%{
\font\de=cmssi12
\font\twelverm=cmr12
\input usual

\def\ANG{Angenent}
\def\Ca{C^\ast}
\def\DB{dual billiards}
\def\DD{Douady}
\def\G{\Gamma}
\def\HTM{half-cylinder twist map}
\def\LL{{\cal L}}
\def\Lip{Lipschitz}
\def\Leb{Lebesgue}
\def\Na{N^\ast}
\def\N{{\cal N}}
\def\QED{  \rlap{$\sqcup$}$\sqcap$ }
\def\Tab{Tabachnikov}
\def\a{\alpha}
\def\arcot{\hbox{arcot}}
\def\bx{{\bar x}}
\def\cD{{\cal D}}
\def\cf{{\it cf.}}
\def\cl{\centerline}
\def\circle{S^1}
\def\d{\; d}
\def\eg{{\it eg.}}
\def\eqdef{\buildrel{\rm def}\over  =}
\def\gp{\gamma^\prime}
\def\g{\gamma}
\def\ie{{\it i.e.}}
\def\integers{{\bbb Z}}
\def\inv{^{-1}}
\def\mG{$\Gamma$}
\def\mo{_{-1}}
\def\naturals{{\bbb N}}
\def\po{periodic orbit}
\def\pp{periodic point}
\def\rationals{{\bbb Q}}
\def\ra{\rightarrow}
\def\reals{{\bbb R}}
\def\plane{{\bbb R}^2}
\def\tCa{{\tilde C}^\ast}
\def\tC{{\tilde C}}
\def\tH{{\tilde H}}
\def\tL{\tilde\Lambda}
\def\tO{\tilde\Omega}
\def\tf{\tilde{f}}
\def\tg{{\tilde\gamma}}
\def\tp{{\theta^\prime}}
\def\tx{\tilde{x}}
\def\tz{{\tilde z}}
\def\t{\theta}
\def\htheta{{\hat \theta}}
\def\ux{\underline{x}}
\def\uy{\underline{y}}
\def\uz{\underline{z}}
\def\va{{\bf \alpha}}
\def\vn{{\bf n}}
\def\vu{{\bf u}}
\def\vv{{\bf v}}
\def\vw{{\bf w}}
\def\words#1{\ \hbox{#1}\ }
\def\xp{{x^\prime}}
\def\yp{y^\prime}
\def\zray{(0,\infty)}
\def\overbar{\bar}
\def\dotp{{\dot p}}
\def\dotx{{\dot x}}
\def\deta{{\dot \eta}}
\def\longfig#1#2#3{%
\midinsert\centerline{\psfig{figure=#1}}\medskip\centerline{%
\hbox to 4 in{\vbox{\hsize=3.9in\noindent {\bf Figure #2:} #3. } }}\endinsert}

\def\fig#1#2#3{%
\midinsert\centerline{\psfig{figure=#1}}\medskip\centerline{%
{\bf Figure #2:} #3. }\endinsert }



\def\sgn{\hbox{sgn}}