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\begin{document}
\begin{titlepage}
\title{Invariants of the length spectrum and spectral invariants of
planar convex domains}
\author{Georgi Popov}
\date{}
\maketitle
\thispagestyle{empty}
\begin{center}
{\bf Institute of Mathematics\\ Bulgarian Academy of Sciences\\ Sofia
1090, Bulgaria}\\ and\\
{\bf Fachbereich Mathematik\\ Technische Hochschule Darmstadt\\
Schlo{\ss}gartenstra{\ss}e 7\\ 6100 Darmstadt}\\[6ex]
\end{center}
\begin{abstract}
This paper is concerned with a
conjecture of V.Guillemin and R. Melrose that the length spectrum
of a strictly convex bounded domain together with the spectra of
the linear Poincar\'{e} maps corresponding to the periodic broken
geodesics
in $\Omega$ determine uniquely the billiard ball map up to a symplectic
conjugation. We consider continuous deformations of bounded
strictly convex domains $\Omega_s,\ s\in [0,1]$,
with smooth boundaries. If the length
spectrum does not change along the deformation, we prove that the
invariant KAM circles of the corresponding billiard ball map $B_s$ with
rotation numbers in a suitable Cantor set of a positive Lebesgue
measure as well
as the restriction of $B_s,\ s\in [0,1]$, on their union are
symplectically equivalent to each other.
We prove as well that the KAM circles and the
restriction of the billiard ball map on them are spectral invariants of
the Laplacian with Dirichlet (Neumann) boundary conditions for suitable
deformations of strictly convex domains.
\end{abstract}
\end{titlepage}
\section{Introduction}
\setcounter{equation}{0}
This paper is concerned with certain length spectrum invariants of
a strictly convex and bounded planar domain $\Omega$ with a smooth
boundary $\partial\Omega$. The motivation for studying such
invariants comes from the inverse spectral problem formulated by
M. Kac \cite{kn:Kac}. It is known \cite {kn:GM1}, \cite{kn:PS}, that the
length spectrum $\lb{\Omega}$ of $\Omega$
is encoded in the spectrum of the Laplace operator
$\Delta$ in $\Omega$
with Dirichlet (Neumann) boundary conditions, and that $\lb{\Omega}$
can be extracted from the spectrum of $\Delta$
by means of the Poisson formula at least for generic domains. In
this connection, V. Guillemin and R. Melrose \cite{kn:GM} formulated
the conjecture that the length spectrum of $\Omega$ and the spectra
of the linear Poincar\`{e} maps of the periodic broken geodesics
of $\Omega$ form together a complete set of symplectic invariants
for the corresponding billiard ball map $B$. As it was mentioned in
\cite{kn:GM}, this conjecture seems to be a little optimistic and
the local version of it is more hopeful.
The first result in this direction was obtained by
Sh.~Marvizi and R.~Melrose \cite{kn:MM} who described
new length spectrum invariants of a strictly convex domain $\Omega$,
studying the asymptotics
of the lengths of the closed broken geodesics approaching the boundary
$\partial\Omega$
. Let us take
$l_{mn}$ arbitrarily in the set $\lb{\Omega;m,n}$ of lengths of
all closed broken geodesics of $\Omega$ with $n$ vertices
and winding number $m$. When $m$ is fixed and $n$
tends to infinity, $l_{mn}$ has an asymptotic expansion
in powers of $n^{-2}$. The corresponding coefficients
$c_{mk},\ k=1,2,\ldots $, do not depend on the choice of $l_{mn}$ in
$\lb{\Omega;m,n}$
and they are length spectrum as well as spectral invariants of $\Omega$
\cite{kn:MM}.
Y.~Colin de Verdi\`{e}re \cite{kn:CV} proved that the labeled length
spectrum and the spectra of the linear Poincar\`{e} maps determine
uniquely the Birkhoff invariants of a closed and elliptic broken
ray in $\Omega \subset {\bf R}^2$. Recently this result was
generated in higher dimensions as well as for contact manifolds by
J.~P.~Francoi\c{c}e and V.~Guillemin \cite{kn:FG}.
R.~de~la~Llave, J.~Marco and R.~Moriy\'{o}n \cite{kn:LMM} proved that
there are no non-trivial deformations of exact symplectic
mappings $B_s,\ s\in [0,1]$,
leaving the period spectrum
fixed when $B_s$ are Anosov's mappings on a compact
symplectic manifold. One of the reasons for symplectic rigidity in
\cite{kn:LMM} is that all periodic points of $B_s$ are hyperbolic and
form a dense set. Although the billiard ball map of a strictly convex
domain is in the opposite situation, conjugation can still be
made on a large part of the domain of $B_s$.
Consider the billiard ball map $B$ corresponding to a strictly
convex domain $\Omega$ with a smooth boundary.
Near the boundary $B$ is an exact symplectic map close to
a completely integrable one. Using that fact V.~Lazutkin \cite{kn:La}
proved that there exists a large family of invariant KAM circles
$\Lambda (\omega )$ of $B$ with rotation numbers $\omega $ in a Cantor
subset
$\Theta $ of
the interval [0,1/2) of positive Lebesgue measure. The
corresponding caustics $C(\omega )$ are strictly
convex
and smooth curves in $\Omega $ accumulating at $\partial \Omega $.
There are two invariants related to any invariant curve
$\Lambda (\omega )$,
namely the length $\ell (\omega )$ of
the caustic
$C(\omega )$ and the Lazutkin parameter $t(\omega )$
\cite{kn:Am}, \cite{kn:La}, (~see also Section 2 ).
\par
The main goal of this paper is to prove that the vector
function
\begin{equation}
\Theta \ni \omega \rightarrow (\ell (\omega ),t(\omega ))
\label{eq:1.1}
\end{equation}
is a length spectrum invariant for continuous deformations of the
domain.
The main result
(see Theorem 2.1) says that the length spectrum determines uniquely the
invariant circles $\Lambda (\omega ),\ \omega \in \Theta $,
as well as the restriction of
$B$ on them (up to a symplectic
conjugation) for continuous deformations of $\Omega $. We prove as well
that the vector function (\ref{eq:1.1}), the invariant
circles $\Lambda (\omega ),\ \omega \in \Theta $
, and the restriction of
$B$ on them are spectral invariants of the Laplacian with Dirichlet
(Neumann) boundary conditions for suitable continuous deformations of
$\Omega $.
Let $\Omega_1$ and $\Omega_2$ be two strictly convex and bounded domains
with smooth boundaries. Suppose that the corresponding
"marked" length spectra
${\cal L}(\Omega_j ,m,n),\ m \le n/2,\ j=1,2$ coincide. We show in
Section 5 that the corresponding vector functions (\ref{eq:1.1})
coincide as well which improves Theorem 3 in \cite{kn:KP}.
The plan of the paper is as follows: In Section 2 we formulate the main
result about the length spectrum invariants of continuous deformations
of a strictly convex domain (see Theorem 2.1). Section 3 is devoted to
a symplectic KAM theorem for the billiard ball map, which is the basic
tool in the proof of the main results. First we introduce action-angle
coordinates $(\theta ,r) \in {\bf T}\times \Gamma ,\ {\bf T} =
{\bf R}/2\pi
{\bf Z}$, for the approximated interpolating Hamiltonian of $B$
where $ \Gamma = (l-\Ge ,l)$ and $2\pi l = \ell = \ell
(0)$ is the length of the boundary $\partial \Omega $. In these
coordinates $B$ is a small perturbation of the completely integrable map
$$
(\theta ,r) \longrightarrow (\theta + (\tau ^0)'(r),r),\
\tau ^0 (r) = - \frac{4}{3}\,\zeta (r)^{3/2},
$$
where $\zeta (l) =0$, the first derivative $\zeta '(l)<0$ and
$\zeta > 0$ in $\Gamma$.
Moreover, ${\bf T}\times \{l\}$ is a component of the boundary
$\partial \Sigma $ of the
phase space $\Sigma $ of $B$. Applying suitable KAM theorem, we find
symplectic coordinates $(\Gp ,I) \in {\bf T}\times \Gamma $ and smooth
functions $K(I)$ and $Q(\Gp ,I)$ in {\bf R}\ and ${\bf T}\times
{\bf R}$
respectively such that
$$
B(\Gp ,I)\ =\ (\Gp + \tau '(I),I)\ +\ Q(\Gp ,I),\
\tau (I) = - \frac{4}{3}\,K(I)^{3/2} ,
$$
where $K(l) =0,\ K '(l)<0$ and $K > 0$ in $\Gamma $, the vector
function $Q$ has a zero of infinite order on
${\bf T}\times E$, and the Cantor set
$E=\{I\in\Gamma :\ \tau '(I)/2\pi \in \Theta \}$ has a positive Lebesgue
measure. In Section 4 we give a simple geometric meaning to the
restriction of $\tau$ on $E$. We prove that the function ${\cal J}(\omega )$
inverse to the frequency mapping $\Gamma \ni I \rightarrow \tau'(I)/2\pi $
is equal to $\ell (\omega )/2\pi $ while $\tau ({\cal J}(\omega )) =
t(\omega )$ for any $\omega \in \Theta $. In particular, the Legendre
transform ${\cal I}(\omega ) = \omega {\cal J}(\omega ) -
\tau({\cal J}(\omega ))$
of $\tau (I)/2\pi $ is given on $\Theta$ by
\begin{equation}
2\pi {\cal I}(\omega )\ =\ \omega \ell (\omega ) - t(\omega ),\
\forall\ \omega \in \Theta .
\label{eq:1.2}
\end{equation}
Theorem 2.1 is proved in Section 5. The main ideas here are:
\par
1. Let $[0,b]\ni s \rightarrow \Omega _s ,\ b>0$, be a continuous
deformation of strictly convex bounded domains with smooth boundaries.
Following an argument due to Birkhoff,
we prove that for any pair $(m,n)\in {\bf Z}^2_+ ,\ m\leq n/2$,
the function
$$
[0,b]\ni s\ \longrightarrow\ T_{mn}(s)\ =\ {\rm max}\, \{t:\ t\in
{\cal L}(\Omega _s,m,n)\},
$$
is continuous, and that ${\cal L}(\Omega _s)$ is a subset of {\bf
R}\ of
Lebesgue measure zero for any $s$. If the length spectrum of $\Omega _s$
is independent of $s$ along the deformation, the continuous function
$T_{mn}(s)$ takes values in the set
${\cal L}(\Omega _s) = {\cal L}(\Omega _0)$ which does not contain
intervals. Hence,
\begin{equation}
T_{mn}(s)\ =\ T_{mn}(0), \ s\in[0,b].
\label{eq:1.3}
\end{equation}
\par
2. Fix $\omega \in \Theta $ and choose a sequence
$(m_j,n_j)\in {\bf Z}^2_+ ,\ j=1,2,... $, such that
$$\mid m_j/n_j-\omega\mid \leq n_j^{-1/2}.$$ We prove that
\begin{equation}
{\cal I}_s(\omega ) = \lim_{j \rightarrow \infty }
(T_{m_j n_j}(s)/n_j),\ \forall s \in [0,b],
\label{eq:1.*}
\end{equation}
\noindent and taking into account (\ref{eq:1.3}) we obtain
$$
{\cal I}_s(\omega )\ =\ {\cal I}_0(\omega ),\ \forall \omega \in
\Theta,\ \forall s \in [0,b].
$$
Since $\Theta$ has no isolated points, differentiating the last
equality with respect to $\omega $ we obtain
\begin{equation}
\ell _s(\omega )\ =\ \ell _0(\omega ),\
t_s(\omega )\ =\ t_0(\omega ),\
\forall \omega \in \Theta,\ \forall s \in [0,b].
\label{eq:1.**}
\end{equation}
Close idea has been used in \cite{kn:Po} to study the length spectrum
invariants of elliptic geodesics in Riemannian manifolds.
\par
Equality (\ref{eq:1.*}) is a consequence of the following estimate (see
Theorem~5~.~1~)~:
\begin{equation}
\mid l_{mn}\ -\ n {\cal I} (m/n) \mid\ \leq C_p\, n^p,\ \forall p>0,
\label{eq:1.4}
\end{equation}
for any $l_{mn} \in {\cal L}(\Omega ,m,n)$ and any pair
$(m,n)\in {\bf Z}^2_+$ satisfying the inequality dist$(m/n,\Theta )
\leq n^{-1/2}$. Here, $C_p$ depends only on $p$ and $\Omega $. In particular,
if we fix $m$, let $n$ go to infinity, and expend ${\cal I}(t)$ in
Taylor series at $t=0$, we obtain Theorem 5.15 in \cite{kn:MM} as a
consequence of (\ref{eq:1.4}) since $0 \in \Theta$ (see Corollary 5.1).
The invariants
$c_{mk}$ of Sh. Marvizi and R. Melrose are explicitly given by the
Taylor coefficients of ${\cal I}(t)$ at $t=0$.
3. We assume that $\Omega_s$ is strictly convex only in a small interval
$[0,b_0),\ b_0 >0$. To show that $\Omega_s$ is strictly convex for any
$s$ in $[0,1]$ along the deformation, we use (\ref{eq:1.**}) as well as
the integral invariants $I^{(k+1)}(0)$ of Sh. Marvizi and R. Melrose
\cite{kn:MM}. In particular, we prove that $I^{(k+1)}(0) = 2\pi {\cal
R}^{(k)}(0),\ k=1,2,\ldots $, where ${\cal R}(t)$ is the function
inverse to $t=K(I)$ and ${\cal R}^{(k)}(0)$ are the corresponding
derivatives at $t=0$.
\par
Section 6 is devoted to spectral
invariants of the Laplacian
$\Delta = - \partial ^2/\partial x_1^2 - \partial ^2/\partial x_2^2$
in $\Omega $ with Dirichlet (Neumann) boundary conditions. We suppose
that $\Omega$ is a strictly convex bounded domain in ${\bf R}^2$ with a
smooth boundary. Then, the spectrum of $\Delta $ consists of non-negative
eigenvalues tending to infinity. V. Guillemin and R. Melrose formulated
in
\cite{kn:GM} the conjecture that the spectrum of the Laplace operator in
$\Omega$ with Dirichlet (Neumann) boundary conditions determines
uniquely the billiard ball map. Partial affirmative answer to this
conjecture is given in Section 6 (see Theorem 6.2).
We prove that the vector
function (\ref{eq:1.1}), the invariant circles $\Lambda (\omega ),\
\omega \in\Theta$ as well as the restriction of the billiard ball map on
them are spectral invariants of the Laplacian for suitable continuous
deformations of a strictly convex domain. In particular, the
corresponding billiard ball maps are conjugated to each other on a large
subset of $\Sigma$ of a positive Lebesgue measure. We investigate
the singularities of the distribution
\begin{equation}
Z(t) = {\rm trace} \cos (t \Delta ^{1/2}) = (1/2) \sum \exp (i\lambda t)
\label{eq:1.5}
\end{equation}
where the sum is taken over all $\lambda $ with
$\lambda ^2$ in the spectrum of $\Delta
$ counted with multiplicity. The singular
support of $Z(t)$ satisfies the Poisson relation
\begin{equation}
{\rm sing.supp.}Z(t)\ \subseteq\ \{T \in {\bf R} :\ \pm T \in \lb{\Omega
}\}\cup \{0\}
\label{eq:1.6}
\end{equation}
\cite{kn:AM},
\cite{kn:PS}. The inverse relation may not always be true,
because singularities created by different closed broken geodesics may
cancel each other. It is known that (\ref{eq:1.6}) turns into
equality in the generic case when all periodic broken geodesics are
non-degenerate and of different lengths \cite{kn:PS}.
Let $(m,n)\in {\bf Z}^2_+$ satisfy the inequality dist$(m/n,\Theta )
\leq n^{-1/2}$.
In Section 6 we prove under the natural condition (\ref{eq:6.3}) that
$$
T_{mn}\ \in\ {\rm sing.supp.}Z
$$
if $n\geq n_0(\Omega )$. For $m=1$ and $n$ sufficiently large this
result has been proved in \cite{kn:MM}. The main idea in \cite{kn:MM} is
to write $Z(t)$ in a \nbd of $T_{1n}$ as a Lagrangian distribution with
a suitable phase function and then to apply a result of Soga. We use
another representation on $Z(t)$ in a \nbd of $T_{mn}$ which is based
on the KAM theorem and the results obtained in Section 5.
