\input amstex
\documentstyle {amsppt}
\magnification \magstep1
\openup3\jot
\NoBlackBoxes
\pageno=1
\define\pp{\delta p}
\define\qq{\delta q}
\define\ph{\varphi}
\def\sk{{\scriptscriptstyle\angle}}
\topmatter
\title
Monotonicity, $\Cal J$-algebra of Potapov and
Lyapunov Exponents
\endtitle
\rightheadtext{Monotonicity and Lyapunov exponents}
\author
Maciej P. Wojtkowski
\endauthor
\affil University of Arizona
\endaffil
\address
Maciej P. Wojtkowski,
Department of Mathematics,
University of Arizona,
Tucson, Arizona 85 721
USA
\endaddress
\email
maciejw\@math.arizona.edu
\endemail
\date
January 15, 1999
\enddate
\abstract
We present a new approach and a generalization of
the estimates of Lyapunov exponents
developed first in \cite{W2} in the symplectic case.
The work of
Lewowicz \cite{L}, Markarian \cite{M},
and our \cite{W1}, \cite{W2}, \cite{W5},
are combined with the $\Cal J$--algebra of Potapov,
\cite{P1},\cite{P2},\cite{P3}.
We obtain a general theory which we then specify to the
symplectic case. The appendix contains a simple
application to the gas of hard spheres.
\endabstract
\endtopmatter
\document
\subhead \S 0. Introduction
\endsubhead
\vskip.4cm
We will discuss a general method for establishing
hyperbolic properties in concrete dynamical systems.
Hyperbolicity is defined in the language of the linearized system (the
derivative in the case of discrete time). It amounts to
exponential growth or decay of solutions of the linearized system.
To capture such a behavior we
use indefinite quadratic forms. If an indefinite quadratic form (more
precisely a field of such forms on the tangent bundle of the phase
space) evaluated on any nonzero solution grows, then it provides us
with a distinguished cone (sector) of positive tangent vectors which
is taken into itself by the linearized dynamics. Once a solution
enters this cone the quadratic form can be used as a measure of
the magnitude of the solution and the growth can be usually
strengthened to exponential growth with the help of recurrence. The
decay of some solutions is obtained by reversing time and using the
absolute value of our quadratic form.
This crucial idea was put forward by Lewowicz \cite{L}
and independently in \cite{W1}. It also appeared in
the study of exponential
dichotomies in linear differential equations,
Coppel, \cite{C}, Samoilenko and Kulik, \cite{Sa-K}.
We will consider a more general situation when solutions of the
linearized system stay in the positive cone forever, once they
enter it, but the form defining the cone is not necessarily
increased.
To illustrate the contents of this paper let us consider the
linear system of differential equations (with periodic
coefficients) of the form.
$$
\aligned
\dot x &= \alpha(t) x + B(t)y, \\
\dot y &= C(t) x + \delta(t)y,
\endaligned \tag{0.1}
$$
where $x\in \Bbb R^m, y \in \Bbb R^m$, the functions
$\alpha$ and $\delta$ have scalar values and $B,C$ are
$m\times m$ matrix valued functions.
We introduce the quadratic form $\Cal Q(x,y) = \langle x, y\rangle$
where $\langle \cdot, \cdot \rangle$ is the arithmetic scalar product
on $\Bbb R^m$. We have
$$
\frac{d}{dt}\Cal Q = \left(\alpha(t) + \delta(t)\right)\langle
x,y\rangle + \langle B(t)y ,y \rangle + \langle C(t)x ,x \rangle.
$$
Under the assumption that the symmetric parts of $B$ and $C$ are
positive definite we obtain the following property
$$
\frac{d}{dt}\Cal Q > 0, \ \ \ \text{if} \ \ \ \Cal Q = 0.
$$
If this property holds we will say that the linear flow
defined by \thetag{0.1} is strictly $\Cal Q$-separated.
We will explore the consequences of this property.
We will establish that if the system \thetag{0.1} is strictly
$\Cal Q$-separated then the linear flow has a dominated
splitting, i.e., there is an invariant splitting of
$\Bbb R^m \times \Bbb R^m$
into $m$-dimensional subspaces
$E^-$ and $E^+$ such that the largest Floquet
multiplier on $E^-$ is strictly smaller (in absolute value)
than the smallest Floquet multiplier on $E^+$.
The quadratic form $\Cal Q$ is negative on $E^-$ and positive on
$E^+$. We will also introduce quantitative description of the
spectral gap.
