### Does every point along the real axis of the M-brot set needl contain a minibrot?

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#### DeusDarker

• Fractal Fanatic
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• Infinite complexity in finite space

#### Does every point along the real axis of the M-brot set needl contain a minibrot?

« on: August 06, 2020, 03:44:04 PM »
Heres a strange thought thats kind of hard for me to articulate so hopefully you get my meaning. as you zoom into the Mandelbrot needle its apparent that minibrots are 'Very' numerous along the real axis. they seem to have gaps between them but when you zoom into the gaps there are smaller minibrots between the bigger minibrots. you can zoom in as far as you want and still reach a minibrot. ive watched many very intense Mandelbrot zoom ins and minibrots still show up. So, lets say you pick any point along the needle, and zoom in forever, are you absolutely guaranteed to hit a minibrot at some point, for every point? if so, wouldnt that imply that every point on the needle (in the real axis) is part of the Mandelbrot set in almost a line like fashion? It would make sense i guess but when looking at the M-brot set overall the needle 'appears' to have gaps, but as you zoom in it seems those gaps are filled by minibrots of descending sizes. Am i wrong? Have i overlooked something? Let me know your thoughts!

• 3f
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#### Re: Does every point along the real axis of the M-brot set needl contain a minibrot?

« Reply #1 on: August 06, 2020, 04:44:20 PM »

#### hobold

• Fractal Fluff
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#### Re: Does every point along the real axis of the M-brot set needl contain a minibrot?

« Reply #2 on: August 06, 2020, 04:59:29 PM »

• 3c
• Posts: 858

#### Re: Does every point along the real axis of the M-brot set needl contain a minibrot?

« Reply #3 on: August 06, 2020, 05:05:30 PM »
There were some related questions in the old forum:

http://www.fractalforums.com/general-discussion-b77/minibrots-at-every-point-of-the-border-of-the-m-set/

http://www.fractalforums.com/mandelbrot-and-julia-set/proximity-of-minibrots/

If there were something else on the x-axis that has area but is not a bulb, one had disproven the (open) hyperbolicity conjecture. And for a single point, one cannot argue that it does not belong to an adjacent minibrot, so you have to prove that there is a whole finite length interval that is a pure line where every non-zero imaginary point close to it is exterior (also open afaik).

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