As a consequence, we show that
the vector function (\ref{eq:1.1}) is a spectral invariant of the
Laplace operator with Dirichlet (Neumann) boundary conditions for
suitable continuous deformations of the domain.
\section{Length spectrum}
\setcounter{equation}{0}
Let $\Omega $ be a bounded domain in ${\bf R}^2$ with a smooth boundary
$\partial \Omega$ . The
length spectrum ${\cal L}(\Omega )$ of $\Omega$ is defined as the
set of lengths of all
periodic generalized geodesics $\tilde{\gamma }$ of $\overline{\Omega}$
( $\tilde{\gamma }$
is the projection on $\overline{\Omega}$ of a
closed generalized bicharacteristic of the Hamiltonian $p(x,\xi ) =
\mid \xi \mid ^{2} - 1$, (cf. \cite{kn:Ho}, Def. 24.3.7)). Suppose in
addition that
$\Omega$ is
strictly convex. Then any generalized bicharacteristic of $\Omega$ is
either a broken bicharacteristic reflecting at the boundary by
the usual law of the geometric optics or it is a gliding ray
traveling along the boundary. Hence,
${\cal L}(\Omega ) = {\cal L}_{b}(\Omega )\cup {\cal L}(\partial \Omega )$
in this
case where ${\cal L}_{b}(\Omega )$ is the set of lengths of all closed
broken
geodesics and ${\cal L}(\partial \Omega ) = \{n\ell : n \in {\bf
N} \},
\ell $ being the length of the boundary.
\par
The broken bicharacteristic flow induces a discrete dynamical
system on the boundary
\par
$$
B:\Sigma\ \longrightarrow\ \Sigma ,\ \Sigma\ =\ \{(x,\xi ) \in
T^{*}\partial \Omega :\ \mid \xi \mid \le 1\},
$$
\noindent called billiard ball map which is defined as follows:
Pick
$\Gr = (x,\xi)$ in $T^{*}\partial\Omega$ with $\mid \xi \mid <
1$ and set $\Gr^{\pm} = \pi ^{\pm}(\Gr) =
(x,e^{\pm}(\Gr))$. Here $e^{\pm}(\Gr)\in ({\bf R}^2)^*$ are unit
covectors such that
$$
\pm \ >\ 0,\
=<\xi,v>,\ \forall v\in T_x\partial\Omega,
$$
$n(x)$ being the inward normal to $\partial\Omega$ at $x$.
Via the canonical inner product in ${\bf R}^2$ we identify $e^{\pm}(\Gr )$
with a vector
$\tilde{e}^{\pm}(\Gr )$
in ${\bf R}^2$. The bicharacteristic
$${\bf R} \ni t\ \longrightarrow\ (x + t\tilde{e}^{+},e^{+})$$
of
$S^*{\bf R} ^2 = \{(y,\eta )\in T^*{\bf R} ^2 :\ \mid \eta\mid =1 \}$
passing through $\pi ^{+}(\Gr)$ intersects
$(T^*{\bf R}^2)_{\mid\partial\Omega}$ at a second point $(y,e^+)$. Define
$\eta
\in T^*_y\partial\Omega$ by the equality
$$
=<\eta ,v>,\ \forall v\in T_y\partial\Omega.
$$
Then $\mid\eta\mid < 1$ and $B$ sends $(x,\xi )$ to
$(y,\eta )$. Moreover, any
point in $\partial \Sigma$ is a fixed point of $B$.
\par
Defined in this way the billiard ball map is exact symplectic
in the interior of $\Sigma $, indeed
\begin{equation}
B^{*}\sigma\ - \sigma\ =\ dT \label{eq:2.1}
\end{equation}
where $\sigma$ is the canonical one-form in $T^{*}\partial \Omega $
and $T(x,y)=\mid x-y \mid$ is the distance between $x$ and $y$ in
${\bf R}^2$, $y$ being the first component of $B(x,\xi ) = (y,\eta )$ (see
Proposition 2.3 in \cite{kn:GM}).
Near the boundary of $\Sigma $ the billiard ball map $B$ is a small
perturbation
of a completely integrable map for which the KAM theorem can be
applied \cite{kn:Mo}, \cite{kn:La}. In particular, there exists a large
family of
invariant circles $\Lambda (\omega)$ of $B$ enumerated by their rotation
numbers $\omega \in \Theta$, where $\Theta$ is a Cantor subset of the
interval (0,1/2] with a positive measure (see Section 3). For each
$\omega\,\in\,\Theta$, denote by $C(\omega)$ the corresponding caustic
in $\Omega$, i.e. the envelope of the rays
$\{x + \tilde{e}(x,\xi): t > 0\}, (x,\xi) \in \Lambda(\omega)$,
issuing from $\Lambda(\omega)$.
Then $C(\omega)$ is a smooth and strictly convex curve in $\Omega$
and the boundary $\partial\Omega$ is an evolute of $C(\omega)$
\cite{kn:La}.
In other words, if we loop a string with a suitable length
$T(\omega)$ around $C(\omega)$, lean a pen against it and draw,
we get $\partial\Omega$. The Lazutkin parameter of $C(\omega )$ is
defined by
$$t(\omega) = T(\omega) - \ell (\omega)$$
where $\ell (\omega)$ is the length of $C(\omega)$.
\par
We consider a continuous deformation
\begin{equation}
[0,1]\ni s\ \longrightarrow\ \Omega_{s} \subset {\bf R} ^2
\label{eq:2.*}
\end{equation}
of bounded domains with smooth boundaries
$\partial\Omega_{s} = \{x^{s}(t): t \in {\bf T}\}$,
such that the mapping
$[0,1] \ni s \longrightarrow x^{s}(\cdot) \in C^{\infty}(
{\bf T},{\bf R}^{2})$
is continuous. For any strictly convex domain $\Omega_{s}$ we denote
by $B_{s}$, $\Lambda_{s}(\omega)$ and $C_{s}(\omega)$ the corresponding
billiard ball map, invariant circle and caustic with a rotation number
$\omega$. Consider the Cantor set $\Theta$ defined by
(\ref{eq:3.7}). According to (\ref{eq:3.17}), the union $\Lambda_{s}$ of
the invariant
circles $\Lambda_{s}(\omega),\ \omega \in \Theta$, is a set of
positive Lebesgue measure in $T^{*}\partial \Omega_{s}$. The main
result in this paper is:
\begin{theo}
Let $[0,1] \in s \longrightarrow \Omega_{s}$ be a continuous deformation
of bounded domains in ${\bf R}^{2}$ with smooth boundaries. Suppose that
$\Omega_{0}$ is strictly convex and
\begin{equation}
\lb {\Omega_{s}} = \lb {\Omega_{0}}\ ,\ s \in [0,1].
\label{eq:2.2}
\end{equation}
Then:
\par
(i) $\Omega_{s}$ is strictly convex for any $s \in [0,1]$,
\par
(ii) there exists a continuous family of smooth exact
symplectic mappings
$$\chi_{s}: T^{*}\partial \Omega_{0} \longrightarrow
T^{*}\partial \Omega_{s}$$
such that
\begin{equation}
\chi_{s}(\Lambda_{0}(\omega ))\ =\ \Lambda_{s}(\omega ),\
\forall\, \omega \in \Theta \label{eq:2.3}
\end{equation}
\begin{equation}
\chi_{s} \circ B_{0} = B_{s}\circ\chi_{s}\
{\em on}\ \Lambda_{0} \label{eq:2.4}
\end{equation}
for any $s \in [0,1]$,
\par
(iii) for any $\omega \in \Theta$ there exists a
continuous family of caustics $[0,1] \longrightarrow C_{s}(\omega)$
in $\Omega_{s}$ and
\begin{equation}
\ell_{s}(\omega) = \ell_{0}(\omega),\
t_{s}(\omega) = t_{0}(\omega),\ s \in [0,1].
\label{eq:2.5}
\end{equation}
\end{theo}
\section{KAM theorem}
\setcounter{equation}{0}
In this section we formulate a symplectic version of the KAM theorem
for a family of exact symplectic mappings depending continuously on
a parameter which will be the basic technical tool in the proof of the
main results. This theorem is close to Theorem 1.1 in \cite{kn:Po}.
First
we consider a continuous deformation of bounded strictly convex domains
$\Omega_{s},\ s \in [0,b],\ b>0$, in ${\bf R}^{2}$, with smooth
boundaries
$\partial\Omega_{s}$ of length $\ell_s$, and introduce action-angle
coordinates for the approximated interpolating Hamiltonians of the
corresponding billiard ball maps $B_{s}$.
\par
Performing a suitable change of the variables in ${\bf R}^{2}$, we consider
$\Omega_{s},\ s \in [0,b]$, as a Riemannian manifold with a base
$\Omega = \Omega_{0}$ and metric $g_{s}$ depending continuously on $s$.
The boundary $\partial\Omega_{s}$ is given by $\partial\Omega$ equipped
with the induced metric $g^{0}_{s}$. The corresponding billiard ball map
$B_{s}$ is defined in the same manner as in Section 2. Its phase
space coincides with the coball bundle
$$\Sigma_{s} = \{(x,\xi) \in T^{*}\partial\Omega :\
g^{0}_{s}(x,\xi) \leq 1\}.$$
Let us
denote
by $\partial\Sigma_{s}^{+}$ one of the two components of the boundary of
$\Sigma_{s}$. Since $\partial\Omega_{s}$ is strictly geodesically
convex, $B_{s}$ can be written as a small perturbation of a completely
integrable map as follows (see \cite{kn:MM}): there exists
a smooth function $\zeta_{s}$
called approximated interpolating Hamiltonian which
defines
$\partial\Sigma_{s}^{+}$ $(\zeta_{s} = 0\ {\rm and}\ \nabla\zeta _{s}
\neq 0$
on $\partial \Sigma_{s}$), $\zeta_{s} \geq 0$ on
$\Sigma_{s}$, and such that in any local coordinates $\Gr =
(x,\xi)$ in a local chart $U$ in $T^{*}\partial\Omega$ we have
\begin{eqnarray}
B_{s}(\varrho) & = &
\exp(-2\zeta_{s}(\varrho)^{1/2}H_{\zeta_{s}})(\varrho)
+ R_{s}(\varrho),\ \varrho\in\Sigma_{s}\cap U,
\label{eq:3.1}\\
R_{s}(\varrho) & = & O(\zeta_{s}^{\infty}(\varrho))\ {\rm at}\
\partial\Sigma_{s}^{+} \cap U .
\label{eq:3.2}
\end{eqnarray}
Here $t \longrightarrow\ \exp(tH_{\zeta_{s}})(\Gr )$ stands for the
integral curve of
the Hamiltonian vector field $H_{\zeta_{s}}$ starting at $\varrho\in
\Sigma_{s}$, $R_s$ is a continuous family of smooth functions in $U$,
and (\ref{eq:3.2}) means that
\begin{equation}
\mid\partial_{x}^{\alpha}\partial_{\xi}^{\beta}R_{s}(x,\xi)\mid \
\leq\ C_{N \alpha\beta }\, \zeta_{s}(x,\xi)^{N},
(x,\xi) \in U, \label{eq:3.3}
\end{equation}
for any indices $\alpha, \beta, N$. Moreover, the mapping
$[0,b] \ni\ s \longrightarrow\ \zeta_{s}(\cdot)\ \in\
C^{\infty}(T^{*}\partial\Omega)$ is continuous.
\par
We are going to describe action-angle coordinates for the Hamiltonian
$\zeta_{s}$. To simplify the notations we drop the index $s$. Denote
by $M_{r}$ the closed curve $\{\zeta = r\}$ in $T^{*}\partial \Omega$
where $r$ varies in a small neighborhood of the origin. For any $\Gr \in
M_r$ consider
the map ${\bf R} \ni \ t\longrightarrow \ \exp(tH_{\zeta _{s}})(\Gr ) \in\
M_{r}$ and
denote by $2\pi \Pi (r)$ its period. Let $S$ be
a section transversal to $M_{0}$ in $\Sigma$. It is equipped with local
coordinates $S\ni \Gr \rightarrow \zeta (\Gr )$.
Denote by ${\cal O}$ the discrete group in ${\bf R}\times S$
generated by
$$
{\bf R}\times S\ \ni\ (t,\zeta(\Gr)) \longrightarrow
(t + 2\pi \Pi (\zeta(\Gr)),\zeta(\Gr)).
$$
Let $({\bf R}\times S)/{\cal O}$ be the corresponding factor space. It is a
symplectic manifold,
$d\zeta \wedge dt$ is a symplectic two-form on it, and the mapping
$$
{\bf R}\times S\ \ni\ (t,\Gr) \longrightarrow
\exp(tH_{\zeta })(\Gr ) \in\ T^{*}\partial\Omega
$$
lifts to a symplectic diffeomorphism from
$({\bf R}\times S)/{\cal O}$ to a neighborhood of $M_{0}$~.
Making suitable symplectic change of the variables
$$
\theta \ = t/\Pi (\zeta),\ r = g(\zeta),
$$
in ${\bf R}\times S$ we can suppose that ${\cal O}$ is generated by
$(\theta,r) \longrightarrow\ (\theta + 2\pi ,r)$ while the symplectic
two-form becomes $d\theta \wedge dr$. It is easy to see that
the first derivative $ g'(\zeta) = - \Pi (\zeta)$ which yields
\begin{equation}
r(\zeta) = - \int_{0}^{\zeta} \Pi (t)\,dt
+ l,\ l = l(0) = \ell /2\pi . \label{eq:3.4}
\end{equation}
Denote by $\zeta (r)$ the function inverse to
$r(\zeta )$. We have obtained
symplectic coordinates $(\theta _{s}(x,\xi), r_{s}(x,\xi))$ in a
\nbd of \boun\ in \cob\ with values in ${\bf T}\times {\bf R} $ such that
\boun\ = $\{r_{s} = l _{s}\},\ l_s = \ell _s/2\pi$ and
$\Sigma _{s} \subset \{r_{s} \geq\ l _{s}\}$.
Fix $\Ge > 0$ and set $\Gamma_s = (l_s - \Ge , l_s),\ {\bf A_s} = {\bf
T}\times \Gamma$.
The exact symplectic map $B_{s}$ is
generated in this coordinates by the function
\begin{equation}
G_{s}(\theta ,r) = - \frac{4}{3}\,\zeta _{s}(r)^{3/2}
+ Q_{s}(\theta,r),\
(\theta ,r) \in\ {\bf A_s},
\label{eq:3.5}
\end{equation}
where
\begin{equation}
\zeta _{s}(l_{s}) = 0,\ \zeta _{s}' (l_{s}) = - 1 /\Pi_{s}(0) < 0,\
\partial_{\theta}^{\alpha}\partial_{r}^{\beta}Q_{s}(\theta,l_{s})
\ = 0,\ \forall\ \theta \in {\bf T},
\label{eq:3.6}
\end{equation}
for any indices $\Ga \geq 0, \beta \geq 0$, and $s \in [0,b]$.
Hereafter we say that $G_{s}$ generates the exact symplectic map $B_{s}$
in ${\bf A_s}$ if ${\rm graph}(B_s)=\{(B_s(x,\xi),(x,\xi):\ (x,\xi)
\in {\bf A}_s \}$ is parameterized by
$$
{\rm graph}(B_{s}) = \{(
\theta ,r - \frac{\partial G_{s}}{\partial \theta }(\theta,r);
\theta - \frac{\partial G_{s}}{\partial r}(\theta,r),r):\
(\theta ,r) \in\ {\bf A_s} \},
$$ where
$$
\mid\frac{\partial ^2 G_{s}}{\partial \theta \partial r}(\theta,r)\mid\
<\ 1 ,\ \forall\ (\theta ,r) \in\ {\bf A_s}.
$$
Multiplying $G_{s}$ by a cut-off
function we can suppose that it is equal to zero for $r \leq\ l_{s}
- 2\Ge /3$. From now on we denote by $B_{s}$ the corresponding modified
exact symplectic mappings. Note that $B_{s}, Q_{s}$, as well as the
exact symplectic mappings $\psi _{s}^{0}$ defined by
$$
(\psi _{s}^{0})^{-1}(x,\xi) = (\theta _{s}(x,\xi), r_{s}(x,\xi))
$$
depend continuously on $s$ in the corresponding $C^{\infty}$ semi-norms.
