Where am I ?

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Offline knighty

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« on: January 22, 2018, 04:27:48 PM »
Where?

Linkback: https://fractalforums.org/share-a-fractal/22/where-am-i/746/

Offline claude

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« Reply #1 on: January 22, 2018, 06:42:45 PM »
here!

Offline knighty

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« Reply #2 on: January 22, 2018, 09:10:41 PM »
Wow! that was fast! :)

Offline claude

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« Reply #3 on: January 22, 2018, 09:20:10 PM »
counted the spirals (about 25?), notice that it looks like seahorse valley with 12 or 13 spirals on either side of the connections, the doubling structure is like that near the antenna of a minibrot, so I chose the minibrot at angled internal address 1 12/25 26 and zoomed to the tip of its antenna

Offline knighty

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« Reply #4 on: January 23, 2018, 11:30:09 AM »
That makes me think about few questions:
How the angled address of a tip of a minibrot looks like? Is it rational or irrational?
Are the "cores" (when removing the decorations) of the Julia sets of the tips of minibrots straight lines or are they slightly broken in general?

Offline claude

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« Reply #5 on: January 23, 2018, 01:58:37 PM »
The angled internal address of the tip of the antenna of the main set is infinite: 1 ->(1/2) -> 2 ->(1/2) -> 3 ->(1/2) -> 4 ->(1/2) -> 5 ->(1/2) -> etc.  For minibrots it'll just be a "tuned" version (add the prefix of the parent island and multiply all periods by a factor, I suspect - never done this calculation in anger so I'm not 100% sure....)

The Julia set of -2 is a straight line segment, so the Julia sets of minibrot tips will have "cores" of straight lines too (though I'm not sure if they will be warped?  I don't think they will have gaps, at least)

Compare with "Example: period 3 island", "Example: period 3 island 1 over 3 bulb", "Example: period 4 island" in https://mathr.co.uk/mandelbrot/julia-dim.pdf (around slide 170 in my pdf viewer).

Offline knighty

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« Reply #6 on: January 23, 2018, 03:18:18 PM »
That document is too cool!

I found a deformed (that's what I was meaning by broken) line near the tip of a minibrot in the elephant valley (picture below).

Sorry, I wanted to say external angle. Of course there are two rays that land on the tip if a minibrot so there are two external angles instead of one.

Offline claude

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« Reply #7 on: January 23, 2018, 06:23:28 PM »
Quote from: knighty
Sorry, I wanted to say external angle. Of course there are two rays that land on the tip if a minibrot so there are two external angles instead of one.

The angles are rational, as the tip is a Misiurewicz point.  You can find them by the tuning algorithm, using the two representations for the tip: .0(1) and .1(0) See https://mathr.co.uk/blog/2013-10-02_islands_in_the_hairs.html

Offline claude

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« Reply #8 on: January 23, 2018, 06:25:46 PM »
I found a deformed (that's what I was meaning by broken) line near the tip of a minibrot in the elephant valley (picture below).
That's an embedded Julia set (and thus a Cantor dust connected to the minbrot with filament decorations), not a Julia set (the dynamical plane instead of the parameter plane).  Confusing terminology!

Offline knighty

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« Reply #9 on: January 24, 2018, 10:33:22 AM »
Thanks a lot. I thought that the tips of the minibrots were not Misiurewicz points.

Offline claude

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« Reply #10 on: January 24, 2018, 04:29:29 PM »
The tips of limits of sequences of discs are Feigenbaum points with irrational external angles.