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The Mandelbrot set is surely one of the most visually fascinating structures in mathematics. The Mandelbrot set 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 the 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
For values of z_{n} close to z* this reduces to 2z*<1. The boundary between the stable and unstable limit points is given by z*=1/2. 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
This equation is a parametric equation for the set of c values. It shows how the points on the circle of radius 1/2 in the z* space map into the c space. For example, z*=½ maps into c=¼, z*=−½ maps into c=−3/4, z*=i½ maps into c=½+i¼ and z*=−i½ maps into c=−½+i¼.
The plot below shows the full set of c values.
And there is the familiar cardioid shape that bounds the main body of the mandelbrot set, as seen below.
The iteration may approach a limit cycle rather than a limit point. For a twoperiod cycle of z_{1}* and z_{2}* the values would have to satisfy the equations
From these equations defining z_{1}* and z_{2}* it follows that
The deviations from the limit values satisfy the equations
Therefore if z_{n+2}z_{2}* is to be less than z_{n}z_{2}* it must be that
For values very close to a cycle pair this reduces to:
For the boundary of the stable set equality prevails; i.e.,
It was previously determined that z_{2}* = −(1 + z_{1}*), therefore the condition for the boundary of stability reduces to:
The allowable values of z_{1} are solutions to the quadratic equation
There are two solutions to this quadratic equation; one corresponding to z_{1}* and the other to z_{2}*.
It is not the solutions for z_{1}* or z_{2}* that are of most interest; it is the set of values of c corresponding to these solutions.
The defining equation
It is known that
This is the locus of a circle of radius ¼, the center of which is shifted one unit to the left on the real axis. This is shown below in a diagram with the values of c corresponding to limit points.
The analysis for 3cycles is more difficult but by same procedures as in the previous it can be established that the boundaries between the stable and unstable cycles are circles of radius 1/8. In general the boundaries between stable and unstable mcycles are circles of radius 1/2^{m}. The diagram below shows the placement of circles having radii of 1/8, 1/16 and 1/32.
There are also easily discernible circles in the Mandelbrot set of radius 1/64.
The limit points z_{1}*, z_{2}* and z_{3}* satisfy the equations
The substitution of the equation for z_{1}* into the equation for z_{2}*² and the substitution of that equation into the equation for z_{3}* produces a polynomial equation of degree 8.
There will be eight solutions to this equation and two pairs of triples which satisfy the conditions for limiting 3cycles for each given value of c. The product of the eight solutions will be equal to the constant term divided by the coefficient of the highest power of z*. This means that the product of the solutions is equal to (c² + c)² + c.
In order to establish the conditions for stability of the 3cycles consider the deviations between z_{n+3} and z_{3}*, z_{n+2} and z_{2}*, z_{n+1} and z_{1}*, and z_{n} and z_{3}*. These deviations satisfy the equations
Successive substitutions establish that if z_{n+3} − z_{3}* is to be less than z_{n} − z_{3}* it is necessary that
For values very close to the limiting 3cycles this reduces to
and for the boundary between the stable and unstable 3cycles
From the defining equations for the limit points it follows that
and therefore
(To be continued.)
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