\par
The billiard ball map $B_s$ is a small perturbation of the completely
integrable mapping generated in ${\bf A_s}$ by $\tau ^0_s(r) = -
\frac{4}{3}\,\zeta
_{s}(r)^{3/2}$. In what follows we apply a symplectic version of the KAM
theorem to $B_s$ which is close to Theorem 1.1 in \cite{kn:Po}. In
contrast to \cite{kn:Po}, the generating function $\tau ^0_s(r)$ has
singularity at $r = l_s$.
As a consequence of (\ref{eq:2.2}) and Lemma 5.2 we easily obtain $\ell
_s = \ell _0$ in $[0,1]$. Indeed, the continuous function $[0,1]\ni s
\rightarrow \ell _s$ takes values in the set ${\cal L}(\Omega _s) =
{\cal L}(\Omega _0)$ which does not contain intervals, according to
Lemma 5.2. Hence, $\ell _s$ does not depend on $s$. To simplify the
notations we set $\Gamma = \Gamma _0$ and ${\bf A} = {\bf A}_s$.
We are going to define the Cantor set $\Theta$. Fix $\sigma
> 1,\ \mu > 0$, and for any $a > 0$ and $N \in {\bf Z}_+$ define the
Cantor set
$\Theta (a,\mu ,N)$ by the small denominator condition as follows:
$$
\Theta (a,\mu ,N) = \{\omega \in {\bf R} :\ \mid \omega k_{1} - k_{2}\mid\
\geq\
\mu a^{N}\mid k\mid ^{-\sigma }\ {\rm for\ any}\ k = (k_{1},k_{2}) \in\
{\bf Z}^{2}\backslash \{0\}\}.
$$
Fix $0 < C < 1$ and $\omega _0 > 0$, and denote
\begin{equation}
\Theta ^{*}(a,\mu ,N)\ =\ \Theta (a,\mu ,N)\cap [Ca,C^{-1}a],\ 0<
a < \omega _0.
\label{eq:3.*}
\end{equation}
Consider the Cantor subset of $[0,\infty )$ defined by
\begin{equation}
\Theta = \cup \{\Theta ^{*}(a,\mu ,N) :\
\ 0 < a \leq \Ge (\mu ,N),\ N \in {\bf Z}_+\}\cup \{0\}
\label {eq:3.7}
\end{equation}
where
$$
\Ge (\mu ,N)\ =\ \Ge _N \mu ^{2/M},\ M = N^{2} + 2N,
$$
while the positive constants $\Ge _N$ will be specified later.
This set is of a positive Lebesgue measure in {\bf R}\ and even
\begin{equation}
\Ge -{\rm meas}(\Theta \cap [0,\Ge))\ \leq\ C_{p}\Ge ^{p},\ 0 < \Ge
< \Ge _{0},
\label {eq:3.8}
\end{equation}
for any $p \geq 1$. The following theorem provides a symplectic normal
form for the family of symplectic mappings $B_{s}$ in a \nbd of
\boun.As above we assume that $B_s$ is generated by a function
$G_s$ in {\bf A} and that $B_s$ coincides with the identity
mapping in ${\bf T}\times [l_0 - \Ge ,l_0 - 2\Ge /3]$.
\begin{theo}
Let $[0,b] \ni s \longrightarrow\ B_s \in C^{\infty}({\bf A},{\bf A})$
be a continuous deformation of exact symplectic mappings. Suppose that
the corresponding generating
functions $G_{s}$ satisfy (\ref{eq:3.5}) and (\ref{eq:3.6}), and $\ell
_s = \ell _0$ for any
$s$. Then there is a Cantor set $\Theta $ defined by (\ref{eq:3.7}) with
suitable $\Ge _N > 0$ and there exist
continuous in $[0,b]$ families of exact symplectic mappings $\psi _s
\in C^{\infty}({\bf T}\times {\bf R}, {\bf T}\times {\bf R} )$ and
functions $K _s \in
C^{\infty}({\bf R}), Q_s^0 \in C^{\infty}({\bf T}\times {\bf R})$ such
that:
\par
(i) $K_s (l_0 ) = 0,\ K_{s}'(l_0) < 0$, $K_s(t) >
0$ in $\Gamma$, and the exact symplectic map
$B_s^0 = \psi _s^{-1} \circ B_s \circ \psi _s$ is generated in
${\bf A}$ by
\begin{equation}
G_s ^0 (\Gp,I) = \tau_s (I) + Q_s^0 (\Gp,I), \
\tau _s (I) = - \frac{4}{3}K_s (I)^{3/2},\ (\Gp,I) \in {\bf A},
\label{eq:3.9}
\end{equation}
where
\begin{equation}
Q_s^0 (\Gp ,I) = 0 \ {\em on}\ {\bf T} \times E_s
\label{eq:3.10}
\end{equation}
and $E_s =\{I \in \Gamma : \ \tau^{'}_{s}(I)/2\pi \in \Theta
\}$
\par
(ii) $K _s ,\ Q_s^0$, and the generating function $S_s (\theta ,I)$ of
$\psi _s$ satisfy the estimates
\begin{eqnarray}
\mid D_I^{\alpha}(K_s (I) - \zeta _s (I))\mid\ + \
\mid D_I^{\alpha}D_{\Gp}^{\beta}Q_s^0 (\Gp,I)\mid\ & \leq\ &
C_{\alpha \beta p} \mid t_0 - I\mid ^p
\label{eq:3.11}\\
\mid D_I^{\alpha}D_{\theta}^{\beta}S_s^0 (\theta,I)\mid & \ \leq\ &
C_{\alpha \beta p} \mid t_0 - I\mid ^p
\label{eq:3.12}
\end{eqnarray}
in ${\bf T} \times [l _0 - \Ge _0,l _0 + \Ge _0],\ \Ge _0 > 0$,
for any $s \in [0,b]$ and any indices $\alpha \geq 0, \ \beta \geq 0 $
and $p>0$.
\end{theo}
\par
The proof of Theorem 3.1 is given in the Appendix. First we construct
exact symplectic mappings conjugating the billiard ball maps $B_s$ in
suitable domains away from the singularity set $\{r=l_0\}$ of $B_s$
and then we patch them together using the
uniqueness of the KAM circle with a given rotation number. An important
role here plays Proposition A.1.
\par
In view of (\ref{eq:3.6}), (\ref{eq:3.11}), and the equality $\ell _s =
\ell _0 $, the frequency map
\begin{equation}
\Gamma \ni I\ \longrightarrow\ \omega = \tau _s '
(I)/2\pi \ \in \ (0,\omega _0), \ \omega _0 > 0,
\label{eq:3.13}
\end{equation}
is invertible if $\Gamma $ is sufficiently small. Denote by ${\cal J} _s
(\omega)$ the inverse map to (\ref{eq:3.13}) in $(0,\omega _0)$.
Then $E_s = {\cal J}_s (\Theta)$
and we have
\begin{equation}
{\cal J} _s (\omega)\ =\ l _0 - c _0
\omega ^2 \ +\ O(\omega ^4),\ l_0 = \ell _0 /2\pi ,\ c_0 =
\frac{1}{4}\Pi_0(0)^{-1} > 0.
\label{eq:3.14}
\end{equation}
Moreover, $
{\cal J} _s (\omega)$ can be extended to a smooth even function
in {\bf R} .
Set $\chi _s = \psi_s^0 \circ \psi _s$.
Since $E_s$ has
no isolated points, (\ref{eq:3.10}) means that $Q_s^0$ has a
zero of infinite order at ${\bf T} \times E _s$. In particular,
\begin{equation}
B_s^0(\Gp ,I)\ =\ (\Gp\ +\ \tau _s '(I),I),\ (\Gp ,I) \in
{\bf T} \times E_s , \label{eq:3.15}
\end{equation}
and
\begin{equation}
[0,b] \ni s\ \longrightarrow\ \Lambda _s (\omega) =
\chi _s ({\bf T} \times\{{\cal J} _s (\omega )\})
\label{eq:3.16}
\end{equation}
is a continuous family of invariant circles of $B _s $ with a rotation
number $\omega \in \Theta $ which accumulate at \boun =
$\chi _s ( {\bf T}
\times\{l _0 \})$ since $l _0 = {\cal J} _s (0)$.
Denote by $\Lambda _s$ the union of the invariant circles $\Lambda _s
(\omega)$, $\omega \in \Theta$, and consider the function
$$
h_s (x,\xi) = K_s (\chi _s ^{-1}(x,\xi))
$$
where $K_s$ is introduced by Theorem 3.1. Since $Q_s^0$ has a zero of
infinite order at $E_s$ and $l _0 \in E_s ,\ h_s$ is an approximated
interpolating hamiltonian of $B_s$. Thus we obtain
\begin{cor}
We have
$$
B_{s}(\varrho) =
\exp(-2h_{s}(\varrho)^{1/2}H_{h_{s}})(\varrho)
+ R_{s}(\varrho),\ \varrho\in\Sigma_{s}\cap U,
$$
in any local chart $U$ in $T^*\partial\Omega$ where
$R_{s}(\varrho)\ \in\ C^{\infty}(U)$
is continuous with respect to $s\in [0,b]$ and
$R_s$ has a zero of infinite order at $\Lambda _s \cap U$.
\end{cor}
\par
We are going to show in Proposition 4.1 that $h_s(\Gr)$ coincides with
$(3 \ell _s (\omega )/4)^{2/3}$
for any $\Gr \in \Lambda _s(\omega)$ and $\omega \in \Theta$
where $\ell _s (\omega)$ is the Lazutkin parameter of the
invariant circle $\Lambda _s (\omega)$. Note that, according to
(\ref{eq:3.8})
\begin{equation}
1 - {\rm meas}(U\cap \Lambda _s) \leq\ C_{N}({\rm meas}(U)) ^{N},
\ s\in [0,b], \label{eq:3.17}
\end{equation}
for any sufficiently small neighborhood $U$ of \boun .
\section{Caustics and Lazutkin's parameter}
\setcounter{equation}{0}
This section is devoted to the geometry of the caustics of a strictly
convex domain. Our aim is to give a simple geometric interpretation
of the function
$$
\Theta \ni \omega\ \longrightarrow\ ({\cal J} _s (\omega),
K_s({\cal J} _s (\omega))
$$
where $K_s$ was introduced by Theorem 2.1 and ${\cal J} _s$ is the
function inverse to the frequency map defined above. To simplify the
notations we drop the index $s$.
Fix $\omega \in \Theta$ and consider the invariant circle
$\Lambda (\omega)$ of $B$ and the corresponding caustic $C(\omega)$ with
a rotation number
$\omega$. Take $\Gr = (x,\xi)$ arbitrarily in $\Lambda(\omega)$.
The projections
$t \longrightarrow x + t\tilde{e}^{\pm}(\Gr)$
of the bicharacteristics of
$S^*{\bf R} ^2 = \{(y,\eta ) \in T^*{\bf R} ^2) :\ \mid \eta\mid =1 \}$
passing through $\pi ^{\pm}(\Gr)$ are tangent to the caustic
$C(\omega )$ which is smooth and strictly convex (see
\cite{kn:La}). Let $y^{\pm} = y^\pm (\Gr )$ be the corresponding
points of tangency. Denote by
$\mid x y^{\pm}\mid $ the distance between $x$ and $y^{\pm}$ and by
$\mid y^-\frown y^+\mid $ the length of the shortest arc in $C(\omega)$
connecting $y^-$ with $y^+$. Lazutkin's parameter of the caustic
$C(\omega )$ is defined by
$$
t(\omega)\ =\ \mid xy^-\mid\ +\ \mid xy^+\mid\ -\
\mid y^-\frown y^+\mid
$$
and it does not depend on the choice of $x \in \partial\Omega$ (see
\cite{kn:La}, \cite{kn:Am}). As above denote by $\ell (\omega )$ the
length of the caustic $C(\omega )$. The main result in this section is:
\begin{prop}
For any $\omega \in \Theta$ we have
\begin{equation}
{\cal J}(\omega)\ =\ \ell (\omega)/2\pi ,\
\tau ({\cal J}(\omega))\ =\ - t(\omega ).
\label{eq:4.1}
\end{equation}
\end{prop}
{\em Proof.} Consider the flow-out
$$
{\cal M} (\omega)\ =\ \{ \exp(tH_g)(\pi ^+(\Gr )):\
\Gr\in\Lambda(\omega),\ 0\leq t\leq q(\Gr )\}
$$
of $\Lambda(\omega)$ with respect to the Hamiltonian $g(y,\eta ) = \mid
\eta \mid - 1,\ (y,\eta )\in T^*{\bf R} ^2$, where $q(\Gr )$ is the time $t$
for which a point starting at $x$ and travelling with unit speed along
the ray
$t \longrightarrow x + t\tilde{e}^+(\Gr)),\ t\geq 0$, reaches
$y^+ \in C(\omega)$. Then $\cal M$ is a Lagrangian submanifold of
$T^*{\bf R}^2$ whose boundary consists of two components, namely
$\{\pi ^+(x,\xi):\ (x,\xi )\in \Lambda (\omega )\}$ and the cosphere
bundle $S^*C(\omega)\ =\ \{(y,\eta )\in T^*{\bf R}^2 :\ y\in
C(\omega),\
\mid \eta\mid = 1\}$. By Stoks' theorem
$$
\ell (\omega)\ =\ \int_{S^*C(\omega)} \eta dy\ =\
\int _{\Lambda(\omega)} \xi dx\ =\ 2\pi {\cal J}(\omega)
$$
since the map $\chi$ conjugating $B$ to its symplectic normal form given
by (\ref{eq:3.9}) and (\ref{eq:3.10})
is exact symplectic. This proves the first part
of the claim.
To prove the second equality in (\ref{eq:4.1}) we use a
symplectic trick which is due to
V. Guillemin and R. Melrose \cite{kn:GM} and Y. Colin
de Verdi\'{e}re \cite{kn:CV}. Denote by $\sigma _0 = Id\Gp $ the
symplectic
one-form in $T^*{\bf T}$. Since $B^0 = \chi ^{-1}\circ B\circ \chi$ is
an exact symplectic map with a generating function
$G^0(\Gp ,I) = \tau (I) + Q^0(\Gp ,I)$ given by Theorem 3.1, it is easy
to see that
\begin{equation}
(B^0)^*\sigma _0\ -\ \sigma _0\ =\ df
\label{eq:4.2}
\end{equation}
where
\begin{equation}
f(\Gp ,I)\ =\ I\tau '(I)\ -\ \tau (I)\ +\ F(\Gp ,I),\
(\Gp ,I)\in {\bf T} \times \Gamma,
\label{eq:4.3}
\end{equation}
and the function
$$
F(\Gp ,I)\ =\ I\frac{\partial Q^0}{\partial I}(\Gp ,I)\ -\
Q^0(\Gp ,I)
$$
has a zero of infinite order on the Cantor set ${\bf T} \times E$ in
view of
(\ref{eq:3.10}). On the other hand, (\ref{eq:2.1}) implies
\begin{equation}
B^*\sigma\ -\ \sigma\ =\ dT^0,\ \sigma\ =\ \xi dx, \label{eq:4.4}
\end{equation}
on $\Sigma$, where $T^0(x,\xi ) = T(x,y(x,\xi))=\mid x-y(x,\xi)\mid $
and $y(x,\xi)$
is the first component of $B(x,\xi)$. Since $\chi $ is exact symplectic
\begin{equation}
\chi ^* \sigma\ -\ \sigma ^0\ =\ d\Phi ,\
\Phi\in C^{\infty}(T^*{\bf T}). \label{eq:4.5}
\end{equation}
Now, (\ref{eq:4.2}), (\ref{eq:4.4}) and (\ref{eq:4.5}) yield together
the following useful equality :
\begin{equation}
f(\Gp ,I)\ =\ T^0(\chi (\Gp ,I))\ +\ \Phi (\Gp ,I)\ -\
\Phi (B^0(\Gp ,I))\ +\ C,\ (\Gp ,I)\in {\bf A}
\label{eq:4.6}
\end{equation}
where $C$ is a constant. Taking $I= l = \ell /2\pi$ we get $C=0$.
