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inegrable billiard tables, inverse spectral problem
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\documentclass[11pt]{article}
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\begin{document}
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\title{ Liouville billiard tables and an inverse spectral result}
\author{G.Popov and P.Topalov\footnote{partially supported by
MESC Grants no. MM-810/98 and MM-1003/00}}
\date{}
\maketitle
\thispagestyle{empty}
\begin{abstract}
\noindent
We consider a class of billiard tables $(X,g)$, where $X$ is a smooth
compact manifold of dimension 2 with smooth boundary $\partial X$
and $g$ is a
smooth Riemannian metric on $X$, the billiard
flow of which is completely integrable.
The billiard table $(X,g)$ is defined by means of a special double
cover with two branched points and it admits a group of isometries
$G \cong {\bf Z}_2 \times{\bf Z}_2$. Its boundary can be
characterized by the string property, namely, the sum of distances
from any point of $\partial X$ to the branched points is constant.
We provide examples of such
billiard tables in the plane (elliptical regions), on the sphere ${\bf
S}^2$, on the hyperbolic space ${\bf H}^2$, and on quadrics.
The main result is that the spectrum of the corresponding
Laplace-Beltrami operator with Robin boundary conditions involving
a smooth function $K$ on $\partial X$ determines uniquely the function
$K$ provided that $K$ is
invariant under the action of $G$ .
\vspace{0.3cm}
\noindent
Subject classification: 58J50, 70H06
\end{abstract}
\section{Introduction}
\setcounter{equation}{0}
This paper is concerned with a class of smooth compact Riemannian
manifolds of dimension two with smooth boundaries,
the billiard flows of which are
completely integrable. We call them {\it Liouville billiard
tables of classical type}. For such billiard
tables we prove that the
following inverse spectral result is true:
Let $(X,g)$ be a closed Riemannian manifolds,
${\rm dim}\, X = 2$, with a $C^\infty$ boundary
$\Gamma := \partial X\ne\emptyset$.
Let $\Delta$ be the ``positive'' Laplace-Beltrami operator on $(X,g)$.
Given a real-valued function $K\in C^{\infty}(\Gamma)$,
we consider the operator $\Delta$ with a domain of
definition
\[
\{u\in H^2(X):\, \frac{\partial u}{\partial n}|_\Gamma = K
u|_\Gamma\}\, ,
\]
where $n(x)$, $x\in \Gamma$, is the inward unit normal to
$\Gamma$ with respect to the metric $g$. We denote this operator by
$\Delta_K$.
This is a selfadjoint operator in $L^2(X)$ with
discrete spectrum
\[
{\rm Spec}\, \Delta_K:=\{ \lambda_1 \le \lambda_2 \le \cdots
\}\, ,
\]
where each eigenvalue $\lambda=\lambda_j$ is repeated according to its
multiplicity, and it solves the spectral problem
\begin{equation}
\left\{
\begin{array}{rcll}
\Delta u\ &=& \ \la u \, \quad \mbox{in}\ $X$\, , \\
\frac{\partial u}{\partial n}|_\Gamma \ &=& \ Ku|_\Gamma \, .
\end{array}
\right.
\label{thespectrum}
\end{equation}
The manifolds we define admit a group of isometries $G$ isomorphic to
${\bf Z}_2\times{\bf Z}_2$. We denote by $\mbox{Symm}_G\, (\Gamma)$
the set of all real-valued functions $K\in C^{\infty}(\Gamma)$ which
are invariant
under the induced action of $G$ on $\Gamma$. Consider the map
\begin{equation}
\mbox{Symm}_G\, (\Gamma)\ \longrightarrow \ {\bf R}^{\bf N}\, ,
\label{themap}
\end{equation}
assigning to each $K\in\mbox{Symm}_G\, (\Gamma)$ the spectrum
${\rm Spec}\, (\Delta_K)$ of the boundary value problem (\ref{thespectrum}).
Guillemin and Melrose \cite{GM} have proved that for elliptical
regions in ${\bf R}^2$ ($\Gamma$ is ellipse)
the map (\ref{themap}) is one-to-one
(injective). We generalize their result for
Liouville billiard tables of classical type (see Definition
\ref{lb-class}).
Our main result is:
\begin{Th}\hspace{-2.mm}.
Let $(X,g)$ be a Liouville billiard table of classical type. Then the
map (\ref{themap}) is one-to-one (injective).
\label{MainTh}
\end{Th}
By billiard table we mean a compact connected Riemannian
manifold $(X,g)$ of dimension 2 with smooth boundary. The
corresponding dynamical system is the billiard flow.
Denote by $H:T^\ast X \rightarrow {\bf R}$
the Hamiltonian corresponding to the metric $g$ via the
Legendre transformation.
It will be shown that Hamiltonian systems corresponding to
Liouville billiard tables are integrable. Hereafter
integrable means that there exists a real-valued function $I\in
C^\infty(T^\ast X \setminus 0)$
which is constant on each (broken)
bicharacteristic of $H$, and such that the differentials
$dH$ and $dI$ are linearly
independent at almost any $\varrho \in T^\ast X\setminus 0$. A (broken)
bicharacteristic $\gamma$ of $H$ is a map
\[
\gamma:\ [0,T)\setminus P \ \longrightarrow \ T^\ast \displaystyle
\mathop X^\circ \setminus
0 ,\
\displaystyle \mathop X ^\circ = X\setminus \Gamma\, ,
\]
where $P$ is either a finite
subset $0 < t_1 < t_2 <\cdots < t_N < T$ of $[0,T)$ or the empty
set, such that on each connected component of $[0,T)\setminus P$ the
curve
$\gamma(t) = (x(t),\xi(t))$ is an integral curve of the Hamiltonian vector
field $X_H$ of $H$ in $T^\ast \displaystyle \mathop X ^\circ \setminus 0$
and for $0 0\, .
\]
Recall from \cite{Hor}, Sect. 24.3 , that for each
$\varrho \in \displaystyle T^\ast\mathop X ^\circ \setminus 0$
there exists at least one generalized bicharacteristic $\gamma$ starting from
$\varrho$ such that any compact arc of $\gamma$
can be approximated uniformly by (broken) bicharacteristics.
Hence, $I$ is constant on any such generalized bicharacteristic.
In other words, $I$ is invariant under the ``generalized billiard
flow''. Note that in the analytic case for each
$\displaystyle \varrho \in T^\ast \mathop X ^\circ
\setminus 0$ there is only one generalized bicharacteristic issuing from
$\varrho$.
Elliptical regions in ${\bf R}^2$ give a particular case
of integrable billiard tables. We shall give several other examples
on the sphere $ {\bf S}^2$, the hyperbolic space ${\bf H}^2$,
and on quadrics.
There is a common property for all of them,
namely, the existence of a special double cover with
two branched points (see Proposition \ref{FactorizationLemma}).
It is more or less known that the corresponding
billiard flows are integrable. The novelty here is that we propose a
general construction for such billiard tables and
we give explicit formulae for the covering maps.
We show also that
the Liouville billiard tables in the examples are of classical type
(Definition \ref{lb-class}). In particular, we obtain that
Theorem 1 holds for all of them.
We prove that the boundary
$\Gamma$ of any Liouville billiard table has the string
property, namely, the sum of distances
from any point of $\Gamma$ to the branched points is constant.
In particular elliptical regions are the only Liouville billiard
tables in ${\bf R}^2$. In addition, a class of the billiard
tables we define satisfies the so-called {\em strong evolution property}
(see Proposition \ref{StrongEvolutionProperty}).
To prove Theorem 1 we use a result of Guillemin and Melrose in
\cite{GM} and \cite{GM1} concerning
the singularities of the distribution
\begin{equation}
Z_K(t)=\displaystyle \sum_{\lambda \in {\rm Spec}\, (\Delta_K)}\, \cos
(t\sqrt \lambda) = {\rm tr}\, (\cos(t\sqrt {\Delta_K})) \, ,
\quad t>0 \, .
\label{trace}
\end{equation}
It is well known that the singular support of $Z_K(t),\ t>0$, is contained
in the length spectrum ${\cal L}(X,g)$ of the corresponding billiard
table. Recall that ${\cal L}(X,g)$ consists of the lengths of all
closed generalized geodesics of $(X,g)$.
