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 The Structure of the Mandelbrot Set for an Irrational 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 number r. Nevertheless the nature of the main body may still be investigated for a real number which might not be rational. The main body of a Mandelbrot set consists of those values of c such that the iteration scheme approaches limits for which

#### zn+1 = zr.

Such a limit point z* would satisfy the equation

#### z* = z*r +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.,

Thus

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

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

#### lim [(znr−z*)/(zn-z*)] = rz*r-1

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

#### |rz*r-1| = 1 or, equivalently |z*| = (1/r)1/(r-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/r)1/(r-1); i.e.,

#### z* = (1/r)1/(r-1)eiφ

The range of φ must be specified. The problem is that for some values of r 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*rwhich reduces to c = (1/r)1/(r-1)eiφ − (1/r)r/(r-1)eirφ

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*r. To obtain all of the values of c the range of φ would have to be such that when rφ is at an integral multiple of 2π so is φ.

#### Thus if r2π = k for k an integer k/r would also have to be an integerand for irrational r this is impossible.

For example, if m=√2 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 r=√2 the radius of the circle for z* has to be (√2/2)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.

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 r=√2 is the area enclosed by the outer curve.

(To be continued.)