San José State University

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Thayer Watkins
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The Symmetries of the General
Structures of 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 the 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, but note that ζw-1=(1/w). Thus

c = ζe − ζweiwφ
c = ζe − ζ(1/w)eiwφ

To simplify the further computations let C=c/ζ. Thus

C = e − (1/w)eiwφ

The conjugate of C is

C* = e−iφ − (1/w*)e−iw*φ

Therefore the squared magniture of C is

|C|² = C*C = 1 − (1/w)ei(w-1)φ − (1/w*)e−i(w*-1)φ + (1/|w|²)ei(w-w*)φ

The derivative of |C|² with respect to φ is

d|C|²/dφ = (i(w-1)/w)ei(w-1)φ − (i(w*-1)/w*)e−i(w*-1)φ
+ (i(w-w*)/|w|²)ei(w-w*)φ

For an extremum this derivative is set equal to zero and thus

(i(w-1)/w)ei(w-1)φ − (i(w*-1)/w*)e−i(w*-1)φ + (i(w-w*)/|w|²)ei(w-w*)φ = 0

For a real w, w*=w, the above condition reduces to sin((w-1)φ) = 0 and thus (w-1)φ=0, π, 2π, 3π etc. For example, for w=3 extrema, cusps and the tips of lobes in alternation, occur for φ=0, π/2, π and 3π/2, as shown below.

For the case of w=4 the extrema occur at 0, π/3, 2π/3, 3π/3=π, 4π/3, 5π/3.

Likewise for w=5 the extrema occur at 0, π/4, π/2, 3π/4, 4π/4=π, 5π/4, 3π/2, 7π/4.

For w=6 the extrema occur at 0, π/5, 2π/5, 3π/5, 4π/5, 5π/5=π, 6π/5, 7π/5, 8π/5, 9π/5.

Fractional Exponents

For w=3/2 the condition for an extreme is sin(φ/2)=0 and thus extrema occur at 0, 2π 4π, 6π, …. These are all the same point. What this may be saying is that the maximum and minimum are the same, which means the function is a constant for all φ; i.e., it is a circle. The plot for w=3/2 is shown below.


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