Internal addresses in the Mandelbrot set and irreducibility of polynomials

Details Internal addresses in the Mandelbrot set and irreducibility of polynomials Internal addresses in the Mandelbrot set and irreducibility of polynomials

1994-11-21

arxiv.org/abs/math/9411238v2

Mathematics

Dynamical Systems

32 page, 5 figures. -- Rewritten and updated

Scientific paper

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. In the early 1990's, Devaney asked the question: how can you tell where in the Mandelbrot set a given external ray lands, without having Adrien Douady at your side? We provide an answer to this question in terms of internal addresses: these 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 (Section 2). 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 quite easily existence theorems for certain kneading sequences and internal addresses in the Mandelbrot set (Section 4); these in turn help to establish some algebraic results about permutations of periodic points and to determine Galois groups of certain polynomials (Section 5).

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