We are ready to prove the second equality in Proposition 4.1. Take
$g^0\in \Lambda (\omega )$ and consider the orbit {\cal g} of $B$
defined by $g^j = B^j g^0 = (x_j,\xi _j),\ j=0,1,...$. Denote by
$\tilde{g}^j = (\Gp ^j, {\cal J}(\omega)) =
\chi ^{-1}(g^j),\ j=0,1,...,$ the
corresponding orbit of $B^0$. For any $k\in {\bf N}$ denote by $m_k$ the
number of rotations
that a point makes moving around $\partial\Omega$ in a
counterclockwise
direction from $x_0$ to $x_k$ and passing successively through all
$x_j,\ j \geq k$. Then
\begin{equation}
t(\omega )\ =\ \lim_{k \rightarrow \infty}(\frac{1}{k} \sum_{j=0}^{k}
T^0(g^j)\ -\ \frac{m_k}{k}\ell (\omega)).
\label{eq:4.7}
\end{equation}
On the other hand, (\ref{eq:3.10}), (\ref{eq:4.3}) and (\ref{eq:4.6})
imply
$$
T^0(g^j)\ =\ 2\pi \omega {\cal J}(\omega) - \tau (
{\cal J}(\omega)) + \Phi (\Gp + 2\pi j\omega ,{\cal J}(\omega))
- \Phi (\Gp + 2\pi (j+1) \omega ,{\cal J}(\omega)).
$$
Hence, the average action on {\cal g} $= (g_0,g_1,...\ )$ is equal to
\begin{equation}
\lim_{k \rightarrow \infty}(\frac{1}{k} \sum_{j=0}^{k}
T^0(g^j))\ =\ 2\pi \omega {\cal J}(\omega) \ -\ \tau (
{\cal J}(\omega)),
\label{eq:4.8}
\end{equation}
the right hand side being just the Legendre transform of $\tau (I)/2\pi
$ times $2\pi$. Moreover,
$$
\lim_{k \rightarrow \infty}(\frac{m_k}{k})\ =\ \omega
$$
while $\ell (\omega ) = 2\pi {\cal J}(\omega)$. Now, (\ref{eq:4.7}) and
(\ref{eq:4.8}) yield together the second equality in (\ref{eq:4.1}).
This completes the proof of the proposition. $\Box$
Consider the approximated interpolating Hamiltonian $h(x,\xi ) =
K(\chi ^{-1}(x,\xi ))$ of $B$ introduced by Corollary 3.2. For $r>0$
small enough we set as in Section 3
$$
M_r\ =\ \{(x,\xi)\in \Sigma :\ h(x,\xi ) = r \}
$$
and denote
\begin{equation}
\nu (r)\ =\ \int_{M_r}^{} dt
\label{eq:4.9}
\end{equation}
where the Poisson bracket
\begin{equation}
\{h,t\}\ =\ H_ht\ =\ 1.
\label{eq:4.10}
\end{equation}
It is easy to show that the set of Taylor coefficients of $\nu (r)$
at $r=0$ is algebraically equivalent to the set of Taylor's coefficients
of $K(I)$ at $I=l$. Indeed, performing a symplectic change of the
variables $(x,\xi ) = \chi (\Gp ,I),\ (\Gp ,I)\in {\bf T} \times \Gamma
$, and using (\ref{eq:4.10}) we easily get
\begin{equation}
K'(I) \nu (K(I))\ =\ 2\pi ,\ I\in E.
\label{eq:4.11}
\end{equation}
Denote by ${\cal R}$ the function inverse to $I \rightarrow K(I)$ and
set $\tilde{E} = \{K(I):\ I\in E\}$. Then $0\in\tilde{E}$ and
(\ref{eq:4.11}) implies
\begin{equation}
\nu (r)\ =\ 2\pi {\cal R}'(r),\ r\in \tilde{E}.
\label{eq:4.11*}
\end{equation}
The Taylor coefficients of $\nu (t)$ at $t=0$, also called
integral invariants, have been investigated by Sh.~Marvizi and
R.~Melrose
\cite{kn:MM}. They are given by integrals on $\partial\Omega$ of certain
polynomials of the curvature $\kappa (x)$ of $\partial\Omega$ and its
derivatives. In particular, (4.6) in \cite{kn:MM} and (\ref{eq:4.11*})
yield together
\begin{eqnarray}
{\cal R}'(0)\ & = & -\frac{1}{\pi} \int_{0}^{\ell}
\kappa (x)^{2/3} dx\\ \label{eq:4.12}
{\cal R}''(0)\ & = & \frac{1}{2160\pi} \int_{0}^{\ell}
(9\kappa (x)^{4/3} \ +\ 8\kappa (x)^{-8/3}
\kappa '(x)^{2})dx.
\label{eq:4.13}
\end{eqnarray}
\section{Marked length spectrum and asymptitics of the average action}
\setcounter{equation}{0}
Fix $b>0$ such that $\partial\Omega _s$ is strictly convex for any
$0 \leq s \leq b$. Consider the set ${\cal G}(\Omega_s,m,n)$ of
periodic
broken geodesics $\gamma$ of $\Omega _s$ with $n$ vertices and winding
number $m \leq n/2$. The marked length spectrum ${\cal L}(\Omega_s,m,n)$
is
defined as the set of lengths of all $\gamma$ in ${\cal
G}(\Omega_s,m,n)$. The set ${\cal L}(\Omega_s,m,n)$ is compact and we
denote
$$
T_{mn}(s)\ =\ {\rm max} \, \{ t:\ t\in{\cal L}(\Omega _s,m,n)\}.
$$
Following an idea due to G.~Birkhoff we easily prove
\begin{lemma}
The set ${\cal L}(\Omega_s,m,n)$ is not empty for any $s\in [0,b]$ and
any integers $n\geq 2,\ m\leq n/2$. The function
$$
[0,b]\ni s\ \longrightarrow\ T_{mn}(s)
$$
is continuous.
\end{lemma}
{\em Proof}. As above consider $\Omega _s$ as $\Omega = \Omega _0$
equipped with a suitable Riemannian metric $\| \cdot \|_s^2$ which
depends continuously on $s$. Denote by $\theta :\ {\bf R} \rightarrow
\partial \Omega$ a smooth covering of $\partial \Omega ,\ \theta (x+1)\
=\ \theta (x),\ x\in {\bf R}$. Consider the function
$$
S_s(x_1, \ldots ,x_n)\ =\ \| \theta (x_1) -\theta (x_2)\|_s^2
\ +\ \ldots \ +\ \| \theta (x_{n}) -\theta (x_{n+1})\|_s^2, \
x_{n+1} = x_1 + m,
$$
and set
$$
M\ =\ \{(x_1, \ldots ,x_n) \in {\bf R} ^n :\ x_1 \leq x_2 \leq \ldots \leq
x_n \leq x_{n+1} = x_1 +m,\ x_{j+1}-x_j \leq 1\}.
$$
Obviously $S_s$ is a
continuous and periodic function in $M$ with a period $e=(1, \ldots
,1)$. Moreover, the factor space $M/{\bf Z}e$ is compact.
The triangle inequality shows that the set
$$
M_s^{max}\ =\ \{p \in M:\ S_s(p) = {\rm max}_M S_s\}
$$
consists only of points internal for $M$. Hence, $S_s$ is smooth on
$M_s^{max}$. Moreover, $\nabla S_ s(p) = 0$ at $p = (x_1,
\ldots
,x_n) \in M_s$ if and only if $\theta (x_1), \ldots, \theta (x_n)$
are successive vertices of a closed broken geodesic of $\Omega _s$ which
belongs to
${\cal G}(\Omega_s,m,n)$. Therefore, the set ${\cal G}(\Omega_s,m,n)$ is
not empty for any $s \in [a,b]$. Moreover,
$$
T_{mn}(s)\ =\ {\rm sup} \, \{S_s(p):\ p\in M\}
$$
is continuous in $s \in [0,b]$. $\Box$
\begin{lemma}
% Lemma 5.2
The Lebesgue measure of $\lb{\Omega_s}$ is zero.
\end{lemma}
{\em Proof}. Let $\gamma $ be a periodic broken geodesic in
${\cal G}(\Omega_s,m,n)$. Then length$(\gamma ) = S_s(p)$ for some
$p \in
M$ such that $\nabla S_s(p) = 0$. Applying Sard's theorem we obtain that
${\cal L}(\Omega_s,m,n)$ has Lebesgue measure zero which proves the
claim. $\Box$
Using (\ref{eq:2.2}), Lemma 5.1 and Lemma 5.2 we easily obtain
\begin{equation}
T_{mn}(s)\ =\ T_{mn}(0),\ s\in [0,b].
\label{eq:5.1}
\end{equation}
Indeed, according to (\ref{eq:2.2}) the continuous function $T_{mn}(s)$
takes values in $\lb {\Omega _0}$ which does not contain intervals in
view of Lemma 5.2. Hence, $T_{mn}(s)$ should be constant in $[0,b]$.
In what follows we shall evaluate the average action on the periodic
orbits of the billiard ball map. To simplify the notations we drop the
index $s$. Consider the set
$\Gamma (m,n)$ of periodic orbits $g = (g_1, \ldots , g_n)$ of $B$
of period $n$ and winding number $m$. Any such orbit gives rise to a
periodic broken geodesic in ${\cal G} (\Omega,m,n)$.
Denote by $L(g)$ the length of the periodic
broken geodesic of $\Omega $ associated with the periodic orbit $g$ of
$B$.
According to (\ref{eq:4.8}), the average action of any orbit
$\{g_0,g_1, \ldots \},\ g_j = B^j(g_0),\ g_0 \in \Lambda (\omega )$,
of $B$ on the invariant circle
$\Lambda (\omega ),\ \omega \in \Theta$, is given by the Legendre
transform
\begin{equation}
{\cal I}(\omega )\ =\ \omega {\cal J}(\omega ) \ -\ \tau (
{\cal J}(\omega ))/2\pi
\label{eq:5.2}
\end{equation}
of $\tau (I)/2\pi $. Note that ${\cal I}(\omega )$ can be extended
to a smooth odd function in {\bf R}\ since ${\cal J}(\omega )$ is
even, $\tau (I) = - \frac{4}{3}K (I)^{3/2},\ K(l_0)=0,\ K'(l_0)<0$ and
(\ref{eq:3.14}) holds. Moreover,
\begin{equation}
2\pi {\cal I}(\omega )\ =\ \omega \ell (\omega ) \ -\ t
(\omega ),\ \forall \omega \in \Theta ,
\label{eq:5.**}
\end{equation}
in view of Proposition 4.1.
Next we impose the following condition on the pair $(m,n) \in
{\bf N} ^2$
\begin{equation}
{\rm dist}(m/n,\Theta )\ \leq\ n^{-1/2}
\label{eq:5.8}
\end{equation}
where ${\rm dist}(z,\Theta )$ is the distance between $z$ and $\Theta $.
Here is the main result in this section:
\begin{theo}
For any $(m,n) \in
{\bf N}^2$ satisfying (\ref{eq:5.8}) and any $g \in \Gamma (m,n)$ we have
\begin{equation}
\mid L(g)/n\ -\ {\cal I}(m/n )\mid\ \leq\
C_N n^{-N},\ \forall\ N>0,
\label{eq:5.9}
\end{equation}
where $C_N$ depends only on $N$ and $\Omega $.
\end{theo}
\par
{\em Proof of Theorem 2.1}. Fix $\omega \in \Theta$ and choose a
sequence $(m_j,n_j) \in {\bf N} ^2, j \in {\bf N} $, such that
$$
\mid \omega - m_j/n_j \mid \ \leq \ n_j^{-1/2},\ j=1,2, \ldots \ .
$$
For any $j \in {\bf N} $ pick a periodic orbit
$g_s^j \in \Gamma _s(m_j,n_j)$ of $B_s$ such that
$$
L_{s}(g_s^j)\ =\ {\rm max}\, \{t:\ t\in {\cal L}(\Omega _s,m_j,n_j)\}\
=\ T_{m_j n_j}(s),\ s \in [0,b].
$$
Theorem 5.1 yields
$$
{\cal I}_s(\omega )\ =\ \lim_{j \rightarrow \infty }
(T_{m_j n_j}(s)/n_j),\ s \in [0,b],
$$
and taking into account (\ref{eq:5.1}) we obtain
$$
{\cal I}_s(\omega )\ =\ {\cal I}_0(\omega ),\ \forall \omega \in
\Theta,\ \forall s \in [0,b].
$$
Since $\Theta$ has no isolated points, differentiating the last
equality with respect to $\omega $ we obtain
$$
{\cal J}_s(\omega )\ =\ {\cal J}_0(\omega ),\
K_s({\cal J}_0(\omega ))\ =\ K_0({\cal J}_0(\omega )),\
\forall \omega \in \Theta,\ \forall s \in [0,b],
$$
which proves (\ref{eq:2.5}). In particular,
\begin{equation}
E_s = E_0,\ K_s(I) = K_0(I),\ \forall\ I \in E_0,
\label{eq:5.5}
\end{equation}
and using Theorem 3.1 we prove (ii), Theorem 2.1 for $s \in [0,b]$.
\par
It remains to show that $\Omega _s$ is strictly convex for any $s \in
[0,1]$. Suppose that $\Omega _s$ is strictly convex for $s C_1 > 0, \ s \in [0,b_0),
$$
for suitable $x_s \in \partial \Omega _s$. Then $f_s(x_s) \leq C_2,\ s
\in [0,b_0)$, and using Taylor's formula and (\ref{eq:5.7}) we obtain
the estimate
$$
\int_{\partial \Omega _s} (\mid f_s(x) \mid ^2 \ +\
\mid f'_s(x) \mid ^2) dx \ \leq\ C_3,\
s \in [0,b_0),
$$
which means that $\{f_s:\ s\in [0,b_0)\}$ is a compact subset of
$L^2(\partial \Omega)$ (we regard $\partial\Omega _s$ as
$\partial\Omega$ equipped with a suitable Riemannian metric). In
particular, $f_{b_0} \in L^2(\partial\Omega)$. On the other hand, the
curvature $k_{b_0}(x) \geq 0$ and it has a zero of at least second
order at a point $x_0 \in \partial\Omega$. Hence,
$$
\mid f_{b_0}(x) \mid \ \geq\ C\mid x-x_0\mid ^{-2/3}
$$
in any local coordinates in a \nbd of $x_0$ in $\partial\Omega$ which
implies $f_{b_0} \not \! \in L^2(\partial\Omega )$. Hence,
$\partial\Omega
_s$ is strictly convex for any $s \in [0,1]$. The proof of Theorem 2.1
is complete. $\Box$
{\em Proof of Theorem 5.1}.
The proof is based on a suitable approximation
of $(B^0)^j(\Gp ,I),\ j\leq n$, where $B^0$ is introduced by Theorem
3.1.
Fix $\Ge _0 > 0$ and consider a \nbd $V = \{ \Gr \in \Sigma
:\ 0 \leq h(\Gr ) \leq \Ge _0\}$ of\ \boun \ in $\Sigma$,
where $h$ is a continuous family of approximated interpolating
Hamiltonians of $B$ introduced by Corollary 3.2.
There exists $\Ge
>0$ such that if $m/n < \Ge $ and $(g_1, \ldots ,g_n) \in \Gamma (m,n)
$, then $g_j \in V$ for each
$j \leq n$. Indeed, denote by $t$ the maximal length of the segments
with end points $x_j$ and $x_{j+1},\ j=1, \ldots ,n$, where $g_j =
(x_j,\xi _j),\ x_{n+1} =
x_1$. Then $t< m \ell _0 /n < \Ge $ which implies $g_j \in V$ if $\Ge
$ is sufficiently small.
Let $\omega
\in \Theta \subset [0,\omega _0],\ \omega _0 < \Ge _0$
and $m/n \leq \Ge _0$.
We have $g_j \in V,\ j \leq n$, for any periodic
orbit $g = (g_1, \ldots , g_n) \in \Gamma (m,n)$.
