Fractals/Iterations in the complex plane/q-iterations

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Julia set drawn by inverse iteration of critical orbit ( in case of Siegel disc )
Periodic external rays of dynamic plane made with backward iteration

Iteration in mathematics refer to the process of iterating a function i.e. applying a function repeatedly, using the output from one iteration as the input to the next.[1] Iteration of apparently simple functions can produce complex behaviours and difficult problems.

One can make inverse ( backward iteration) :

  • of repeller for drawing Julia set ( IIM/J)[2]
  • of circle outside Jlia set (radius=ER) for drawing level curves of escape time ( which tend to Julia set)[3]
  • of circle inside Julia set (radius=AR) for drawing level curves of attract time ( which tend to Julia set)[4]
  • of critical orbit ( in Siegel disc case) for drawing Julia set ( probably only in case of Goldem Mean )
  • for drawing external ray

Repellor for forward iteration is attractor for backward iteration



One can iterate ad infinitum. Test tells when one can stop

  • bailout test for forward iteration

Target set or trap[edit]

Target set is used in test. When zn is inside target set then one can stop the iterations.


Dynamic plane f_0 for c=0[edit]

Equipotential curves (in red) and integral curves (in blue) of a radial vector field with the potential function \phi(x,y) = \sqrt{x^2+y^2}

Lets take c=0, then one can call dynamical plane f_0 plane.

Here dynamical plane can be divided into :

  • Fatou set
  • Julia set =  \{  z : |z| = 1 \}

Fatou set consist from 2 subsets :

  • interior of Julia set = basin of attraction of finite attractor =  \{  z : |z| < 1 \}
  • exterior of Julia set = basin of attraction of infinity =  \{  z : |z| > 1 \}

Forward iteration[edit]

The 10 first powers of a complex number inside the unit circle
Exponential spirals
Principle branch of arg

z = r e^{i \theta} \,

where :

  • r is the absolute value or modulus or magnitude of a complex number z = x + i
  • \theta is the argument of complex number z (in many applications referred to as the "phase") is the angle of the radius with the positive real axis. Usually principal value is used


\theta = \arg(z) = \operatorname{atan2}(y,x) =
\arctan(\frac{y}{x}) & \mbox{if } x > 0 \\
\arctan(\frac{y}{x}) + \pi & \mbox{if } x < 0  \mbox{ and } y \ge 0\\
\arctan(\frac{y}{x}) - \pi & \mbox{if } x < 0 \mbox{ and } y < 0\\
\frac{\pi}{2} & \mbox{if } x = 0 \mbox{ and } y > 0\\
-\frac{\pi}{2} & \mbox{if } x = 0 \mbox{ and } y < 0\\
\mbox{indeterminate } & \mbox{if } x = 0 \mbox{ and } y = 0.


f_0(z) = z^2 = (r e^{i \theta})^2 = r^2  e^{i 2 \theta}\,

and forward iteration :[5]

f^n_0(z) =  r^{2^n}  e^{i 2^n \theta}\,

Forward iteration gives forward orbit = list of points {z0, z1, z2, z3... , zn} which lays on exponential spirals.[6] [7]

Escape test[edit]

If distance between point z of exterior of Julia set and Julia set is :

distance(z, J_c) = 2^{-n} 

then point escapes ( measured using bailout test and escape time )

|z_n| > ER 

after :

See here for the precision needed for escape test

Backward iteration[edit]

Backward iteration of complex quadratic polynomial with proper chose of the preimage

Every angle α ∈ R/Z measured in turns has :

  • one image = 2α mod 1 under doubling map
  • "two preimages under the doubling map: α/2 and (α + 1)/2." [9]. Inverse of doubling map is multivalued function.

Note that difference between these 2 preimages

\frac{\alpha}{2} - \frac{\alpha +1}{2} = \frac{1}{2}

is half a turn = 180 degrees = Pi radians.