We will use the language of
matrix cocycles rather than that of differential equations
and we will obtain estimates of Lyapunov exponents.
The results can be readily translated into the language of
differential equations.
The system \thetag{0.1} with $\alpha = \delta \equiv 0,
B(t) \equiv I$ and symmetric positive definite $C(t)$
appears in the study of geodesic flows on manifolds
of negative sectional curvature. In this situation
we obtain the uniform expansion on $E^+$ and the uniform
contraction on $E^-$, i.e., the Anosov property.
More generally in the symplectic case, if a system
is $\Cal Q$-separated then it is Anosov.
The symplectic case was studied earlier in \cite{W2},\cite{W5},
where a special case of the estimates of Theorem 3.10
was obtained.
This method of estimating Lyapunov exponents for the
Hamiltonian dynamics
has two major applications, the Boltzmann-Sinai gas of
hard spheres \cite{Ch-S},\cite{W2}, \cite{W5},\cite{S-Sz}, and the
systems of falling balls in 1-dimension, \cite{W3},\cite{W4},
\cite{S1},\cite{W6},\cite{W7}.
In the Appendix we present an application of the estimates
of Lyapunov exponents from Theorem 3.10
to the gas of hard spheres.
The primary motivation for this paper was our
study of the geodesic flows on Weyl manifolds,
which we call W-flows, \cite{W9},\cite{W8}. The linearization
of these flows has (in appropriate coordinates) the form
\thetag{0.1} with $\delta \equiv 0,
B(t) \equiv I$, and $C(t)$ not necessarily symmetric.
\vskip.4cm
\subhead \S 1. The $\Cal J$--algebra of Potapov and
contraction/expansion properties
\endsubhead
\vskip.4cm
Let us consider an $n$-dimensional linear space
$V$ over $\Bbb R$ with a chosen pseudo Euclidean structure, i.e., a
nondegenerate quadratic form
$\Cal J$
with the positive index of inertia equal to $p$ and the negative
index of inertia equal to $q$, $p + q = n, p\geq 1, q\geq 1$.
Depending on the sign of $\Cal J$
we have positive and negative vectors. We denote by
$$ \Cal C_{\pm} = \{ v\in V | \pm\Cal J(v) > 0 \}\cup \{0\} $$
the open cones of, respectively,
positive and negative vectors (with the zero
vector included), and by $\Cal C_{0}$ their common
boundary, $\Cal C_{0} = \{ v\in V | \Cal J(v) = 0 \}$.
A linear subspace is called positive (negative), if it is
contained in $\Cal C_+ $ ($\Cal C_- $ ).
The maximal dimension of a positive subspace is $p$ and of a
negative subspace $q$. Let $P$ be a positive subspace
of dimension $p$ (a maximal positive subspace)
and let $Q$ be the $\Cal J$--orthogonal complement of $P$.
The dimension of $Q$ is $q$ and it is by necessity a
negative subspace. We can introduce an auxiliary scalar product
in $V$ by using $\Cal J$ in $P$, $-\Cal J$ in $Q$ and
declaring $P$ and $Q$ to be orthogonal. We choose coordinates
in $P = \Bbb R^p$ and $Q = \Bbb R^q$ in such a way that
$\Cal J$ and $-\Cal J$ become the arithmetic scalar products
in $\Bbb R^p$ and $\Bbb R^q$ respectively. As a result we obtain
the following coordinate representation:
$V = \Bbb R^p \times \Bbb
R^q$ and for $ v = (v_1,v_2), v_1 \in \Bbb R^p, v_2 \in \Bbb R^q $
$$ \Cal J (v) =\langle Jv, v \rangle = v_1^2 - v_2^2,\ \
\text{where}
\ \ J = \left[\matrix I_p & 0\\ 0& -I_q\endmatrix\right] $$
$I_p, I_q$ are identity matrices in $\Bbb R^p$ and $\Bbb R^q$
respectively. We will use freely this coordinate representation
when we find it convenient. It is important though for discussing
diffeomorphisms and flows on manifolds that the following theory
depends on the choice of the form $\Cal J$ alone.
