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Internal Addresses of the Mandelbrot Set and Galois Groups of Polynomials

Internal Addresses of the Mandelbrot Set and Galois Groups of Polynomials 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Arnold Mathematical Journal Springer Journals

Internal Addresses of the Mandelbrot Set and Galois Groups of Polynomials

Schleicher, Dierk
Arnold Mathematical Journal , Volume 3 (1) – Aug 2, 2016
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References (14)

Publisher
Springer Journals
Copyright
Copyright © 2016 by Institute for Mathematical Sciences (IMS), Stony Brook University, NY
Subject
Mathematics; Mathematics, general
ISSN
2199-6792
eISSN
2199-6806
DOI
10.1007/s40598-016-0042-x
Publisher site
See Article on Publisher Site

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.

Journal

Arnold Mathematical Journal – Springer Journals

Published: Aug 2, 2016

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