Suppose now that there is an ``invariant circle'' $S$
of the billiard ball map $B$ in the co-ball bundle $B^\ast \Gamma$
such that the rotation
number of the map $B:S \rightarrow S$ is rational. Then the
generalized geodesics issuing from $S$ are all closed and we denote by
$\ell$ the corresponding common minimal length. Assume that $S$ is a
``clean'' submanifold of $B^\ast \Gamma$ and that there are no other
closed geodesics with the same length $\ell$. Then the integral
\begin{equation}
M \ =\ \int\limits_{S}\frac{K}{\cos\phi}d\mu\, .
\label{sp-invariant}
\end{equation}
can be recovered from the leading term of
the asymptotic expansion of $\sigma(t) = Z_K(t) - Z_0(t)$
at $t=\ell$, where $Z_0$ is the trace (\ref{trace}) corresponding to
the Neumann problem (see \cite{GM}, Theorem 4.2.).
Here $\phi$ is the angle between the initial vector of the
corresponding geodesic issuing form $S$
and the inward normal to $\Gamma$
at the initial point of the geodesic. The measure
$\mu$ on $S$ coincides (up to multiplication with a constant) with
the Leray form.
To prove Theorem 1 we look for an infinite sequence of such
``invariant circles'' $S_j$ approaching the glancing manifold as
$j\to \infty$ (the
boundary $\Gamma$ is strictly geodesically convex). The main
difficulty is to find $S_j$ so that the corresponding lengths $\ell_j$ are
all ``simple'' in the length spectrum of $(X,g)$,
which means that if $\gamma$ is a
closed generalized geodesic of length $\ell_j$ then $\gamma$ is a
broken geodesic issuing from $S_j$.
To do this we use essentially the properties of
the corresponding billiard ball map (see
Sect. 4). In this way, we recover the integral
invariants $M_j$ on $S_j$ in (\ref{sp-invariant})
from the spectrum of $\Delta_K$.
Moreover, we obtain a simple formula for
$M_j$ in terms of the functions $f$ and $q$
defining the Liouville billiard table (see (\ref{invariant})).
This allows us to recover $K$ from the sequence $M_j$.
An interesting and difficult question is if similar
results are valid for dimensions
$\ge 3$. A construction for integrable billiard tables close to the
one in this paper could be done for dimensions $\ge 3$. The corresponding
results will be published elsewhere.
The paper is organized as follows: In Sect. 2 we give a general
construction of Liouville billiard tables. In Sect. 3 we consider
several examples on ${\bf S}^2$, ${\bf H}^2$ and on quadrics. In Sect. 4
we investigate the corresponding billiard ball map. The proof of the
main theorem is given in Sect. 5.
\section{Liouville billiard tables}\label{LB-section}
\setcounter{equation}{0}
In the present section we define a class of
completely integrable billiard tables of dimension $2$.
The construction we propose is influenced from the
classification theorems of {\it Liouville surfaces} given in
\cite{Kiyo1,Kiyo2,Kolokol,B-F,M-T}
and the classical examples of integrable billiards described
in Section \ref{Examples} (see
\cite{Birkhoff, Poritsky, Bolotin, Drizzi, Veselov, C-S, Tabach}).
By definition, a Liouville surface is a complete $2$-dimensional
Riemannian manifold without boundary the geodesic flow of which admits
a quadratic in velocities integral functionally independent of the
energy integral. The idea to use special covers first appears in
\cite{Whittaker,Kolokol} (see Section \ref{Examples} and \cite{B-F} for
a complete list of references).
Anyway, to our best knowledge,
similar construction of integrable billiard tables has not been
documented in the literature.
We consider two functions $f\in C^{\infty}({\bf R})$, $f(x+1)=f(x)$,
and $q\in C^{\infty}([-N,N])$, $N>0$, such that
\begin{itemize}
\item[(H$_1$)] $f$ is even, $f>0$ if $x\notin {1\over 2}{\bf Z}$, and
$f(0)=f(1/2)=0$;
\item[(H$_2$)] $q$ is even, $q<0$ if $y\ne 0$, $q(0)=0$ and $q^{''}(0)<0$;
\item[(H$_3$)] $f^{(2k)}(l/2)=(-1)^kq^{(2k)}(0)$, $l=0,1$,
for every natural $k\in{\bf N}$.
\end{itemize}
In particular,
if $f\sim \sum_{k=1}^{\infty}\ f_kx^{2k}$ is the Taylor expansion of $f$
at $0$, then, by (H$_3$), the Taylor expansion of $q$ at $0$
is $q\sim \sum_{k=1}^{\infty}\ (-1)^k f_kx^{2k}$.
Consider the quadratic forms
\begin{eqnarray}
dg^2&=&(f(x)-q(y))(dx^2+dy^2)\label{themetric}\\
dI^2&=&(f(x)-q(y))(q(y)dx^2+f(x)dy^2)\label{theintegral}
\end{eqnarray}
defined on the cylinder $C= {\bf T}^1\times[-N,N]$,
${\bf T}^1\eqdef{\bf R}/{\bf Z}$.
The involution $\sigma : (x,y)\mapsto(-x,-y)$ induces an involution
of the cylinder $C$, that will be denoted by
$\sigma$ as well. We identify the points $m$ and $\sigma(m)$
on the cylinder and denote by $\C\eqdef C/\sigma$ the topological quotient
space. Let $\pi : C\to\C$ be the corresponding projection.
A point $x\in C$
is called {\em singular} if $\pi^{-1}(\pi(x))=x$, otherwise it is
a {\em regular} point of $\pi$. Obviously, the singular points are
$F_1=\pi(0,0)$ and $F_1=\pi(1/2,0)$.
Denote by ${\bf D}^2$ the unit disk $\{x_1^2+x_2^2\le 1\}$ in ${\bf R}^2$.
\begin{Prop}\hspace{-2.mm}.\label{FactorizationLemma}
Suppose that $f$ and $g$ satisfy {\em (H$_1$)$\div$(H$_3$)}.
Then the quotient space $\C$ is homeomorphic to
the unit disk ${\bf D}^2$.
There exist an unique differential structure on $\C$ such that the
projection $\pi: C\to\C$ is a smooth map, $\pi$ is a local diffeomorphism
in the regular points, and the push-forward $\pi_*g$ gives a smooth
Riemannian metric.
The push-forward of the form $I$ is also smooth.
\end{Prop}
\begin{Remark}\hspace{-2.mm}.\label{explanation}
The uniqueness of the differential structure on $\C$ means that if
$(\C,{\cal D}_1)$ and $(\C,{\cal D}_2)$ are two differential structures
on the quotient space $\C$ satisfying the conditions of
Proposition \ref{FactorizationLemma},
then the identity map $\mathop{\rm id} : \C\to\C$ is a diffeomorphism of
the manifolds $(\C,{\cal D}_1)$ and $(\C,{\cal D}_2)$.
In this way we obtain an unique Riemannian manifold, that we denote
by $(X,\pi_\ast g)$.
\end{Remark}
We denote by $\Gamma$ the boundary $\partial X$
of $X$. We give the following:
\begin{Def}\hspace{-2.mm}.
The billiard table $(X,\pi_*g)$
is said to be Liouville.
\end{Def}
To prove Proposition \ref{FactorizationLemma} we need the next simple lemma
which will be useful also in the next section.
Denote by $V\subset{\bf R}^2$ a neighborhood of the origin and by
$U$ the square
$(-\varepsilon,\varepsilon)\times(-\varepsilon,\varepsilon)$,
$\varepsilon>0$ in ${\bf R}^2$. We denote by $(x,y)$ and $(r,s)$ the
coordinates in $U$ and $V$ respectively.
\begin{Lemma}\hspace{-2.mm}.\label{smoothforward}
Let $\Phi : U\to V$ be a smooth mapping such that
$\sigma^\ast \Phi=\Phi$ and
$\Phi^{-1}(0,0)=(0,0)$. Suppose that for each
$(r,s)\ne (0,0)$, the set $\Phi^{-1}(r,s)$ consists of
exactly two
points and the differential $d\Phi|_{(x,y)}$ is non-degenerate at each
of them.
Consider the quadratic form (\ref{themetric}), where $f$ and $q$
are smooth functions on the interval $(-\varepsilon,\varepsilon)$,
$f\ge 0$ and $q\le 0$ and suppose that
the push-forward $d{\tilde g}^2=\Phi_\ast(dg^2)$ is a well-defined
smooth Riemannian metric on $\Phi(U)$. Then the next conditions are satisfied
\begin{itemize}
\item[(A$_1$)] $f$ is even, $f>0$ if $x\ne 0$, $f(0)=0$, $f''(0)>0$,
\item[(A$_2$)] $q$ is even, $q<0$ if $y\ne 0$, $q(0)=0$,
\item[(A$_3$)] $f^{(2k)}(0)=(-1)^k q^{(2k)}(0)$, $k\in{\bf N}$.
