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\markboth{\rm\bf W. Jung: The Mandelbrot set}
{\rm\bf W. Jung: The Mandelbrot set}
\date{December 31, 1996.}
\author{Wolf Jung\\Inst.f.Reine u.Angew.Mathematik, RWTH Aachen\\
Templergraben 55, D-52062 Aachen, Germany \\
jung@iram.rwth-aachen.de}
\title{Some explicit formulas for components of the Mandelbrot set}
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\begin{document}
\maketitle
\begin{abstract} \noindent
Let $f$ be a rational function, which has $k$ \mbox{$n$-cycles} under
iteration. By using the symmetry of the underlying equation of degree
$k\cdot n$, it is reduced to equations of degree $k$ and $n$. This is
explained in terms of Galois theory.\\
The \mbox{3-} and \mbox{4-cycles} of $f_c=z^2+c$ are obtained explicitly.
This yields the corresponding multiplier, which maps hyperbolic components
of the Mandelbrot set conformally onto the unit disk.
\end{abstract}
\section{Introduction}
For a rational function $f:\hat{\C} \to \hat{\C}$, denote the \mbox{$n$-th}
iterate by $f^n$. $z_0$ is in the Julia set of $f$, if the sequence
$(f^n(z))$ is not normal in any neighborhood of $z_0$. We consider the
family of quadratic polynomials $f_c(z)=z^2+c$. The Mandelbrot set $\M$
contains those $c \in \C$, such that the Julia set of $f_c$ is connected.
Since 0 is the only critical point of the polynomial $f_c$, $c \in \M$
iff $(f^n(0))$ is bounded \cite{cg}.
A \mbox{$n$-cycle} of $f_c$ consists of distinct points $z_1\dots z_n$
with $f_c(z_1)=z_2,\, \dots,\\ f_c(z_n)=z_1$. The corresponding multiplier is
$\l={f^n_c}'(z_1)=2^n z_1\cdot z_2\cdots z_n$. The cycle is attracting,
if $|\l|<1$. The set of those $c$, such that $f_c$ has an attracting
\mbox{$n$-cycle}, is a union of components of $\M$ (called hyperbolic).
These are mapped conformaly onto the unit disk by $\l$.
It is well known that $\l$ is an algebraic function, with
$(\l/2)^2 - \l/2 + c = 0$ for $n=1$ and $\l = 4(c + 1)$ for $n=2$ \cite{sm}.
We describe an algorithm to obtain these functions for every $n$, and give
the results for period 3 and 4. Define the polynomials $g_n(z,\,c)$
recursively by $f^n_c(z) - z = \prod_{d|n}g_d(z,\,c)$, then the zeros of $g_n$
are the $n-$periodic points of $f_c$.
For $n \ge 3$, the degree of $g_n$ is at least 6. In general, only polynomial
equations of degree 4 or less can be solved explicitly, but $g_n$
satisfies the symmetry relation $g_n(z,\,c)=0\Rightarrow g_n(f_c(z),\,c)=0$,
which is used to reduce the equation. The resulting algorithm is best
understood in terms of Galois theory.
\section{The cycles and multipliers of $f_c$}
Except for some values of $c$, at which a bifurcation occurs, $g_n$ has
$k\cdot n$ simple zeros. These form $k$ $n-$cycles
$z_1^{(j)}\dots z_n^{(j)}$ with $f_c(z_i^{(j)}) = z_l^{(j)}$,
$l=i+1\ ({\rm mod}\ n)$. This suggests the following algorithm:
Define $s_n(z,\, c,\, a)=z+f_c(z)+\dots+f_c^{n-1}(z)-a$.
Then $s_n(z_i^{(j)},\, c,\, a)=s_n(z_l^{(j)},\, c,\, a)$, thus $a$ can be
chosen such that the greatest common divisor of $g_n$ and $s_n$ is of
degree $n$. We perform Euklid's algorithm with $g_n$ and $s_n$. The
remainder with degree $4$.
In the case of $n=3$ or $n=4$, the formulas of Theorem \ref{cycles34}
show that $G \cong S_k \,\imath\, C_n$ in general, \mbox{i.e.}, if $c$ is
transcendental, or equivalently, if $\Q(c)$ is understood as the field of
rational functions in one variable.
\section{Summary}
If $f$ is rational function with $k$ \mbox{$n$-cycles}, the underlying
equation of degree $k\cdot n$ is reduced to one equation of degree $k$ and
$k$ equations of degree $n$. The first is solvable explicitly at least if
$k\le 4$, while the latter equations are always solvable, as
Galois theory shows.
For $f_c=z^2+c$, our algorithm yields the conformal mappings of
hyperbolic components of the Mandelbrot set onto the unit disk.
It also yields explicit results for other low-degree rational functions,
\mbox{e.g.,} the 3 \mbox{2-cycles} of a third-degree polynomial and the 8
\mbox{3-cycles} of $z - \frac{z^3 - 1}{3z^2}$ can be obtained.
\begin{thebibliography}{99}
\bibitem{cg} L. Carleson and T. Gamelin, {\em Complex Dynamics}, Springer,
New York 1993.
\bibitem{pm} P. Morandi, {\em Field and Galois Theory}, Springer,
New York 1996.
\bibitem{sm} N. Steinmetz, {\em Rational Iteration}, de Gruyter, Berlin 1993.
%\bibitem{}
\end{thebibliography}
\vspace{1cm}
This paper is available by anonymous ftp from work1.iram.rwth-aachen.de \\
(134.130.161.65) in the directory /pub/papers/jung/ or from\\
http://www.iram.rwth-aachen.de/\symbol{126}jung/index.html\\
as a \LaTeX~2.09 tex dvi or ps file jg-96-2.* .
\end{document}