# Intro

Boundary of Mandelbrot set consist of :[1]

• boundaries of primitive and satellite hyperbolic components of Mandelbrot set including points :
• Parabolic (including 1/4 and primitive roots which are landing points for 2 parameter rays with rational external angles = biaccesible ).
• Siegel ( a unique parameter ray landing with irrational external angle)
• Cremer ( a unique parameter ray landing with irrational external angle)
• Boundary of M without boundaries of hyperbolic components with points
• non-renormalizable (Misiurewicz with rational external angle and other).
• finitely renormalizable (Misiurewicz and other).
• infinitely renormalizble (Feigenbaum and other). Angle in turns of external rays landing on the Feigenbaum point are irrational numbers
• boundaries of non hyperbolic components ( we believe they do not exist but we cannot prove it ) Boundaries of non-hyperbolic components would be infinitely renormalizable as well.

Boundary of Mandelbrot set as an image of unit circle under ${\displaystyle \Psi _{M}\,}$

# Drawing boundaries

implicit polynomial curves that converge to the boundary of the Mandelbrot set = lemniscates

Methods used to draw boundary of Mandelbrot set :[2]

## How to draw whole M set boundary

### Jungreis function ${\displaystyle \Psi _{M}\,}$

Boundary of Mandelbrot set as an image of unit circle under Jungreis function

Description :

• Drawing Mc by Jungreis Algorithm [8][9]
• Maxima CAS src code [10]

Python code

#!/usr/bin/env python
"""
Python code by Matthias Meschede 2014
http://pythology.blogspot.fr/2014/08/parametrized-mandelbrot-set-boundary-in.html
"""
import numpy as np
import matplotlib.pyplot as plt

nstore = 3000  #cachesize should be more or less as high as the coefficients
betaF_cachedata = np.zeros( (nstore,nstore))
def betaF(n,m):
"""
This function was translated to python from
http://fraktal.republika.pl/mset_jungreis.html
It computes the Laurent series coefficients of the jungreis function
that can then be used to map the unit circle to the Mandelbrot
set boundary. The mapping of the unit circle can also
be seen as a Fourier transform.
I added a very simple global caching array to speed it up
"""

nnn=2**(n+1)-1
return betaF_cachedata[n,m]
elif m==0:
return 1.0
elif ((n>0) and (m < nnn)):
return 0.0
else:
value = 0.
for k in range(nnn,m-nnn+1):
value += betaF(n,k)*betaF(n,m-k)
value = (betaF(n+1,m) - value - betaF(0,m-nnn))/2.0
betaF_cachedata[n,m] = value
return value

def main():
#compute coefficients (reduce ncoeffs to make it faster)
ncoeffs= 2400
coeffs = np.zeros( (ncoeffs) )
for m in range(ncoeffs):
if m%100==0: print '%d/%d'%(m,ncoeffs)
coeffs[m] = betaF(0,m+1)

#map the unit circle  (cos(nt),sin(nt)) to the boundary
npoints = 10000
points = np.linspace(0,2*np.pi,npoints)
xs     = np.zeros(npoints)
ys     = np.zeros(npoints)
xs = np.cos(points)
ys = -np.sin(points)
for ic,coeff in enumerate(coeffs):
xs += coeff*np.cos(ic*points)
ys += coeff*np.sin(ic*points)

#plot the function
plt.figure()
plt.plot(xs,ys)
plt.show()

if __name__ == "__main__":
main()


## How to draw boundaries of hyperbolic components

Boundaries of hyperbolic components for periods 1-6 as a solutions of boundary equations

Methods of drawing boundaries:

• solving boundary equations :
• using method by Brown,Stephenson,Jung . It works only up to period 7-8 [11]
• using resultants [12]
• parametrisation of boundary with Newton method near centers of components[13] [14]. This methods needs centers, so it has some limitations,
• Boundary scanning : detecting the edges after detecting period by checking every pixel. This method is slow but allows zooming. Draws "all" components
• Boundary tracing for given c value. Draws single component.
• Fake Mandelbrot set by Anne M. Burns : draws main cardioid and all its descendants. Do not draw mini Mandelbrot sets. [15]

