mating polynomial Julia sets

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Online claude

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« on: January 16, 2020, 08:26:44 AM »
Arnaud Chéritat has some nice videos of polynomial mating:
https://www.math.univ-toulouse.fr/~cheritat/MatMovies/

Wolf Jung has a nice pre-print "The Thurston Algorithm for quadratic matings"
In particular Chapter 5 "Implementation of slow mating" describes the mating algorithm.
https://arxiv.org/abs/1706.04177
There is also code accompanying the paper, but I didn't find it easy to understand.

"Workshop on polynomial matings" 8-11 juin 2011, Toulouse has some nice papers
http://www.numdam.org/issues/AFST_2012_6_21_S5/
in particular:

"On The Notions of Mating" Carsten Lunde Petersen & Daniel Meyer
has a brief section on "Cheritat movies" about how to draw the things which also says "see the paper by Cheritat in this volume", that paper seems to be:

"Tan Lei and Shishikura’s example of non-mateable degree 3 polynomials without a Levy cycle" Arnaud Chéritat
has a comment in Figure 4's caption saying the visualization method "will be explained in a forthcoming article".

Private communication with Arnaud Chéritat revealed that that article was never published.

But I think I figured it out.

Linkback: https://fractalforums.org/fractal-mathematics-and-new-theories/28/mating-polynomial-julia-sets/3273/

Offline gerrit

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« Reply #1 on: January 16, 2020, 03:46:43 PM »
Is it possible to explain what "polynomial mating" is without all the dense math, dividing by "~", gnomic maps and what not?

Online claude

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« Reply #2 on: January 16, 2020, 05:36:05 PM »
It's gluing Julia sets together such that the external rays match up, then shrinking all the rays to points.  Or that's how I understand it.  Don't know how the maths works in detail, just enough to implement it based on the papers....

My code is here: https://code.mathr.co.uk/mating
Blog post coming soon.

Online claude

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Offline gerrit

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« Reply #4 on: January 16, 2020, 11:29:59 PM »
Can you explain what is "pulling back the critical orbits as described in Wolf Jung's preprint"?
I can't understand a word of Wolf Jung's preprint.

Online claude

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« Reply #5 on: January 17, 2020, 01:41:43 PM »
I only understand parts of chapter 5, but that is enough to implement it.

start with the p and q, complex numbers
calculate orbits p_i = 0, p, p^2 + p, ...  and q_i = 0, q, q^2 + q, ...
pick a number of steps per path, S
for step s in [0..S) initialize radius R(s) decay from some large R_1^2 to R_1
initialize x_i(0, s) = p_i / R(s) and y_i(0, s) = R(s) / q_i  (this is an approximation for large R, for smaller R see the paper)
for n = 0 to iteration count
 for s in 0 to S-1
  for z in x, y
    z_i(n + 1)(s) = F(z_{i + 1}(n, s)) chosing sign of square root such that z_i(n+1)(s) is nearest to z_i(n+1)(s-1); (with wrapping to previous n and s=S-1 if s=0)
    F is defined by equation 22 in Wolf Jung's paper, it is sqrt((1-b)/(1-a) (z-a)/(z-b)) where a and b are x_1(n,s), y_1(n,s)

for more precise details see my code, I tried to keep it quite straightforward with some comments - there is some subtlety required to deal with z near infinity as well as z near 0
« Last Edit: January 17, 2020, 02:00:17 PM by claude, Reason: typo »

Offline Adam Majewski

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« Reply #6 on: January 17, 2020, 07:14:58 PM »

Offline Adam Majewski

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« Reply #7 on: January 18, 2020, 07:37:07 AM »
My code is here: https://code.mathr.co.uk/mating
Blog post coming soon.

The code is so short.
Very complicated problem and so short code !!!
Cool.

Thx

Online claude

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« Reply #8 on: January 18, 2020, 09:09:40 PM »
The code is so short.
It does not yet check for "homotopy violations", as I don't know what they are, but it could lead to wrong results if they occur (afaict).


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