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  \begin{center}
	Type III hyperbolic components in $V_{3}$.
	
	Mary Rees, University of Liverpool.
	\end{center}
\newenvironment{utheorem}{\par\noindent\textbf{Theorem}\,\,\em}{\rm}	
   
Every quadratic rational map with a period $3$ critical point is 
represented up to M\"obius conjugacy by
$$h_{a}(z)={(z-a)(z-1)\over z^{2}}.$$
The critical points are $0$ and $c_{2}(a)=\displaystyle{{2a\over a+1}}$ and $0$ 
is of period $3$. In fact:
$$h_{a}(0)=\infty ,\ \ h_{a}(\infty )=1,\ \ h_{a}(1)=0,$$
$$h_{a}(c_{2}(a)) =-{(a-1)^{2}\over 4a}=v_{2}(a),\ \ 
h_{a}^{-1}(0)=\{ 1,a\} ,\ \ h_{a}^{-1}(1)=\left \{ \infty ,{a\over 
a+1}\right \} .$$
Write
$$V_{3}=\{ h_{a}:a\in \mathbb C,\ a\neq 0\} .$$


A {\em{hyperbolic component}} is a connected (open) set in which 
$\lim _{n\to \infty }h_{a}^{n}(c_{2}(a)) $ is 
an attractive periodic orbit, which is either 
$\{0,1,\infty \} $ (for types II and III) or a different periodic 
orbit (for type IV). Each hyperbolic component $H$ in $V_{3}$ has a unique 
{\em{centre}} $a_{c}=a_{c}(H)$ such that the set 
$\{ h_{a_{c}}^{n}(c_{2}(a_{c}):n>0\}  $ is finite. Thus, 
$c_{2}(a_{c})$ is either periodic under $h_{a}$ (for types II and IV) or 
strictly preperiodic (for type III). There are only two type II 
hyperbolic components in $V_{3}$, with centres $\pm 1$. There are 
infinitely many of types III and IV. 

A type III hyperbolic component wiith centre $a_{*}$ is 
{\em{of preperiod $m$}} if  $c_{2}(a_{c})$ is of preperiod $m$ under $h_{a}$. 
This happens if and only if 
$a_{c}\neq 0$, $\pm1$ and $a_{c}$ satisfies one of the equations:
$$h_{a}^{m}(c_{2}(a))=a{\rm{\ or\ }}{a\over a+1}.$$
This happens if and only if $a_{c}$ is a nontrivial root ($a\neq 0$, $\pm 
1$) of $p_{m}(a)-aq_{m}(a)$ 
or $(a+1)p_{m}(a)-aq_{m}(a)$, where
$$p_{0}(a)=2a,\ \ q_{0}(a)=a+1,$$
$$p_{m+1}(a)=(p_{m}(a))^{2}-(a+1)q_{m}(a)+a(q_{m}(a))^{2},$$
$$q_{m+1}(a)=(p_{m}(a))^{2}.$$
In both cases, the number of nontrivial roots is
$$2^{m}.{23\over 21}+O(1),$$
where the $O(1)$ term is period $6$ in $m$. 
\begin{utheorem} All roots are simple.\end{utheorem}
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%\newenvironment{utheorem}{\par\noindent\textbf{Theorem}\,\,\em}{\rm}
    A full description of the dynamics in all type III hyperbolic 
 components in $V_{3}$ can be found in \cite{R1}. It is enough to 
 describe the dynamics of the centres of the hyperbolic components. As usual (or 
 always) in dynamics, variation of dynamics with parameter is defined 
 in terms of dynamics in particular dynamical planes. In this case, we 
 use the three polynomials up to M\"obius conjugacy in $V_{3}$, the two 
 {\em{rabbit polynomials}}, given up to M\"obius conjgacy by 
 $a=a_{0}$, $\overline{a_{0}}$ and the {\em{aeroplane polynomial}} 
 given by $a=a_{1}$. Here, $a=a_{0}$, $\overline{a_{0}}$, $a=a_{1}$ 
 are the roots of
 $$c_{2}(a)=v_{2}(a)$$
 A {\em{fundamental domain for $V_{3}$}} is chosen to make 
 the description of dynamics of  each hyperbolic component unique. The 
 description of dynamics of (the centre of) each type III hyperbolic component 
 of preperiod $\leq m$ is then given by the following.
 \begin{description}
     \item[.] $a_{*}=a_{0}$, $\overline{a_{0}}$ or $a_{1}$.
     \item[.] a point $x$ in $h_{a_{*}}^{-m}(\{ 0,1,\infty \} )$. In the case of the aeroplane polynomial this point  has a 
     symbolic coding by letters $L_{j}$, $R_{j}$ ($j=1$, $2$ or $3$) $BC$ ,$UC$ 
     and $C$.
 