Let $\tilde{g} = (\tilde{g}_1, \ldots ,
\tilde{g}_n),\ \tilde{g} = (\Gp _j,I_j) = \chi ^{-1}(g_j)$, where $\chi
$ has been introduced in Section 3. Then $\tilde{g}$ is a periodic
orbit of $B^0$
of period $n$ and winding number $m$, and $\tilde{g} _j \in {\bf T}
\times
\Gamma, \ \Gamma = (l _0 - \Ge,l _0)$. According to Theorem 3.1,
the map $B^0$ has the form
\begin{equation}
B^0 (\Gp ,I) = (\Gp + \tau '(I) + Q^0_1(\Gp ,I), I+ Q^0_2(\Gp ,I))
\label{eq:5.*}
\end{equation}
where $Q^0_1$ and $Q^0_2$ have a zero of infinite order at ${\bf T}
\times E$ and
$E = {\cal J}(\Theta ) \subset [l_0 -\Ge , l_0] $.
Consider the sets
$$
V_n = \{ \xi \in (\frac{1}{2n},\omega _0):\ {\rm dist}(\xi , \Theta )
\leq
2 n^{-1/2}\},\ J_n = {\cal J}(V_n).
$$
There exists $C_0>0$ such that
\begin{equation}
{\rm dist}(I,E)\ \leq\ C_0 n^{-1/2} \ {\rm and}\ l _0 -I
\geq C_0^{-1}n^{-2},\ \forall I\in V_n,\ \forall n\in {\bf Z}_+ ,
\label{eq:5.100}
\end{equation}
where $E={\cal J}(\Theta )$.
\begin{prop}
For any integers $\Ga \geq 0,\ \beta \geq 0$, and $N \geq 1$,
there exists a constant $C_{N \Ga \beta }$ such that
\begin{equation}
\mid \partial ^{\Ga}_{\Gp}\partial ^{\beta}_I ((B^0)^j(\Gp ,I)
- (\Gp + j\tau
' (I),I))\mid\ \leq\ C_{N \Ga \beta }\, n^{-N},\ 1 \leq j \leq n,\
\forall n \in {\bf N},
\label{eq:5.110}
\end{equation}
in $(\Gp , I)\in {\bf T} \times J_n $.
\end{prop}
{\em Proof}. Set $U(\Gp ,\xi ) = (\Gp ,{\cal J}(\xi )),\ (\Gp ,\xi )
\in {\bf T} \times {\bf R}$, and consider the map $B = U^{-1}\circ B^0
\circ U$ in ${\bf T}\times (0,\omega _0)$. We have
$$
B(\Gp ,\xi ) = (\Gp + 2\pi \xi , \xi ) + R(\Gp ,\xi )
$$
where $R=(R_1,R_2)$ can be extended as a smooth mapping in ${\bf T}
\times {\bf R}$ through $\xi = 0$. Indeed, we have
$$
R_1(\Gp ,\xi ) = Q_1^0(\Gp ,{\cal J}(\xi )),\
R_2(\Gp ,\xi ) = \frac{1}{2\pi}\tau '({\cal J}(\xi ) +
Q_2^0(\Gp ,{\cal J}(\xi )) - \xi
$$
where $Q_j^0,\ j=1,2$, are given by (\ref{eq:5.*}), they are smooth in
${\bf T}\times {\bf R}$, and have a zero of infinite order at ${\bf T}
\times E$. On the other hand, the singularity of $\tau '$ at $I = l$
is described by
$$
\tau '(I) = - 2K(I)^{1/2}K'(I),\ K(l)=0,\ K'(l) < 0,
$$
and we prove easily that $R_j$ are smooth at $\xi = 0$
as $l \in E$.
Moreover,
\begin{equation}
\mid \partial ^{\Ga}_{\Gp}\partial ^{\beta}_{\xi}R_j(\Gp ,\xi )\mid\
\leq\ C_{N\Ga \beta}\, n^{-3N},\ j=1,2,
\label{eq:5.10}
\end{equation}
for any $(\Gp ,\xi ) \in {\bf T}\times {\bf R}$ such that ${\rm
dist}(\Theta ,\xi )\leq 3n^{-1/2}$. Set
$$
B^j(\Gp,\xi ) = (\Phi _j (\Gp ,\xi ),\Xi _j(\Gp ,\xi )),\ (\Gp ,\xi )\in
V_n,\ j\leq n.
$$
First we prove by induction with respect to $j\leq n$ the inequalities
$$
\mid \Xi _j - \xi \mid \leq jn^{-4},\
\mid \Phi _j - \Gp - 2\pi j \xi \mid \leq \pi j^2n^{-4},
$$
\begin{equation}
{\rm dist}(\Xi _j,\Theta )\leq 2n^{-1/2} + jn^{-4} < 3n^{-1/2},\
j\leq n,
\label{eq:5.11}
\end{equation}
for any $(\Gp ,\xi )\in {\bf T} \times V_n$ and $n\geq n_1$ where
$n_1$ is sufficiently large. In the same way, making use the
third
inequality in (\ref{eq:5.11}) as well as (\ref{eq:5.10}) we obtain
$$
\mid \partial ^{\Ga}_{\Gp}\partial ^{\beta}_{\xi}(\Xi _j(\Gp ,\xi ) -
\xi) \mid\ +\
\mid \partial ^{\Ga}_{\Gp}\partial ^\beta _{\xi}(\Phi _j(\Gp ,\xi ) -
\Gp - 2\pi j \xi )\mid\ \leq\ C_{N \Ga \beta}\, n^{-3N},\ j\leq n,\
n \geq 1,
$$
in $(\Gp ,\xi )\in {\bf T} \times V_n$
for any nonnegative integers $\Ga ,\, \beta $, and $N$.
Conjugating $B$ with $U$
and using the estimate
$$
\mid \partial _I^{\Ga} \tau (I)\mid \ \leq\ C_\Ga n^{2\Ga},\ I \in
J_n,
$$
which follows from (\ref{eq:5.100}) we complete the proof of Proposition
5.1. $\Box$
Consider the set $W_n$ of all $(\Gp ', \Gp ) \in {\bf
R} ^2$ such that
$$
( \Gp ' - \Gp )/ 2\pi n\ \in\ V_n.
$$
Set
$$
P^0(\Gp ,I) = (B^0)^n(\Gp ,I) = (\Gp + n \tau '(I),I) + Q(\Gp ,I)
$$
where $Q=(Q_1,Q_2)$ is a smooth function in ${\bf T}\times {\bf
R}$.
Using Proposition 5.1 we solve the equation
$$
\Gp ' = \Gp + n\tau ' (I) + Q_1(\Gp ,I)
$$
with respect to $I \in J_n$ when $(\Gp ',\Gp)\in W_n$ and $n$ is
sufficiently large. This equation is equivalent to
\begin{equation}
I = {\cal J}((\Gp ' - \Gp )/2\pi n) + Q_3(\Gp ',\Gp ,I),\
I \in J_n,\ (\Gp ',\Gp)\in W_n
\label{eq:5.12}
\end{equation}
where
$$
\mid \partial _{\Gp '} ^\Ga \partial _\Gp ^\beta \partial _I^p
Q_3(\Gp ',\Gp ,I)\mid\
\leq\ C_{N \Ga \beta p}\, n^{-N},\ (\Ga ,\beta ,p) \in {\bf Z}_+^3,
$$
in $W_n\times J_n$.
Hence, (\ref{eq:5.12}) can be solved by successive iterates for $n \geq
n_0$ and $n_0$ sufficiently large.
Denote by
$I(\Gp ', \Gp ),\ (\Gp ',\Gp )\in W_n,\ n\geq n_0$, the solution of
(\ref{eq:5.12}). Then
\begin{equation}
\mid I(\Gp ',\Gp ) - {\cal J}((\Gp ' - \Gp )/2\pi n)\mid\ \leq\
C_N \, n^{-N},\ (\Gp ',\Gp ) \in W_n.
\label{eq:5.13}
\end{equation}
In particular, ${\rm graph}(P^0)$ can be
parameterized over ${\bf T} \times J_n $ by
$(\Gp ',\Gp) \in W_n$ as follows:
\begin{equation}
{\rm graph}(P^0)\ =\ \{(\Gp ',\frac{\partial H_n}{\partial\Gp '}(\Gp
',\Gp
),
\Gp , - \frac{\partial H_n}{\partial\Gp }(\Gp ',\Gp )):\
(\Gp ',\Gp ) \in W_n\}
\label{eq:5.14}
\end{equation}
where $H_n$ is a smooth and $2\pi$ - periodic function function on ${\bf
R}^2$ satisfying the equality
$$\frac{\partial H_n}{\partial\Gp}(\Gp ',\Gp ) = I(\Gp ',\Gp ).$$
Choose $m \in {\bf N}$ and suppose that (\ref{eq:5.8}) holds for the pair
$(m,n) \in {\bf N}^2,\ n \geq n_0$. Then $( \Gp + 2\pi m,\Gp)\in W_n$ for any
$\Gp \in {\bf R}$. Set
\begin{equation}
h(\Gp )\ =\ h_{mn}(\Gp )\ =\ H_n(\Gp + 2\pi m, \Gp ),\ \Gp \in {\bf R}.
\label{eq:5.15}
\end{equation}
The function $h(\Gp)$ is smooth and $2\pi$-periodic in {\bf R}.
According to (\ref{eq:5.14}) there is one - one correspondence between
the critical points of $h_{mn}$ in ${\bf T}$ and the fixed points of
$P^0$ in ${\bf T}\times \Gamma$ which is given by
\begin{equation}
{\rm Crit}(h_{mn}) \ni \Gp\ \longrightarrow\ g_1(\Gp ) =
(\Gp ,I(\Gp + 2\pi m, \Gp )) \in {\rm Fix}(P^0).
\label{eq:5.16}
\end{equation}
Then
$$
{\rm Crit}(h_{mn}) \ni \Gp\ \longrightarrow\ g(\Gp ) = (g_1(\Gp ),\ldots
, g_n(\Gp )) \in \Gamma (m,n),
$$
$$
g_j(\Gp )=B^{j-1}g_1(\Gp ),\ j=1,\ldots ,n,
$$
is a one - one correspondence between the critical points of $h_{mn}$
and the periodic orbits of $B$ in $\Gamma
(m,n)$, and (\ref{eq:5.13}) implies
\begin{equation}
\mid I - {\cal J}(m/n)\mid\ \leq\
C_N \, n^{-N},
\label{eq:5.21}
\end{equation}
at any periodic point $(\Gp ,I)$ of $B^0$ of period $n$ and winding
number $m$.
Take $g = (g_1, \ldots ,g_n) \in \Gamma (m,n)$ and denote as before
$(\Gp _j,I_j) = \chi ^{-1}(g_j)$.
Using (\ref{eq:4.6}) we obtain
$$
\mid L(g)/n - {\cal J}(m/n)\mid\
\leq\ \frac{1}{n}\sum_{j=1}^{n} \mid f(\Gp _j,I_j) - {\cal I}(m/n)\mid .
$$
On the other hand,
$$
\mid f(\Gp _j,I_j)\ -\ {\cal I}(m/n)\mid\
\leq\ \mid \tau (I_j)\ -\ \tau ({\cal J}(m/n))\mid \\ +\
\mid I_j\tau '(I_j)\ -\ {\cal J}(m/n)\tau '({\cal J}(m/n))\mid\ +\
C_Nn^{-N}.
$$
We evaluate the right hand side of the inequality above using Taylor's
formula. We have $\mid \tau '(I)\mid \leq C$ in $\Gamma$. Moreover,
$$
\tau '(I_j) - \tau '({\cal J}(m/n))\ =\ (I_j - {\cal J}(m/n))\tau
''(\tilde{I}_j)
$$ where $$
\mid \tilde{I}_j\ -\ {\cal J}(m/n) \mid\ \leq\
\mid I_j - {\cal J}(m/n) \mid\ \leq\ C_Nn^{-N}
$$
according to (\ref{eq:5.21}). Then
$$
\mid \tau ''(\tilde{I}_j)\mid\ \leq\ \mid K (\tilde{I}_j)^{-1/2}
K'(\tilde{I}_j)\mid\ +\ C\ \leq\ Cn^2.
$$
Hence,
$\mid f(\Gp _j,I_j) - {\cal I}(m/n)\mid
\leq\ C_Nn^{-N}$
which completes the proof of Theorem 5.1. $\Box$
The function $h$ introduced by (\ref{eq:5.15}) is going to play an
important role in Section~6. Note that it is uniquely determined by
(\ref{eq:5.14}) and (\ref{eq:5.15})
up to a constant and we normalize it
by taking $H_n(\Gp
_0, \Gp _0 + 2\pi m) = L(g(\Gp _0))$ where $\Gp _0$ is a point in
${\rm Crit}(h_{mn})$. Then we obtain
\begin{lemma}
We have
$$
h_{mn}(\Gp )\ =\ L(g(\Gp )),\ \forall \Gp \in {\rm Crit}(h_{mn}).
$$
\end{lemma}
{\em Proof}. Taking into account (\ref{eq:4.4}) we get
$$
(B^n)^*(\sigma ) - \sigma\ =\ dT_n,\ T_n = \sum_{j=0}^{n}(B^j)^*T^0.
$$
On the other hand,
$$
(P^0)^*(Id\Gp ) - Id\Gp \ =\ dH_n,\ P^0 = (\chi ^{-1}\circ B \circ
\chi)^n,
$$
in ${\bf T}\times J_n$, and
$$
\chi ^*(\sigma ) - Id\Gp \ =\ d\Phi,\ \Phi \in C^{\infty}({\bf T}\times
{\bf R}),
$$
since $\chi $ is exact symplectic. Set $T^0_n(\Gp ,I) = T_n(\chi (\Gp
,I))$.
Taking into account the equalities above as well as the normalization of
$H_n$ we easily obtain
\begin{equation}
H_n(\Gp ',\Gp ) = T^0_n(\Gp , I(\Gp ',\Gp )) - \Phi (P
(\Gp , I(\Gp ',\Gp )) ) + \Phi (\Gp , I(\Gp ',\Gp )) ,\
(\Gp ',\Gp )\in W_n.
\label{eq:5.17}
\end{equation}
Using (\ref{eq:5.15}) and
(\ref{eq:5.16}) we complete the proof of the lemma. $\Box$
The following result is a generalization of Theorem 5.15 in \cite{kn:MM}
\begin{cor}
Fix $m\in {\bf Z}_+$ and pick arbitrarily $g_{mn} \in \Gamma (m,n)$.
Then
$$
\mid L(g_{mn})\ - \ \sum_{k=0}^{N} c_{mk} n^{-2k} \mid\ \leq\
C_{Nm} n^{-2N-2}
$$
where
$$
c_{mk}\ =\ \frac{m^{2k+1}}{(2k+1)!} {\cal I}^{(2k+1)}(0),\ k=0,1,... .
$$
\end{cor}
Since $0 \in \Theta$, the pair $(m,n)$ satisfies
(\ref{eq:5.8}) if $m$ is fixed and $n > n_1(m)$.
Let us expand ${\cal I}(t)$ in Taylor series at $t=0$.
The derivatives
${\cal I}^{(2k)}(0) = 0$ since the function ${\cal I}(t)$ is odd.
Applying Theorem 5.1 we prove the assertion.
{\em Remark 5.1}.
Since $0 \in \Theta $,
we can write the coefficients $c_{mk}$ explicitly in terms of the
Taylor
series of $t=0$ of the restriction of the function ${\cal I}(\omega)$ on
$\Theta$,
which is given by (\ref{eq:5.**}).
Moreover, the relation between $c_{mk}$ and
the integral invariants of Marvizi and Melrose is
explicitly given by (\ref{eq:4.11}) and (\ref{eq:5.2}).
Consider two strictly convex domains $\Omega _1$ and $\Omega _2$.
