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the Fully General Mandelbrot Sets |
Fully general means that the Mandelbrot sets are generated from iteration schemes of the form
where c, z and w are complex numbers. (The integers and other 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 z_{0}=0. A significant subset 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.,
Now consider the ratio ρ_{n}=[z_{n}^{w} - z*^{w}]/[z_{n+1}-z*]
The absolute value of the (n+1)th deviation, |z_{n+1}-z*|, will be less than that of the n-th deviation,|z_{n}-z*|, if |ρ_{n}|<1.
For values of z_{n} close to z* this reduces to limit of |ρ_{n}| as |z_{n}| approaches z*. That limit can be evaluated using l'Hospital's Rule.
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
The question is what are the values of c which give those limit points. Those values of c are simply
Note that
Thus
To simplify the expressions let (1/w)^{w-1} be denoted as ζ. Thus
This equation is a parametric equation for the set of c values. It shows how the points on the circle of radius ζ in the z* space map into the c space.
The plot below shows the full set of c values for the case of w=2.
This is the familiar cardioid shape that bounds the main body of the mandelbrot set.
The plot for the case of w=3 is:
This is the double lobed shape that bounds the main body of the cubic mandelbrot set.
For the case of w=4
For w=5 and w=6:
For the fractional value of w=3/2:
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 w=3/2 is the area enclosed by the outer curve.
For example, if w=√2 the range of φ must be from 0 to ∞.
For r=√2 the radius of the circle for z* has to be (1/√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.
Again 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 w=√2 is the area enclosed by the outer curve.
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 or strictly complex values this is impossible.
For example, if w=i=√(−1) then k would have to be such that (1/i)k would also have to be an integer; an impossibility. 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 numerator and denominator of the exponent by the conjugate of the denominator, -1+i; i.e.,
This means z* is on a circle of radius (1/0.455934)=2.1933. Since c = z*−z*^{i} the curve for the boundary between the values of c giving stable and unstable limit points is given by
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.
The iteration may approach a limit cycle rather than a limit point.
(To be continued.)
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