Equation defining centers

One gets equation :

${\displaystyle F_{p}(0,c)=0\,}$

Centers of components are roots of above equation.

Because ${\displaystyle z=0\,}$ one can remove z from these equations :

• for period 1 : z^2+c=z and z=0 so ${\displaystyle c=0}$
• for period 2 : (z^2+c)^2 +c =z and z=0 so ${\displaystyle c^{2}+c=0}$
• for period 3 : ((z^2+c)^2 +c)^2 +c = z and z=0 so ${\displaystyle (c^{2}+c)^{2}+c=0}$

Here is Maxima functions for computing above functions :

P[n]:=if n=1 then c else P[n-1]^2+c;

Reduced equation defining centers

Set of roots of above equation contains centers for period p and its divisors. It is because :

${\displaystyle F_{p}(0,c)=\prod _{m|p}G_{m}(0,c)\,}$

where :

• ${\displaystyle G\,}$ is irreducible divisors ^[14] of ${\displaystyle F_{p}\,}$ ^[15]
• capital Pi notation of iterated multiplication is used
• ${\displaystyle m|p\,}$ means here : for all divisors of ${\displaystyle p\,}$ ( from 1 to p ). See table of divisors.

For example, :

${\displaystyle F_{1}=c=G_{1}\,}$

${\displaystyle F_{2}=c*(c+1)=G_{1}*G_{2}\,}$

${\displaystyle F_{3}=c*(c^{3}+2*c^{2}+c+1)=G_{1}*G_{3}\,}$

${\displaystyle F_{4}=c*(c+1)*(c^{6}+3*c^{5}+3*c^{4}+3*c^{3}+2*c^{2}+1)=G_{1}*G_{2}*G_{4}\,}$

So one can find irreducible polynomials using :

${\displaystyle G_{p}(0,c)={\frac {F_{p}(0,c)}{\prod _{m|p,m<p}G_{m}(0,c)}}\,}$

Here is Maxima function for computing ${\displaystyle G\,}$ :

GiveG[p]:=
block(
[f:divisors(p),
t:1], /* t is temporary variable = product of Gn for (divisors of p) other than p */
f:delete(p,f), /* delete p from list of divisors */
if p=1
then return(Fn(p,0,c)),
for i in f do t:t*GiveG[i],
g: Fn(p,0,c)/t,
return(ratsimp(g))
)$

Here is a table with degree of ${\displaystyle F\,}$ and ${\displaystyle G\,}$ for periods 1-10 and precision needed to compute coefficients of these functions (the roots can be calculated with lower precision, provided you work with the unexpanded form ${\displaystyle (\ldots (c^{2}+c)^{2}+c\ldots )^{2}+c}$).

period ${\displaystyle p\,}$	${\displaystyle \deg(F_{p}(0,c))\,}$	${\displaystyle \deg(G_{p}(0,c))\,}$	${\displaystyle fpprec\,}$
1	1	1	16
2	2	1	16
3	4	3	16
4	8	6	16
5	16	15	16
6	32	27	16
7	64	63	32
8	128	120	64
9	256	252	128
10	512	495	300
11	1024	1023	600

Where:

• fpprec is the number of significant decimal digits for arithmetic on bigfloat numbers^[16]

Here is a table of greatest coefficients.

period ${\displaystyle p\,}$	${\displaystyle log10(gc_{p})\,}$	${\displaystyle gc_{p}\,}$
1	0	1
2	0	1
3	1	2
4	1	3
5	3	116
6	4	5892
7	11	17999433372
8	21	106250977507865123996
9	44	16793767022807396063419059642469257036652437
10	86	86283684091087378792197424215901018542592662739248420412325877158692469321558575676264
11	179	307954157519416310480198744646044941023074672212201592336666825190665221680585174032224052483643672228833833882969097257874618885560816937172481027939807172551469349507844122611544

Precision can be estimated as bigger than size of binary form of greatest coefficient ${\displaystyle log_{2}(gc)\,}$ :

${\displaystyle fpprec>log_{2}(gc)\,}$

period ${\displaystyle p\,}$	${\displaystyle \deg(G_{p}(0,c))\,}$	${\displaystyle \log _{2}(gc_{p})\,}$	${\displaystyle fpprec\,}$
1	1	1	16
2	1	1	16
3	3	2	16
4	6	2	16
5	15	7	16
6	27	13	16
7	63	34	32
8	120	67	64
9	252	144	128
10	495	286	300
11	1023	596	600

