## Institute for Mathematical Sciences

## Preprint ims94-19

** E. Lau and D. Schleicher**
* Internal Addresses in the Mandelbrot Set and Irreducibility of Polynomials.*

Abstract: For the polynomials $p_c(z)=z^d+c$, the periodic points of periods dividing $n$ are the roots of the polynomials
$P_n(z)=p_c^{\circ n}(z)-z$, where any degree $d\geq 2$ is
fixed. We prove that all periodic points of any exact period
$k$ are roots of the same irreducible factor of $P_n$ over
$\cz(c)$. Moreover, we calculate the Galois groups of these
irreducible factors and show that they consist of all
permutations of periodic points which commute with the
dynamics. These results carry over to larger families of maps,
including the spaces of general degree-$d$-polynomials and
families of rational maps.
Main tool, and second main result, is a combinatorial
description of the structure of the Mandelbrot set and its
degree-$d$-counterparts in terms of internal addresses
of hyperbolic components. Internal addresses interpret kneading
sequences of angles in a geometric way and answer Devaney's
question: ``How can you tell where in the Mandelbrot a
given rational external ray lands, without having Adrien Douady
at your side?''

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