|San José State University|
& Tornado Alley
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 Z1 and Z2. 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 Z1 and Z2 are solutions to the equation
For w≠2 there may be more than two solutions, in which case Z1 and Z2 would be any pair of the set of solutions.
The deviations from the limit values satisfy the following three equations
The ratio |zn+2 − Z2|/|zn − Z2| can be expressed in the form
Then zn+2 − Z2 can be replaced by its equal value of zn+1w − Z1w. In the second term of the product zn+1 − Z1 can be replaced by its equal value of znw − Z2w.
Thus in order for |zn+2 − Z2| to be less than |zn − Z2| 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 Z1 and Z2 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
This equation can be solved for Z1 as a function of φ, say Z1(φ).
Since for this case
(To be continued.)
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