Pathfinding in the Mandelbrot set - Revisited

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Offline wbarry

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« Reply #15 on: July 14, 2019, 05:00:28 PM »
Re: "higher degree monomials in the parameter-space"
The short answer, I don't know. I'm not even certain I understand your questions.

It does appear, after a whopping 15 minutes of zooming, that the higher order M-sets record the paths around their minibrots.
But that is purely "I looked a tiny bit and it seems true", which in math terms means nothing.

For your more complicated questions, I have no idea. I have a 30-year-old undergraduate degree in math.
I did take calculus from Heinz-Otto Peitgen.
And he invited Mandelbrot to give a lecture at UCSC where he explained how his ideas came about.
We also viewed the first Mandelbrot Zoom video ever made (by Pietgen & Richter).
But I can't pretend I understood all of what they said.
I remember pretty pictures.
I was probably stoned.
Walter Barry
Mandelbrot Evolution

Offline gerrit

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« Reply #16 on: July 14, 2019, 09:31:40 PM »
Does this recording of paths also apply to other monomials of higher degree or to polynomials in general (or is the z²-Mandelbrot the "master fractal"? And only to the parameter-space or also to the dynamical entity?
Yes (eg, sort-of (usually (only simple roots) just reproduces quadratic when zooming), the former only (Julia set records path to that point in p-space, nothing recorded in J-space zoom).

Offline claude

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« Reply #17 on: July 15, 2019, 03:43:04 PM »
Does this recording of paths also apply to other monomials of higher degree
yes, power 3 has tripling of features (3x, 9x, 27x, ...) etc.
also applies (albeit sometimes distorted) to some non-analytic fractals like the Burning Ship and variations

Offline quadralienne

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« Reply #18 on: September 27, 2019, 02:33:15 PM »
I've been playing around with xaos with __float128 and getting some super deep zooms ... in this one I think I started in the seahorse valley, then the 3-way fork you can see in the bottom left, then some big-circle spirals (across from the seahorse valley), then ... well I lost track but it's cool to see them all together.

Offline marcm200

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« Reply #19 on: September 28, 2019, 10:46:39 AM »
An interesting paper about the address of the centers of (some) hyperbolic components of the Mset:

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Determination of Mandelbrot Set's Hyperbolic Component Centres
Alvarez, Romera, Pastor, Montoya
Chaos, Solitons and Fractals Vol 8 No 01, 1998

The authors look at the orbit of the critical point and note whether an iterate is on the left or the right side of the critical point. That sequence of symbols (L, R) determines a hyperbolic center. (Technically they do the opposite: Provided a sequence of (non-ending) Ls and Rs they solve an iterated equation (c² +- c) +- c ... using L,R as a plus or minus sign in the formula.)

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