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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