# Introduction

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.

# applications

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

# Notes

• Iteration is allways on the dynamic plane.
• There is no dynamic on the parametr plane.
• Mandelbrot set carries no dynamics. It is a set of parameter values.
• There are no orbits on parameter plane, one should not draw orbits on parameter plane.
• Orbit of critical point is on the dynamical plane

# step

• integer
• fractional
• Continuous Iteration of Dynamical Maps[5][6]. This continuous iteration can be discretized by the finite element method and then solved—in parallel—on a computer.

# decomposition

Move during iteration in case of complex quadratic polynomial is complex. It consists of 2 moves :

• angular move = rotation ( see doubling map)
• radial move ( see external and internal rays, invariant curves )
• fallin into target set and attractor ( in hyperbolic and parabolic case )

# direction

## backward

Backward iteration or inverse iteration

### Peitgen

 /* Zn*Zn=Z(n+1)-c */
Zx=Zx-Cx;
Zy=Zy-Cy;
/* sqrt of complex number algorithm from Peitgen, Jurgens, Saupe: Fractals for the classroom */
if (Zx>0)
{
NewZx=sqrt((Zx+sqrt(Zx*Zx+Zy*Zy))/2);
NewZy=Zy/(2*NewZx);
}
else /* ZX <= 0 */
{
if (Zx<0)
{
NewZy=sign(Zy)*sqrt((-Zx+sqrt(Zx*Zx+Zy*Zy))/2);
NewZx=Zy/(2*NewZy);
}
else /* Zx=0 */
{
NewZx=sqrt(fabs(Zy)/2);
if (NewZx>0) NewZy=Zy/(2*NewZx);
else NewZy=0;
}
};
if (rand()<(RAND_MAX/2))
{
Zx=NewZx;
Zy=NewZy;
}
else {Zx=-NewZx;
Zy=-NewZy; }


### Mandel

Here is example of c code of inverse iteration using code from program Mandel by Wolf Jung

/*

gcc i.c -lm -Wall
./a.out

iPeriodChild = 1 , c = (0.250000, 0.000000); z = (-0.0000000000000000, -0.5000000000000000)
iPeriodChild = 2 , c = (-0.750000, 0.000000); z = (-0.0000000000000001, 0.3406250193166067) z = 0.000000000000000  -0.340625019316607 i
iPeriodChild = 3 , c = (-0.125000, 0.649519); z = (-0.2299551351162812, -0.1413579816050057) z = -0.229955135116281  -0.141357981605006 i
iPeriodChild = 4 , c = (0.250000, 0.500000); z = (-0.2288905993372874, -0.0151096456992674)
iPeriodChild = 5 , c = (0.356763, 0.328582); z = (-0.1990400075391210, 0.0415980651776321)
iPeriodChild = 6 , c = (0.375000, 0.216506); z = (-0.1727194378627304, 0.0675726990190151)
iPeriodChild = 7 , c = (0.367375, 0.147184); z = (-0.1530209385352789, 0.0799609106267383)
iPeriodChild = 8 , c = (0.353553, 0.103553); z = (-0.1386555899358813, 0.0860089512209437)
iPeriodChild = 9 , c = (0.339610, 0.075192); z = (-0.1281114080080390, 0.0889429110652104) z = -0.128111408008039  +0.088942911065210 i

*/

#include <stdio.h>
#include <math.h> // M_PI; needs -lm also
#include <complex.h>

/* find c in component of Mandelbrot set

uses code by Wolf Jung from program Mandel
see function mndlbrot::bifurcate from mandelbrot.cpp
http://www.mndynamics.com/indexp.html

*/
double complex GiveC(double InternalAngleInTurns, double InternalRadius, unsigned int Period)
{
//0 <= InternalRay<= 1
//0 <= InternalAngleInTurns <=1
double t = InternalAngleInTurns *2*M_PI; // from turns to radians
double Cx, Cy; /* C = Cx+Cy*i */

switch ( Period ) // of component
{
case 1: // main cardioid
break;
case 2: // only one component
Cx = InternalRadius * 0.25*cos(t) - 1.0;
Cy = InternalRadius * 0.25*sin(t);
break;
// for each iPeriodChild  there are 2^(iPeriodChild-1) roots.
default: // higher periods : to do, use newton method
Cx = 0.0;
Cy = 0.0;
break; }

return Cx + Cy*I;
}

/* mndyncxmics::root from mndyncxmo.cpp  by Wolf Jung (C) 2007-2014. */

// input = x,y
// output = u+v*I = sqrt(x+y*i)
complex double GiveRoot(complex double z)
{
double x = creal(z);
double y = cimag(z);
double u, v;

