Naming bulbs on the Mandelbrot set

Question

Can anyone point me to an article or webiste that explains exactly how the bulbs of the Mandelbrot set are named. I know there are bulbs that have the names "p/q" for every set of co-prime integers. Are these just the names of bulbs that come off the bulbs of the main cardiod? How does one name the bulbs that come off those bulbs - the third-tier bulbs (so to speak)? And how does one name bulbs that come off all the infinitely many "little Mandelbrots"? Do these ones have names at all?

Answer 1

The first thing to note is that each of the hyperbolic components (either cardioid-like or disk-like) is associated to a period, which is a positive integer. The biggest cardioid has period $1$, the biggest disk has period $2$.

The $p/q$ bulb (without further qualification) is one attached to the period $1$ cardioid, but each of those bulbs has its own child bulbs. The fraction $p/q$ corresponds to the internal angle measured in turns, where $0=1$ corresponds to the root (the cusp for cardioid-like components, bond point to parent for disk-like components). The period of a $p/q$ child is $q$ times the period of its parent.

A good introduction to the $p/q$ bulbs is:

"The Mandelbrot Set, the Farey Tree, and the Fibonacci Sequence" Robert L. Devaney The American Mathematical Monthly Vol. 106, No. 4 (Apr., 1999), pp. 289-302

Our goal in this paper is to explain and make precise several "folk theorems" involving the Mandelbrot set and the Farey tree.

Folk theorem 1: You can recognize the $p/q$ bulb by locating the "smallest" spoke in the antenna and determining is location relative to the principle spoke.

Folk theorem 2: To obtain the largest bulb between two given bulbs, we simply add the corresponding fractions by adding the numerators and adding the denominators.

It covers some properties of external angles and rays which are prerequisites of:

"Internal addresses in the Mandelbrot set and Galois groups of polynomials" Dierk Schleicher https://arxiv.org/abs/math/9411238

Internal addresses are a convenient and efficient way of describing the combinatorial structure of the Mandelbrot set, and of giving geometric meaning to the ubiquitous kneading sequences in human-readable form. A simple extension, angled internal addresses, distinguishes combinatorial classes of the Mandelbrot set and in particular distinguishes hyperbolic components in a concise and dynamically meaningful way.

For examples, see Figure 2 on page 11 of the PDF. The angled internal address of the end of a finite chain of child bulbs $p_j/q_j$, $j \in 1, 2, \ldots, k$ would be: $$1 \xrightarrow{p_1/q_1} q_1 \xrightarrow{p_2/q_2} q_1 q_2 \xrightarrow{p_3/q_3} \ldots \xrightarrow{p_k/q_k} \prod_{j=1}^k q_j$$

Robert Munafo lists some naming systems in his Encyclopedia of the Mandelbrot set https://www.mrob.com/pub/muency/analyticalnamingsystem.html He calls the bulbs that are directly attached to the period $1$ cardioid secondary continental mu-atoms and names them by their internal angle $p/q$. His R2 naming system can name other things too, not just bulbs.