Let $B_j,\ j=1,2$ be the corresponding billiard ball maps. Choosing the
constant $\omega _0 $ in (\ref{eq:3.6}) sufficiently small we obtain for
any $\omega \in \Theta$ an invariant curve $\Lambda _j(\omega )$ of
$B_j,\ j=1,2$. The following statement is a discrete version of Theorem
2.1.
\begin{theo}
Let $\Omega _j,\ j=1,2$ be strictly convex domains in ${\bf R} ^2$ and
$${\cal L}(\Omega _1,m,n) \ =\ {\cal L}(\Omega _2,m,n),\ \forall (m,n),\
m\leq n/2.$$ Then
$$ \ell _1(\omega )\ =\ \ell _2(\omega ),\
t _1(\omega )\ =\ t _2(\omega ), \forall\ \omega \in \Theta, $$
and there exists an exact symplectic mapping $\chi :\Sigma _1
\rightarrow \Sigma _2$ such that
$$
\chi (\Lambda _1(\omega ))\ =\ \Lambda _2(\omega )\ {\it and}\
\chi\circ B_1\ =\ B_2 \circ\chi\ {\it on}\ \Lambda _1 (\omega),\
\forall\
\omega\in\Theta.
$$
\end{theo}
Theorem 5.2 follows from Theorem 5.1 and the arguments at the end of the
proof of Theorem 2.1.
\section{Spectral invariants}
\setcounter{equation}{0}
Let $\Omega $ be a bounded domain in $R^2$
with a smooth boundary $\partial \Omega$. Consider the Laplacian $\Delta
$ in
$\Omega$ with Dirichlet (Neumann) boundary conditions, and the related
distribution $Z(t)$ defined by (\ref{eq:1.5}).
Denote by ${\cal G}(\Omega )$ the set of all periodic generalized
geodesics of $\Omega $ and by
${\cal G}(\Omega ;m,n)$ the set of the periodic broken geodesics in
$\Omega $ corresponding to the periodic orbits of $B$ in $\Gamma
(m,n)$. Let ${\cal L}_{mn}(\Omega )$ be the set of lengths of all
periodic broken geodesics in ${\cal G}(\Omega )\backslash {\cal G}
(\Omega,m,n)$. In order to prevent cancellation of the singularity
of $Z(t)$ created
by the geodesics of maximal length $T_{mn}$ in ${\cal G}(\Omega )$,
we impose the following condition
\begin{equation}
T_{mn} \not\!\in\ {\cal L}_{mn}(\Omega ).
\label{eq:6.3}
\end{equation}
This condition is satisfied for generic domains $\Omega $ (see
\cite{kn:PS}).
\begin{theo}
Let $\Omega $ be a strictly convex bounded domain in ${\bf R}^2$ with a
smooth boundary. Choose $(m,n) \in {\bf N} ^2$ such that ${\rm
dist}(m/n,\Theta)\leq n^{-1/2}$ and suppose
(\ref{eq:6.3}) to be fulfilled. Then
\begin{equation}
T_{mn} \in {\rm sing.supp.}(Z)
\label{eq:6.4}
\end{equation}
holds if $n \geq n_0$ and $n_0 = n_0(\Omega )$ is sufficiently large.
\end{theo}
This statement generalizes Theorem 6.4 in \cite{kn:MM} where it has
been proved for $m=1$ and $n$ sufficiently large. The main idea in
\cite{kn:MM} is to find suitable representation of $Z(t)$ in a \nbd of
$T_{mn}$ as a Fourier integral on $\partial \Omega $ (see Proposition
6.11 in \cite{kn:MM}) and then to apply a result of Soga
\cite{kn:So}. When $m$ is fixed and
$n\geq n_1(m)$ is sufficiently large
Proposition 6.11 in \cite{kn:MM} still holds.
In the general case when $(m,n)$ satisfies (\ref{eq:5.8}) and $n \geq
n_0(\Omega )$, we use another representation of $Z(t)$ in a \nbd of
$T_{mn}$ which is close to that obtained in Proposition 5.4,
\cite{kn:Po}. Consider the function $h(\Gp ) = h_{mn}(\Gp ), \Gp \in
{\bf T}$, defined by (\ref{eq:5.15}). For any $z \in {\bf C}$ denote by
$\Re z$ its real part.
\begin{prop}
Suppose that $(m,n)$ satisfies (\ref{eq:5.8}) and $n \geq n_0$. If $n_0
= n_0(\Omega )$ is sufficiently large, we have
$$
Z(t)\ =\ \exp (i\pi \mu _n) \int _0^{\infty}\, \int _{T}\, \Re (\exp
(i \tau (t-h(\Gp ))) a(\Gp ,\tau )) d\Gp d\tau\ +\ \tilde{Z}(t)
$$
where $\tilde{Z}(t)$ is smooth at $t = T_{mn}$, $\mu _n$ is a Maslov's
index, and $a(\Gp ,\tau )$ is a classical symbol of order one with
respect to $\tau $. Moreover, $a(\Gp ,\tau )=0$ for $\tau \leq 1$ and
the principal part of $a$ is equal to $a_1(\Gp )\tau$ for $\tau \geq 2$
where $a_1 > 0$ on {\bf T}.
\end{prop}
Applying Lemma 5.5 in \cite{kn:Po} to the oscillatory integral given by
Proposition 6.1 we prove Theorem 6.1.
{\em Proof of Proposition 6.1.} Consider the fundamental solution
$E(t,x,y)$ of the mixed problem
$$
(D_t^2\ -\ D_x^2)E\ =\ 0,\ E_{\mid x\in\partial\Omega}\ =\ 0,\
E(0,x,y)\ =\ \delta (x-y),\ (D_t E)(0,x,y)\ =\ 0,
$$
where $D_t = -i\partial /\partial t,\ D_x^2 = D_{x_1}^2 + D_{x_2}^2,\
D_{x_j} = -i\partial /\partial x_j,\ j=1,2$. The distribution $E$ is
just the kernel of the operator $\cos (t \sqrt{\Delta })$.
Denote by $E^\pm$ the Schwartz kernel of the operator
$\exp (\mp it \sqrt{\Delta })$ and consider
$$
Z^\pm (t)\ =\ \int_{\Omega } E^\pm (t,x,x)dx.
$$
Then, $Z(t)=Z^+(t) + Z^-(t)$ and $Z^- = \overline{Z^+}$ in a
distribution sense where $\overline{z}$ is the complex conjugated number
to $z\in {\bf C}$. According to (6.14), \cite{kn:MM}, we have
$$
Z^+ (t)\ =\ \int_{\partial\Omega } K^+ (t,x,x)dx\ +\ Z_1^+(t)
$$
where $Z_1^+$ is smooth at $T_{mn}$ and $K^+$ belongs to the
H\"{o}rmander's class $I^{-1/4}({\bf R} \times \partial\Omega \times \partial
\Omega ; C'_+ )$ of Lagrangian distributions associated with the
Lagrangian manifold
$$
C'_+ = \{(t,x,y;\tau ,\xi ,\eta )\in T^*(
{\bf R} \times \partial\Omega \times \partial\Omega ):\ t = T_n(y,-\eta
/\tau
),\ (x, \xi/\tau ) = B^n(y,-\eta/\tau ),\ \tau > 0 \}
$$
where $T_n(y,\eta)$ is introduced in Lemma 5.3.
Choose neighborhoods $V_1 \subset V_2 \subset C'_+$ of the set
$$
C_{mn} = \{(t,x,y;\tau ,\xi ,\eta )\in C'_+;\ (y,-\eta /\tau ) \in
\Gamma (m,n)\}.
$$
Without loss of generality we can suppose that the complete symbol of
$K_+$ vanishes outside $V_2$ while its principal symbol is a positive
function in $V_1$ modulo a Maslov's factor. Denote by $C_0'$ a
Lagrangian submanifold of
$T^*(\partial\Omega \times \partial\Omega )$ associated with the graph
of $B^n$
$$
C_0' = \{(x,y;\xi ,\eta )\in
T^*(\partial\Omega \times \partial\Omega );\
(x, \xi ) = B^n(y,-\eta ),\ (y,-\eta )\in \Sigma \},
$$
and let $V_1^0 \subset V_2^0 \subset C'_0$ be neighborhoods of the set
$$
\{(x,y; \xi ,\eta )\in C'_0;\ (y,-\eta ) \in \Gamma
(m,n)\}.
$$
As in Lemma 5.7, \cite{kn:Po}, we find a Fourier oscillatory integral
$R(x,y,\tau )$ of the class
$I^0(\partial\Omega \times \partial\Omega ; C'_0,\tau )$ such that
$$
K_+(t,x,y)\ = \int e^{-it\tau } R(x,y,\tau )d\tau.
$$
Since $B^n$ is an
exact symplectic mapping, the Liouville class of $C_0'$ in
$H^1(C_0';{\bf R} )$ given by the restriction of the canonical symplectic one
form of $T^*(\partial\Omega \times \partial\Omega )$ on $C_0'$ is
trivial. In this case there is a complete analogy between the theory of
the Lagrangian distributions and the Fourier oscillatory integrals (see
\cite{kn:Du}). The only difference
is that the principal symbol
of a Fourier oscillatory integral associated with $C_0'$ has an
additional Liouville factor $\exp( i\tau f(\Gr )),\ \Gr \in C_0'$, where
$f$ is given by the restrictions on $C_0'$ of
suitable phase functions generating $C_0'$. In our case, the principal
symbol of $R$ is equal to $\exp(i\tau T_n(y,-\eta ) + i\pi \mu _n)$
times
a positive function in $V_1^0$, the complete symbol of $R$ vanishes
outside $V_2^0$ and $R(x,y,\tau ) = O_N(\tau ^{-N})$ as $\tau
\rightarrow
- \infty $ for any $N>0$.
Denote by $R(\tau )$ the Fourier integral operator with a large
parameter $\tau $ whose Schwartz kernel is $R(x,y,\tau )$. Using Theorem
3.1 (here $s$ is fixed) we shall conjugate $R(\tau )$ to a Fourier
integral operator on {\bf T} with a large parameter $\tau $ and phase
function $h_{mn}(\Gp )$. Denote by $C_1'$ the Lagrangian manifold
$$
C_1' = \{(x,\Gp;\xi ,I )\in
T^*(\partial\Omega \times {\bf T};\
(x,\xi ) = \chi (\Gp, -I ) \}.
$$
and observe that the Maslov bundle of $C_1'$ is trivial over ${\bf Z}_4$
since the projection
$$
C_1' \ni (x,\Gp;\xi ,I ) \longrightarrow (x,\Gp) \in
\partial\Omega \times {\bf T}
$$
is a diffeomorphism. Indeed, we can replace $\chi$ by $\chi\circ\psi$
where $\psi (\Gp ,I) = (\Gp +CI,I),\ C>0$. Then
$$
(\chi\circ\psi )^{-1}(x,\xi )\ =\ (\Phi (x,\xi ) + C\Xi (x,\xi ), \Xi
(x,\xi )),\ \partial\Xi /\partial\xi \neq 0\ {\rm on}\ {\bf T}\times
\{l\},
$$
and we can solve the equation $\Gp = \Phi (x,\xi ) + C\Xi (x,\xi )$ with
respect to $\xi $ if $C$ is sufficiently large.
As in \cite{kn:Po}, Lemma 5.8, (see also \cite{kn:CV1}),
we easily obtain
\begin{lemma}
Let $\Psi (\tau )$ be a pseudodifferential operator with a large
parameter
$\tau$ with symbol equal to one in ${\bf T}\times V_n$ and equal to zero
outside a \nbd of this set in ${\bf T}\times \Gamma$. There exists a
Fourier integral operator $A(\tau )$ of the class $I^0(
\partial \Omega \times {\bf T}, C_1',\tau)$ such that
$$
A^*(\tau )A(\tau )\ =\ \Psi (\tau ),
$$
the principal symbol of $A(\tau )$ is equal to one on the lifting of
${\bf T}\times V_n$ in $C_1'$ and the complete symbol of $A(\tau )$
vanishes outside a small \nbd of it.
\end{lemma}
The operator
$$
R_1(\tau )\ =\ A^*(\tau )R(\tau )A(\tau ):L^2({\bf T}) \
\longrightarrow\ L^2({\bf T})
$$
has a distribution kernel $R_1(\Gp ',\Gp ,\tau )$ in
$I^0({\bf T}\times {\bf T}, C_2',\tau)$ where
$$
C_2' = \{(\Gp ',\Gp ;I' ,I )\in
T^* \, {\bf T}^2;\ (\Gp ',I' ) = P^0(\Gp, -I ) \},\ P^0 = (B^0)^n.
$$
According to (\ref{eq:5.14}), $C_2'$ is generated by $H_n(\Gp ',\Gp )
+ C
,\ (\Gp ',\Gp )\in W_n$ where $C$ is constant. Hence we get
$$
R_1(\Gp ',\Gp ,\tau )\ =\ \exp (i\tau (H_n(\Gp ',\Gp ) + C) +i\pi \mu
_n)
\, b(\Gp ',\Gp ,\tau )
$$
where $b = 0$ for $\tau < 1$ and
$b(\Gp ',\Gp ,\tau ) = b_1(\Gp ',\Gp ) \tau + b_0(\Gp ',\Gp )
+ \ldots $ is a classical symbol of order one as $\tau \rightarrow
+\infty$,
$b=0$ outside a \nbd of $W_n$, and $b_0$ is a positive function in
$W_n$. On the other hand, comparing the Liouville factors of $R(x,y,\tau
)$ and $R_1(\Gp ',\Gp ,\tau )$ on $C_{mn}$, we obtain as in
Section 5.4 in \cite{kn:Po} and Section 3.2 in \cite{kn:CP} that $C=0$.
The $L^2$-trace of $R(\tau )$ is equal to
$$
{\rm trace}\, R(\tau )\ =\ {\rm trace}\, R_1(\tau )\ =\
\int_{\bf T} \exp (i\tau h(\Gp )+i\pi \mu _n)
\, b(\Gp + 2\pi m,\Gp ,\tau )\, d \Gp\, ,
$$
which completes the proof of Proposition 6.1. $\Box$
Using Theorem 6.1 and certain arguments form Section 5 we prove that the
invariant circles $\Lambda _s(\omega )$ and the restriction of the
billiard ball map on them are spectral invariants of the Laplacian for
suitable continuous deformations of a strictly convex domain. Define
${\cal T}_{mn}(s)$ as $T_{mn}(s)$ when $\Gamma
_s(m,n)$ is not empty and set ${\cal T}_{mn}(s) = 0$ otherwise.
Fix $n_1 > 0$. We say that the deformation (\ref{eq:2.*})
satisfies the condition $({\cal R})$, if for any $(m,n) \in {\bf
N} ^2$ fixed
so that
$$
{\rm dist}(m/n,\Theta )\ \leq\ n^{-1/2},\ n\geq n_0,
$$
the relation
$$
{\cal T}_{mn}(s)\ \not\!\in\ {\cal L}_{mn}(\Omega _s)
$$
is fulfilled for $s$ in a dense subset of $[0,1]$. Note that by
definition
${\cal T}_{mn}(s)$ does not belong to the length spectrum of $\Omega _s$
if the set $\Gamma _s(m,n)$ is empty. Moreover, using arguments from
\cite{kn:PS} it could be proved that $({\cal R})$ is
generic for continuous deformations of the domain.
\begin{theo}
Let $\Omega _0$ be a strictly convex bounded domain with a smooth
boundary.
Suppose that (\ref{eq:2.*}) is a continuous deformation of $\Omega _0$
satisfying $({\cal R})$ and that
\begin{equation}
{\rm Spec}(\Delta _s)\ =\ {\rm Spec}(\Delta _0),\ 0 \leq s \leq 1.
\label{eq:6.5}
\end{equation}
Then:
\par
(i) $\Omega_{s}$ is strictly convex for any $s \in
[0,1]$,
\par
(ii) there exists a continuous family of smooth exact
symplectic mappings
$$\chi_{s}: T^{*}\partial \Omega_{0} \longrightarrow
T^{*}\partial \Omega_{s}$$
such that
$$
\chi_{s}(\Lambda_{0}(\omega ))\ =\ \Lambda_{s}(\omega ),\
{\it and}\
\chi_{s} \circ B_{0} = B_{s}\circ\chi_{s}\
{\it on}\ \Lambda_{0}(\omega)
$$
for any $\omega \in \Theta_{0}$ and any $s \in [0,1]$,
\par
(iii) for any $\omega \in \Theta_{0}$ there exists a
continuous family of caustics $[0,1] \longrightarrow C_{s}(\omega)$
in $\Omega_{s}$ and
$$
\ell_{s}(\omega) = \ell_{0}(\omega),\
t_{s}(\omega) = t_{s}(\omega),\ s \in [0,1].
$$
\end{theo}
{\em Proof}. Take $b>0$ such that $\Omega_s$ are strictly convex for any
$s\in [0,b]$. Then ${\cal T}_{mn}(s) = T_{mn}(s)$ is continuous in
$[0,b]$. Fix $\omega \in \Theta $ and suppose that the pair $(m,n)\in
{\bf N}$ satisfies (\ref{eq:5.9}). We are going to prove that
\begin{equation}
T_{mn}(s) \ \in \ {\rm sing.supp.}Z_0,\ \forall\ s\in [0,b].
