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Internal Addresses of the Mandelbrot Set and Galois Groups of Polynomials

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Abstract

We describe an interesting interplay between symbolic dynamics, the structure of the Mandelbrot set, permutations of periodic points achieved by analytic continuation, and Galois groups of certain polynomials. 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 (Sects. 3 and 4). 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. This combinatorial description of the Mandelbrot set makes it possible to derive existence theorems for certain kneading sequences and internal addresses in the Mandelbrot set (Sect. 6) and to give an explicit description of the associated parameters. These in turn help to establish some algebraic results about permutations of periodic points and to determine Galois groups of certain polynomials (Sect. 7). Through internal addresses, various areas of mathematics are thus related in this manuscript, including symbolic dynamics and permutations, combinatorics of the Mandelbrot set, and Galois groups.

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Notes

  1. If several dynamic ray pairs of equal period \(S_{k+1}\) separate \(P_c(\vartheta _k,\vartheta '_k)\) from z, take the one which minimizes \(|\vartheta '_{k+1}-\vartheta _{k+1}|\). (There is an analog to Lavaurs’ Lemma in dynamical planes which says that all candidate ray pairs have to land at the same periodic point; this is not hard to prove, but we do not need it here.)

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Acknowledgments

Much of this work goes back to joint discussions with Eike Lau and the joint preprint (Lau and Schleicher 1994). We would like to express our gratitude to him. We also gratefully acknowledge helpful and interesting discussions over the years with Christoph Bandt, Henk Bruin, Dima Dudko, John Hubbard, Karsten Keller, Misha Lyubich, John Milnor, Chris Penrose, and many others, and we thank Simon Schmitt for his help with the pictures.

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Schleicher, D. Internal Addresses of the Mandelbrot Set and Galois Groups of Polynomials. Arnold Math J. 3, 1–35 (2017). https://doi.org/10.1007/s40598-016-0042-x

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