Thayer Watkins
Silicon Valley
& Tornado Alley

The Structure of the Mandelbrot Set
for a Complex Exponent

The Mandelbrot sets are the sets of complex numbers c such that the iteration scheme

zn+1 = znm + c
for a postive integer m

is bounded when starting from the point z0=0. When m is a positive integer this is appropriate. If m has a rational number value there are problems. There are still more problems if the exponent is an irrational real number. This is an attempt to investigate the nature of the main body when the exponent is a complex number w. The main body of a Mandelbrot set consists of those values of c such that the iteration scheme approaches limits for which

zn+1 = zw.

Such a limit point z* would satisfy the equation

z* = z*w +c

For any c there there might be 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*w + c


|zn+1-z*| will be less than |zn-z*| if the limit of |znw−z*|/|zn-z*| as zn→z* is less than unity.

The limit may be evaluated using L'Hospital's Rule; i.e.,

lim [(znw−z*)/(zn-z*)] = wz*w-1

The boundary between the stable and unstable limit points is therefore where

|wz*w-1| = 1
or, equivalently
|z*| = |(1/w)1/(w-1)|

This is equivalent to the condition that z* is on the circle in the complex plane centered at the origin with a radius of |(1/w)1/(w-1);| i.e.,

z* = |(1/w)1/(w-1)|e

The range of φ must be specified. The problem is that for some values of w the relationships involved may be multiple valued instead of single valued as in the case of the exponent being an integer.

The values of c for the boundary between the stable and unstable limit points are given by:

c = z* - z*w
which reduces to
c = |(1/w)1/(w-1)|e − |(1/)w/(w-1)|eiwφ

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 component values this is impossible.

For example, if w=i=√(−1) then k would have to be such that (√2/2)k would also have to be an integer; an impossibility for √2 or any other irrational number. 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 denominator of the exponent by the conjugate of -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 2.1933 (1/0.455934). 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.

This is indicates that the main body of the Mandelbrot set for w=i is essentially the area enclosed by a circle of radius 2.1933. This is because the exponentiation by the imaginary unit results in terms involving e which very quickly become insignificant as φ increases. Even the points which are not on the 2.1933 radius circle are enclosed within that circle.

(To be continued.)

HOME PAGE OF applet-magic
HOME PAGE OF Thayer Watkins