\label{eq:6.6}
\end{equation}
Take $s_0\in [0,b]$ and choose a sequence $s_j$ tending to $s_0$ such
that $T_{mn}(s_j) \not\!\in {\cal L}_{mn}(\Omega _{s_j})$. Theorem 6.1
implies
$$
T_{mn}(s_j) \in {\rm sing.supp.}Z_{s_j}.
$$
On the other hand, $Z_s(t) \equiv Z_0(t)$ for any $s$ in view of
(\ref{eq:6.5}) and we get (\ref{eq:6.6}) since sing.supp.$Z_0$ is a
closed set and $T_{mn}(s) $ is continuous. Using the Poisson relation
(\ref{eq:1.6}) we obtain
$$
T_{mn}(s) \in {\cal L}(\Omega _0),\ \forall\ s\in [0,b].
$$
Hence,
$$
T_{mn}(s) \ =\ T_{mn}(0),\ \forall\ s \in [0,b],
$$
and as in Theorem 2.1 we complete the
proof of Theorem 6.2. $\Box$
\appendix
\section*{Appendix}
\setcounter{equation}{0}
\newtheorem{theoa}{Theorem A}[section]
\newtheorem{lemmaa}{Lemma A}[section]
\newtheorem{propa}{Proposition A}[section]
\newtheorem{rema}{Remark A}[section]
\newtheorem{coro}{Corollary A}[section]
\renewcommand{\theequation}{A.\arabic{equation}}
We are going to prove Theorem 3.1. As in Section 3 we fix $N \in {\bf Z
}_+$ and $ \mu > 0$, and denote
$$
\Ge (\mu ,N) = \Ge _N \mu ^{2/M},\ M = N^{2} + 2N,
$$
where the positive constants $\Ge _N$ will be specified later.
Fix $0> 1$ - sufficiently large, set
$\Gamma _a =[l _0 - C_1 a^2,l _0 - C_1^{-1}a^2]$, and denote
$$
\Theta ^0(a,\mu ,N) = \Theta (a,\mu ,N) \cap (C_0 a,C_0^{-1}a),\ 0
< a < \omega _0,
$$
where $C$ is fixed in (\ref{eq:3.*}). The following result is a
counterpart of Theorem 1.1, \cite{kn:Po}:
\begin{theoa}
Let $[0,b] \ni s \longrightarrow\ B_s \in C^{\infty}({\bf A},{\bf
A})$ be a
continuous deformation of exact symplectic mappings satisfying the
assumptions of Theorem 3.1 for any $s$ in $[0,b]$.
Then for any $N \in {\bf Z}_+ $ there is $\Ge _N > 0$,
and for any $a \in (0,\Ge (\mu ,N))$ there exist
continuous in $s \in [0,b]$ families of exact symplectic mappings $\chi
_{sa}
\in C^{\infty}({\bf T}\times {\bf R}, {\bf T}\times {\bf R} )$
and functions $K_{sa} \in
C^{\infty}({\bf R}), Q_{sa} \in C^{\infty}({\bf T}\times {\bf R})$ such that
$K_{sa}(I) > 0$ in $\Gamma$ and:
\par
(i) the exact symplectic mappings
$B_{sa}^0 = \chi ^{-1}_{sa} \circ B_s \circ \chi _{sa}$ is generated in
${\bf A}$ by
\begin{equation}
G_{sa}^0 (\Gp,I) = \tau_{sa} (I) + Q_{sa}^0 (\Gp,I), \
\tau _{sa}(I) = - \frac{4}{3}K_{sa} (I)^{3/2},\ (\Gp,I) \in {\bf
A},
\label{eq:A.1}
\end{equation}
and
\begin{equation}
Q_{sa}^0 (\Gp ,I) = 0 \ {\em on}\ {\bf T} \times E_{sa}
\label{eq:A.2}
\end{equation}
where $E_{sa} =\{I \in \Gamma : \ \tau^{'}_{sa}(I)/2\pi \in \Theta
^0(a,\mu,N)\} \subset \Gamma _a$,
\par
(ii) $\chi _{sa} = 0$ outside ${\bf T} \times \Gamma _a$
, and $K_{sa},\ Q_{sa}^0$, and the
generating function $S_{sa} (\theta ,I)$ of $\chi _{sa}$ satisfy
the estimates
\begin{eqnarray}
\mid D_I^{\alpha}(K_{sa} (I) - \zeta _s (I))\mid +
\mid D_I^{\alpha}D_{\Gp}^{\beta}Q_{sa}^0 (\Gp,I)\mid & \leq &
C_{\beta N} \mid t_0 - I\mid ^{N-\alpha}
\label{eq:A.3}\\
\mid D_I^{\alpha}D_{\theta}^{\beta}S_{sa}^0 (\theta,I)\mid & \leq &
C_{\beta N} \mid t_0 - I\mid ^{N-\alpha}
\label{eq:A.4}
\end{eqnarray}
in ${\bf T} \times [l _0 - \Ge _0,l _0 + \Ge _0],\ \Ge _0 > 0$,
for $s \in [0,b]$ and any indices $0 \leq \alpha \leq N, \ \beta
\geq 0$, where $C_{\beta N}$ depend neither on $a$ nor on $s$.
\end{theoa}
We point out that $K_{sa},\ Q_{sa},\ \chi _{sa}$, as well as the Cantor
set
$E_{sa}$
in Theorem A.1 depend on the parameters $a$ and $s$ but the constants
$C_{\beta N}$ in (\ref{eq:A.3}) and (\ref{eq:A.4}) are independent of
$a$ and $s$.
The proof of Theorem A.1 is close to that of Theorem 1.1 in
\cite{kn:Po}
and we are going only to sketch it. It is based on Theorem A,
\cite{kn:Poe} and
on an idea of R. Douady \cite{kn:Do} (see also Appendix, \cite{kn:Po})
to transform the initial problem for symplectic mappings to a similar
problem for Hamiltonian systems.
First we write the generating function $G_s(\theta ,r)$ of $B_s$ in the
form
\begin{equation}
G_{s}(\theta ,r) = - \frac{4}{3}\,\zeta _{s}^0(r)^{3/2}
+ R_{s}(\theta,r),\
(\theta ,r) \in\ {\bf A},
\label{eq:A.5}
\end{equation}
where $\zeta _{s}^0(r)$ is the Taylor polynomial of $\zeta _{s}(r)$ at
$r=l _0$ up to order $M_1 = N^2 +4N+4$ while $R_s$ satisfies
\begin{equation}
\mid\partial_{\theta}^{\alpha}\partial_{r}^{\beta}R_{s}(\theta,r)
\mid\ \leq\ C\mid r - l _0 \mid ^{M_1 -\beta},\
(\theta ,r) \in {\bf A},
\label{eq:A.6}
\end{equation}
for any indices $\Ga \geq 0, \beta \geq 0$, and $s \in [0,b]$, and $R_s$
depends continuously on $s$ in $C^{\infty}({\bf A})$. As in Section 3,
we suppose that $R_s(\theta,r) = 0$ for $r \in [l_0 -\Ge ,l_0 -
2\Ge/3]$. Fix $d= a^2$ and set
$${\bf D} = \Gamma \times (-\Ge _0,\Ge_0),\
{\bf D}_d = (l_0 - C_2d,l _0 - C_2^{-1}d) \times (-C_2d,C_2d)
$$
where $C_2 > C_1 > 1$. Define $\tilde{{\bf D}}_d$ the same way as
${\bf D}_d$ with
a constant $C_3 > C_2$ and set
$$
\tilde{\bf A} = {\bf T} ^2 \times {\bf D} ,\
\tilde{\bf A}_d = {\bf T} ^2 \times \tilde{{\bf D} }_d.
$$
Denote $y=(y_1,y_2) \in {\bf T} ^2,\ \eta = (\eta _1,\ \eta _2)
\in {\bf D}
,\ y_1
=\theta,\ \eta _1 = r$ and
$\tau _s^0(\eta ) = - \frac{4}{3}\,\zeta _{s}^0(\eta _1)^{3/2}$. As in
\cite{kn:Do}, \cite{kn:Po}, we first construct a Hamiltonian
$\tilde{H}_s(y,\eta)$ in $\tilde{{\bf A}}$ close to $\tilde{H}_s^0(\eta )
= 2\pi \eta
_2 + \tau _s^0(\eta _1)$ and such that the Poincar\'{e} map
corresponding to the Hamiltonian flow $F_s^t(\Gr)$ of $\tilde{H}_s$ on
the level surface $\{\tilde{H}_s = 0\}$ coincides with $B_s$. Set
$$
{\bf A'}\ =\ \{(y,\eta )\in \tilde{\bf A}:\ \tilde{H}_s(\eta ) = 0,\
y_2=0\}
$$
and denote by $\imath _s: {\bf A} \longrightarrow {\bf A'}$ the
inclusion map
$$
\imath _s(y_1,\eta _1)\ =\ (y_1,0,\eta _1,-\tau _s^0(\eta _1)/2\pi ).
$$
Taking into account (\ref{eq:A.5}) and (\ref{eq:A.6}) we prove as in
\cite{kn:KP} and \cite{kn:Po}
\begin{lemmaa}
There exists a continuous in $[0,b]$
family of Hamiltonians $\tilde{H}_s \in C^{\infty}(\tilde{{\bf
A}})$
such that
$$
\mid D_y^{\alpha} D_{\eta}^{\beta} (\tilde{H}_s(y,\eta )\ -\
\tilde{H}_s^0(\eta ))\mid \ \leq\ C_{\alpha\beta}
\mid \eta _1\mid ^{M_1-\mid\beta\mid},
$$
$$
\tilde{H}_s(y,\eta )\ =\ H^0_s(\eta )\ {\it in\ a\ neighborhood\ of}\
{\bf A'} {\it as\ well\ as\ outside}\ {\bf \tilde{A}}_d ,
$$
$$
B_s\ =\ \imath _s^{-1} \circ F_s^1 \circ \imath _s.
$$
\end{lemmaa}
Set $H_s^0 = \tilde{H}_s^0 + (\tilde{H}_s^0)^2,\
H_s ^{\prime} = \tilde{H}_s + (\tilde{H}_s)^2$. Next we apply a KAM
theorem to the pair $H_s^0,\ H_s^{\prime}$, which is a variant of
Theorem A, \cite{kn:Poe}, proved by J. P\"{o}schel (see also Theorem 5.4
in
\cite{kn:KP}). Fix $p > M_1$. As in \cite{kn:Poe} denote by $\sigma
_{\gamma}
(y,\eta)$ the map $(y,\eta) \longrightarrow (y,\gamma \eta)$ and by
$\norm{\cdot}{p,\tilde{A}_d}$ the respective H\"{o}lder norms of
the functions in $\tilde{A}_d$ as well as
$$
\norm{f}{p,\tilde{A}_d;\gamma}\ =\
\norm{f\circ \sigma_{\gamma}}{p,\sigma _{\gamma}^{-1}(\tilde{A}_d)}
$$
Denote by $\tilde{\Gamma}$ the intersection of a \nbd of the point $\{(l
_0,0)\}$ in {\bf C} with
$\{\eta \in {\bf C}:\ \Re\, \eta _1 < l _0 \}$, and set $d=a^2$ and
$$
{\bf D}_d + \Gr\ =\ \{z \in \tilde{\Gamma}:\ \mid z-\eta \mid \leq
\Gr\ {\rm for\ some}\ \eta \in \tilde{{\bf D}} _d\}
$$
\par
Next fix $\sigma
> 1,\ \mu > 0$, as in Section 3, set $\gamma _d = \mu d^{N/2} = \mu
a^N$, and consider the Cantor set
$$
\Theta ^d = \{\omega \in {\bf R}^2 :\ \mid <\omega ,k> \mid\ \geq\
\gamma _d \mid k\mid ^{-\sigma }\ {\rm for \ any}\ k = (k_{1},k_{2})
\in\
{\bf Z} ^{2}\backslash \{0\}\}.
$$
Fix $s_0 \in [0,b]$. The following KAM theorem is a variant of Theorem A
in \cite{kn:Poe}.
\begin{theoa}
Let $[0,b] \ni s \longrightarrow H_s^0(\eta )$ be a continuous family of
analytic functions in $\tilde{\Gamma}$ such that
\begin{equation}
\norm{\partial H_s^0/\partial \eta ^2}{{\bf D} _{d+\Gr}}\ ,\
\norm{((\partial H_s^0/\partial \eta ^2)^{-1}}{{\bf D} _{d+\Gr}}\
\leq\ C d^{-1/2},\ d \in (0,d_0),
\label{eq:A.7}
\end{equation}
where $\Gr = cd$, $00$, and assume the map
$\partial H_s^{\prime}/\partial \eta :\ \tilde{\Gamma} \rightarrow
{\bf C}^2$ to be invertible.
\par
For any fixed $\lambda > \sigma +1 > 2$, and $\alpha > 1,\ \alpha
\not\!\in \Lambda = \{i/\lambda + j :\ i,j \geq 0\ {\it integer}\}$,
there is a positive $\Ge$ independent of $d$ and $s$ such
that if
$H_s^{\prime} \in C^{\infty}(\tilde{{\bf A}})$ is continuous in $[0,b]$, and
\begin{equation}
\norm{H_s^{\prime} - H_s^0}{p,\tilde{{\bf A}}_d;\gamma _d}\ \leq\
\gamma _d^2 d^2 \Ge ,\ p = \alpha \lambda + \lambda + \sigma ,
\label{A.8}
\end{equation}
then:
\par
(i) for any $d \in (0,d_0)$ there is a neighborhood $U(s_0)$ of
$s_0$ and continuous with respect to $s \in U(s_0)$
families of functions $S_{sd} \in C^{\infty}({\bf \tilde{A}}),\
H_{sd} \in C^{\infty}({\bf R}^2)$
such that $\tilde{S}_{sd}=0$ outside ${\bf \tilde{A}}_d$ and
$$
\partial ^q_{\xi}(H_s^{\prime}(y,\xi - \nabla _y \tilde{S}_{sd}(y,\xi
))\ - \ H_{sd}(\xi ))\ =\ 0\ ,\ 0 \leq q \leq 1,
$$
on ${\bf T}^2 \times \tilde{E}_{sd}$ where
$$
\tilde{E}_{sd}\ =\ \{\xi \in {\bf D} _{d} :\ \nabla _{\xi}H_{sd}
(\xi )/2\pi \in \Theta ^d \},
$$
\par
(ii) for any $\beta \geq \alpha$,
$$
\norm{\tilde{S}_{sd}}{\tilde{\beta},{\tilde{\bf A}}_{d};\gamma _d}\
\leq\ C_{\beta }\gamma_d^{-1} d^{-(\beta +1)/2}
\norm{H_s^{\prime} - H_s^0}{\tilde p}
$$
where $\tilde{p} = \beta \lambda + \lambda + \sigma$ and
$\tilde{\beta} = \beta -(\lambda - \sigma )/\lambda $ is not in
$\Lambda $.
\end{theoa}
The proof of Theorem A.2 is similar to that of Theorem A in
\cite{kn:Poe} (see also Theorem 5.4 and the Appendix in \cite{kn:KP})
and we omit it. The continuity of $\tilde{S}_{sd}$ and $H_{sd}$
with respect to $s \in U(s_0)$ in the
corresponding $C^{\infty}$ spaces follows
from the arguments in A.2, Appendix, in \cite{kn:Po}.
