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In the Mandelbrot set, all points of the main carodioid are asymptotic (that is, the iterations of c^2 + c approach a constant). In contrast, it seems that all bulbs have a periodicity greater than 1, that is, the iterations settle into a cycle with a certain period.

There are several questions to be asked here:

  1. Is it the case that the largest bulb off any bulb has a periodicity that is double the periodicity of its parent bulb. For instance, see:

Mandelbrot set (left) with zoomed in area (right), periods of bulbs written on in blue

All bulbs that I've tried this with, this seems to be the case. Any proof or reason would be welcome.

  1. It appears that the second largest bulb has a periodicity that is triple that of its parent bulb - see below.

enter image description here

Similarly, I had conjectured that for any n, the nth largest bulb has a periodicity that is n+1 times larger than its parent bulb. (this is not counting the negative unreal half of the Mandelbrot set, as it is a symmetrical to the positive unreal half) However, I soon realised that there were two different period 5 bulbs off the main cardioid, of different sizes:

enter image description here

enter image description here

Similarly, there were two bulbs of period 10 off the '2' bulb, and two bulbs of period 15 off the '3' bulb. Any thoughts? What makes a number, like 5, get more bulbs of different sizes?

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  • $\begingroup$ First google hit: math.stackexchange.com/questions/419983/…. Also, you should really ask a specific question, rather than several. $\endgroup$
    – Noah Schweber
    Jul 5, 2017 at 4:07
  • $\begingroup$ math.stackexchange.com/questions/2347952/… I answered one of the questions here $\endgroup$
    – Claude
    Jul 6, 2017 at 2:42
  • $\begingroup$ Reasoned mathematical arguments need definitions to proceed. How do you define the periodicity or size of a "bulb"? $\endgroup$
    – hardmath
    Jul 6, 2017 at 4:41
  • $\begingroup$ @hardmath. Good question. I would define a 'bulb' as a part of the Mandelbrot set which is connected to the rest of the set through only one point. 'periodicity': as the number of iterations approaches infinity, how many iterations go by between getting two numbers which are, say, less than 10^-10 magnitude apart from each other. 'size': area $\endgroup$
    – Thomas Delaney
    Jul 7, 2017 at 2:24

1 Answer 1

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Mu-ency's enumeration of features page states:

There is one secondary continental mu-atom for each rational number between 0 and 1. The secondary continental mu-atoms of period n correspond to the rational numbers with n in the denominator. Because of this, the number of secondary continental mu-atoms is equal to Euler's Totient function:

$$\phi(n) = n - \sum_{f:\gcd(n,f)=1}1$$

That is, take n and subtract 1 for every number that is relatively prime to n. This sequence is Sloane's sequence A0010. It starts:

0, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, 6, 8, 8, 16, 6, 18, 8, 12, 10, 22, 8, 20, 12, 18, 12, 28, 8, 30, ...

Because of period scaling, each mu-atom has children that have the same distribution as the secondary continental mu-atoms, but with scaled-up periods. [...]

The encyclopedia pages have many references to further properties that might be of interest.

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