\documentstyle{article}
\renewcommand{\baselinestretch}{1.3}
\input epsf
\begin{document}
\title{Classical mechanics for the circle-step potential}
\author{}
\maketitle
\section{Trajectories going through the step}
We describe here the structure of the manifolds where the trajectory
allways passes the step and continue on the other side. Many points
in the phase space then only have a short future and past because the
trajectory is pruned (forbidden) because of total reflection. The
other pruning mechanisms results from trajectories which are reflected
from the circle at the points $x=0$, $y=\pm 1$, and from a smooth
folding of the phase space which are more difficult to directly relate
to a simple geometrical cause.
A picture of unstable and stable manifolds are given in
figure~\ref{f_e_2_manif}. We have in this figure also drawn a number
of curves correspoinding to singular trajectories. Curves $R_1$ and
$R_2$ are the trajectories which when going into the right semicircle
will be on the border of having a total reflection the next time it
hits the $x=0$ line. The area $R$ between these two border curves do
then not contain any part of the unstable manifold. The manifolds
approaches the curves $R_1$ and $R_2$ from the outside and stop
discontinous on the curve. This is analogous to the way manifolds
changes in the corner of a disk billiard, or the wedge
billiard~\cite{kai_thesis}. For this parameter value there is only
trajectories which bounce {\em once} in the semicircle that can get a
total reflection. For higher parameter values there may be several
total reflection regions.
% ********* figure~\ref{f_e_2_manif} *************
\begin{figure}
\epsfbox{figs/f_e_2_color.ps}
\caption[]{\sf \label{f_e_2_manif} The
stable and unstable manifolds of the map.}
\end{figure}
% ********* figure~\ref{f_e_2_orb_corner} *************
\begin{figure}
\epsfbox{figs/f_e_2_orb_corner.ps}
\caption[]{\sf \label{f_e_2_orb_corner}
Trajectories bouncing on both sides and on the corner point illustrating
how one gets a sharp folding of the phase space here
}
\end{figure}
The other curves drawn in the figure, labelled $B_1$, $B_2$, $B_3$,
\dots $B_n$, \ldots, are the trajectories which move into the right
semicircle and bounces $n$ times in the semicircle before it hits the
point $(x=0,y=-1)$. At this point the manifolds are continuous but
change direction. This is analogous to the way manifolds changes at
tangent orbits in dispersive billiards and at the singular point in
the stadium billiard~\cite{kai_thesis}. In figure~\ref{f_e_2_orb_corner}
we show how trajectories fold around this singular point. The dotted
curve bounces on the right side and the dashed curve bounces on the
left side of the corner point. After the bounce only the dotted curve
crosses the corner point curve and at the next bounce both curves are
on the left side of the corner bouncing trajectory.
The third mechanism in the flow is a smooth folding of the phase space
visable in figure~\ref{f_e_2_manif} as a smooth S-shapes in the
manifolds. In figure~\ref{f_e_2_orb_pair} we have drawn three
trajectories starting from three different points of the same fold of
the unstable manifold but on three different parts of the fold
separated by a folding point (a primary homoclinic tangency).
Toghether with each trajectory we have drawn a neighbor trajectory
starting with a slightly larger initial angle $\alpha$. We have to
follow the trajectory for a rather long time before we find that the
orientation of the two neighbor trajectories are the same for the
uppermost and downmost trajectories, while the one in the middle have
a different orientation. This is a similar mechanism as we found for
the corner singularity, but here we have to singular points and the at
the singular points the manifolds changes smoothly, not a sharp change
in the direction of the manifolds. The singular points are here
homoclinic tangencies between the stable and unstable manifold.
This is the mechanism which creates stable orbits, and we do find the
stable orbits in this region. This structure is typical for smooth
systems and billiard systems which are not completely chaotic, se
e.g.~\cite{freddy,cardioid,kai_stefan,...}.
There is a smooth S-shape folding for each region separated with the
$B_n$ curves and each has to be threated separately. In each $B$
region we then get two partition curves which are a connected curve
through the primary homoclinic tangencies. We denote these curves
$H^a_0$, $H^b_0$, $H^a_1$, $H^b_1$, $H^a_2$, $H^b_2$, \ldots,
$H^a_n$, $H^b_n$.
% ********* figure~\ref{f_e_2_orb_pair} *************
\begin{figure}
\mbox{ \epsfbox{figs/f_e_2_orb_pair1.ps}
\epsfbox{figs/f_e_2_orb_pair2.ps}}
\epsfbox{figs/f_e_2_orb_pair3.ps}
\caption[]{\sf \label{f_e_2_orb_pair}
Trajectories with neighbor trajectories from the lower, middle and
upper part of region A. The orientation between two neighbor
trajectories are different in the middle part compared with the
upper and lower case where the trajectories end.
(a) $y_0=0.3593$, $\alpha=-0.3161$,
(b) $y_0=0.3592$, $\alpha=-0.1953$,
(c) $y_0=0.3624$, $\alpha=-0.1441$.
}
\end{figure}
To construct a symbolic dynamics we now have to introduce different
symbols for each region separated by curves through the singular
orbits. We choose to give these a simple enumbering such that
$s=0$ is the region with $y