots of w=±1 but puts the polynomial into a more amenable form of

w^2m + (w^m+1-w^m-1)/(re^iφ) −1 = 0
and with division by w^m the even more amenable form
w^m−w^−m + (w-w^-1)/(re^iφ) = 0

This above form can be used for obtaining the numerical solutions. There are (m-1) different components. These can be obtained by starting with the (m-1) bifurcation points; i.e.,

z*_k = r exp^i(k+½)θ₀
and hence
w_k = r^½ exp^{i(k+½)(θ₀/2)}
for k=0,1,..(m-2)

where θ₀ = 2π/(m-1) and r=(1/m)^1/(m-1).

These values for w are entered into the polynomial and the values of φ to which they correspond are determined, say {φ_k}. The polynomial is then solved for values of φ in the range [φ_k,φ_k+2π]. The solutions for w are converted into values for z*₁ and then into values for c.

(To be continued.)

For more on the structure of other mandelbrot sets Click here.