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Author Topic:  How does an Mset boundary point know which cycle length to take?  (Read 260 times)

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Offline marcm200

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I feel this is a very basic question, but I seem to be unable to find an answer for (neither by myself nor searching):

The main cardioid is period 1 in its interior, the period-2 bulb, of course, has a period-2 attracting cycle - and their touching point at c=-0.75 is a parabolic period 1. The point belongs with the same "right" to both bulb's boundaries. How/why did it choose to be period-1?

And on the other side, the point c=-1.25 transferring from period-2 to the next bulb. This point is period-2 parabolic.

Is it always the smaller period when two bulbs touch? Or no rule?

In this sense, does the boundary of the main cardioid form a closed curve when considering only points with period-1? And the period-2 bulb then has a "hole" at -0.75.

How many holes has the boundary of a hyperbolic bulb with period, let's say 2000 - is there one smaller periodic bulb attaching, and all others are of larger cycle lengths?




Linkback: https://fractalforums.org/noobs-corner/76/how-does-an-mset-boundary-point-know-which-cycle-length-to-take/3714/
« Last Edit: August 23, 2020, 09:05:07 AM by marcm200 »

Offline hobold

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Re: How does an Mset boundary point know which cycle length to take?
« Reply #1 on: August 10, 2020, 06:46:24 PM »
I don't have an authoritative source to offer. But I remember that the period length of any disk is some integer multiple of the connected parent's period length. Cardioids are the only natural exception. And the "Main Cardioid" is main because it is the starting anchor, so to speak, with the shortest possible period length of 1.

So the touching point between any two connected components will naturally "choose" the parent's period length, because that allows this point to also satisfy (the n times longer) periodicity of the child component.

Offline Adam Majewski

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Re: How does an Mset boundary point know which cycle length to take?
« Reply #2 on: August 10, 2020, 09:03:16 PM »
Do you mean how one cycle changes to another cycle thru touching point = root = parabolic point = bond?
https://commons.wikimedia.org/wiki/File:Bifurcation1-2.png
see image and comments/links


HTH

Offline marcm200

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Re: How does an Mset boundary point know which cycle length to take?
« Reply #3 on: August 10, 2020, 09:23:11 PM »
So I can view the Mset as a collection of cardioids of minibrots as the principal components with some period, and all primary, secondary and so on bulbs attached to them have a period length larger than their parent (be it a bulb or a cardioid). In consequence all cardioid's boundaries are closed curves w.r.t. to periodic length (attracting or indifferent), and all bulbs' boundaries have exactly one hole. Interesting! Thanks for the clarification.



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