  \includegraphics[width=12cm]{aerocode}

    % $$\epsfbox{aerocode.0}$$
     \item[.] A path from $c_{2}(a_{*}$ to 
     $x$, which, apart from endpoints, does not intersect
     $Y_{m}(a_{*})= h_{a_{*}}^{-m}(\{ 0,1,\infty \} )\cup \{ 
     c_{2}(a_{*})$ up to homotopy fixing $Y_{m}(a_{*})$. 
     \end{description}
     
     A path is called a {\em{capture path}} if it intersects the Julia 
     set of $h_{a_{*}}$ in exactly one point. A hyperbolic component 
     is called a {em{capture}} if it can be describe by a capture path.
\begin{utheorem} The number of capture type III hyperbolic components is 
$\geq \lambda 2^{m}$ and $\leq {22\over 21}.2^{m}+O(1)$, for some 
$\lambda >1$.\end{utheorem}

It can be considered a folklore-result that $\lambda >{6\over 7}$


\begin{thebibliography} {99}
\bibitem{R1} Rees, Mary: A Fundamental Domain for $V_{3}$. Preprint, May 2006.
\end{thebibliography}
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\begin{tabular}{|p{1cm}|p{25cm}|p{24cm}|}
    \hline 
    $m$ &\ nontrivial roots of $p_{m}(a)-aq_{m}(a)$\ &\ 
    nontrivial roots of $(a+1)p_{m}(a)-aq_{m}(a)$\\
\hline
\ 1&\ $2$,  & $3=2+1$.\\
\ & both rabbit captures & $2$ rabbit captures, one aeroplane capture.\\
\ & \begin{tabular}{p{6cm}}\includegraphics[width=4cm]{m=1rabcap1.pdf}\end{tabular}&\begin{tabular}{p{6cm}p{6cm}}
\includegraphics[width=6cm]{m=1rabcap2}&\includegraphics[width=9cm]{m=1aeroplane}\end{tabular}\\
%\ &$\epsfbox{m=1rabcap1.0}$&$\epsfbox{m=1rabcap2.0}\epsfbox{m=1aeroplane.0}$\\ 
\hline
\ 2&\ $5=4+1$.& \ $5=4+1$.\\
\ &  $4$ rabbit captures,  one aeroplane capture.&\  
$4$ rabbit captures, one aeroplane capture.\\
\ & \begin{tabular}{p{4cm}p{4cm}p{4cm}}\includegraphics[width=4cm]{m=2rabcap1}& \includegraphics[width=4cm]{m=2rabcap2}& \includegraphics[width=5.5cm]{m=2aero1}\end{tabular}&
\begin{tabular}{p{4cm}p{4cm}p{6cm}}\includegraphics[width=4cm]{m=2rabcap3}&\includegraphics[width=4cm]{m=2rabcap4}&\includegraphics[width=6cm]{m=2aero2}\end{tabular}\\ 
%\ &\ $\epsfbox{m=2rabcap1.0}\epsfbox{m=2rabcap2.0}$ $\epsfbox{m=2aero1.0}$&
%\ $\epsfbox{m=2rabcap3.0}\epsfbox{m=2rabcap4.0}$ $\epsfbox{m=2aero2.0}$\\
\hline
\ 3&\ $7=6+1$. &\ $8=6+1+1$. \\ 
\ & $6$ rabbit captures, one aeroplane capture.&\ $6$ rabbit 
captures, one aeroplane capture.\\
\ &\ &\ {\em{First noncapture}}, using symbolic code $BCL_{1}R_{1}R_{2}C$\\
\hline
\ &\ &\ \includegraphics[width=6cm]{m=3aero}\\
%\hline
%\ &\ &\ $\epsfbox{m=3aero.0}$\\
\hline
\ 4&\ $18=14+4$ &\  $18=14+3+1$.\\ 
