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Arnold Mathematical Journal

, Volume 3, Issue 1, pp 1–35 | Cite as

Internal Addresses of the Mandelbrot Set and Galois Groups of Polynomials

  • Dierk Schleicher
Research Contribution
  • 353 Downloads

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.

Keywords

Internal address Mandelbrot set Symbolic dynamics Kneading sequence Admissibility Permutation  Analytic continuation Galois group 

Mathematics Subject Classification

37F20 30D05 37E15 37F10 37F45 12F10 

Notes

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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Copyright information

© Institute for Mathematical Sciences (IMS), Stony Brook University, NY 2016

Authors and Affiliations

  1. 1.Jacobs University, Research IBremenGermany

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