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The General Structures of
the Fully General Mandelbrot Sets

Fully general means that the Mandelbrot sets are generated from iteration schemes of the form

zn+1 = znw + c

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

zn+1 = zn.

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.,

zn+1 = znw + c
z* = z*² + c
Subtraction gives
zn+1-z* = znw - z*w

Now consider the ratio ρn=[znw - z*w]/[zn+1-z*]

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

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

lim ρn = 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
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)ei(w-1)φ

Thus

c = (1/w)1/(w-1)e(1-(1/w)ei(w-1)φ)
or, equivalently
c = (1/w)1/(w-1)e − (1/w)w/(w-1)e

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

c = ζe − ζweiwφ

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 − (2.1933)ie
= 2.1933e − eiln(2.1933)e
= 2.1933[cos(φ)+isin(φ)] − [cos(ln(2.1933))+isin(ln(2.1933))]e
= [2.1933cos(φ)−cos(ln(2.1933))e]
+ i[2.1933sin(φ)−sin(ln(2.1933))]e]

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.)


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