Thayer Watkins
Silicon Valley
& Tornado Alley

The Structure of the Mandelbrot Set
for a Fractional 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 fractional value there are problems. Nevertheless the nature of the main body may still be investigated for a fractional exponent. The main body 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*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.,

zn+1 = znm + c
z* = z*m + c


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

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

lim [(znm−z*)/(zn-z*)] = mz*m-1

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

|mz*m-1| = 1
or, equivalently
|z*| = (1/m)1/(m-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/m)1/(m-1); i.e.,

z* = (1/m)1/(m-1)e

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

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

c = z* - z*m
which reduces to
c = (1/m)1/(m-1)e − (1/m)m/(m-1)eimφ

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 the case for z*m. To obtain all of the values of c the range of φ must be such that when mφ is at an integral multiple of 2π so is φ.

Thus if m2π = k for k an integer
k/m must also be an integer.

For example, if m=3/2 then k be such that (2/3)k is also an integer. That will be the case if k=3. So the range of φ must be from 0 to 6π.

For m=3/2 the radius of the circle for z* has to be (2/3)²=4/9. The diagram below shows the curve for the values of c which correspond to the boundary between the stable and unstable limit point.

When the radius of the circle for z* is reduced to 0.4 from 4/9 the circle of the corresponding c values falls within the boundary curve.

This is indicates that the main body of the Mandelbrot set for m=3/2 is the area enclosed by the outer curve.

(To be continued.)

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