\proclaim{Definition 1.1}
A linear nonsingular operator $S: V \to V$ is
\roster
\item
$\Cal J$--separated, if $S\Cal C_+ \subset \Cal C_+$,
\item
strictly $\Cal J$--separated, if
$S(\Cal C_0 \cup \Cal C_+)\subset \Cal C_+ $,
\item
$\Cal J$--monotone, if $\Cal J(Sv ) \geq \Cal J(v)$ for every
$v\in V$,
\item
strictly $\Cal J$--monotone, if $\Cal J(Sv ) > \Cal J(v)$ for every
$v\in V, v \neq 0$,
\item
$\Cal J$--isometry, if $\Cal J(Sv ) = \Cal J(v)$ for every
$v\in V$,
\endroster
\endproclaim
Note that the five classes of operators
are semigroups with respect to composition. The inclusions between them
are as follows
$$
\aligned
&(1) \ \ \supset \ \ (3) \ \ \supset \ \ (5)\\
& \cup \ \ \ \ \ \ \ \ \ \cup \\
& (2) \ \ \supset \ \ (4)
\endaligned
$$
We extend the definitions of properties 1 - 5
to one parameter subgroups of linear operators
(elements of the Lie algebra $gl(n)$) in the natural way.
An element $G$ of $gl(n)$ has property (i) if the operator
$e^{tG}$ has property (i) for all $t > 0$.
Let us further define a
$\Cal J$--symmetric operator $S$ as
a linear operator on $V$
such that $\langle JS \cdot, \cdot \rangle$ is a symmetric
bilinear form on $V$.
Potapov, \cite{P1},\cite{P2},\cite{P3}, obtained fundamental results
about the structure and
properties of $\Cal J$--monotone operators, which he called
$\Cal J$--noncontractive. Potapov developed his theory
for a pseudo unitary form $\Cal J$ rather than pseudo Euclidean,
but we will concentrate on the real case. We will comment
along the way on the complex versions of theorems.
We will formulate a series of propositions
about the $\Cal J$-algebra of Potapov. We will prove
only the infinitesimal versions. The (harder) proofs in the general
case can be found in the papers of Potapov,
\cite{P1},\cite{P2},\cite{P3}.
1. If $S$ is $\Cal J$--separated then the set of positive
numbers $r$ such that $\frac 1r S$ is $\Cal J$--monotone
is a closed interval $[r_-,r_+], r_- > 0$, possibly degenerating to
a point.
2. If $S$ is strictly $\Cal J$--separated then the set of positive
numbers $r$ such that $\frac 1r S$ is strictly $\Cal J$--monotone
is an open interval $(r_-,r_+), r_- > 0$.
These properties are the consequence of the theorem of K\"uhne.
\proclaim{Theorem 1.2}
If a symmetric bilinear form $F(\cdot,\cdot)$ is nonnegative on
$\Cal C_0$ then
$$
\inf_{v\in \Cal C_+}\frac{F(v,v)}{\langle Jv,v\rangle} \geq
\sup_{u\in \Cal C_-}\frac{F(u,u)}{\langle Ju,u\rangle},
$$
and for every number $r$ between the infimum and the supremum
$ F(v,v) \geq r \langle Jv,v\rangle$,
for all $v$.
\endproclaim
\demo{Proof}
Suppose to the contrary that there is a positive
vector $v_0 \in \Cal C_+$ and a negative vector
$u_0 \in \Cal C_-$ which violate the inequality, i.e.,
assuming $\langle Jv_0,v_0\rangle = 1,
\langle Ju_0,u_0\rangle = -1$ we have
$F(v_0,v_0) + F(u_0,u_0) < 0$.
We can find the angle $\alpha$ such that
the linear combinations
$v_1 = \cos\alpha v_0 + \sin\alpha u_0$
and $v_2 = -\sin\alpha v_0 + \cos\alpha u_0$ are in
$\Cal C_0$. Then
$F(v_1, v_1) \geq 0, \ F(v_2,v_2) \geq 0$.
We obtain the contradiction
$$
F(v_1,v_1) + F(v_2,v_2) = F(v_0,v_0) + F(u_0,u_0) \geq 0.
$$
\qed\enddemo
The proof of the pseudo unitary version of Theorem 1.2
can be found in \cite{P2}.
3. Every $\Cal J$--separated operator $S$ has a unique representation
$S = RU$, where $U$ is a $\Cal J$--isometry and $R$ is
$\Cal J$--symmetric and has only positive eigenvalues.
The operator $R$ is equal to $(SJS^*J)^{\frac 12}$ and it
is called the modulus of $S$. Note that although the adjoint $S^*$
depends on the auxiliary Euclidean structure in $V$ the modulus
$R$ depends on the form $\Cal J$ alone.
