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for a Fractional Exponent |
The Mandelbrot sets are the sets of complex numbers c such that the iteration scheme
is bounded when starting from the point z_{0}=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
Such a limit point z* satisfies the equation
For any c there is a limit point z*; i.e., such that if z_{0}=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 z_{0}=0 will approach them.
Consider the deviations of the iteration values from the corresponding limit point; i.e.,
Thus
The limit may be evaluated using L'Hospital's Rule; i.e.,
The boundary between the stable and unstable limit points is therefore where
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.,
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:
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.
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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