# Intro

Exterior or complement of filled-in Julia set is :

${\displaystyle A_{f_{c}}(\infty )=K(f_{c})^{C}=F_{\infty }}$

It can be analysed using

• escape time (simple but gives only radial values = escape time ) LSM/J,
• distance estimation ( more advanced, continuus, but gives only radial values = distance ) DEM/J
• Boettcher coordinate or complex potential ( the best )

" The dynamics of polynomials is much better understood than the dynamics of general rational maps" due to the Bottcher’s theorem[1]

## Superattracting fixed points

For complex quadratic polynomial there are many superattracting fixed point ( with multiplier = 0 ):

• infinity ( It is always is superattracting fixed point for polynomials )
• ${\displaystyle z_{s}=0\,}$ is finite superattracting fixed point for map ${\displaystyle f_{0}\,}$
• ${\displaystyle z_{s}=0\,}$ and ${\displaystyle z_{s}=-1\,}$ are two finite superattracting fixed points for map ${\displaystyle f_{-1}\,}$

## Description

Near[2] super attracting fixed point (for example infinity) the behaviour of discrete dynamical system :

${\displaystyle z_{n+1}=f_{c}(z_{n})=z_{n}^{2}+c\,}$

based on complex quadratic polynomial ${\displaystyle f_{c}(z)=z^{2}+c\,}$ is similar to

${\displaystyle w_{n+1}=f_{0}(w_{n})=w_{n}^{2}\,}$

based on ${\displaystyle f_{0}(w)=w^{2}\,}$

It can be treated as one dynamical system viewed in two coordinate systems :

• easy ( w )
• hard to analyse( z )

[3]

In other words map ${\displaystyle f_{c}\,}$ is conjugate [4] to map ${\displaystyle f_{0}\,}$ near infinity. [5]

## History

In 1904 LE Boettcher  :

• solved Schröder functional equation[6][7] in case of supperattracting fixed point[8]
• "proved the existence of an analytic function ${\displaystyle \phi (z)\sim z}$ near ${\displaystyle \infty }$ which conjugates the polynomial with ${\displaystyle z^{d}}$, that is ${\displaystyle \phi \circ p(z)=\phi (z)^{d}}$" ( Alexandre Eremenko ) [9]

## Names

• ${\displaystyle w\,}$ is Boettcher coordinate
• ${\displaystyle \Phi _{c}\,}$ is Boettcher function
• Boettcher Functional Equation [10][11]: ${\displaystyle \Phi _{c}(f_{c}(z))=\Phi _{c}(z)^{2}\,}$

where :

${\displaystyle w=\Phi _{c}(z)\,}$

# Complex potential on the parameter plane

Complex potential

# Complex potential on dynamical plane

Complex potential or Boettcher coordinate has :

• radial values ( real potential ) LogPhi = CPM/J
• angular values ( external angle ) ArgPhi

Both values can be used to color with 2D gradient.

To compute Boettcher coordinate ${\displaystyle w\,}$ use this formula [12]

${\displaystyle w=\Phi _{c}(z)=z*\prod _{n=1}^{\infty }\left({\frac {f_{c}^{n}(z)}{f_{c}^{n}(z)-c}}\right)^{1/2^{n}}\,}$

It looks "simple", but :

## LogPhi - Douady-Hubbard potential - real potential - radial component of complex potential

### CPM/J

Potential of filled Julia set
Diagram of potential computed with 2 methods : simple and full

Note that potential inside Kc is zero so :

Pseudocode version :

if (LastIteration==IterationMax)
then potential=0    /* inside  Filled-in Julia set */
else potential= GiveLogPhi(z0,c,ER,nMax); /* outside */


It also removes potential error for log(0).

#### Full version

Math (full) notation : [13]

${\displaystyle LogPhi_{c}(z)=ln|z|+\sum _{n=1}^{\infty }{\frac {1}{2^{n}}}ln|1+{\frac {c}{(f_{c}^{n-1}(z))^{2}}}|}$

Maxima (full) function :

GiveLogPhi(z0,c,ER,nMax):=
block(
[z:z0,
logphi:log(cabs(z)),
fac:1/2,
n:0],
while n<nMax and abs(z)<ER do
(z:z*z+c,
logphi:logphi+fac*log(cabs(1+c/(z*z))),
n:n+1
),
return(float(logphi))
)$ #### Simplified version The escape rate function of a polynomial f is defined by : ${\displaystyle G_{f}(z)=\lim _{n\rightarrow \infty }{\frac {1}{2^{n}}}log^{+}|f^{n}(z)|\,}$ where : ${\displaystyle log^{+}=max(log,0)}$ "The function Gp is continous on C and harmonic on the complement of the Julia set. It vanishes identically on K(f) and as it has a logarithmic pole at infinity, it is a it is the Green's function for C/ K(f)." ( Laura G. DeMarco) [14] Math simplified formula : ${\displaystyle SLogPhi_{c}(z)={\frac {log(f^{n}(z))}{2^{n}}}\,}$ Maxima function : GiveSLogPhi(z0,c,e_r,i_max):= block( [z:z0, logphi, fac:1/2, i:0 ], while i<i_max and cabs(z)<e_r do (z:z*z+c, fac:fac/2, i:i+1 ), logphi:fac*log(cabs(z)), return(float(logphi)) )$


