Mandelbrot probability

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Offline gerrit

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« on: April 19, 2018, 02:03:32 AM »
Here's something I don't get. Suppose you zoom in on the M-set on the real axis somewhere near -2, say a random point between -2 and -1.99.
No matter how far you zoom, you'll always see a simple line with a few vertical lines.

There are minis between -2 and -1.99 so there is an uncountable infinite number of points that are in a mini. On the other hand, there is only a countable infinite number of Misiurewizc points so the probability of hitting one of those is zero.

So if the random point is not hitting a Misiurewicz point, what is it hitting and why do we never land inside a mini?

Linkback: https://fractalforums.org/fractal-mathematics-and-new-theories/28/mandelbrot-probability/1239/

Offline pauldelbrot

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« Reply #1 on: April 19, 2018, 02:24:21 AM »
There are also not-inside-mini points with irrational external angles on the spike, and on any filament, and there are uncountably infinitely many of those. The limit points of regular sequences of increasing-period minibrots tend to be Misiurewicz, as do their spike tips and cusps, but the limit points of ergodic sequences of increasing-period minibrots will be irrational points, sort of like how infinite repeating decimals produce rational numbers, but infinite nonrepeating ones produce irrational (and mostly transcendental) ones.


Offline gerrit

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« Reply #3 on: April 19, 2018, 05:46:04 PM »
Thanks that explains it, so I'm hitting Feigenbaum points. Nice explanation for how there can be uncountably many.
Seems they make poor references with usual glitch detection methods, as per my latest KF bug report.


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