"... to draw the boundaries of hyperbolic components using Newton's method. That is, take a point in the hyperbolic component that you are interested in (where there is an attracting cycle), and then find a curve along which the modulus of the multiplier tends to one. Then you will have found an indifferent parameter. Now you can similarly change the argument of the multiplier, again using Newton's method, and trace this curve. Some care is required near "cusps". " Lasse Rempe-Gillen[16]

### solving boundary equations

/*
c functions using complex type numbers
computes c from  component  of Mandelbrot set */
complex double Give_c( int Period,  int p, int q , double InternalRadius )
{

complex double w;  // point of reference plane  where image of the component is a unit disk
complex double c; // result
double t; // InternalAngleInTurns

t  = (double) p/q;
t = t * M_PI * 2.0; // from turns to radians

w = InternalRadius*cexp(I*t); // map to the unit disk

switch ( Period ) // of component
{
case 1: // main cardioid = only one period 1 component
c = w/2 - w*w/4; // https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/Mandelbrot_set/boundary#Solving_system_of_equation_for_period_1
break;
case 2: // only one period 2 component
c = (w-4)/4 ; // https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/Mandelbrot_set/boundary#Solving_system_of_equation_for_period_2
break;
// period > 2
default:
printf("higher periods : to do, use newton method \n");
printf("for each q = Period of the Child component  there are 2^(q-1) roots \n");
c = 0.0; //

break; }
return c;
}


#### System of 2 equations defining boundaries of period ${\displaystyle n\,}$ hyperbolic components

• first defines periodic point,
• second defines indifferent orbit ( multiplier of periodic point is equal to one ).

${\displaystyle {\begin{cases}F(n,z,c)=z\\abs(\lambda (z))=1\end{cases}}}$

Because stability index ${\displaystyle abs(\lambda )\,}$ is equal to radius of point of unit circle ${\displaystyle abs(w)\,}$:

${\displaystyle abs(\lambda )=abs(w)\,}$

so one can change second equation to form [17] :

${\displaystyle \lambda =w\,}$

It gives system of equations :

${\displaystyle {\begin{cases}F(n,z,c)=z\\\lambda =w\end{cases}}}$

It can be used for :

• drawing components ( boundaries, internal rays )
• finding indifferent parameters ( parabolic or for Siegel discs )

Above system of 2 equations has 3 variables : ${\displaystyle z,c,w\,}$ ( ${\displaystyle n\,}$ is constant and multiplier ${\displaystyle \lambda \,}$ is a function of ${\displaystyle z,c\,}$). One have to remove 1 variable to be able to solve it.

Boundaries are closed curves : cardioids or circles. One can parametrize points of boundaries with angle ${\displaystyle t\,}$ ( here measured in turns from 0 to 1 ).

After evaluation of ${\displaystyle w=l(t)\,}$ one can put it into above system, and get a system of 2 equations with 2 variables ${\displaystyle z,c\,}$.

Now it can be solved

For periods:

• 1-3 it can be solved with symbolical methods and give implicit ( boundary equation) ${\displaystyle b_{p}(w,c)=0\,}$ and explicit function (inverse multiplier map) : ${\displaystyle c=\gamma _{p}(w)\,}$
• 1-2 it is easy to solve [18]
• 3 it can be solve using "elementary algebra" ( Stephenson )
• >3 it can't be reduced to explicitly function but

##### Solving system of equation for period 1

Here is Maxima code :

(%i4) p:1;
(%o4) 1
(%i5) e1:F(p,z,c)=z;
(%o5) z^2+c=z
(%i6) e2:m(p)=w;
(%o6) 2*z=w
(%i8) s:eliminate ([e1,e2], [z]);
(%o8) [w^2-2*w+4*c]
(%i12) s:solve([s[1]], [c]);
(%o12) [c=-(w^2-2*w)/4]
(%i13) define (m1(w),rhs(s[1]));
(%o13) m1(w):=-(w^2-2*w)/4


where

  w:exp(2*%pi*%i*t);



and

(%i15) display2d:false;
(%o15) false
(%i16) 2*w;
(%o16) 2*%e^(2*%i*%pi*t)
(%i17) w*w;
(%o17) %e^(4*%i*%pi*t)



