# Combinatorial description of the Mandelbrot set

I have a very naïve question: can one find anywhere a combinatorial description of the Mandelbrot set?

Let me try to be a bit more precise: is it possible to encode each of its "bulbs" by some sort of finite sequence of numbers (or more complicated combinatorial data), and then give a simple combinatorial description of how each of these bulbs connects to the other ones, and (if possible) some sort of formula giving the equation of each bulb in terms of its "code"?

I suppose the mathematical community in general should know the answer. A partial construction of this sort is given here: https://en.wikipedia.org/wiki/Mandelbrot_set#Main_cardioid_and_period_bulbs ; and hints of a bigger picture are given for example in this answer: The deep significance of the question of the Mandelbrot set's local connectedness? . But there is so much literature about the Mandelbrot set that searching for a precise reference is somewhat hard.

Could by chance anyone point me to some book that systematically expounds the answer to that question?

In fact, more generally, I would like to learn "how the Mandelbrot set works". So basically, I am looking for some nice introductory book to the Mandelbrot set. I have tried to ask a more specific question (whose answer, in my opinion, should occupy a central place in such a book), but it is necessarily somewhat vague - precisely because I know very little about this subject. So what reading would you recommend?

• Many years ago I tried to generate such a language to intuitively describe where in the Mandelbrot set you were based on following bulbs and filaments starting from the largest cardioid, but I never completed it. Oct 9, 2021 at 22:39

## 4 Answers

I can suggest:

• several papers with Dirk Schleicher as (co)author, such as:

• the unpublished book (in progress? abandoned? I don't know) Symbolic Dynamic of Quadratic Polynomials by Henk Bruin, Alexandra Kaffl & Dierk Schleicher, of which there is one version here and a longer (more recent) one here, from which the picture below is taken,

• and the book Invariant Factors, Julia Equivalences and the (Abstract) Mandelbrot Set by Karsten Keller.

Q: Can one find anywhere a combinatorial description of the Mandelbrot set?

This could be one such description, from

• Dierk Schleicher, Internal addresses in the Mandelbrot set and Galois groups of polynomials, Arnold Mathematical Journal volume 3, pages 1–35 (2017), doi:10.1007/s40598-016-0042-x, arXiv:math/9411238

We describe the interplay between symbolic dynamics, the structure of the Mandelbrot set, permutations of periodic points achieved by analytic continuation, and Galois groups of certain polynomials. This combinatorial description of the Mandelbrot set makes it possible to derive existence theorems for certain kneading sequences and internal addresses.

• You may compare this picture with the original source that I linked in my answer. Oct 14, 2021 at 18:21

Douady, A.; Hubbard, J. H. Étude dynamique des polynômes complexes. Partie I. Publications Mathématiques, 84-2. http://sites.mathdoc.fr/PMO/feuilleter.php?id=PMO_1984

Douady, A.; Hubbard, J. H. Étude dynamique des polynômes complexes. Partie II. With the collaboration of P. Lavaurs, Tan Lei and P. Sentenac. Publications Mathématiques d'Orsay, 85-4. Université de Paris-Sud, Département de Mathématiques, Orsay, 1985. http://sites.mathdoc.fr/PMO/feuilleter.php?id=PMO_1985

Remark. This was previously announced in CR 294, 1982 from which I reproduce the image.

In addition to Gro-Tsen's answer (that I chose to accept since it seemed to be the most complete one) and the other two answers (that are certainly also very helpful), let me also mention a website that I recently stumbled upon and that also seems relevant to this question: http://mrob.com/pub/muency.html, "The Encyclopedia of the Mandelbrot Set". It features in particular a naming system for various features of the Mandelbrot set ("R2"), which is precisely the kind of thing I was looking for.