San José State University |
---|
applet-magic.com Thayer Watkins Silicon Valley & Tornado Alley USA |
---|
Sets Associated with Limit 2-Cycles |
Consider Mandelbrot generated by the following iteration
where c is a complex number and w is as well.
The value of z may approach a limit cycle of period 2 in which it value oscillates between Z_{1} and Z_{2}. This means that
Subtracting the second equation from the first gives
Let the common value of each side of the equation be denoted as λ. Thus
Thus Z_{1} and Z_{2} are solutions to the equation
For w≠2 there may be more than two solutions, in which case Z_{1} and Z_{2} would be any pair of the set of solutions.
The deviations from the limit values satisfy the following three equations
The ratio |z_{n+2} − Z_{2}|/|z_{n} − Z_{2}| can be expressed in the form
Then z_{n+2} − Z_{2} can be replaced by its equal value of z_{n+1}^{w} − Z_{1}^{w}. In the second term of the product z_{n+1} − Z_{1} can be replaced by its equal value of z_{n}^{w} − Z_{2}^{w}.
Thus in order for |z_{n+2} − Z_{2}| to be less than |z_{n} − Z_{2}| it is necessary that
At the boundary between the stable and unstable values of c the inequality becomes an equality so
Now consider the limits as n→∞. Both terms go to 0/0 so l'Hospital's rule applies and hence
Thus the condition to be satisfied for a limit 2-cycle is
The complex numbers whose absolute value is equal to 1 are those given by exp(iφ) for 0≤φ≤2π. Thus
As stated previously, the values of Z_{1} and Z_{2} are solutions to the equation
When w is a positive integer the above equation is a monic polynomial. The product of the roots of a monic polynomial is equal to the constant term in the equation. For w=2 then
Also for a monic polynomial the sum of the roots is equal to the coefficient of the term with the next to largest exponent. For w=2 this is 1 and for w>2 it is equal to 0. For w=2 then
For w=2
Thus
This equation can be solved for Z_{1} as a function of φ, say Z_{1}(φ).
Since for this case
(To be continued.)
HOME PAGE OF Thayer Watkins |