Second equation contains only one variable, one can eliminate this variable. Because boundary equation is simple so it is easy to get explicit solution

m1(w):=-(w^2-2*w)/4


So

  m1(t) := block([w], w:exp(2*%pi*%i*t), return ((2*w-w^2)/4));
g(t) := float(rectform(t));


Math equation:

 ${\displaystyle c_{t}={\frac {2e^{2\pi it}-e^{4\pi it}}{4}}}$

##### Solving system of equation for period 2

Here is Maxima code using to_poly_solve package by Barton Willis:

(%i4) p:2;
(%o4) 2
(%i5) e1:F(p,z,c)=z;
(%o5) (z^2+c)^2+c=z
(%i6) e2:m(p)=w;
(%o6) 4*z*(z^2+c)=w
(%i7) e1:F(p,z,c)=z;
(%o7) (z^2+c)^2+c=z
to_poly_solve([e1, e2], [z, c]);
(%o10) C:/PROGRA~1/MAXIMA~1.1/share/maxima/5.16.1/share/contrib/topoly_solver.mac
(%o11) [[z=sqrt(w)/2,c=-(w-2*sqrt(w))/4],[z=-sqrt(w)/2,c=-(w+2*sqrt(w))/4],[z=(sqrt(1-w)-1)/2,c=(w-4)/4],[z=-(sqrt(1-w)+1)/2,c=(w-4)/4]]
(%i12) s:to_poly_solve([e1, e2], [z, c]);
(%o12) [[z=sqrt(w)/2,c=-(w-2*sqrt(w))/4],[z=-sqrt(w)/2,c=-(w+2*sqrt(w))/4],[z=(sqrt(1-w)-1)/2,c=(w-4)/4],[z=-(sqrt(1-w)+1)/2,c=(w-4)/4]]
(%i14) rhs(s[4][2]);
(%o14) (w-4)/4
(%i16) define (m2 (w),rhs(s[4][2]));
(%o16) m2(w):=(w-4)/4


explicit solution :

m2(w):=(w-4)/4

##### Solving system of equation for period 3

For period 3 ( and higher) previous method give no results (Maxima code) :

(%i14) p:3;
e1:z=F(p,z,c);
e2:m(p)=w;
to_poly_solve([e1, e2], [z, c]);
(%o14) 3
(%o15) z=((z^2+c)^2+c)^2+c
(%o16) 8*z*(z^2+c)*((z^2+c)^2+c)=w
(%i17)
(%o17) C:/PROGRA~1/MAXIMA~1.1/share/maxima/5.16.1/share/contrib/topoly_solver.mac
algsys' cannot solve - system too complicated.
#0: to_poly_solve(e=[z = ((z^2+c)^2+c)^2+c,8*z*(z^2+c)*((z^2+c)^2+c) = w],vars=[z,c])
-- an error.  To debug this try debugmode(true);


I use code by Robert P. Munafo[19] which is based on paper of Wolf Jung.

One can approximate period 3 components with equations [20] :

(%i1) z:x+y*%i;
(%o1)                              %i y + x
(%i2) w:asinh(z);
(%o2)                           asinh(%i y + x)
(%i3) realpart(w);
(%o3)
2    2
2    2     2      2  2 1/4     atan2(2 x y, - y  + x  + 1)      2
log((((- y  + x  + 1)  + 4 x  y )    sin(---------------------------) + y)
2
2    2
2    2     2      2  2 1/4     atan2(2 x y, - y  + x  + 1)      2
+ (((- y  + x  + 1)  + 4 x  y )    cos(---------------------------) + x) )/2
2
(%i4) imagpart(w);
2    2
2    2     2      2  2 1/4     atan2(2 x y, - y  + x  + 1)
(%o4) atan2(((- y  + x  + 1)  + 4 x  y )    sin(---------------------------)
2
2    2
2    2     2      2  2 1/4     atan2(2 x y, - y  + x  + 1)
+ y, ((- y  + x  + 1)  + 4 x  y )    cos(---------------------------) + x)



#### Boundary equation

The result of solving above system with respect to ${\displaystyle c\,}$ is boundary equation,

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

where ${\displaystyle b_{p}(w,c)\,}$ is boundary polynomial.