\end{itemize}
Moreover, there exist neighborhoods $V_0\subset\Phi(U)$ and
$W_0\subset{\bf R}^2=\{(u,v)\}$ of the origin in ${\bf R}^2$ and
a diffeomorphism $K : V_0\to W_0$,
such that the map ${\tilde K}$ given by the
diagram
$$
\begin{array}{ccccc}
& &\Phi^{-1}(V_0) &\subset&U\\
&\stackrel{\tilde K}\swarrow &\downarrow\lefteqn{\Phi}& &
\downarrow\lefteqn{\Phi} \\
W_0&\stackrel{K}{\longleftarrow}&V_0 &\subset&\Phi(U)
\end{array}
$$
satisfies ${\tilde K}(x,y)\eqdef(K\circ\Phi)(x,y)=(x^2-y^2,2xy)$.
In other words, setting
$z=x+iy$ and $w=u+iv$ we obtain ${\tilde K}(z)=z^2$.
Conversely, if the mapping $\Phi$ is given by
$\Phi : z\mapsto w=z^2$ and $(A_1)\div(A_3)$ are satisfied,
then $\Phi_\ast g$ is well-defined and smooth,
and the push-forward of the quadratic form (\ref{theintegral})
is also well-defined and smooth.
\end{Lemma}
{\em Proof.}
Suppose first that
the push-forward $d{\tilde g}^2$ is well-defined and smooth. Then
$\sigma^\ast(\Phi^\ast{\tilde g})=\Phi^\ast{\tilde g}$, and
$\sigma^\ast g=g$. The last equality shows that $f(-x)-q(-y)=f(x)-q(y)$,
hence, $f$ and $q$ are even functions. Moreover, $f(0)=q(0)=0$ and
$f>0$ if $x\ne 0$, and $q<0$ if $y\ne 0$, since $d{\tilde g}^2$ is a
Riemannian metric.
On the other hand, it is clear that
the push-forward $\tilde g$ is well-defined and smooth on
$\Phi(U)\setminus(0,0)$, provided $f$ and $q$ are even.
Suppose that $\tilde g$ is smooth in $\Phi(U)$. Consider a
conformal coordinate system
$\{(u',v')\}$ in a neighborhood $V_0$ of the point $(r,s)=(0,0)$. In
other words,
$d{\tilde g}^2=\mu(u',v')(du'^2+dv'^2)$ in these coordinates.
Let $K_1 : V_0\to W_0'$ be
the corresponding transition function, $K_1(0,0)=(0,0)$, and
${\widetilde K}_1\eqdef K_1\circ\Phi$.
Since both $(x,y)$ and $(u',v')$ are conformal local coordinates of the
metric $dg^2$ in $U\setminus(0,0)$, the mapping
${\widetilde K}_1$ is conformal in $U\setminus(0,0)$.
Taking the right orientation and introducing the complex variables
$z= x+iy$ and $w = u'+iv'$, we identify ${\widetilde K}_1$ with a
holomorphic function $p(z)$.
Then $p(-z)=p(z)$, and we have $p(z)=p_1(z^2)$, where
$p_1(w)$ is holomorphic and $p_1(0)=0$.
If $\frac{dp_1}{dw}(0)=0$ then for each $w_0'\ne 0$ close to $0$ the
preimage
$p^{-1}(w_0')$ would consist
of more than two points which is not allowed by assumption.
Hence, $\frac{dp_1}{dw}(0)\ne 0$ and shrinking $W_0'$ if necessary we
obtain a
biholomorphic map $p_1: W_0\to W_0'$, where
$W_0$ is a neighborhood of the origin in $\bf C$.
It is clear that in the coordinates $w=u+iv$ (with transition function
$w=p_1^{-1}(w')$) the map $\Phi$ is given by $z\mapsto w=z^2$.
Let $d{\tilde g}^2=\la(u,v)(du^2+dv^2)=\la(w,{\bar w})dwd{\bar w}$ be
the Riemannian metric ${\tilde g}$ in the new chart.
We have $g=\Phi^\ast{\tilde g}=4\la(w,{\bar w})|z|^2dzd{\bar z}$.
Hence,
\begin{equation}\label{obstacle}
f(x)-q(y)=4\la(u,v)(x^2+y^2),
\end{equation}
where $\la(u,v)$ is a smooth positive function,
$u=x^2-y^2$, $v=2xy$.
Let
\[
\la\sim\la^{(1)}+(\la^{(2)}_1u+\la^{(2)}_2v)+\cdots\ ,\
f\sim f_0+f_1x^2+\cdots\ , \
q\sim q_0+q_1y^2+\cdots
\]
be the Taylor
expansion of the corresponding functions in the points $u=v=0$, and
$x=0$ and $y=0$ respectively. Substituting these series in
(\ref{obstacle}) and comparing the coefficients of the homogeneous
terms of $x$ and $y$ we obtain that $f_0=q_0$,
$f_1=-q_1=4\la^{(1)}>0$ and $f_k=(-1)^kq_k$. Therefore, conditions
$(A_1)\div(A_3)$ are satisfied.
Conversely, suppose that $(A_1)\div(A_3)$ are satisfied
and let $\Phi$ be given by $z\mapsto w=z^2$. It is clear that
the metric $d{\tilde g}^2=\la(u,v)(du^2+dv^2)$ is well-defined and
smooth on $\Phi(U)\setminus 0$ and
$4\la(u,v)=\frac{f(x)-q(y)}{x^2+y^2}$.
We have
\[
4\la(u,v)=f_1+f_2(x^2-y^2)+o(|z|^2) =f_1+f_2u+o(|w|),
\]
where $\la(0,0)=f_1/4$.
Hence, $\la$ is differentiable at $(u,v)=(0,0)$.
The proof that $\la$ is smooth is straightforward.
The same arguments show that the push-forward
$\Phi_\ast I$ gives a smooth form in a neighborhood of the zero, and
we complete the proof of
Lemma \ref{smoothforward}. \finishproof
A variant of the last lemma is proved in \cite{Kiyo3}.
\noindent{\em Proof of Proposition \ref{FactorizationLemma}.}
The first statement of the proposition is obvious.
The points $A=(0,0)$ and $B=(1/2,0)$ on the cylinder $C$ are fixed points
of the involution $\sigma$. Denote by $D^2_r$ the disk $\{x^2+y^20$ (see \cite{Whittaker}).
The images of the coordinate lines $y=y_0=\const\ne 0$ are
confocal ellipses on the plane $\{(u,v)\}$ with foci $F_1=(-\ep,0)$ and
$F_2=(\ep,0)$,
while the images of the coordinate lines $x=x_0=\const\ne\frac{k}{2}$,
$k\in{\bf Z}$, are confocal hyperbolas with the same foci $F_1$ and $F_2$.
The mapping $\Phi$ is a double cover of the plane ${\bf R}^2$ with
branched points $F_1$ and $F_2$.
Let $ds^2=du^2+dv^2$ be the Euclidean metric on ${\bf R}^2$. The pull-back
$g:= \Phi^{*}s$ of the metric $s$ has a
Liouville form
\begin{equation}\label{themetr-ellipse}
dg^2=(f(x)-q(y))(dx^2+dy^2),
\end{equation}
where $f(x)= 4\ep^2\pi^2\sin^2 2\pi x$ and $q(y)= -4\ep^2\pi^2\sinh^2
2\pi y$.
The quadratic form
\begin{equation}\label{theint-ellipse}
dI^2\ :=\ (f(x)-q(y))(q(y)dx^2+f(x)dy^2)
\end{equation}
is a first integral of the geodesic flow of the ``metric'' $g$. Note that the
quadratic forms $g$ and $I$ vanish at the points $(\frac{k}{2},0)$,
$k\in{\bf Z}$. It follows from Proposition \ref{smoothforward},
\ref{Integrability} and Lemma \ref{FactorizationLemma} that the
push-forward $\widetilde I$
of the form $I$ gives a smooth integral of the elliptical
billiard tables in the interior of the ellipses $\{y=\const\ne 0\}$.
Moreover, ${\widetilde I}$ is an integral of any billiard table,
the boundary of which consists of curves from the confocal family
described above.
\subsection{Integrable billiard tables on the sphere.}\label{S^2}
Integrable billiard tables on ${\bf S}^2$ and ${\bf H}^2$ have been
considered earlier in \cite{C-S,Bolotin}. Here we give explicit formulae
for the corresponding branched covers.
