The Potential Function and The Real Iteration Number
The Julia set for is the unit circle, and on the outer Fatou domain, the potential function is defined by . The equipotential lines for this function are concentric circles. As we have, where is the sequence of iteration generated by z. For the more general iteration, it has been proved that if the Julia set is connected (that is, if c belongs to the (usual) Mandelbrot set), then there exist a biholomorphic map between the outer Fatou domain and the outer of the unit circle such that . This means that the potential function on the outer Fatou domain defined by this correspondence is given by:
This formula has meaning also if the Julia set is not connected, so that we for all c can define the potential function on the Fatou domain containing ∞ by this formula. For a general rational function such that ∞ is a critical point and a fixed point, that is, such that the degree m of the numerator is at least two larger than the degree n of the denominator, we define the potential function on the Fatou domain containing ∞ by:
where d = m - n is the degree of the rational function.
If N is a very large number (e.g. 10100), and if k is the first iteration number such that, we have that, for some real number, which should be regarded as the real iteration number, and we have that:
where the last number is in the interval [0, 1).
For iteration towards a finite attracting cycle of order r, we have that if z* is a point of the cycle, then (the r-fold composition), and the number (> 1) is the attraction of the cycle. If w is a point very near z* and w' is w iterated r times, we have that . Therefore the number is almost independent of k. We define the potential function on the Fatou domain by:
If is a very small number and k is the first iteration number such that, we have that for some real number, which should be regarded as the real iteration number, and we have that:
If the attraction is ∞, meaning that the cycle is super-attracting, meaning again that one of the points of the cycle is a critical point, we must replace by (where w' is w iterated r times) and the formula for by:
And now the real iteration number is given by:
For the colouring we must have a cyclic scale of colours (constructed mathematically, for instance) and containing H colours numbered from 0 to H-1 (H = 500, for instance). We multiply the real number by a fixed real number determining the density of the colours in the picture, and take the integral part of this number modulo H.
The definition of the potential function and our way of colouring presuppose that the cycle is attracting, that is, not neutral. If the cycle is neutral, we cannot colour the Fatou domain in a natural way. As the terminus of the iteration is a revolving movement, we can, for instance, colour by the minimum distance from the cycle left fixed by the iteration.
Read more about this topic: Julia Set
Famous quotes containing the words potential, function, real and/or number:
“The traditional American husband and father had the responsibilities—and the privileges—of playing the role of primary provider. Sharing that role is not easy. To yield exclusive access to the role is to surrender some of the potential for fulfilling the hero fantasy—a fantasy that appeals to us all. The loss is far from trivial.”
—Faye J. Crosby (20th century)
“Every boy was supposed to come into the world equipped with a father whose prime function was to be our father and show us how to be men. He can escape us, but we can never escape him. Present or absent, dead or alive, real or imagined, our father is the main man in our masculinity.”
—Frank Pittman (20th century)
“... the big courageous acts of life are those one never hears of and only suspects from having been through like experience. It takes real courage to do battle in the unspectacular task. We always listen for the applause of our co-workers. He is courageous who plods on, unlettered and unknown.... In the last analysis it is this courage, developing between man and his limitations, that brings success.”
—Alice Foote MacDougall (1867–1945)
“The rising power of the United States in world affairs ... requires, not a more compliant press, but a relentless barrage of facts and criticism.... Our job in this age, as I see it, is not to serve as cheerleaders for our side in the present world struggle but to help the largest possible number of people to see the realities of the changing and convulsive world in which American policy must operate.”
—James Reston (b. 1909)