v  = sqrt(x*x + y*y);

if (x > 0.0)
{ u = sqrt(0.5*(v + x)); v = 0.5*y/u; return  u+v*I; }
if (x < 0.0)
{ v = sqrt(0.5*(v - x)); if (y < 0.0) v = -v; u = 0.5*y/v; return  u+v*I; }
if (y >= 0.0)
{ u = sqrt(0.5*y); v = u; return  u+v*I; }

u = sqrt(-0.5*y);
v = -u;
return  u+v*I;
}

// from mndlbrot.cpp  by Wolf Jung (C) 2007-2014. part of Madel 5.12
// input : c, z , mode
// c = cx+cy*i where cx and cy are global variables defined in mndynamo.h
// z = x+y*i
//
// output : z = x+y*i
complex double InverseIteration(complex double z, complex double c)
{
double x = creal(z);
double y = cimag(z);
double cx = creal(c);
double cy = cimag(c);

// f^{-1}(z) = inverse with principal value
if (cx*cx + cy*cy < 1e-20)
{
z = GiveRoot(x - cx + (y - cy)*I); // 2-nd inverse function = key b
//if (mode & 1) { x = -x; y = -y; } // 1-st inverse function = key a
return -z;
}

//f^{-1}(z) =  inverse with argument adjusted
double u, v;
complex double uv ;
double w = cx*cx + cy*cy;

uv = GiveRoot(-cx/w -(cy/w)*I);
u = creal(uv);
v = cimag(uv);
//
z =  GiveRoot(w - cx*x - cy*y + (cy*x - cx*y)*I);
x = creal(z);
y = cimag(z);
//
w = u*x - v*y;
y = u*y + v*x;
x = w;

//if (mode & 1) // mode = -1
//  { x = -x; y = -y; } // 1-st inverse function = key a

return x+y*I; // key b =  2-nd inverse function

}

// make iPeriod inverse iteration with negative sign ( a in Wolf Jung notation )
complex double GivePrecriticalA(complex double z, complex double c, int iPeriod)
{
complex double za = z;
int i;
for(i=0;i<iPeriod ;++i){

za = InverseIteration(za,c);
//printf("i = %d ,  z = (%f, %f) \n ", i,  creal(z), cimag(z) );

}

return za;
}

int main(){

complex double c;
complex double z;
complex double zcr = 0.0; // critical point

int iPeriodChild;
int iPeriodChildMax = 10; // period of
int iPeriodParent = 1;

for(iPeriodChild=1;iPeriodChild<iPeriodChildMax ;++iPeriodChild) {

c = GiveC(1.0/((double) iPeriodChild), 1.0, iPeriodParent); // root point = The unique point on the boundary of a mu-atom of period P where two external angles of denominator = (2^P-1) meet.
z = GivePrecriticalA( zcr, c, iPeriodChild);
printf("iPeriodChild = %d , c = (%f, %f); z = (%.16f, %.16f) \n ", iPeriodChild, creal(c), cimag(c), creal(z), cimag(z) );
}

return 0;
}


# Test

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

• bailout test for forward iteration

# Target set or trap

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

# Planes

## Parameter plane

"Mandelbrot set carries no dynamics. It is a set of parameter values. There are no orbits on parameter plane, one should not draw orbits on parameter plane. Orbit of critical point is on the dynamical plane"

## Dynamic plane ${\displaystyle f_{0}}$ for c=0

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

Lets take c=0, then one can call dynamical plane ${\displaystyle f_{0}}$ plane.

Here dynamical plane can be divided into :

• Julia set = ${\displaystyle \{z:|z|=1\}}$
• Fatou set which consists of 2 subsets :
• interior of Julia set = basin of attraction of finite attractor = ${\displaystyle \{z:|z|<1\}}$
• exterior of Julia set = basin of attraction of infinity = ${\displaystyle \{z:|z|>1\}}$

### Forward iteration

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

${\displaystyle z=re^{i\theta }\,}$

where :

• r is the absolute value or modulus or magnitude of a complex number z = x + i
• ${\displaystyle \theta }$ 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

${\displaystyle r=|z|={\sqrt {x^{2}+y^{2}}}.\,}$

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

so

${\displaystyle f_{0}(z)=z^{2}=(re^{i\theta })^{2}=r^{2}e^{i2\theta }\,}$

and forward iteration :[7]

${\displaystyle f_{0}^{n}(z)=r^{2^{n}}e^{i2^{n}\theta }\,}$

Forward iteration gives forward orbit = list of points {z0, z1, z2, z3... , zn} which lays on exponential spirals.[8] [9]

#### Chaos and the complex squaring map

The informal reason why the iteration is chaotic is that the angle doubles on every iteration and doubling grows very quickly as the angle becomes ever larger, but angles which differ by multiples of 2π radians are identical. Thus, when the angle exceeds 2π, it must wrap to the remainder on division by 2π. Therefore, the angle is transformed according to the dyadic transformation (also known as the 2x mod 1 map). As the initial value z0 has been chosen so that its argument is not a rational multiple of π, the forward orbit of zn cannot repeat itself and become periodic.