Denote by ${\bf S}^2= \{x^2+y^2+z^2=1\}$ the unit sphere embedded
in the Euclidean space ${\bf R}^3$.
The metric $g_1$ is the restriction of the Euclidean one
$dg_0^2=dx^2+dy^2+dz^2$ on ${\bf S}^2$. The coordinates $\{(x,y)\}$ of the
unit disk ${\bf D}^2_1= \{x^2+y^2<1\}$ give a parameterization of the positive
half ($z>0$) of the sphere ${\bf S}^2$ and we can rewrite the metric $g_1$
in these coordinates. Let us consider the branched cover
of the unit disk
$\Phi_1 : {\bf R}/2\pi{\bf Z}\times (-\frac{1}{k},\frac{1}{k})\to{\bf D}^2_1$
given by
\[
\left\{
\begin{array}{ccc}
x&=&\frac{k}{\sqrt{1+k^2}}\sqrt{1+v^2}\cos u,\\
y&=&kv\sin u,
\end{array}
\right.
\]
where $k$ is a positive constant.
The pull-back ${\widetilde g}_1\eqdef\Phi_1^{*}g_1$ has Liouville form
\[
d{\widetilde g}_1^2=k^2(f(u)-q(v))\left(\frac{du^2}{1+k^2\sin^2u}+
\frac{dv^2}{(1+v^2)(1-k^2v^2)}\right),
\]
where $f(u)= \sin^2u$ and $q(v)= -v^2$. After an obvious change of
the variables the metric ${\widetilde g}_1$ takes form (\ref{themetric}).
Propositions \ref{FactorizationLemma}, \ref{Integrability} and
Lemma \ref{smoothforward} show that the interior of any curve
$\{v=\const>0\}$ is a Liouville billiard table. It is easy to see
that all these billiard tables are of classical type.
\subsection{Integrable billiard tables on the Hyperbolic space.}
Consider the hyperboloid of two sheets ${\bf H}^2= \{-x^2-y^2+z^2=1\}$
embedded in the Minkowski space ${\bf R}^{2,1}$.
The metric $g_{-1}$ of the Hyperbolic space is the restriction of
the Minkowski metric $dg_0^2=dx^2+dy^2-dz^2$ on ${\bf H}^2$.
The coordinates $\{(x,y)\}$ give a parameterization of the positive
sheet ($z>0$) of the hyperboloid ${\bf H}^2$. Consider the branched
cover
of the plane
$\Phi_{-1} : {\bf R}/2\pi{\bf Z}\times{\bf R}\to{\bf R}^2$
given by
\[
\left\{
\begin{array}{ccc}
x&=&\frac{k}{\sqrt{1-k^2}}\sqrt{1+v^2}\cos u,\\
y&=&kv\sin u,
\end{array}
\right.
\]
where $k$ is a constant in the interval $00\}$, $n(x)$ being the unit inward normal to $\Gamma$
at $x$.
The corresponding billiard ball map $B$ is defined as
follows (\cite{Birkhoff}):
Take $(x,\xi)$ in the interior of $B^\ast \Gamma$ and denote
by $\xi_+ \in \Sigma_x^+$ the co-vector uniquely determined by
$\xi_+|_{T_x \Gamma} = \xi$. Consider the integral
curve $\exp(tX_{\tilde H})(x,\xi_+)$, of the Hamiltonian vector field
$X_{\tilde H}$ starting at $(x,\xi_+)$. If it intersects transversally
$S^\ast X|_\Gamma$ at a time $t_1>0$ and lies entirely in the
interior $\displaystyle {S^\ast } \mathop X ^\circ$ of $S^\ast X$
for $t\in (0,t_1)$, we set
$(y,\eta_-)=J(x,\xi_+) =\exp(t_1X_{\tilde H})(x,\xi_+)$,
and define $B(x,\xi)=(y,\eta)$,
where $\eta = \eta_-|_{T_y \Gamma}$.
We denote by $\widetilde B^\ast \Gamma$ the set
of all such points $(x,\xi)$. As in \cite{kn:MM} we can write $B$ in an
invariant form as follows. Consider the pull-back
$\omega_1$ in $T^{*}X|_\Gamma$ of
the symplectic form $\omega$ in $T^{*}X$ via the inclusion map.
The projection along the characteristics of $\omega_1$ induces a
smooth map $\pi_1: S^{*}X|_\Gamma \to B^{*}\Gamma$ and
we denote by
$\displaystyle \pi_1^+: \mathop {B^{*} }^\circ \Gamma \to \Sigma^+$ an inverse
map to $\pi_1$. Then we can write $B=\pi_1\circ J \circ \pi_1^+$.
In this way we obtain a symplectic map
$B:\widetilde B^\ast \Gamma \rightarrow B^\ast \Gamma $ which is
analytic if the billiard table is analytic.
We extend $B$ as the identity mapping on the
boundary $S^\ast \Gamma$ of $B^\ast \Gamma$.
\subsection{The phase portrait of the integral}\label{Portrait}
Suppose now that $(X,{\tilde g})$ is a Liouville billiard table where
${\tilde g}=\pi_*(g)$.
We identify the boundary $\Gamma$ of a Liouville billiard table
with the circle $\{(x,N):\, x\in{\bf T}^1\}$
on the cylinder $C$ with
coordinates $\{(x,y)\}$.
Consider the natural coordinates $\{(x,y,p_1,p_2)\}$ of the
cotangent bundle $T^{*}C$ and set $\Gamma_N = \{y=N\}\subset T^{*}C$.
Then $\Gamma_N$ is diffeomorphic to $T^{*} X|_{\Gamma}$ and using
(\ref{hamilton}) we identify
\[
S^\ast X|_\Gamma \cong
\{(x,N,p_1,p_2)\in \Gamma_N: p_1^2 + p_2^2 = f(x) -
q(N)\}\, .
\]
Moreover, we
identify $T^\ast \Gamma$ with $T^\ast {\bf T}^1$ via the inclusion map
$(x,p_1) \longmapsto (x,N;p_1,0)$.
The billiard ball map $B : {\widetilde B}^\ast\Gamma\to B^\ast\Gamma$
preserves the symplectic form $\om_0 = dp_1\wedge dx$ of $T^\ast \Gamma$.
It is easy to see that the characteristics of the pull-back
$\omega|_{\Gamma_N}$ to $\Gamma_N$ of
the symplectic form $\om=dp_1\wedge dx+dp_2\wedge dy$
are spanned by the vectors $\frac{\partial}{\partial p_2}$.
Then $\pi_1(x,N,p_1,p_2) = (x,p_1)$ on $S^\ast X|_\Gamma$,
\[
B^\ast \Gamma \cong \{(x,p_1)\in
T^\ast{\bf T}^1: p_1^2 - f(x) + q(N)\le 0\}\, ,
\]
and the map
$\pi_1^+:B^\ast \Gamma \to \Sigma^+$ is given by
\[
(x,p_1)\longmapsto \left(x,N;p_1,-\sqrt{f(x)-q(N)-p_1^2}\, \right)\, .
\]
Consider the function ${\cal I} = I\circ \pi_1^+$ on $B^\ast \Gamma$,
where $I$ is the integral (\ref{integral}). In the coordinates
$\{(x,p_1)\}$
it is given by
\begin{equation}
{\cal I}(x,p_1)=f(x)-p_1^2\, .
\label{integral-bill}
\end{equation}
By construction, ${\cal I}$ is a smooth function in
$ \displaystyle\mathop {B^\ast}^\circ \Gamma$
and it is analytic if the Liouville billiard is of
classical type.
\begin{Lemma}\hspace{-2.mm}.
The function ${\cal I}(x,p_1)=f(x)-p_1^2$ is constant on the trajectories
of the billiard ball map $B$.
\end{Lemma}
The lemma follows immediately from Proposition \ref{Integrability}.
It is easy to see that
$q(N)\le {\cal I}(x,p_1)\le f(x)$.
Fix a value $h$ of the integral ${\cal I}(x,p_1)$,
$q(N)\le h \le \max f$ and denote by $S_h=\{{\cal I}(x,p_1)=h\}$
the corresponding
level set in $B^\ast \Gamma$.
The set of glancing points of the
billiard ball map coincides with the set of constant level
$\{{\cal I}(x,p_1)=q(N)\}$.
Consider now the critical points of the integral ${\cal I}(x,p_1)$ in $B^\ast
\Gamma$. Using (H$_1$)
(and (H$_5$) if the Liouville billiard table is of classical type),
we easily obtain:
\begin{Prop}\hspace{-2.mm}. \label{Morse}
The critical points of the integral ${\cal I}(x,p_1)$ are given by
$P_i=(x_i,0)$,
where $x_i$ are the critical points of $f$.
