• January 18, 2021, 09:24:47 PM

### Author Topic:  algebraic number theory and the Mandelbrot set  (Read 291 times)

0 Members and 1 Guest are viewing this topic.

#### claude

• 3f
• Posts: 1735
##### algebraic number theory and the Mandelbrot set
« on: August 05, 2018, 06:32:41 PM »
Centers of hyperbolic components are known to be algebraic numbers, but I found they are in fact algebraic integers (their minimal polynomial is monic with integer coefficients).  https://arxiv.org/pdf/math/9711213.pdf shows this is well known already.  Same with Misiurewicz points, see: https://math.stackexchange.com/questions/2872942/the-degree-of-the-multiplier-of-misiurewicz-points

Parabolic points on the boundary of the period 1 cardioid at rational internal angles are general algebraic numbers (but not algebraic integers):
https://sagecell.sagemath.org/?z=eJwrVLBVMOLlKgBSprxcVUAqNU7DSEFLoSATSHgCcaGCvkKBJpAAKksGylcp6CpUxYE4ermZeQX5OZUamgAF0w_o&lang=sage
Same for period 2 circle:
https://sagecell.sagemath.org/?z=eJwrVLBVMOLlKgBSprxcyUBK11BBWyE1TsNIQUuhIBNIeAJxoYK-QoEmkDABKtLLzcwryM-p1NAEAIddDbQ=&lang=sage
I haven't figured out how to generalize this to components with higher period yet.
It seems that all bond points from one component (checked p=1 and p=2) to its period (P*) children at Q/P internal angle share a minimal polynomial of degree (number of children), which is P-1 for prime P

There are no comments for this topic. Do you want to be the first?

### Similar Topics

###### Jean-Francois Colonna - Number Theory

Started by Alef on Share a fractal

1 Replies
353 Views
September 21, 2018, 08:34:27 PM
by gerrit
###### Actual Implementation of Perturbation Theory for the Mandelbrot Set

Started by LionHeart on Fractal Mathematics And New Theories

7 Replies
985 Views
November 08, 2020, 04:47:25 AM
by LionHeart
###### Exploring The Number 3

Started by Tas_mania on Fractal Image Gallery

0 Replies
371 Views
November 29, 2017, 10:40:39 PM
by Tas_mania
###### How to Find An Optimal Number of Maximum Iterations

Started by Byte11 on Fractal Mathematics And New Theories

3 Replies
552 Views
December 24, 2017, 01:26:47 AM
by Byte11
###### Experiment - Using A Random Number Generator To Choose Fractals

Started by Tas_mania on Mandelbulber

2 Replies
654 Views
December 22, 2019, 01:16:12 PM
by daudvyd