# Structure of Mandelbrot bug antennae

I have analysed the structure of of Mandelbrot set and I have understood something, but I still have some questions, mainly about antennae and little bugs.

Mandelbrot set consists of

• the main cardioid,
• heads, that are deformed copies of the whole bug,
• and antennae (First part of the visible antenna (for example: from ~(-1.4 + 0i) to period 2 Misiurewicz point at ~(-1.543689 + 0i), from ~(-0.109 - 0.895i) to period 3 Misiurewicz point at ~(-0.101 - 0.956i) and from ~(-0.0176 - 0.796i) to period 3 Misiurewicz points at ~(0.00164 - 0.822i) and ~(-0.0000831 - 0.793i)) is the image of the antenna of the bug, image of which the head is. It is not a part of the antenna defined in this point by me.).

The antennae come out of each image of period 1 Misiurewicz point, where from the end of image of antenna coming out of period n head come n-1 antennae (so this image of period 1 Misiurewicz point is period n Misiurewicz point). (Infinitely many images of antennae come out of each head and each of this images has infinitely many ends. Each of the ends is image of period 1 Misiurkiewicz point.)

Main heads going out of the main cardioid have simply periods equal to consecutive natural numbers ... 7, 6, 5, 4, 3, 2, 3, 4, 5, 6, 7 ... . Besides, between each two neighbouring heads, there is a head with period being the sum of periods of the two heads (I see that my description of secondary heads periods is not unambiguous, but the pattern is obvious, so I hope that you understand it).

Antennae consist of the main little bug, the part before it and antennae coming out of the main little bug, one from each image of period 1 Misiurkiewicz point. Both parts of antennae seem to have Misiurewicz points with each period that appear in heads and images on heads (in little bugs and in main heads), from which they are coming out, but I can't recognize how they are organised. Each antenna comes from image of head with period 2 in image of head with period 2 in ..., so period 2 Misiurewicz points should be common, but they are hard to find. The most important one inside an antenna as I define it, seems to be ~(-1.839286755214161 + 0i).

Structure of copies of antennae in antennae seems to emphasize the copies that correspond to the copy part of which they are.

Main little bug of main antenna coming from main heads with period n have period n+1. How can one determine periods of other little bugs?

Antennae coming out of a little bug dominate in groups of 2k

• 1+1: part before little bug and antenna from head with period 2 (head 2)
• 2: 2 1st antennae from heads 3
• 4: 2 1st antennae from copy of head 3 in head 2 (head 2 3) and 2 1st antennae from head 4
• 8: 2 1st antennae from head 3 in main little bug from head 2, 2 2nd antennae from head 5=2+3, 2 2nd antennae from head 3 and 2 1st antennae from head 5

What is the pattern, except that 1st antennae from head n come in the group of 2n-2? The fact that the antenna from the period 5 little bug nearer to the end comes in group of 16 and antenna from the one nearer period 3 little bug in group 32 seems to say something about the structure of parts of antenna.

Maybe you know also some strict formulas for the self similarities with deformed copies. For example Julia sets consist of two images of itself coming from two branches of square root function.

I'd like you to correct my description of this structure, answer my questions and supplement it.

• See papers by G Pastor, for example : researchgate.net/publication/…
Commented Feb 27, 2015 at 16:12
• @Adam: Than you. Finally I tried to look at the whole paper and I saw something about the structure of antennae. Now I have to understand it. Commented Aug 2, 2015 at 14:30
• Ok. If you need software see : mathr.co.uk/blog/mandelbrot.html and mndynamics.com/indexp.html.
• chaotic part = main antenna = a shrub of $$F_{1/2}$$ family