The point $P_i$ is non-degenerate in Morse sense if and only if
$f^{''}(x_i)\ne 0$.
If $f^{''}(x_i)>0$, then $P_i$ is non-degenerate of index $1$
(``hyperbolic'' point).
If $f^{''}(x_i)<0$, then $P_i$ is non-degenerate of index $2$
(``elliptic'' point).
The points $A_1 = (0,0)$ and $A_2 = (1/2,0)$ are non-degenerate of
index $1$. Each $h\in (q(N),0)$ is a regular value of the
integral ${\cal I}(x,p_1)$ and the corresponding
level set $S_h$ consist of two circles. The level set $S_0$ consists of
the critical points $A_1$ and $A_2$ and four different arcs
connecting them and containing only
regular points of the integral ${\cal I}(x,p_1)$.
Moreover, if the Liouville billiard table
is of classical type then
each
$h\in (0,f(1/4))$ is a regular value of ${\cal I}(x,p_1)$,
the corresponding level set $S_h$ consists of two circles, and
$S_{f(1/4)}$ consists of two non-degenerate critical points
of index $2$.
\end{Prop}
If $f(x)-h>0$ for $x\in [a,b]$,
then $p_1=\pm\sqrt{f(x)-h}$ is a parameterization
of $S_h$, and we set $S_h^{\pm}[a,b] = \{(x,\pm\sqrt{f(x)-h}): x\in [a,b]\}$.
In a neighborhood of these curves we can introduce local
coordinates $\{(x,{\cal I})\}$.
The pull-back of $\omega_0$ to $S_h^{\pm}[a,b]$
has the form
\[
\om_0\ =\ dp_1\wedge dx\ =\
\pm\left(\frac{dx}{2\sqrt{f(x)-{\cal I}}}\right)\wedge d{\cal I}.
\]
We consider the following 1-form on $S_h$:
\[
\la_h\eqdef\left\{
\begin{array}{l}
\frac{dx}{\sqrt{f(x)-h}},\ p_1>0,\\
-\frac{dx}{\sqrt{f(x)-h}},\ p_1<0.
\end{array}
\right.
\]
If $h$ is a regular value of the integral ${\cal I}(x,p_1)$,
the form $\la_h$ gives a smooth $1$-form on $S_h$
which is invariant under the Hamiltonian flow of ${\cal I}$.
$\la_h$ is called {\it Leray's form}.
It is easy to see that the Leray's form $\la_h$ is invariant with
respect to any symplectic transformation defined in a tubular
neighborhood of $S_h$ and preserving the function
${\cal I}(x, p_1)$.
We supply $S_h$ with orientation by means of the Leray's form.
\subsection{Rotation function for Liouville billiard tables of
classical type}\label{Sec:details}
Fix a regular value $h$ of the integral ${\cal I}(x,p_1)$ and consider
a connected component $S_h^0 \cong {\bf T}^1$ of the level set $S_h$.
Suppose that the billiard ball map $B$ is defined on $S_h^0$
and preserves it inducing a diffeomorphism $B|_{S_h^0}:S_h^0 \to S_h^0$.
The Leray's form $\la_h$ provides $S_h^0$ with a smooth positive
measure which is invariant with respect to the map $B|_{S_h^0}$.
By means of $\la_h$ we introduce
a periodic coordinate $\{s \mod \mu_0\}$ ($\mu_0=\int_{S_h^0}\la_h$) on
$S_h^0$ such that $ds=\la_h$. It is clear that $B|_{S_h^0}$ takes
$s$ to $s+\mu$ where $\mu$ is a constant on $S_h^0$.
The number $\rho|_{S_h^0}=\mu/\mu_0$ is the {\em rotation number} of the
map $B|_{S_h^0}$.
The rotation number $\rho|_{S_h^0}$ depends on the choice of
the orientation on $S_h^0$ and it is defined modulo $1$.
It is clear that $\rho|_{S_h^0}=\int_{\widehat{PQ}}\la_h/\int_{S_h^0}\la_h$,
were $P$ is an arbitrary point on $S_h^0$, $Q=B(P)$, and $\widehat{PQ}$ is
an arc in $S_h^0$ connecting $P$ and $Q$.
We will prove (see Proposition \ref{RotFunc1})
that there is at most a finite set $\Xi(q)$
of values $h_1,...,h_l\in(q(N),0)$ such that the billiard ball map $B$
is well defined on $S_h$ for each $h\in (q(N),0)\setminus \Xi(q)$ but it
is not defined on the level sets $S_{h_i}$. Geometrically this means
that the geodesics issuing from the sets $S_{h_i}$ do not reach
the boundary $\Gamma$ again and they stay forever in
the interior of the billiard table.
If $h\in(q(N),0)\setminus\Xi(q)$ then the billiard ball map $B$
preserves the connected components of the level sets $S_h$.
Moreover, the involution
$(x,p_1) \stackrel{\imath}{\to} (x,-p_1)$ interchanges the connected
components of $S_h$ and we have $\imath B\imath = B^{-1}$.
In particular, the
rotation number of the restriction of $B$ to each of the components of
$S_h$ is the same and we
denote it by $\rho^-(h)$.
If $00$ for each $\tau \ge 0$, and there exist $\tau_1> \tau(h)>0$
such that $dy/d\tau (\tau(h))=0$,
$dy/d\tau (\tau)<0$ for $\tau\in[0,\tau(h))$, $dy/d\tau (\tau)>0$ in
$\tau\in(\tau(h),\tau_1]$ and $y(\tau_1)=N$. By the same argument to any
other solution $y_0$ of $q(y)=h$ correspond caustics for $T_h$ such that
the geodesics tangent to $\{y=y_0\}$ never reach the boundary
($y(\tau)$
oscillates between two different zeros of $q(y)=h)$.
Consider the rotation function $\rho^{-}(h)$, $q(N) h_k$.
If such a point $N_{k+1}$ doesn't exist we set
$\Xi(q)=\{h_1,\ldots,h_k\in(q(N),0)\}$.
\begin{Prop}\hspace{-2.mm}.\label{RotBihav1}
The rotation function $\rho^{-}(h)$ is strictly increasing in a
neighborhood of the point $q(N)$ and $\lim\limits_{h\to h(N)+0}\rho^{-}(h)=0$.
The rotation function is analytic on the set
$(q(N),0)\setminus\Xi(q)$, and
$\lim\limits_{h\to h_i\pm 0}\rho^{-}(h)=+\infty$ for every $h_i\in\Xi(q)$.
\end{Prop}
\noindent{\em Proof.}
It is clear that $y_m(h)\to N$ as $h\to q(N)+0$, hence,
$\lim\limits_{h\to q(N)+0}\rho^-=0$.
We have
\[
\tau(h) = \int\limits_{y_m(h)}^N\frac{dy}{\sqrt{h-q(y)}}
=-2\int\limits_0^{\sqrt{h-q(N)}}\frac{dy}{dq}(h-w^2)dw,
\]
where $w^2=h-q(y)$.
Hence,
\[
\tau^{'}(h)=-\frac{dy}{dq}(q(N))\frac{1}{\sqrt{h-q(N)}}
-2\int\limits_0^{\sqrt{h-q(N)}}\frac{d^2y}{dq^2}(h-w^2)dw.
\]
This shows that $\tau^{'}(h)\to+\infty$ as $h\to q(N)+0$.
We have
\[
\frac{d\rho^-}{dh}(h)=
\frac{2\tau^{'}(h)\int\limits_0^1\frac{dx}{\sqrt{f(x)-h}}-
\tau(h)\int\limits_0^1\frac{dx}{(f(x)-h)^{3/2}}}
{\left(\int\limits_0^1\frac{dx}{\sqrt{f(x)-h}}\right)^2}.
\]
Therefore, $(\rho^{-})^\prime (h)>0$ if $h$ is sufficiently close to $q(N)$.
Consider the closest to $N$ critical point $N_1$ of the function $q$
and the critical value $h_1=q(N_1)$.
It is clear that $q(y)$ is strictly monotone on the interval $[N_1,N]$.
Moreover, in view of (H$_6$), there exists $\delta >0$ such that
\[
\frac{dy}{dq}(h)=\frac{R(h)}{\sqrt{h_1-h}}\, , \quad h \in
(h_1-\de,h_1]\, ,
\]
where the function $R(h)$ is continuous
and $R(h)
-2c_1\int\limits_0^{\sqrt{h-q(N_1+\de)}}\frac{1}{\sqrt{(h_1-h)+w^2}}dw=
c_1\ln(h_1-h)+O(1).