It defines exact coordinates of hyperbolic componenets for given period ${\displaystyle p\,}$.

It is implicit equation.

Period 1

One can easly compute boundary point c

${\displaystyle c=c_{x}+c_{y}*i}$

of period 1 hyperbolic component ( main cardioid) for given internal angle ( rotation number) t using this code by Wolf Jung[21]  

t *= (2*PI); // from turns to radians cx = 0.5*cos(t) - 0.25*cos(2*t); cy = 0.5*sin(t) - 0.25*sin(2*t); 

Period 2

 

t *= (2*PI); // from turns to radians cx = 0.25*cos(t) - 1.0; cy = 0.25*sin(t); 

#### Solving boundary equations

Solving boundary equations for various angles gives list of boundary points.

# size

## size of the component

-- size-estimate.hs
import Data.Complex (Complex((:+)), polar)
import System.Environment (getArgs)

lambdabeta :: Int -> Complex Double -> (Complex Double, Complex Double)
lambdabeta period c = (lambda, beta)
where
zs = take period . iterate (\z -> z * z + c) $0 lambdas = drop 1 . map (2 *)$ zs
lambda = product lambdas
beta = sum . map recip . scanl (*) 1 $lambdas sizeorient :: Int -> Complex Double -> (Double, Double) sizeorient period c = polar . recip$ beta * lambda * lambda
where
(lambda, beta) = lambdabeta period c

main :: IO ()
main = do
[p, x, y] <- getArgs
(size, orient) = sizeorient period (re :+ im)
putStrLn $"period = " ++ show period putStrLn$ "re     = " ++ show re
putStrLn $"im = " ++ show im putStrLn$ "size   = " ++ show size
putStrLn \$ "orient = " ++ show orient


# distinguish cardioids from pseudocircles

method of distinguish cardioids from pseudocircles is described in : Universal Mandelbrot Set by A. Dolotin. The relevant part is section 5, in particular equation 5.8. In the paper ${\displaystyle \gamma }$ is defined implicitly in equation 5.1,

${\displaystyle \gamma ={\frac {\partial }{\partial z}}f(z,c)}$

Equation 5.8 then becomes:

${\displaystyle \epsilon =-{\frac {1}{{\frac {\partial }{\partial c}}F{\frac {\partial }{\partial z}}F}}\left({\frac {{\frac {\partial }{\partial c}}{\frac {\partial }{\partial c}}F}{2{\frac {\partial }{\partial c}}F}}+{\frac {{\frac {\partial }{\partial c}}{\frac {\partial }{\partial z}}F}{{\frac {\partial }{\partial z}}F}}\right)}$

When epsilon is:

• near 0 then you have a cardioid
• near 1 then you have a circle

Code:

# compute t from c

Function describing relation between parameter c and internal angle t :

${\displaystyle c=c(t)={\frac {\lambda }{2}}\left(1-{\frac {\lambda }{2}}\right)={\frac {e^{\Pi *t*i}}{2}}-{\frac {e^{2*\Pi *t*i}}{4}}}$

It is used for computing c point of boundary of main cardioid

To compute t from c one can use Maxima CAS:

(%i1) eq1:c = exp(%pi*t*%i)/2 -  exp(2*%pi*t*%i)/4;

%i %pi t     2 %i %pi t
%e           %e
(%o1)                    c = ---------- - ------------
2             4
(%i2) solve(eq1,t);
%i log(1 - sqrt(1 - 4 c))        %i log(sqrt(1 - 4 c) + 1)
(%o2) [t = - -------------------------, t = - -------------------------]
%pi                              %pi



So :

 f1(c):=float(cabs( -  %i* log(1 - sqrt(1 - 4* c))/%pi));
f2(c):=float(cabs( -  %i* log(1 + sqrt(1 - 4* c))/%pi));

`