If you don't check if orbit is not bounded ( escapes, bailout test) then use this Maxima function :

GiveSLogPhi(z0,c,e_r,i_max):=
block(
[z:z0, logphi, fac:1/2, i:0],
while i<i_max and cabs(z)<e_r do
(z:z*z+c,
fac:fac/2,
i:i+1 ),
if i=i_max
then logphi:0
else logphi:fac*log(cabs(z)),
float(logphi)
)$ C version : double jlogphi(double zx0, double zy0, double cx, double cy) /* this function is based on function by W Jung http://mndynamics.com */ { int j; double zx=zx0, zy=zy0, s = 0.5, zx2=zx*zx, zy2=zy*zy, t; for (j = 1; j < 400; j++) { s *= 0.5; zy = 2 * zx * zy + cy; zx = zx2 - zy2 + cx; zx2 = zx*zx; zy2 = zy*zy; t = fabs(zx2 + zy2); // abs(z) if ( t > 1e24) break; } return s*log2(t); // log(zn)* 2^(-n) }//jlogphi  Euler version by R. Grothmann ( with small change : from z^2-c to z^2+c) :[15] function iter (z,c,n=100) ... h=z; loop 1 to n; h=h^2+c; if totalmax(abs(h))>1e20; m=#; break; endif; end; return {h,m}; endfunction x=-2:0.05:2; y=x'; z=x+I*y; {w,n}=iter(z,c); wr=max(0,log(abs(w)))/2^n;  ### Level Sets of potential = pLSM/J Here is Delphi function which gives level of potential : Function GiveLevelOfPotential(potential:extended):integer; var r:extended; begin r:= log2(abs(potential)); result:=ceil(r); end;  ### Level Curves of potential = equipotential lines = pLCM/J ## ArgPhi - External angle - angular component of complex potential One can start with binary decomposition of basin of attraction of infinity. The second step can be using ${\displaystyle \Psi _{c}\,}$ polar coordinate system and Psi_c for c=-2 ### period detection How to find period of external angle measured in turns under doubling map : Here is Common Lisp code : (defun give-period (ratio-angle) "gives period of angle in turns (ratio) under doubling map" (let* ((n (numerator ratio-angle)) (d (denominator ratio-angle)) (temp n)) ; temporary numerator (loop for p from 1 to 100 do (setq temp (mod (* temp 2) d)) ; (2 x n) modulo d = doubling) when ( or (= temp n) (= temp 0)) return p )))  Maxima CAS code : doubling_map(n,d):=mod(2*n,d);  /* catch-throw version by Stavros Macrakis, works */ GivePeriodOfAngle(n0,d):= catch( block([ni:n0], for i thru 200 do if (ni:doubling_map(ni,d))=n0 then throw(i), 0 ) )$

/* go-loop version, works */
GiveP(n0,d):=block(
[ni:n0,i:0],
block(
loop,
ni:doubling_map(ni,d),
i:i+1,
if i<100 and not (n0=ni) then go(loop)
),
if (n0=ni)
then i
else 0
);

/* Barton Willis while version without for loop , works */
GivePeriod(n0,d):=block([ni : n0,k : 1],
while (ni : doubling_map(ni,d)) # n0 and k < 100 do (
k : k + 1),


## Lamination of dynamical plane

Lamination of rabbit Julia set

Here is long description

# References

1. ON THE NOTIONS OF MATING by CARSTEN LUNDE PETERSEN AND DANIEL MEYER
2. Neighbourhood in wikipedia
3. The work of George Szekeres on functional equations by Keith Briggs
4. Topological conjugacy in wikipedia
5. How to draw external rays by Wolf Jung
6. Schröder equation in wikipedia
7. Lucjan Emil Böttcher and his mathematical legacy by Stanislaw Domoradzki, Malgorzata Stawiska
8. L. E. Boettcher, The principal laws of convergence of iterates and their aplication to analysis (Russian), Izv. Kazan. fiz.-Mat. Obshch. 14) (1904), 155-234.
9. Mathoverflow : Growth of the size of iterated polynomials
10. Böttcher equation at Hyperoperations Wiki
11. wikipedia : Böttcher's equation
12. How to draw external rays by Wolf Jung
13. The Beauty of Fractals, page 65
14. Holomorphic families of rational maps: dynamics, geometry, and potential theory. A thesis presented by Laura G. DeMarco
15. Euler examples by R. Grothmann
16. A. Douady, “Algorithms for computing angles in the Mandelbrot set,” in Chaotic Dynamics and Fractals, M. Barnsley and S. G. Demko, Eds., vol. 2 of Notes and Reports in Mathematics in Science and Engineering, pp. 155–168, Academic Press, Atlanta, Ga, USA, 1986.
17. Thierry Bousch : De combien tournent les rayons externes? Manuscrit non publié, 1995
18. Program Mandel by Wolf Jung
19. Explanation by Wolf Jung
20. Modular arithmetic in wikipedia
21. Square root of complex number gives 2 values so one has to choose only one. For details see Wolf Jung page
22. c program by Curtis McMullen (quad.c in Julia.tar.gz)
23. Quadratische Polynome by Matjaz Erat
24. wikipedia : Complex_quadratic_polynomial / planes / Dynamical_plane
25. wikipedia : Line segment