\]
Therefore, $\lim\limits_{h\to h_1-0}\rho^{-}(h)=+\infty$.
Analogously we prove $\lim\limits_{h\to h_1+0}\rho^{-}(h)=+\infty$.
The same arguments can be applied to any of the points of $\Xi(q)$.
This completes the proof of Proposition \ref{RotBihav1}. \finishproof
\vspace{0.5cm}
\noindent{\bf Case B).} Suppose that $00$ there are at most finitely many $L\in(0,T)$ such
that $L$ is the length of a closed broken geodesic issuing from
$S_0$.
\end{Lemma}
\noindent
{\em Proof of Proposition \ref{StringProperty}.}
Consider a broken geodesic $\gamma$ in $X$ starting from $F_1$ and denote by
$P\in \Gamma$ its first point of contact with the boundary. Suppose
that $\gamma$ is different from $\gamma_1$. Then the
intersection points of $\gamma$ with
$B^\ast\Gamma$ lie in $S_0$ and they are different from the critical
points $A_1$ and $A_2$. In the coordinates $(x,y)$ in $C$ we have
$F_1=(0,0)$, $F_2=(1/2,0)$, and we can suppose that
$P=(x_1,N)$ with $0< x_1< 1/2$ (the case $P=(x_1,-N)$, $0< x_1< 1/2$,
is treated in the same way). Then $\gamma$ is given by
the solution $\bar\gamma(\tau)= (x(\tau),y(\tau))$
of the system
$$
\left\{
\begin{array}{ccc}
\frac{dx}{d\tau}&=&\sqrt{f(x)},\\
\frac{dy}{d\tau}&=&\pm\sqrt{-q(y)},
\end{array}
\right.
$$
such that $\lim_{\tau\to -\infty}\bar\gamma(\tau )=(0,0)$, and there
exists $\tau_0\in {\bf R}$ such that $y(\tau_0)=N$, $y'(\tau)>0$ for
$\tau<\tau_0$ and $y'(\tau)<0$ for $\tau>\tau_0$. We have to prove
that $\lim_{\tau\to +\infty}\bar\gamma(\tau )=(1/2,0)$. Using that
$-q(y)= Cy^2(1 + O(y^2))$, $C>0$, near $y=0$, we obtain that
$\displaystyle\lim_{\tau\to +\infty}y(\tau )=0$.
In the same way we prove that
$\displaystyle\lim_{\tau\to +\infty}x(\tau )=1/2$. Differentiating
with respect to $P\in \Gamma$ we prove that the sum of distances from
$P\in \Gamma$ to $F_1$ and $F_2$ is
$|\displaystyle\widehat{F_1P}| +|\displaystyle\widehat{F_2 P}| =
C_0$, where $C_0>0$ is a positive constant. \finishproof
Suppose that (H$_4$) holds.
As we have seen in \ref{Sec:details} (Case A), the caustics of
the billiard trajectories can be identified with the curves
$\{y=y(h)\}$ on $C$, where $y(h)$ is a positive solution of
the equation $q(y)=h$ and $h$ is a fixed real number in
$(q(N),0)$.
If $h$ is close to $q(N)$, the curve $\{y=y(h)\}$ (defined uniquely)
is a boundary of another Liouville billiard table $(
X_h,g|_{X_h})$,
$X_h \subset X$, defined by functions $f_h$ and
$q_h$ such that $f_h \equiv f$ and $q_h$
is given by the restriction of
the function $q$ on the interval $[-y(h),y(h)]$.
It is clear that $X_h$ and $X$ share the same caustics
$\Ga_{h'}=\{y=y(h')\} \subset X_h$, $h0$ such that
$l(P_1,h)>C_{\ep}$ for each $h>q(N)+\ep$.
\end{itemize}
\end{Lemma}
\noindent{\em Proof. }
Suppose first that $h<0$.
It follows from $\ref{eq:caustic}$ that
\begin{equation}
\begin{array}{rcll}
l(P_1,h)&\eqdef&\int\limits_0^{\tau_0}
\left\{(f(x)-q(y))\left(\left(\frac{dx}{d\tau}\right)^2+
\left(\frac{dy}{d\tau}\right)^2\right)\right\}^{\frac{1}{2}}d\tau\\
&=&\int\limits_0^{\tau_0}
\{(f(x)-q(y))(|f(x)-h|+|h-q(y)|)\}^{\frac{1}{2}}d\tau\\
&=&\int\limits_0^{\tau_0}(f(x)-q(y))d\tau=\int\limits_0^{\tau_0}f(x)d\tau-
\int\limits_0^{\tau_0}q(y)d\tau\\
&=&\int\limits_{x_1}^{x_2}\frac{f(x)}{\sqrt{f(x)-h}}dx-
2\int\limits_{y_m(h)}^N\frac{q(y)}{\sqrt{h-q(y)}}dx\\
&\ge&-2\int\limits_{y_m(h)}^N\frac{q(y)}{\sqrt{h-q(y)}}dy.
\end{array}
\label{LengthFormula}
\end{equation}
Similarly, if $h>0$, then
$l(P_1,h)\ge-\int\limits_{-N}^N\frac{q(y)}{\sqrt{h-q(y)}}dy$.
The integral
$l_0^{-}(h)\eqdef-2\int\limits_{y_m(h)}^N\frac{q(y)}{\sqrt{h-q(y)}}dy$
is a positive continuous function of $h$ on $[q(N)-\ep,0]\setminus\Xi(q)$.
Moreover, it can be easily seen that
$\lim\limits_{h\to h_i\pm 0}l_0^{-}=\infty$ and
$\lim\limits_{h\to 0-0}l_0^{-}=l_0>0$.
The integral
$l_0^{+}(h)\eqdef-\int\limits_{-N}^N\frac{q(y)}{\sqrt{h-q(y)}}dy$ is
a positive continuous function of $h$ on $[0,f(1/4)]$.
This completes the proof of Lemma \ref{Lengths}. \finishproof
Let $h\notin\Xi(q)$ be a regular value of the integral
${\cal I}(x,p_1)$ and let
the rotation number $\rho^{\pm}(h)\in {\bf Q}$ be rational.
Then for each $\nu \in S_h$ there is a closed broken
bicharacteristic issuing from $\nu$ and we denote by $\gamma_\nu$ the
corresponding primitive broken closed geodesic.
It follows from the definition of the rotation number (see the
beginning of Sect. 4) that the number
of vertices of $\gamma_\nu$ is independent of the choice of $\nu$
on $S_h$ and the length function $\nu \to L(\gamma_\nu)$ is
continuous in $S_h$.
Since the length spectrum ${\cal L}(X,g)$ (the set of lengths of all closed
generalized geodesics) of the billiard table $(X,g)$ has measure $0$
in ${\bf R}$, we obtain that the continuous function
$\nu \to L(\gamma_\nu)\in {\cal L}(X,g)$
is constant on each connected component of $S_h$. If $h<0$ and the broken
closed geodesic $[0,L(\gamma)]\ni t \to \gamma(t)$ issues from one of
the components of $S_h$
then $[0,L(\gamma)]\ni t \to \gamma(L(\gamma)-t)$ issues from the other
component. In particular, the primary length of the closed broken
geodesics issuing from $S_h$ is constant and we denote it by $l(h)$.
The same is true for $h\in (0,f(1/4))$ since $B$ interchanges the
connected components of $S_h$ in this case.
We consider now the closed broken geodesics issuing from $S_h$ for
$h$ close to $0$ and to $f(1/4)$. Denote by
${\cal L}_b(X,g)$ the set of lengths $L(\gamma)$
of all closed generalized geodesics $\gamma$ having at least one common point
with the boundary. By definition the set $L(\gamma)\, {\bf N}^\ast$ is
contained also in ${\cal L}_b(X,g)$.
\begin{Lemma}\hspace{-2.mm}.\label{Lengths2}
There exist neighborhoods $U_0$ and $U_1$ in ${\bf R}$ of $0$ and $f(1/4)$
respectively such that for each $T>0$ there are at most finitely many
$L \in {\cal L}_b(X,g)\cap (0,T)$ such that $L$ is the length of
a closed broken geodesic issuing from $S_h$ with $h\in
U_0 \cup U_1$.
\end{Lemma}
\noindent{\em Proof. }
We shall first prove the Lemma for
$h$ in a neighborhood of 0. In view of Lemma \ref{singularlevel} we can exclude
the singular level $S_0$ from our consideration.