4. For a strictly $\Cal J$--separated $S$ there is
a $\Cal J$--isometry $W$ such that the modulus $R$ of $S$ is
diagonalized by $W$, i.e., $W^{-1}RW$ is
the diagonal matrix $\left[\matrix R_+ & 0\\ 0&
R_-\endmatrix\right]$. Moreover $R_+ \geq r_+I_p$ and $ 0 <
R_- \leq r_-I_q$, where $r_+ > r_- > 0$ were introduced
in 2.
5. For a $\Cal J$--separated $S$ the $p$ largest eigenvalues of the
modulus $R$ are not smaller than $r_+$ and the $q$ smallest eigenvalues
do not exceed $r_-$. Let us denote these eigenvalues by
$$ r_p^+ \geq \dots \geq r_2^+\geq r_1^+ \geq r_+ \geq
r_- \geq r_{1}^- \geq r_{2}^- \geq \dots \geq r_q^- >0.
$$
Moreover we have $r_1^+ = r_+$ and $r_1^- = r_-$.
Clearly a $\Cal J$--separated $S$ is (strictly) $\Cal J$--monotone
if and only if $r_1^+ = r_+ \geq 1 \ (> 1)$ and
$r_1^- = r_- \leq 1 \ (< 1)$.
The proofs depend on the celebrated theorem of Potapov,
that $S$ is $\Cal J$--separated if and only if $S^*$ is
$\Cal J$--separated.
The infinitesimal version is quite simple. By Theorem 1.2
$G \in gl(n)$ is $\Cal J$--separated if and only if
there is a number $t$ (not necessarily positive)
such that $G^*J + JG \geq tJ $.
Multiplying that inequality by $J$ on the left and on the right
we obtain $JG^* + GJ \geq tJ $.
The
infinitesimal versions of the claims 3 -- 5
are equally simple to prove.
First of all any element $G$ of the Lie algebra $gl(n)$ can be split
uniquely into the $\Cal J$--symmetric
$G_s = \frac 12(G + JG^*J)$ and
the $\Cal J$--antisymmetric $G_a = \frac 12(G - JG^*J)$.
The $\Cal J$--antisymmetric elements of $gl(n)$
are the infinitesimal
$\Cal J$--isometries. If $G$ is strictly
$\Cal J$--separated then $ 2JG_s = G^*J + JG > tJ$.
It follows that the eigenvalues of $G_s$ are real and
they are bigger than $\frac t2$ if the eigenvector is a positive vector
and smaller than $\frac t2$ if the eigenvector is a negative vector.
(To establish that the spectrum of $G_s$ is real
we need to use the pseudo unitary $\Cal J$-form in the complexified
space. It goes through exactly as in the case of symmetric matrices
and we leave it to the reader.)
If $G_s$ is strictly $\Cal J$--separated then
it is diagonalizable by a $\Cal J$--isometry.
Indeed it must have an invariant subspace $P$ which
is maximal positive and an invariant subspace $Q$ which
is maximal negative.
(It is easier to think about $S = e^{\epsilon G_s}$.
$S$($S^{-1}$) takes maximal positive(negative) subspaces into maximal
positive(negative) subspaces. Hence by some kind of fixed point
theorem there must be invariant subspaces.)
Similarly to the case of symmetric matrices the subspaces
$P$ and $Q$ are $\Cal J$--orthogonal. Hence there is a
$\Cal J$--isometry $U$ such that $$U^{-1}G_sU =
\left[\matrix G_+ & 0\\ 0&
G_-\endmatrix\right].$$ Moreover $G_\pm$ are symmetric.
So we can further diagonalize them. Also $G_+ > \frac 12t I_p$ and
$G_- < \frac 12 tI_q$. Indeed since the defining property
for a $\Cal J$-isometry $U$ is $U^{-1} = JU^*J$, we have
$$
2J(U^{-1}G_sU) = U^*2JG_sU \geq tU^*JU = tJ.
$$
We proceed with the development of further properties of
$\Cal J$-separated operators; we will freely use properties 1 -- 5.
For a $d$--dimensional positive linear subspace $L \subset \Cal C_+$
the restriction of $\Cal J$ is a scalar product.
If $S$ is $\Cal J$--separated then $SL \subset \Cal C_+$
is also a positive subspace with the scalar product
furnished by $\Cal J$. Hence the subspaces $L$ and $SL$ are
equipped with (unoriented) volume elements, which we call the
$\Cal J$--volumes.
Let $\alpha_d(L;S)$ be the
coefficient of the $\Cal J$--volume expansion
for the linear map $S$ restricted to $L$.
