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Structures of 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 the 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, but note that ζ^{w-1}=(1/w). Thus
To simplify the further computations let C=c/ζ. Thus
The conjugate of C is
Therefore the squared magniture of C is
The derivative of |C|² with respect to φ is
For an extremum this derivative is set equal to zero and thus
For a real w, w*=w, the above condition reduces to sin((w-1)φ) = 0 and thus (w-1)φ=0, π, 2π, 3π etc. For example, for w=3 extrema, cusps and the tips of lobes in alternation, occur for φ=0, π/2, π and 3π/2, as shown below.
For the case of w=4 the extrema occur at 0, π/3, 2π/3, 3π/3=π, 4π/3, 5π/3.
Likewise for w=5 the extrema occur at 0, π/4, π/2, 3π/4, 4π/4=π, 5π/4, 3π/2, 7π/4.
For w=6 the extrema occur at 0, π/5, 2π/5, 3π/5, 4π/5, 5π/5=π, 6π/5, 7π/5, 8π/5, 9π/5.
For w=3/2 the condition for an extreme is sin(φ/2)=0 and thus extrema occur at 0, 2π 4π, 6π, …. These are all the same point. What this may be saying is that the maximum and minimum are the same, which means the function is a constant for all φ; i.e., it is a circle. The plot for w=3/2 is shown below.
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