Consider the pair $H_s^0 ,\ H_s ^{\prime}$ defined above. Obviously,
$H_s^0$ satisfies (\ref{eq:A.7}) if $c$ is sufficiently small. Fix
$\lambda > \sigma +1$ and $\beta > 0$ such that
$\tilde{\beta} = \beta -(\lambda - \sigma )/\lambda $ is not in
$\Lambda $ and $2N-1 < \tilde{\beta} \leq 2N$. Set
$p = \beta \lambda + \lambda + \sigma$.
Lemma A.2 implies
$$
\norm{H_s^{\prime} - H_s^0}{p,\tilde{\bf A}_d;\gamma _d}\ \leq\
C_N a^{M_1}\ \leq \ \gamma _d^{2} d^{2}\Ge ,\ M_1 = N^2+4N+4,
$$
if $d=a^2,\ a \leq \Ge_N \mu ^{2/M},\ M=N^2+2N$, and $\Ge _N =
(\Ge/C_N)^{1/M}$ is
sufficiently small. Hence, we can apply Theorem A.2. The corresponding
function $\tilde{S}_{sd}$ satisfies the estimate
$$
\norm{\tilde{S}_{sd}}{\tilde{\beta},{\tilde{\bf A}}_{d};\gamma _d}\
\leq\ C\gamma_d^{N}
$$
which implies
\begin{equation}
\mid\partial_{\theta}^p \partial_{\xi}^{q} \tilde{S}_{sd}
(\theta ,\xi)\mid\ \leq\
C_N a^{N}\ ,\ \mid q \mid \leq N.
\label{eq:A.9}
\end{equation}
Moreover, $\tilde{S}_{sd} = 0$ outside a \nbd of ${\bf T}\times
{\bf D}_d$ in
$\tilde{\bf A}_d$ and it
generates an exact symplectic transformation if $d$ is sufficiently
small. As in Appendix, \cite{kn:Po} we complete the proof of Theorem
A.1
We are going to patch together the exact symplectic mappings $\chi
_{sa}$. Fix $N_j \in {\bf Z}_+$ and
$a_j \in (0,\Ge (\mu,M_j)),\ M_j = N_j^2 + 2 N_j,\ j=1,2$,
and consider the corresponding functions $K_{sj}$ and $S_{sj}$ given by
Theorem A.1. Let $\chi _{sj}$ be the exact symplectic mapping
with a generating function $S_ {sj}
$. Denote by ${\cal J}_{sj}(\omega )$ the
inverse to the frequency mapping $\Gamma \ni I \rightarrow \tau
_{sj}'(I)/2\pi $.
\begin{propa}
We have
\begin{equation}
{\cal J}_{s1}(\omega ) \ =\ {\cal J}_{s2}(\omega ),\
K_{s1}({\cal J}_{s1}(\omega ))\ =\ K_{s2}({\cal J}_{s1}(\omega ))
\label{eq:A.10}
\end{equation}
\begin{equation}
\chi _ {s1}
(\theta,{\cal J}_{s1}(\omega )) \ =\ \chi _{s2}(\theta +r({\cal
J}_{s1}(\omega)),\, {\cal J}_{s1}(\omega )),
\label{eq:A.11}
\end{equation}
for any $\omega \in \Theta^{1,2}:=
\Theta ^0(a_1,N_1)\cap \Theta ^0(a_2,N_2)$ where
$ r_{sj}(I)=S_{s1}(0,I)-S_{s2}(0,I)$.
\end{propa}
{\em Proof}. The proof is close to that of Proposition A.5 in
\cite{kn:Po}. To simplify the notations we drop the index $s$.
Suppose that $\Theta^{1,2}$ is not empty. Set $B_j = \chi_j^{-1}\circ B
\circ \chi _j$ and consider $\psi = \chi_1^{-1}\circ\chi_2$. Then $B_2 =
\psi^{-1}\circ B_1\circ \psi$ and (\ref{eq:4.2}) implies
\begin{equation}
B_j^*\sigma_0 \ -\ \sigma_0\ =\ df_j,\ \sigma_0 = Id\Gp,\ j=1,2,
\label{eq:A.12}
\end{equation}
where
\begin{equation}
f_j(\Gp,{\cal J}_j(\omega)) \ =\ 2\pi {\cal I}_j(\omega),\
\omega \in \Theta ^0(a_j,N_j),\ j=1,2,
\label{eq:A.13}
\end{equation}
according to (\ref{eq:4.3}). On the other hand,
\begin{equation}
\psi^*\sigma_0 \ -\ \sigma_0\ =\ d\phi
\label{eq:A.14}
\end{equation}
where $\phi$ is a smooth function in {\bf A}. As in the proof of
(\ref{eq:4.6}) we deduce from (\ref{eq:A.12}) and (\ref{eq:A.14}) that
\begin{equation}
f_2(\Gp,I) \ =\ f_1(\psi (\Gp,I)) + \phi (\Gp,I) - \phi (B_2(\Gp,I)) +
C,\ (\Gp,I)\in {\bf A},
\label{eq:A.15}
\end{equation}
where $C$ is a constant. As the invariant circles $\Lambda(\omega)$ are
uniquely determined by their rotation numbers $\omega \in \Theta$, we
have
\begin{equation}
\psi({\bf T}\times{\cal J}_2(\omega))\ =\
{\bf T}\times{\cal J}_1(\omega),\ \omega \in \Theta^{1,2}.
\label{eq:A.16}
\end{equation}
Using (\ref{eq:A.13}) and (\ref{eq:A.16}) we obtain
$$
2\pi{\cal J}_2(\omega) \ =\ 2\pi{\cal J}_1(\omega) +
\phi (\Gp,{\cal J}_2(\omega)) - \phi
(\Gp + \omega , {\cal J}_2(\omega))
+ C,\ (\Gp,\omega)\in {\bf T}\times \Theta^{1,2},
$$
which implies
$$
2\pi{\cal J}_2(\omega) \ =\ 2\pi{\cal J}_1(\omega) + C,\
\omega\in \Theta^{1,2}.
$$
Differentiating the last equality with respect to $\omega$ we prove
(\ref{eq:A.10}).
According to (\ref{eq:A.10}) and (\ref{eq:A.16}) we have
$$
\psi({\bf T}\times{\cal J}_2(\omega))\ =\
{\bf T}\times{\cal J}_2(\omega),\ \omega \in \Theta^{1,2},
$$
and we obtain
$$
\nabla_\theta S_1(\theta , {\cal J}_2(\omega))\ =\
\nabla_\theta S_2(\theta , {\cal J}_2(\omega)),\
(\theta,\omega)\in {\bf T}\times \Theta^{1,2}.
$$
This proves (\ref{eq:A.11}). $\Box$
{\em Proof of Theorem 3.1}. First we fix $N$ in ${\bf Z}_+$. Take $a_j
=C_0^j,\ \gamma
_j=\mu_0a_j^M,\ M=N^2+2N,\ \mu _0 <\mu$, and denote by $K_{sj}(I)$ and
$S_
{sj}
(\theta ,I)$ the corresponding functions given by Theorem A.1 for
$a=a_j,\ j=1,2,\ldots $. Replacing $S_{sj}(\theta ,I)$ by
$S_{sj}(\theta ,I)-S_ {sj}
(0 ,I)$ we can suppose that $S_{sj}(0,I)=0$. Using
Proposition A.1, (\ref{eq:A.3}) and (\ref{eq:A.4}), and applying a
suitable Whitney extension theorem we find smooth functions ${\cal
J}_ s^N
(\omega ),\ K_ s^N(I)$ and $ S_ s^N(\theta ,I)$ in $\Gamma $ and ${\bf
T}\times
\Gamma $ respectively such that
$$
{\cal J}_ s^N(\omega ) \ =\ {\cal J}_{sj}(\omega ),\
K_ s^N({\cal J}_s^N(\omega ))\ =\ K_{sj}({\cal J}_s^N(\omega ))
$$
$$
S_ s^N(\theta,{\cal J}_s^N(\omega )) \ =\ S _{sj}(\theta
,\, {\cal J}_s^N(\omega )),
$$
for any $(\theta ,\omega )\in {\bf A}^j \equiv {\bf T}\times (\Theta
^0(a_j,\mu _0,N)\cap
[a_{j+1},a_j)]$. Denote by $\chi _ s^N$ the exact symplectic mapping
generated by $S_ s^N$ and by $(\Gp ,I)$ the corresponding symplectic
coordinates in ${\bf T}\times {\bf R}$. Then $B^0_s = (\chi
_s^N )^{-1}\circ B_s\circ
\chi _ s^N$ is generated by $\tau _s^N(I) + Q
_s^N
(\Gp ,I),\ \tau _s^N(I) = -
\frac{4}{3}K _ s^N
(I)^{3/2}$ where $Q_s^N(\Gp ,I)$ is a smooth function and
$Q_s^N(\Gp ,{\cal J}_s^N(\omega ))$
has a zero of
infinite order on each ${\bf A}^j,\ j=1,2,... $ . Fix
$\mu _0 = \mu C_0^N$ and take $a \in [a_{j+1},a_j],\ d=a^2$.
Then
$\gamma _j < \gamma _{j-1} < \gamma _d$ and the Cantor set $\Theta
^*(a,\mu ,N)$ is contained in the union of the sets
${\bf A}^p,\ p=j-1,j$. Hence, $Q^0(\Gp
,{\cal J}_s^N(\omega ))$ has a zero of infinite order on
$$
\Theta _N = \cup \{ \Theta ^*(a,\mu ,N):\ 0 N_1 \geq 1$ pick $a$ in $(0,\Ge (\mu ,N_2)]$ and set
$\gamma _j = \mu a ^{N_j}$. Then $\gamma _2 < \gamma _1$ and we obtain
$$
\Theta _{N_1}\cap (0,\Ge (\mu ,N_2)] \subset \Theta _{N_2}.
$$
Using Proposition A.1 as well as (\ref{eq:A.3}) and
(\ref{eq:A.4})
we obtain smooth functions
(in the sense of Whitney)
${\cal J}_ s (\omega ),\
K_ s({\cal J}_ s (\omega )
)$ and $ S_ s(\theta ,{\cal J}_ s (\omega ))$ in $\Theta $ and ${\bf
T}\times \Theta$ respectively such that
$$
{\cal J}_ s(\omega ) \ =\ {\cal J}_s^N(\omega ),\
K_ s({\cal J}_s(\omega ))\ =\ K_s^N({\cal J}_s(\omega ))
$$
$$
S_ s(\theta,{\cal J}_s(\omega )) \ =\ S _s^N(\theta
,\, {\cal J}_s(\omega )),
$$
for any $(\theta ,\omega )\in {\bf T}\times \Theta $. Denote by
$\tilde{S}_s(\theta,\omega)$ and $\tilde{\cal J}_s(\omega)$
suitable smooth Whitney extensions of
$\tilde{S}_s(\theta,{\cal J}_s(\omega)),\
(\theta,\omega)\in {\bf T}\times \Theta$ and ${\cal J}_s(\omega),\
\omega\in\Theta$, which depend continuously on $s$. Let
$\tilde{K}'_s(I)$ be the
function inverse to $\omega \rightarrow {\cal J}_s(\omega)$. Taking
$\chi _s$ to be the exact symplectic mapping generated by
$S_s(\theta ,I) = \tilde{S}_s(\theta,\tilde{K}'_s(I))$
and using (\ref{eq:A.3}) and (\ref{eq:A.4}) we
complete the proof of Theorem 3.1. $\Box$
{\em Acknowledgments}. This paper has been written under the support of
the Alexander von Humboldt foundation.
The main part of it was prepared during my stay in TH Darmstadt
and I would
like to thank Hans-Dieter Alber for the hospitality and the stimulating
discussions. I would like to thank J\"{u}rgen P\"{o}schel for the
helpful discussions about KAM theory.
\begin{thebibliography}{99}
\bibitem{kn:Am} Amiran, E.: Caustics and evolutes for convex planar
domains, J. Diff. Geometry, {\bf 28}, 2, 345-358(1988).
\bibitem{kn:AM} Anderson K, Melrose, R.: The propagation of
singularities along gliding rays. Invent. Math. {\bf 41}, 197-232
(1977).
\bibitem{kn:CP} Cardoso, F., Popov, G.: Rayleigh quasimodes in linear
elasticity. Comm. Partial Diff. Equations (to appear).
\bibitem{kn:CV} Colin de Verdi\`{e}re, Y.: Sur les longueurs des
trajectoires p\'{e}riodiques d'un billiard. In: Dazord P., Desolneux -
Moulis N. (eds.) G\'{e}om\'{e}trie Symplectique et de Contact: Autour
du Th\'{e}or\`{e}me de Poincar\'{e}-Birkhoff. Travaux en Cours,
S\'{e}m. Sud-Rhodanien de G\'{e}om\'{e}trie III, pp. 122-139, Paris:
Herman 1984.
\bibitem{kn:CV1} Colin de Verdi\`ere, Y.: Quasi-modes sur les
varietes Riemanniennes. Invent. Math. {\bf 43}, 15 - 52(1977).
\bibitem{kn:Do} Douady, R.: Une d\'{e}monstration directe de
l'\'{e}quivalence des th\'{e}or\`{e}mes de tores invariants
pour diff\'{e}omorphismes et champs de vecteurs. C.R.Acad.Sci.Paris,
Ser.A. {\bf 295}, 201-204(1982).
\bibitem{kn:Du} Duistermaat, J.: Oscillatory integrals, Lagrange
immersions and infolding of singularities. Comm. Pure Appl.
Math.{\bf 27}, 207-281 (1974).
\bibitem{kn:FG} Fran\c{c}oise, J.P., Guillemin, V.: On the period
spectrum of a symplectic map. Journal of Functional Analysis. {\bf 100},
317-358 (1991).
\bibitem{kn:GM} Guillemin, V., Melrose R.: A cohomological invariant of
discrete dynamical systems,
\bibitem{kn:GM1} Guillemin, V., Melrose R.: The Poisson summation
formula for manifolds with boundary, Advances in Math. {\bf 32},
204-232(1979).
\bibitem{kn:Ho} H\"{o}rmander, L.: The Analysis of Linear Partial
Differential Operators III, IV. Berlin - Heidelberg - New York:
Springer 1985.
\bibitem{kn:Kac} Kac M.: Can one hear the shape of a drum. Amer. Math.
Soc. Monthly. {\bf 73}, 4, Part II 1-23 (1966).
\bibitem{kn:KP} Kovachev, V., Popov, G.: Invariant tori for the
billiard ball map. Trans. Amer. Math. Soc. {\bf 317}, 45-81(1990).
\bibitem{kn:La} Lazutkin, V.: The existence of caustics for a
billiard problem in a convex domain. Math. USSR Izvestija. {\bf 7},
185-214(1973).
\bibitem{kn:LMM} de la Llave, R., Marco, J., Moriy\'{o}n. R.:
Canonical perturbation of Anosov systems and regularity results
for the Livsic cohomology equations. Annals of Math. {\bf 123},
537-611(1986).
\bibitem{kn:MM} Marvizi, Sh., Melrose, R.: Spectral invariants
of convex planar regions. J.Differ.Geom. {\bf 17}, 475-502(1982).
\bibitem{kn:Mo} Moser, J.: On invariant curves of area preserving
mappings of an annulus. Nachr. Akad. Wiss. G\"{o}tingen Math. Phys.
1-20 (1962).
\bibitem{kn:PS} Petkov, V., Stojanov, L.: Geometry of reflecting
rays and inverse spectral results. John Wiley \& Sons 1992.
\bibitem{kn:Poe} P\"{o}schel, J.: Integrability of Hamiltonian
systems on Cantor sets. Comm.Pure Appl.Math. {\bf 35}, 653-696(1982).
\bibitem{kn:Po1} Popov, G.: Glancing hypersurfaces and length
spectrum invariants. Preprint (1990).
\bibitem{kn:Po} Popov, G.: Length spectrum invariants of
Riemannian manifolds. Mathematische Zeitschrift (to
appear).
\bibitem{kn:So} Soga, H.: Oscillatory integrals with degenerate
stationary points and their application to the scattering theory,
Comm. Partial Diff. Equations, {\bf 6}, 273-287 (1981).
\end{thebibliography}
\end{document}