Suppose that there is an infinite sequence $\{x_j\}_{j=1}^\infty$ with
$\displaystyle \lim_{j\to\infty}x_j =0$ such that the geodesics issuing
from the level sets $S_{x_j}= \{{\cal I}(x,p_1)=x_j\} \subset
B^\ast \Gamma$ are all closed ($\rho^{\pm}(x_j) \in {\bf Q}$)
and $\displaystyle \lim_{j\to\infty}l(x_j) = T_0$. Denote be
$n_j$ the number of vertices of the primitive closed geodesic
issuing from $\nu \in S_{x_j}$, i.e.
the smallest positive integer such that
$B^{n_j}(\nu) = (\nu)$, which is independent
of $\nu$.
Since the set of lengths $l(x_j)$, $j\in {\bf N}$, is bounded,
using Lemma 4.9 b), we obtain that
the set $\{n_j\}_{j\in {\bf N}}$,
is bounded as well. Hence, we can suppose that $n_j=n$ does not depend
on $j\ge 1$.
Choose sufficiently small
neighborhoods $W_0\subset W$ in $B^\ast \Gamma$
of one of the critical points, say $A_1$,
such that $B^n(W_0) \subset W$.
Then $B^n$ is
analytic in $W_0$ and $B^n(q,p)=(q,p)$ for each $(q,p)\in S_{x_j}\cap W_0$.
On the other hand, by Proposition \ref{Morse}, $A_1$ is
a non-degenerate critical point of ${\cal I}(q,p)$ of index $1$
(hyperbolic), and we can provide $W$ with analytic local coordinates
$(q,p)\in {\bf R}^2$ such that $(q,p)(A_1)=(0,0)$ and ${\cal I}(q,p) = q^2 -
p^2$ in $W$. Then
there is an open cone $\Gamma_0$ with a vertex at $(0,0)$ such that
any ray in $\Gamma_0$ starting from the origin intersects infinitely
many level curves $S_{x_j}\cap W_0$. This implies $B^{n}(q,p)=(q,p)$
in $W_0$. Since $B$ is analytic in the connected component $U$ of $A_1$
in the complement of $\cup_{j=1}^l S_{h_j}$, $h_j\in \Xi(q)$,
in $B^\ast \Gamma$, we obtain
$B^{n}(\nu)=\nu$ for each $\nu \in U$. Moreover, the length of the
closed geodesic corresponding to the periodic orbit $\{\nu,
B(\nu),\ldots, B^{n-1}(\nu)\}$ of $B$ does not depend on $\nu \in U$,
and we denote it by $T$.
Taking into account Lemma \ref{Lengths}, a), we obtain that
$\Xi(q)=\emptyset$. Hence, $B^{n}$
is the identity mapping in $B^\ast \Gamma$.
This contradicts to the strict geodesic
convexity of $\Gamma$.
The same is true in the case when $h$ is close to $f(1/4)$ since
the corresponding critical point is non-degenerate elliptic. \finishproof
We investigate now the lengths of the closed broken
geodesics approximating $\Gamma$ with winding number $m=1$.
Given a value $h$ of the integral ${\cal I}(x,p_1)$, close to $q(N)$, and such
that $\rho^-(h)$ is rational, we denote by $l(h)$ the length of the
primitive closed broken geodesics issuing from $S_h$.
Consider a sequence $\{p_k\}_{k=1}^{\infty}$
tending to $q(N)$ and such that
$\rho^{-}(p_k)=\frac{1}{k}$, $k\ge M_0$,
where $M_0 \gg 1$ is a fixed natural number.
\begin{Lemma}\hspace{-2.mm}.\label{Popov}
There is $M_0 \gg 1$ such that for each $k\ge M_0$,
\[
l(p_k) L(\Ga)$.
\end{Lemma}
{\em Proof.}
Set $t_0 = L(\Gamma)/2\pi$ and
denote by $(r, \varphi)\in [t_0 -\delta_0,t_0]\times ({\bf R}/2\pi {\bf Z})$
the ``action-angle'' coordinates corresponding to
the smooth foliation of invariant ``circles'' $S_h$ for $h \ge q(N)$ and
close to $q(N)$. The billiard ball map is given by
$B(r,\varphi) = (r, \varphi + \mu(r))$ for
$r \in [t_0 -\delta_0,t_0]$, where $\mu$ is
smooth in the open interval and continuous in the closed one.
Taking into account that $\Gamma$ is strictly
geodesically convex, we are going to
show that $\mu$ is given by the derivative $\mu(r)= \tau'(r)$,
where $\tau(r)=- Q(r)^{3/2}$ for $r\in [t_0-\delta_0,t_0]$, $Q$ is
smooth in a neighborhood of $r=t_0$ and $Q(t_0)=0$, $Q'(t_0)<0$.
To this end we
make use of an approximate interpolating Hamiltonian $\zeta$ of
$B$ \cite{kn:MM}. The function $\zeta$ is smooth in a neighborhood of
$t_0$, $\zeta(t_0,\varphi)=0$, $\partial\zeta/\partial
r (t_0,\varphi)<0$ and $R(r,\varphi)=B(r,\varphi) - \exp
(H_{-\zeta^{3/2}})(r,\varphi)$ can be extended as a smooth function
across $t_0$ and it vanishes to infinite order at
$r=t_0$. Here
\[
\exp(H_{-\zeta^{3/2}})(r,\varphi)=
\left(r,\varphi - {3\over 2}\zeta(r,\varphi)^{1/2}{\partial
\zeta \over{\partial r}} (r,\varphi) + O((t_0-r))\right)
\]
stands for the
time-one-flow of the Hamiltonian $-\zeta^{3/2}$. We are going to prove
that $\zeta(r,\varphi) = \zeta(r,0)(1 + R_0(r,\varphi))$, where
$R_0$ is smooth and vanishes to infinite order at
$r=t_0$. Obviously, we have
$\mu(r)=- {3\over 2}\zeta(r,\varphi)^{1/2}{\partial
\zeta \over{\partial r}} (r,\varphi)(1 + O((t_0-r)^{1/2})$. On the
other hand,
$B^\ast \zeta = \zeta + O(\zeta^\infty)$, and we get
\[
\zeta (r, \varphi + \mu(r))
= \zeta(r,\varphi)
+ O((t_0-r)^\infty) \, .
\]
Expanding the smooth function $\zeta$ in Fourier series
$\zeta(r,\varphi) =
\sum_{k\in {\bf Z}} f_k(r)e^{ik\varphi}$, we obtain
\[
f_k(r)(e^{ik\mu(r)} -1)= O_k((t_0-r)^\infty)
\]
which implies
$f_k(r)= O_k((t_0-r)^\infty)$ for $k\neq 0$. Since $f_k$ is smooth we get
$(d^jf_k/dr^j)(t_0)=0$ for each $j\ge 0$, $k\neq 0$. Hence
$\zeta(r,\varphi) - \zeta(r,0)$ vanishes to
infinite order at $r=t_0$ and
$ Q(r) = \zeta(r,0) + O((t_0-r)^\infty)$ is smooth in
$[t_0 -\delta_0,t_0]$.
Given an invariant circle $S_h$ of $B$ with a rational rotation number
$\rho^-(h) =\rho = m/n$, where $m,n\in {\bf N}$, $(m,n)=1$, and $\rho$ is
close to $0$, we consider the common primitive length $\ell(\rho)
=l(h)$ of the
family of broken geodesics issuing from $S_h$.
Using a simple argument about the
symplectic invariance of the
length spectrum (see \cite{Po}, Sect. 4, (4.2)-(4.6), with $Q_0=0$),
we obtain that
\[
\frac{\ell(\rho)}{2\pi m}\ =\ \frac{{\cal S}(\rho)}{\rho}\, ,
\]
where
${\cal S}(\rho)$ is the Legendre transformation of $\tau(r)/2\pi$.
In other words,
\[
{\cal S}(\rho) \ =\
r(\rho)\rho - \tau(r(\rho))/2\pi\, ,
\]
where
$[0,\varepsilon_0]\ni\rho \rightarrow r(\rho)\in
[t_0-\delta_0,t_0]$, $r(0)=t_0$, is the inverse function to the
``frequency map'' $r \rightarrow \tau'(r)/2\pi$.
It is easy to see that
\[
{d\over{d\rho}} \left ({{\cal S}(\rho)\over \rho} \right ) =
{\tau(r(\rho))\over {2\pi \rho^2}}\ < \ 0 \, ,
\]
which implies the firs statement.
