# Mandelbrot set: periodicity of secondary and subsequent bulbs as multiples of their parent bulbs

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:

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.

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:

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?

• First google hit: math.stackexchange.com/questions/419983/…. Also, you should really ask a specific question, rather than several. Jul 5, 2017 at 4:07
• math.stackexchange.com/questions/2347952/… I answered one of the questions here Jul 6, 2017 at 2:42
• Reasoned mathematical arguments need definitions to proceed. How do you define the periodicity or size of a "bulb"? Jul 6, 2017 at 4:41
• @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 Jul 7, 2017 at 2:24

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.