Jungreis algorithm gives inverse of Boettcher function.

From Douaddy-Hubbard theory it is known that that M

It is proved by constructing a analytic homeomorphism (bijection) F of D

c = Psi_M (w)

for c: c belongs to M

So one can transform complement (exterior) of closed unit disk to complement (exterior) of Mandelbrot set

Douaddy-Hubbard" construction does not lend itself to computation. "

Jungreis "gives alternative construction which does."

It allows to compute points of external rays R(angle) and equipotential lines in w-plane using definitions:

R(angle) = radius * e

where 1 < radius <= infinity

E(radius) = radius * e

where 0 < angle <= 1 [ turns]

and after that transform it to c-plane using Psi_M and draw.

Isn't it beautifull ?

(Here I will skip theory and will give only final result)

Because function Psi_M is analytic one can compute it using power series

Psi_M(w)= w + Sum ( b

where 0 <=m <=infinity

It is sufficiant to compute some ( finite and not very large) number of terms.

Jungreis uses 4095 terms of power series to achieve accurate ploting result.

First one should compute coefficients of Psi_M.

list of coefficients of the mapping exterior of unit disk to the exterior of the Mandelbrot set

Set D is closed unit disk

Set D

D

Hera are the circles with center=0 and radius: 1, 1.0001, 1.05, 1.1, 1.15, 1.2, 1.25, 1.3, 1.4, 1.5

and ninty equaly spaced radial lines ( external rays of angles = p/90 : 1<=p<=90 measured in turns ).

These circles are equipotential lines for ....

It is the initial image for transformation ( see image in Jungreis paper)

and here is a program that draws this image ( in delphi for win32).

c,z,w are complex numbers

k is integer number

let P

P

P

( iteration gives set of points {z

named orbit)

Mandelbrot set M = { c : P

Set M

On this image M

It is made using boolean escape time ( Mandel ) algorithm.

( translation of maple code by G. A. Edgar

------------------( function version)-------------------------------

u(n,k):=block([j],

if 2^n-1=k then 1

else if 2^n-1>k then sum(u(n-1,j)*u(n-1,k-j),j,0,k)

else if 2^(n+1)-1>k then 0

else (u(n+1,k)-sum(u(n,j)*u(n,k-j),j,1,k-1))/2

);

a(j):=u(0,j+1);

w(n,m):=block

([j],

if n=0 then 0

else a(m-1)+w(n-1,m)+sum(a(j)*w(n-1,m-j-1),j,0,m-2)

);

b(m):= if m=0 then -1/2

else -w(m,m+1)/m;

---------------( array version)--------------------------------------

u[n,k]:=block([j],

if 2^n-1=k then 1

else if 2^n-1>k then sum(u[n-1,j]*u[n-1,k-j],j,0,k)

else if 2^(n+1)-1>k then 0

else (u[n+1,k]-sum(u[n,j]*u[n,k-j],j,1,k -1))/2);

a[j]:=u[0,j+1];

w[n,m]:=block

([j],

if n=0 then 0

else a[m-1]+w[n-1,m]+sum(a[j]*w[n-1,m-j-1],j,0,m-2)

);

b[m]:= if m=0 then -1/2

else -w[m,m+1]/m;

thx to G. A. Edgar and Richard J. Fateman for help (:-))

betaF(n,m):=block

(

[nnn:2^(n+1)-1],

if m=0

then 1.0

else if ((n>0) and (m < nnn)) then 0.0

else (betaF(n+1,m)- sum(betaF(n,k)*betaF(n,m-k),k,nnn,m

-nnn)-betaF(0,m-nnn))/2.0

);

b(m):=betaF(0,m+1);

Robert P. Munafo have noticed that : (above) " maxima code does not gain the efficiency of

remembering previously computed values (as the "option remember" directive does in Maple)"

I have asked Richard J. Fateman for help, and he said:

"replace u(n,k) by u[n,k] everywhere, and it will remember values, because u is now

an array instead of a function." RJF

so here is latest version:

betaF[n,m]:=block

(

[nnn:2^(n+1)-1],

if m=0

then 1.0

else if ((n>0) and (m < nnn)) then 0.0

else (betaF[n+1,m]- sum(betaF[n,k]*betaF[n,m-k],k,nnn,m

-nnn)-betaF[0,m-nnn])/2.0

);

b(m):=betaF[0,m+1];

One can do a list with:

for m:1 thru 20 step 1 do display(b(m));

or

for i:0 thru 64 do ( print(sconcat("b[", i, "]=", b(i))) );

list of coefficients

- Thx for Rafael de la Llave and Pau Atela for help and many informations. (:-))
- Irwin Jungreis: The uniformization of the complement of the Mandelbrot set. Duke Math. J. 52, no. 4 (1985), 935–938
- John H. Ewing and Glenn Schober On the coefficients of the mapping to the exterior of the Mandelbrot set. Michigan Math. J. 37, iss. 2 (1990), 315–320
- John Ewing : Can We See the Mandelbrot Sef ? THE COLLEGE MATHEMATICS JOURNAL VOL. 26, NO. 2, MARCH 199
- D Allingham: Conformal Mappings And The Area Of The Mandelbrot Set
- Bifurcation of Dynamic Rays in Complex Polynomials of Degree Two, Atela, P., Ergod Th & Dynam Sys (1991) 12, 401-423
- Weisstein, Eric W. "Mandelbrot Set." From MathWorld-
- Laurent Series by Robert P. Munafo
- computation of the coefficients b(j) in maple code by G. A. Edgar
- Delphi program that computes coeeficients unfinished
- maxima code
- list of coefficients of the mapping exterior of unit disk to the exterior of the Mandelbrot set
- evaluation of maple code - discussin on sci.math.symbolic ( also Mathematica code)
- A054670 at The On-Line Encyclopedia of Integer Sequences
- A054671
- Drawing the Mandelbrot Set by Jason Eckert

Feel free to e-mail me!

Author: Adam Majewski

adammaj1-at-o2-dot-pl

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