Images and preimages under doubling map d
\alpha d^1(\alpha) d^{-1}(\alpha)
\frac{1}{2} \frac{1}{1} \left \{ \frac{1}{4} , \frac{3}{4} \right \}
\frac{1}{3} \frac{2}{3} \left \{ \frac{1}{6} , \frac{4}{6} \right \}
\frac{1}{4} \frac{1}{2} \left \{ \frac{1}{8} , \frac{5}{8} \right \}
\frac{1}{5} \frac{2}{5} \left \{ \frac{1}{10} , \frac{6}{10} \right \}
\frac{1}{6} \frac{1}{3} \left \{ \frac{1}{12} , \frac{7}{12} \right \}
\frac{1}{7} \frac{2}{7} \left \{ \frac{1}{14} , \frac{4}{7} \right \}

On complex dynamical plane backward iteration using quadratic polynomial f_c

f_c(z) = z^2 + c

gives backward orbit = binary tree of preimages :

z \,

 -\sqrt{z-c} ,  +\sqrt{z-c} \,

 -\sqrt{-\sqrt{z-c} -c} ,  +\sqrt{-\sqrt{z-c} -c}, -\sqrt{+\sqrt{z-c} -c}, +\sqrt{+\sqrt{z-c} -c} \,

One can't choose good path in such tree without extra informations.

Not that preimages show rotational symmetry ( 180 degrees)

Dynamic plane for f_c[edit]

Level curves of escape time[edit]

Preimages of circle under fc

Julia set by IIM/J[edit]

In escape time one computes forward iteration of point z.

In IIM/J one computes:

  • repelling fixed point[10] of complex quadratic polynomial Z_0=\beta_c \,
  • preimages of Z_0\, by inverse iterations

Z_{n-1} = \sqrt{Z_n - C}

Because square root is multivalued function then each Z_{n}\, has two preimages Z_{n-1}\,. Thus inverse iteration creates binary tree.

Root of tree[edit]

  • repelling fixed point[11] of complex quadratic polynomial Z_0=\beta_c \,
  • - beta
  • other repelling periodic points ( cut points of filled Julia set ). It will be important especially in case of the parabolic Julia set.

"... preimages of the repelling fixed point beta. These form a tree like

                    beta                                            -beta
   beta                         -beta                    x                     y

So every point is computed at last twice when you start the tree with beta. If you start with -beta, you will get the same points with half the number of computations.

Something similar applies to the preimages of the critical orbit. If z is in the critical orbit, one of its two preimages will be there as well, so you should draw -z and the tree of its preimages to avoid getting the same points twice." (Wolf Jung )

Variants of IIM[edit]

  • random choose one of two roots IIM ( up to chosen level max). Random walk through the tree. Simplest to code and fast, but inefficient. Start from it.
    • both roots with the same probability
    • more often one then other root
  • draw all roots ( up to chosen level max)[12]
    • using recurrence
    • using stack ( faster ?)
  • draw some rare paths in binary tree = MIIM. This is modification of drawing all roots. Stop using some rare paths.
    • using hits table ( while hit(pixel(iX,iY)) < HitMax )[13]
    • using derivative ( while derivative(z) < DerivativeMax)[14]

Examples of code :

Compare it with:


  1. wikipedia : Iteration
  2. Inverse Iteration Algorithms for Julia Sets by Mark McClure
  3. Complex iteration by Microcomputadoras
  4. On rational maps with two critical points by John Milnor, fig. 5
  5. Real and imaginary parts of polynomial iterates by Julia A. Barnes, Clinton P. Curry and Lisbeth E. Schaubroeck. New York J. Math. 16 (2010) 749–761.
  6. Complex numbers by David E Joyce
  7. Powers of complex numbers from Suitcase of Dreams
  8. Parabolic Julia Sets are Polynomial Time Computable by Mark Braverman
  10. wikipedia : repelling fixed point
  11. wikipedia : repelling fixed point
  12. Fractint documentation - Inverse Julias
  13. Image and c source code : IIMM using hit limit
  14. Exploding the Dark Heart of Chaos by Chris King