Editing Julia set (section)

==The potential function and the real iteration number==
The Julia set for <math>f(z) = z^{2}</math> is the unit circle, and on the outer Fatou domain, the ''potential function'' ''φ''(''z'') is defined by ''φ''(''z'') = log|''z''|. The equipotential lines for this function are concentric circles. As <math>|f(z)| = |z|^{2}</math> we have

:<math>\varphi(z) = \lim_{k\to\infty} \frac{\log|z_k|}{2^k},</math>

where <math>z_k</math> is the sequence of iteration generated by ''z''. For the more general iteration <math>f(z) = z^2 + c</math>, 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 ''ψ'' between the outer Fatou domain and the outer of the unit circle such that <math>|\psi(f(z))| = |\psi(z)|^{2}</math>.<ref>{{cite journal |first1=Adrien |last1=Douady |first2=John H. |last2=Hubbard |year=1984 |title=Etude dynamique des polynômes complexes |journal=Prépublications mathémathiques d'Orsay |volume=2 |postscript=;}} &nbsp; {{cite journal |title=[''op.cit.''] |journal=Prépublications mathémathiques d'Orsay|year=1985 |volume=4}}</ref> This means that the potential function on the outer Fatou domain defined by this correspondence is given by:

:<math>\varphi(z) = \lim_{k\to\infty} \frac{\log|z_k|}{2^k}. </math>

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 ''f''(''z'') 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:

:<math>\varphi(z) = \lim_{k\to\infty} \frac{\log|z_k|}{d^k},</math>

where ''d'' = ''m'' − ''n'' is the degree of the rational function.<ref name="Peitgen1986">{{cite book |title=[[The Beauty of Fractals]] |last=Peitgen |first=Heinz-Otto |author2=Richter Peter |year=1986 |publisher=Springer-Verlag |location=Heidelberg |isbn=0-387-15851-0 }}</ref>

If ''N'' is a very large number (e.g. 10<sup>100</sup>), and if ''k'' is the first iteration number such that <math>|z_k| > N</math>, we have that

:<math>\frac{\log|z_k|}{d^k} = \frac{\log(N)}{d^{\nu(z)}},</math>

for some real number <math>\nu(z)</math>, which should be regarded as the '''real iteration number''', and we have that:

:<math>\nu(z) = k - \frac{\log(\log|z_k|/\log(N))}{\log(d)},</math>

where the last number is in the interval [0,&nbsp;1).

For iteration towards a finite attracting cycle of order ''r'', we have that if <math>z^*</math> is a point of the cycle, then <math>f(f(...f(z^*))) = z^*</math> (the ''r''-fold composition), and the number

:<math>\alpha = \frac{1}{\left |(d(f(f(\cdots f(z))))/dz)_{z=z^*} \right |} \qquad ( > 1)</math>

is the ''attraction'' of the cycle. If ''w'' is a point very near <math>z^*</math> and ''w''&prime; is ''w'' iterated ''r'' times, we have that

:<math>\alpha = \lim_{k\to\infty} \frac{|w - z^*|}{|w' - z^*|}.</math>

Therefore, the number <math>|z_{kr} - z^*|\alpha^{k}</math> is almost independent of ''k''. We define the potential function on the Fatou domain by:

:<math>\varphi(z) = \lim_{k\to\infty} \frac{1}{(|z_{kr} - z^*|\alpha^{k})}.</math>

If ε is a very small number and ''k'' is the first iteration number such that <math>|z_k - z^*| < \epsilon</math>, we have that

:<math>\varphi(z) = \frac{1}{(\varepsilon \alpha^{\nu(z)})}</math>

for some real number <math>\nu(z)</math>, which should be regarded as the real iteration number, and we have that:

:<math>\nu(z) = k - \frac{\log(\varepsilon/|z_k - z^*|)}{\log(\alpha)}.</math>

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 ''α'' by

:<math>\alpha = \lim_{k\to\infty} \frac{\log|w' - z^*|}{\log|w - z^*|},</math>

where ''w''&prime; is ''w'' iterated ''r'' times and the formula for ''φ''(''z'') by:

:<math>\varphi(z) = \lim_{k\to\infty} \frac{\log(1/|z_{kr} - z^*|)}{\alpha^k}. </math>

And now the real iteration number is given by:

:<math>\nu(z) = k - \frac{\log(\log|z_k - z^*|/\log(\varepsilon))}{\log(\alpha)}.</math>

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 <math>\nu(z)</math> 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.