Further let $\sigma_d(S)$ be the infimum of
$\alpha_d(L;S)$ over all positive $d$--dimensional
subspaces $L$. The important property of the coefficient $\sigma_d$
is the supermultiplicativity: for $\Cal J$--separated $S_1$ and $S_2$
we have
$$
\sigma_d(S_1S_2) \geq \sigma_d(S_1)\sigma_d(S_2).\tag{1.1}
$$
Using the notation introduced
in 5. we have
\proclaim{Proposition 1.3}
$\sigma_d(S) = r_1^+r_2^+ \dots r_d^+$.
\endproclaim
\demo{Proof}
The operator $C_\epsilon =
\left[\matrix (1+\epsilon)I_p & 0\\ 0&
(1-\epsilon)I_q\endmatrix\right]$
is strictly
$\Cal J$--monotone for $0 < \epsilon < 1$. If $S$ is $\Cal J$--separated
then $C_\epsilon S$ is strictly $\Cal J$--separated.
Hence by the continuity of the modulus $R$ and
the coefficient $\alpha_d(L;S)$ as functions of $S$,
it is sufficient to prove the Proposition for a strictly
$\Cal J$--separated operator $S$. For such operators we
can diagonalize the modulus as formulated in 4.
Since $S$ differs from its modulus $R$ by a $\Cal J$--isometry $U$
we get that $\alpha_D(L;S)$ is equal to $\alpha_d(UL ; R)$.
Every $d$--dimensional positive subspace $L$ can be included in
a $p$--dimensional positive subspace $T$ (maximal positive subspace).
The subspace $T$ is the graph of the linear mapping $T: \Bbb R^p \to
\Bbb R^q$ such that $T^*T < I_p$. We will slightly abuse the notation
denoting by the same letter $T$ the linear map and its graph.
In the coordinate representation $T$ is a
$q\times p$ matrix.
Using $v_1 \in \Bbb R^p$ as a
coordinate on $T$ we obtain that the scalar product $\Cal J$
on $ T$ is equal to $\langle (I_p -T^*T)v_1, v_1 \rangle$
and the action
of $R$ on $T$ is given by $v_1 \mapsto R_+v_1$. The image space
$RT$ is the graph of the linear map $R_-TR_+^{-1}$.
To estimate the coefficients of $\Cal J$--volume expansion
under the action of the restriction of $R$ to the subspace $T$ we
need to estimate the singular values of
$$
(I_p - R_+^{-1}T^*R^2_-TR^{-1}_+)^{\frac 12}R_+ (I_p - T^*T)^{-\frac
12},$$
i.e., we need to estimate the eigenvalues of
$$
(I_p - T^*T)^{-\frac 12} (R_+^2 - T^*R_-^2T)(I_p - T^*T)^{-\frac 12}
\geq (I_p - T^*T)^{-\frac 12} R_+^2 (I_p - T^*T)^{-\frac 12}.
$$
The $i$-th eigenvalue in ascending order
of the last matrix cannot be smaller than
$(r_i^+)^2$.
It gives us the desired estimate
$ \alpha_d(L,S) \geq r_1^+r_2^+ \dots r_d^+$.
To get the equality for the infimum we need to consider the
subspace $T = 0$.
\qed \enddemo
Proposition 1.3 allows the interpretation of the eigenvalues
of the modulus as pseudo Euclidean
singular values. We will call them simply
positive and negative singular values
of the $\Cal J$--separated operator $S$ (note that both positive
and negative singular values are positive numbers).
In particular they are not changed if we
compose $S$ on the left and/or on the right with $\Cal J$--isometries.
In the pseudo unitary case we consider $d$-dimensional complex
subspaces which have $2d$-dimensional real $\Cal J$-volume.
Essentially the same proof will give us
$\sigma_d(S) = (r_1^+r_2^+ \dots r_d^+)^2$.
\proclaim{Corollary 1.4}
For $\Cal J$-separated $S_1$ and $S_2$ we have
$$
r_1^+(S_1S_2) \geq r_1^+(S_1)r_1^+(S_2) \ \ \ \text{and} \ \ \
r_1^-(S_1S_2) \leq r_1^-(S_1)r_1^-(S_2).
$$
\endproclaim
\demo{Proof}
The first part follows immediately from Proposition 1.3.
For a $\Cal J$-separated $S$ the operator $S^{-1}$ is
$(-\Cal J)$-separated and the positive pseudo Euclidean singular values of
$S^{-1}$ (with respect to $-\Cal J$) are equal to
$\frac 1{r_1^-(S)}, \dots , \frac 1{r_q^-(S)}$.