More formally, the iteration can be written as:

${\displaystyle \qquad z_{n+1}=z_{n}^{2}}$

where ${\displaystyle z_{n}}$ is the resulting sequence of complex numbers obtained by iterating the steps above, and ${\displaystyle z_{0}}$ represents the initial starting number. We can solve this iteration exactly:

${\displaystyle \qquad z_{n}=z_{0}^{2^{n}}}$

Starting with angle θ, we can write the initial term as ${\displaystyle z_{0}=\exp(i\theta )}$ so that ${\displaystyle z_{n}=\exp(i2^{n}\theta )}$. This makes the successive doubling of the angle clear. (This is equivalent to the relation ${\displaystyle z_{n}=\cos(2^{n}\theta )+i\sin(2^{n}\theta )}$.)

#### Escape test

If distance between:

• point z of exterior of Julia set
• Julia set ${\displaystyle J_{c}}$

is :

${\displaystyle distance(z,J_{c})=2^{-n}}$


then point z escapes (= it's magnitude is greate then escape radius = ER):

${\displaystyle |z_{n}|>ER}$


after :

• ${\displaystyle n}$ steps in non-parabolic case
• ${\displaystyle 2^{n}}$ steps in parabolic case [10]

### Backward iteration

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

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

Note that difference between these 2 preimages

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

is half a turn = 180 degrees = Pi radians.

 ${\displaystyle \alpha }$ ${\displaystyle d^{1}(\alpha )}$ ${\displaystyle d^{-1}(\alpha )}$ ${\displaystyle {\frac {1}{2}}}$ ${\displaystyle {\frac {1}{1}}}$ ${\displaystyle \left\{{\frac {1}{4}},{\frac {3}{4}}\right\}}$ ${\displaystyle {\frac {1}{3}}}$ ${\displaystyle {\frac {2}{3}}}$ ${\displaystyle \left\{{\frac {1}{6}},{\frac {4}{6}}\right\}}$ ${\displaystyle {\frac {1}{4}}}$ ${\displaystyle {\frac {1}{2}}}$ ${\displaystyle \left\{{\frac {1}{8}},{\frac {5}{8}}\right\}}$ ${\displaystyle {\frac {1}{5}}}$ ${\displaystyle {\frac {2}{5}}}$ ${\displaystyle \left\{{\frac {1}{10}},{\frac {6}{10}}\right\}}$ ${\displaystyle {\frac {1}{6}}}$ ${\displaystyle {\frac {1}{3}}}$ ${\displaystyle \left\{{\frac {1}{12}},{\frac {7}{12}}\right\}}$ ${\displaystyle {\frac {1}{7}}}$ ${\displaystyle {\frac {2}{7}}}$ ${\displaystyle \left\{{\frac {1}{14}},{\frac {4}{7}}\right\}}$

On complex dynamical plane backward iteration using quadratic polynomial ${\displaystyle f_{c}}$

${\displaystyle f_{c}(z)=z^{2}+c}$

gives backward orbit = binary tree of preimages :

${\displaystyle z\,}$

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

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

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

Not that preimages show rotational symmetry ( 180 degrees)

For other functions see Fractalforum[12]

## Dynamic plane for ${\displaystyle f_{c}}$

### Level curves of escape time

Preimages of circle under fc

### Julia set by IIM/J

In escape time one computes forward iteration of point z.

In IIM/J one computes:

• repelling fixed point[13] of complex quadratic polynomial ${\displaystyle Z_{0}=\beta _{c}\,}$
• preimages of ${\displaystyle Z_{0}\,}$ by inverse iterations

${\displaystyle Z_{n-1}={\sqrt {Z_{n}-C}}}$

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

#### Root of tree

• repelling fixed point[16] of complex quadratic polynomial ${\displaystyle Z_{0}=\beta _{c}\,}$
• - 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                         -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

• 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)[17]
• 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.

Examples of code :

Compare it with: