### algebraic number theory and the Mandelbrot set

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#### 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

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