Fractal Related Discussion > Noob's Corner

Period doubling in minibrots

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marcm200:
(From time to time, some questions about the Mandelbrot occur to me, that seem to be very basic - but for which I cannot find the answers myself (or by a (non-exhaustive) search). Maybe someone here knows of some references towards those topics?

1. Area: Due to x-axis symmetry, any hyperbolic component that fully lies in the negative imaginary part of the complex plane has a mirrored twin in the positive part - and those two have obviously the same area.

Aside from those, are there other pairs or even larger groups of components, that have the same area (equal or different periods)?

2. Period doubling: On the real axis, period doubling for the bulb-y components occur up to some seed value. Does period-doubling also occur in the case of minibrots?

If so: Suppose I connect the hyperbolic centers of the minibrot's cardioid with period p and the period-doubled 2p-component with a straight line and extend it outwards. Does then period-doubling follow that line? Or will it be angled?

EDIT: Thread renamed from "Basic questions about the Mandelbrot set" on Feb 4,2021.

youhn:
Interesting questions.

Some half answers from another noob;

1. All the minibrots not on the x-axis seem to have (more or less) distorted reflectional symmetry and all spokes are also slightly different in exact shape, so strictly speaking I don't think there is absolute reflective symmetry to be found other than around the x-axis (the real numbers). And how to define "area" in this case? The mandelbrot is 1 connected area with only 1 boundary (which can never be found exactly, but the rough shape is very clear for any observer after only a "few" iteration). If you were able to freely add any boundary line in the set, to separate areas, I'm pretty sure it's in theory doable to divide the whole into equally big areas. You would need infinite time to carry out the full work though.
(if my assumptions are incorrect, I'm happy to learn)

2. I have no idea, online orbit visualisations cannot zoom in enough for a simple check. I would guess " yes!".

(no number). No, see answer 1. The line would more or less bend into a curve, depending on the location.

marcm200:

--- Quote from: youhn on January 15, 2021, 05:14:46 PM ---And how to define "area" in this case?

--- End quote ---

With area I am referring to the area of a specific hyperbolic component only. For periods 1 and 2, the boundaries can be expressed as simple curves and the area is calculable - and for period 3, Giarrusso et. al gave an explicit trigonometric formulation ("A parameterization of the period 3 hyperbolic components of the Mandelbrot set"), so one could in principle integrate to get the area (although all my attempts to do so, failed, but that's a different story).

youhn:
Uhm, ok. That's a bit outside my comfortable knowledge zone. Though I did find some references. It appears there are 3 methods for p=3 and no known methods for p=4 :

See the summary given on https://en.wikipedia.org/wiki/File:Mandelbrot_set_Components.jpg