where c, z and w are complex numbers. (The integers and other real numbers are included within the set of complex numbers.) The Mandelbrot set for a given w is the set of complex numbers c such that the iteration scheme is bounded when starting from the point z₀=0. A significant subset of a Mandelbrot 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*^w +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^w + c
z* = z*² + c
Subtraction gives
z_n+1-z* = z_n^w - z*^w

Now consider the ratio ρ_n=[z_n^w - z*^w]/[z_n+1-z*]

The absolute value of the (n+1)th deviation, |z_n+1-z*|, will be less than that of the n-th deviation,|z_n-z*|, if |ρ_n|<1.

For values of z_n close to z* this reduces to limit of |ρ_n| as |z_n| approaches z*. That limit can be evaluated using l'Hospital's Rule.

lim ρ_n = wz*^w-1
for all n

Thus the boundary between the stable and unstable limit points is given by |z*|=(1/w)^1/(w-1). Such limit points are given by the equation

z* = (1/w)^1/(w-1)e^iφ
for 0≤φ≤2π

The question is what are the values of c which give those limit points. Those values of c are simply

c = z* - z*^w = z*(1 − z*^w-1)

Note that

z*^w-1 = (1/w)e^i(w-1)φ

Thus

c = (1/w)^1/(w-1)e^iφ(1-(1/w)e^i(w-1)φ)
or, equivalently
c = (1/w)^1/(w-1)e^iφ − (1/w)^w/(w-1)e^wφ

To simplify the expressions let (1/w)^w-1 be denoted as ζ. Thus

c = ζe^iφ − ζ^we^iwφ

This equation is a parametric equation for the set of c values. It shows how the points on the circle of radius ζ in the z* space map into the c space.

Integer Exponents

The plot below shows the full set of c values for the case of w=2.

This is the familiar cardioid shape that bounds the main body of the mandelbrot set.

The plot for the case of w=3 is:

This is the double lobed shape that bounds the main body of the cubic mandelbrot set.

For the case of w=4

For w=5 and w=6:

Fractional Exponents

For the fractional value of w=3/2:

When the radius of the circle for z* is reduced to 0.9 of the radius of the boundary the circle of the corresponding c values (the green curve) falls within the boundary curve. This is indicates that the main body of the Mandelbrot set for w=3/2 is the area enclosed by the outer curve.

For an Irrational Exponent

For example, if w=√2 the range of φ must be from 0 to ∞.

For r=√2 the radius of the circle for z* has to be (1/√2)^1/(√2-1). The diagram below shows in red the curve for the values of c which correspond to the boundary between the stable and unstable limit point.

Again when the radius of the circle for z* is reduced to 0.9 of the radius of the boundary the circle of the corresponding c values (the green curve) falls within the boundary curve.

This is indicates that the main body of the Mandelbrot set for w=√2 is the area enclosed by the outer curve.

Imaginary Exponents

When φ ranges from 0 to 2π the value of z* at φ=2π is the same as at φ=0, but that is not necessarily the case for z*^w. To obtain all of the values of c the range of φ would have to be such that when wφ is at an integral multiple of 2π so is φ.

Thus if w2π = k for k an integer k/w would also have to be an integer and for w with irrational or strictly complex values this is impossible.

For example, if w=i=√(−1) then k would have to be such that (1/i)k would also have to be an integer; an impossibility. So the range of φ must be from 0 to ∞.

For w=i the radius of the circle for z* has to be |(1/i)^1/(i-1)|. This may be evaluated by multiplying the numerator and denominator of the exponent by the conjugate of the denominator, -1+i; i.e.,

|(1/i)^1/(i-1)| = |(1/i)^-(1+i)/2| = |(1/i)^-1/2||(1/i)^-i/2|
= 1×|iⁱ|^½ = (0.2078758..)^½ = 0.455934

This means z* is on a circle of radius (1/0.455934)=2.1933. Since c = z*−z*ⁱ the curve for the boundary between the values of c giving stable and unstable limit points is given by

c = 2.1933e^iφ − (2.1933)ⁱe^-φ
= 2.1933e^iφ − e^iln(2.1933)e^-φ
= 2.1933[cos(φ)+isin(φ)] − [cos(ln(2.1933))+isin(ln(2.1933))]e^-φ
= [2.1933cos(φ)−cos(ln(2.1933))e^-φ]
+ i[2.1933sin(φ)−sin(ln(2.1933))]e^-φ]

The diagram below shows in red the curve for the values of c which correspond to the boundary between the stable and unstable limit points.

Limit cycles

The iteration may approach a limit cycle rather than a limit point.

(To be continued.)