\ &\ $14$ rabbit captures, $4$ aeroplane captures.&\ $14$ rabbit captures,
$3$ aeroplane captures.
Second noncapture. 
 with code $BCL_{1}R_{1}^{2}R_{2}C$.\\
 \ &\ &\ \includegraphics[width=6cm]{m=4aero}\\
 % \ &\ &\ $\epsfbox{m=4aero.0}$\\
\hline
%\end{tabular}
%\end{textblock*}
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%\begin{tabular}{|p{1cm}|p{14cm}|p{14cm}|}
   % \hline 
   % $m$ &\ nontrivial roots of $p_{m}(a)-aq_{m}(a)$\ &\ 
   % nontrivial roots of $(a+1)p_{m}(a)-aq_{m}(a)$\\
%\hline
\ 5&\ $35=28+6+1$ &\ $36=28+7+1$\\
\ &\ $28$  rabbit captures,  $6$ aeroplane captures {\em{up to 
equivalence}},  
$2$ with same path endpoint, one of these represented by noncapture in 
fundamental domain. One other noncapture.
&\ $28$ rabbit captures, $7$ aeroplane captures, one noncapture. 
\\
\ &\ \includegraphics[width=6cm]{m=5aero1}&\ \includegraphics[width=6cm]{m=5aero4}\\
\ &\  \begin{tabular}{p{6cm}p{1cm}p{6cm}}\includegraphics[width=6cm]{m=5aero2}&\  $\cong $&\includegraphics[width=6cm]{m=5aero3}\end{tabular}&\ \\
\ &\ \includegraphics[width=6cm]{m=5aero5}&\ \\
%\ &\ $\epsfbox{m=5aero1.0}$&\ $\epsfbox{m=5aero4.0}$\\
%\ &$\epsfbox{m=5aero2.0}$ $\cong $ $\epsfbox{m=5aero3.0}$&
%\ \\
%\ &\ $\epsfbox{m=5aero5.0}$&\ \\
\hline
\ 6&\ $69=54+14+1$\ &\ $69=54+11+4$\ \\
\ &\ $54$ rabbit captures, $14$ aeroplane captures, 
$2$ with same path endpoint, both  of these represented by noncaptures 
in the fundamental domain. One other noncapture. &
\ $54$ rabbit captures, $11$ aeroplane captures, $4$ noncaptures.\\
\ &\ \begin{tabular}{p{6cm}p{1cm}p{6cm}}\includegraphics[width=6cm]{m=6aero9}&\  $\cong $\ &\includegraphics[width=6cm]{m=6aero1}\end{tabular} &\  \begin{tabular}{p{6cm}p{6cm}}
\includegraphics[width=6cm]{m=6aero5}&\ \includegraphics[width=6cm]{m=6aero6}\end{tabular}\\
\ &\ \begin{tabular}{p{6.5cm}p{1cm}p{6.5cm}}\ 
\includegraphics[width=5.5cm]{m=6aero2}&\ $\cong $&\includegraphics[width=5.5cm]{m=6aero3}\end{tabular}& \  
\begin{tabular}{p{6cm}p{6cm}}\includegraphics[width=6cm]{m=6aero7}&\includegraphics[width=6cm]{m=6aero8}\end{tabular}\\
\ &\ \includegraphics[width=6cm]{m=6aero4}&\ \\
%\ &\ $\epsfbox{m=6aero9.0}$ $\cong $ $\epsfbox{m=6aero1.0}$&\ $\epsfbox{m=6aero5.0}$\\
%\ &$\epsfbox{m=6aero2.0}$ $\cong $ $\epsfbox{m=6aero3.0}$&\ 
%$\epsfbox{m=6aero6.0}$\\
%\ &\ $\epsfbox{m=6aero4.0}$&\ $\epsfbox{m=6aero7.0}$\\
%\ &\ &\  $\epsfbox{m=6aero8.0}$\\
\hline
\end{tabular}
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