To prove the second one, we observe that
\[
r(\rho) = t_0 - C\rho^2(1 + O(\rho))\ ,\ \tau(r(\rho))/\rho =
- C_0\rho^2 (1 + O(\rho))\, ,\ C, C_0>0,\ \mbox{as}\ \rho \to +0 .
\]
Then $\ell(\rho) = 2\pi m \, {\cal S}(\rho)/\rho =
m(L(\Gamma) - \rho^2 s(\rho))$, where $s$ is
smooth in a neighborhood of $0$ and independent of $m$,
which completes the
proof of the Lemma. \finishproof
\section{Proof of the main theorem}
\setcounter{equation}{0}
First we recall some facts about the singularities of the distribution
$Z_K$ given by (\ref{trace}).
It is known that the singular support of $Z_K(t),\ t>0$, is contained
in the length spectrum ${\cal L}(X,g)$ of the corresponding billiard
table. We denote also by $Z_0(t)$ the corresponding distribution for the
Laplace-Beltrami operator with Neumann boundary conditions. As in
\cite{GM} and \cite{GM1} we consider the distribution
$\sigma(t)=Z_K(t) - Z_0(t)$, $t>0$.
We observe that the singular support of $\sigma$ on ${\bf R}_+$ is
contained in the set ${\cal L}_b(X,g)$ which consists of the lengths
of all closed generalized geodesics having at least one common point
with the boundary. More precisely, we prove that the contribution in
$\sigma$ of
any closed geodesic $\gamma$ lying entirely in
$\displaystyle\mathop X^\circ$ is
$C^\infty$.
Indeed, let $B$ be a
pseudo-differential operator of order $0$ the wave-front of which is
contained in a small conic neighborhood of $\gamma$ in $T^\ast
\displaystyle\mathop X^\circ \setminus \{0\}$
and let $\zeta\in C_0^\infty ({\bf R})$ have
support in a small neighborhood of $L (\gamma)$.
Consider the distribution
\[
\sigma_B(t) = {\rm tr}\, (\cos(t\sqrt {\Delta_K})\, B) -
{\rm tr}\, (\cos(t\sqrt {\Delta_D})\, B)\, ,
\]
where $\Delta_D$ is the Laplace-Beltrami operator in $X$ with Dirichlet
boundary conditions. Since the parametrices of $\cos(t\sqrt
{\Delta_K})B$ and $\cos(t\sqrt {\Delta_D})B$ differ by a smooth function
for $t\in [-T,T]$, $T>L (\gamma)$, choosing the
wave-front set of $B$ sufficiently small, we obtain that $\sigma_B$ is
a smooth function in that interval.
Consider now a regular level $S_h$ of the integral ${\cal I}$,
where $h\notin\Xi(q)$ is
a fixed real number in the interval $q(N)0$ in a neighborhood of $h=q(N)$
(see Proposition \ref{RotBihav1}), which implies that $S_h$ is a
{\it clean submanifold} for the iterated billiard ball map $B^n$.
If there are no other broken geodesics of length $l(h)$,
using Theorem 4.2, \cite{GM}, we recover from the leading term of
the asymptotic expansion of $\sigma(t)$ at $t=l(h)$ the integral
\[
M(h)\eqdef\int\limits_{S_h}\frac{K}{\cos\phi}d\mu_{S_h}\, .
\]
In the last formula $\phi$ is the angle between the initial vector of the
corresponding geodesic issuing form $S_h$
and the inward normal to the boundary of
the billiard at the initial point of the geodesic. The measure
$\mu_{S_h}$ on $S_h$ coincides (up to multiplication with a constant) with
the Leray form $\la_h$ defined in Sect. 4.1.
It is easy to see that $\cos\phi=\sqrt{\frac{h-q(N)}{f(x)-q(N)}}$.
Therefore,
\begin{equation}\label{invariant}
M(h)=\frac{1}{\sqrt{h-q(N)}}
\int\limits_0^1\frac{K(x)}{\sqrt{f(x)-h}}\sqrt{f(x)-q(N)}\;dx
\end{equation}
is a spectral invariant.
\begin{Lemma}\hspace{-2.mm}.\label{sequence}
There exists a strictly monotone sequence $\{p_k\}_{k=1}^{\infty}$
such that
\begin{itemize}
\item[a)] $p_k\to q(N)+0$ as $k\to\infty$;
\item[b)] the geodesics issuing from $S_{p_k}$ are closed and
$l(p_k)\ne L(\Gamma)$;
\item[c)] the primitive closed geodesics issuing
from $S_{p_k}$ are the only closed broken geodesics
in $X$ with length $l(p_k)$.
\end{itemize}
\end{Lemma}
\noindent{\em Proof.}
The lemma follows from the properties of the
Liouville billiard tables proved in the previous sections.
Given $h>q(N)$ close to $q(N)$ and such that $\rho^-(h)$ is
rational, we
consider the closed broken geodesics $\gamma$ issuing from $S_{h}$
and denote by $L(\gamma)$ the length of $\gamma$
(then $h=I(\gamma(t),\dot\gamma(t)))$.
Using Lemma \ref{Popov} we choose $M_0$ such that
the geodesics $\gamma$ issuing from $S_{p_k}$, $p_k = 1/k$, $k\ge
M_0$,
are the only closed geodesics satisfying
\[
L(\gamma) \in [l(p^0), L(\Ga))\ ,\
h=I(\gamma(t),\dot\gamma(t)) \in (q(N),p^0]\ ,\ p^0 =p_{M_0}.
\]
Obviously the sequence $\{p_k\}_{k\ge M_0}$ satisfies items $a)$ and
$b)$ of Lemma \ref{sequence}.
In view of Lemma \ref{Lengths}
there exist $\de_i>0$ $(i=1,...,l)$ such that for each
$h\in (h_i-\de_i,h_i+\de_i)$ the length $l(h)>L(\Ga)$.
Denote by $K$ the set $U_0\cup U_1\cup_j(h_j-\de_j,h_j+\de_j)$
where $U_0$ and $U_1$ are the neighborhoods from Lemma \ref{Lengths2}.
To prove $c)$ it is sufficient to show that there are only
finitely many $k_1,...,k_r\in (p^0,f(1/4)]\setminus K$ such that
$\rho^\pm(k_j)\in{\bf Q}$ and $l(k_j)\in [l(p^0),L(\Gamma))$.
Taking $\ep$ such that $0<\ep\le p^00$ such
that if the number of the vertices of a closed geodesic $\gamma$
is greater than
$J_0$ and $h=I(\gamma(t),\dot\gamma(t)) \in (p^0,f(1/4)]$,
then $L(\gamma)>L(\Ga)$.
Therefore, if $h\in(p^0,f(1/4)]\setminus K$,
$l(h)0$, using the
identity $V_1(h) \equiv 0$,
we get for $h<0$
\[
0\ =\ K_0\, \int_0^1 \,
\frac{ds}{\sqrt{s^2
-h}}\, + \, O(1)\ =\ - \frac{K_0}{2}\, \log(-h) + O(1)\, ,
\]
which implies $K_0=0$. In the same way we prove that $K(1/4)=0$.
Then we have
\[
V_1(h) = \int\limits_0^{f(1/4)}
\frac{{\widetilde K}(t)}{\sqrt{t-h}}\;dt\equiv 0
\]
where $t=f(x)$ and ${\widetilde K}(t)=K(x)\sqrt{f(x)-q(N)}/f^{'}(x)$
is continuous on the interval $[0,f(1/4)]$.
Finally, the arguments used in \cite{GM} show that $K\equiv 0$.
Indeed, differentiating the function $V_1(h)$ with respect to $h$ at
the point $h=q(N)$ we obtain
\[
\int\limits_0^{f(1/4)}\frac{{\widetilde K}(t)}{\sqrt{t-q(N)}}
(t-q(N))^{-k}\;dt=0,
\]
for $k=0,1,...$.
The last equality and the Stone-Weierstrass theorem show that
${\widetilde K}\equiv 0$.
This proves Theorem \ref{MainTh}.
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\vspace{0.5cm}
\noindent
G. P.:
Universit\'e de Nantes, D\'epartement de Math\'ematiques,
UMR 6629 du CNRS, 2, rue de la Houssini\`ere, BP 92208, 44072 Nantes
Cedex 03, France
\vspace{0.5cm}
\noindent
P.T.:
Institute of Mathematics, BAS,
Acad.G.Bonchev Str., bl.8,
Sofia 1113,
Bulgaria
\end{document}
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