Here is Maxima code for ${\displaystyle gc\,}$

period_Max:11;
/* ----------------- definitions -------------------------------*/
/* basic function  = monic and centered complex quadratic polynomial 
http://en.wikipedia.org/wiki/Complex_quadratic_polynomial
*/
f(z,c):=z*z+c $
/* iterated function */
fn(n, z, c) :=
 if n=1 	then f(z,c)
 else f(fn(n-1, z, c),c) $
/* roots of Fn are periodic point of  fn function */
Fn(n,z,c):=fn(n, z, c)-z $
/* gives irreducible divisors of polynomial Fn[p,z=0,c] */
GiveG[p]:=
block(
[f:divisors(p),t:1],
g,
f:delete(p,f),
if p=1 
then return(Fn(p,0,c)),
for i in f do t:t*GiveG[i],
g: Fn(p,0,c)/t,  
return(ratsimp(g))
 )$
/* degree of function */
GiveDegree(_function,_var):=hipow(expand(_function),_var);  
log10(x) := log(x)/log(10);
/* ------------------------------*/
file_output_append:true; /* to append new string to file */
grind:true;  
for period:1 thru period_Max step 1 do
(
g[period]:GiveG[period], /* function g */
d[period]:GiveDegree(g[period],c), /* degree of function g */
cf[period]:makelist(coeff(g[period],c,d[period]-i),i,0,d[period]), /* list of coefficients */
cf_max[period]:apply(max, cf[period]), /* max coefficient */
disp(cf_max[period]," ",ceiling(log10(cf_max[period]))),
s:concat("period:",string(period),"  cf_max:",cf_max[period]),
stringout("max_coeff.txt",s)/* save output to file as Maxima expressions */
);

color is proportional to magnitude of zn

• color is proportional to magnitude of zn^[17]
• Parameter plane is scanned for points c for which orbit of critical point vanishes ^[18]
• YouTube video : Mandelbrot Oscillations ^[19]

color shows in which quadrant zn lands

ninth-decomposition. Image and src code

This is radial nth-decomposition of exterior of Mandelbrot set ( compare it with n-th decomposition of LSM/M )

4 colors are used because there are 4 quadrants :

• re(z_n) > 0 and im(z_n) > 0 ( 1-st quadrant )
• re(z_n) < 0 and im(z_n) > 0 ( 2-nd quadrant )
• re(z_n) < 0 and im(z_n) < 0 ( 3-rd quadrant )
• re(z_n) > 0 and im(z_n) < 0 (4-th quadrant ).

"... when the four colors are meeting somewhere, you have a zero of q_n(c), i.e., a center of period dividing n. In addition, the light or dark color shows if c is in M." ( Wolf Jung )

Here is fragment of c code for computing 8-bit color for point c = cx + cy * i  :

/* initial value of orbit = critical point Z= 0 */
                       Zx=0.0;
                       Zy=0.0;
                       Zx2=Zx*Zx;
                       Zy2=Zy*Zy;
                       /* the same number of iterations for all points !!! */
                       for (Iteration=0;Iteration<IterationMax; Iteration++)
                       {
                           Zy=2*Zx*Zy + Cy;
                           Zx=Zx2-Zy2 +Cx;
                           Zx2=Zx*Zx;
                           Zy2=Zy*Zy;
                       };
                       /* compute  pixel color (8 bit = 1 byte) */
                       if ((Zx>0) && (Zy>0)) color[0]=0;   /* 1-st quadrant */
                       if ((Zx<0) && (Zy>0)) color[0]=10; /* 2-nd quadrant */
                       if ((Zx<0) && (Zy<0)) color[0]=20; /* 3-rd quadrant */
                       if ((Zx>0) && (Zy<0)) color[0]=30; /* 4-th quadrant */

One can also find cpp code in function quadrantqn from class mndlbrot in mndlbrot.cpp file from source code of program Mandel by Wolf Jung^[20]

The differences from standard loop are :

• there is no bailout test (for example for max value of abs(zn) ). It means that every point has the same number of iterations. For large n overflow is possible. Also one can't use test Iteration==IterationMax
• Iteration limit is relatively small ( start for example IterationMax = 8 )

See also code in Sage^[21]

using Gaussian wave function method ^[22]

mandelbrot-solver

It is a package for Ubuntu A solver for Mandelbrot polynomials based on MPSolve. A polynomial rootfinder that can determine arbitrary precision approximations with guaranteed inclusion radii. It supports exploiting of polynomial structures such as sparsisty and allows for polynomial implicitly defined or in some non standard basis. This binary, provided as an example of custom polynomial implementation in the MPSolve package, uses the particular structure of this family of polynomials to develop an efficient solver that exhibit experimental O(n^2) complexity.

This package contains the specialization of mpsolve to Mandelbrot polynomials.

example :

mandelbrot-solver
Usage: mandelbrot-solver n [starting_file]

Parameters: 
 - n is the level of the Mandelbrot polynomial to solve

 - starting_file is an optional file with the approximations that shall be 
                 use as starting points.
a@zelman:~$ mandelbrot-solver 2
 (-1.22561166876654e-01,  7.44861766619744e-01)
 (-1.75487766624669e+00,  0.00000000000000e+00)
 (-1.22561166876654e-01, -7.44861766619744e-01)
a@zelman:~$ mandelbrot-solver 1
 (-1.00000000000000e+00,  0.00000000000000e+00)
a@zelman:~$ mandelbrot-solver 3
 ( 2.82271390766914e-01,  5.30060617578525e-01)
 (-1.56520166833755e-01,  1.03224710892283e+00)
 (-1.94079980652948e+00,  1.26679600613393e-16)
 (-1.00000000000000e+00,  0.00000000000000e+00)
 (-1.31070264133683e+00, -7.22823506473338e-18)
 (-1.56520166833755e-01, -1.03224710892283e+00)
 ( 2.82271390766914e-01, -5.30060617578525e-01)

It seems that:

 level = period -1

and then results are centers.