The second part follows now from the first applied to
$S_2^{-1}S_1^{-1}$ and $-\Cal J$.
\qed\enddemo
Let us consider the manifold $\Cal L_+$ of all $p$-dimensional subspaces
contained in $\Cal C_+$, It can be identified with the
manifold of $q\times p$ matrices $T$ such that $T^*T < I_{p}$.
This manifold carries the Riemannian metric of the symmetric space
\cite{Py},
$$
dT^2 = tr\left( (I_{p} - T^*T)^{-1} dT^* (I_{q} - TT^*)^{-1}dT\right).
\tag{1.2}
$$
To get \thetag{1.2} we need to observe that the $\Cal J$--isometries
act transitively on $\Cal L_+$. The isometry that takes
$T$ into the zero subspace $\Bbb R^p \times \{0\}$ is given by
$$
\left[\matrix (I_p -T^*T)^{-\frac 12} & (I_p -TT^*)^{-\frac 12} T^*\\
-(I_q -TT^*)^{-\frac 12}T & (I_q -TT^*)^{-\frac 12}
\endmatrix\right] =
\left[\matrix (I_p -T^*T)^{-\frac 12} & -T^*(I_q -TT^*)^{-\frac 12} \\
T(I_p -T^*T)^{-\frac 12} & (I_q -TT^*)^{-\frac 12}
\endmatrix\right]^{-1}.
$$
Hence the metric $tr(dT^*dT)$ at $0$ is transported to
$T$ as
$$tr((I_p -T^*T)^{-\frac 12} dT^*(I_q -TT^*)^{-1}dT(I_p -T^*T)^{-\frac
12}).
$$
A $\Cal J$--separated operator $S$ maps naturally $\Cal L_+$
into itself. This mapping is a contraction.
More precisely we have
\proclaim{Theorem 1.5}
For a $\Cal J$--separated operator $S$ and $T_1,T_2 \in \Cal L_+$
$$
dist(ST_1,ST_2) \leq \frac {r_1^-}{r_1^+} dist(T_1,T_2),
$$
and the contraction coefficient is sharp.
\endproclaim
\demo{Proof}
By continuity it is sufficient to prove the theorem for a
strictly $\Cal J$--separated operator $A$.
We take advantage of the polar form of $S = RU$ where $R$ is the
modulus and $U$ is the $\Cal J$--isometry. Hence it is enough to show
the contraction for an operator of the form described in 4.
We will establish the infinitesimal contraction property.
The action of this operator on $\Cal L_+$ is given by
$T \mapsto R_- T R_+^{-1}$. The derivative of this action is
$dT \mapsto R_-dTR_+^{-1}$. Hence we need to compare the
spectra of
$$
\aligned
&(I_p -T^*T)^{-\frac 12} dT^*(I_q -TT^*)^{-1}dT(I_p -T^*T)^{-\frac 12}
\ \ \text{ and }\\
&(I_p -R_+^{-1}T^*R_-^2TR_+^{-1})^{-\frac 12}
R_+^{-1}dT^*R_-(I_q -R_- T R_+^{-2}T^*R_-)^{-1}R_-dTR_+^{-1}
(I_p -R_+^{-1}T^*R_-^2TR_+^{-1})^{-\frac 12}.
\endaligned
$$
We will need the following
\proclaim{Lemma 1.6}
For nonsingular symmetric matrices $X,Y$
and a symmetric matrix $Z$, if
$X^2 \leq Y^2$ then the spectrum of $YZY$ is not smaller than the spectrum
of $XZX$, i.e., the $i$-th eigenvalue
(in ascending order) of $XZX$ does not exceed the
$i$-th eigenvalue of $YZY$.
\endproclaim
Note that we do not claim that $XZX \leq YZY$, which is
not guaranteed by our assumptions.
\demo{Proof}
The eigenvalues of $XZX$ and $YZY$ can be obtained as min-max of the Rayleigh
quotients
$$
\frac {\langle XZX v, v\rangle}{\langle v,v\rangle} =
\frac {\langle Z u, u\rangle}{\langle X^{-2}u,u\rangle} \leq
\frac {\langle Z u, u\rangle}{\langle Y^{-2}u,u\rangle} =
\frac {\langle YZY w, w\rangle}{\langle w,w\rangle},
$$
for $u = Xv$ and $ u = Yw$.
\qed\enddemo
The last operator has the same spectrum as
the operator
$$R_+^{\frac 12}(R_+^2 -T^*R_-^2T)^{-\frac 12}
R_+^{-\frac 12}dT^*(R_-^{-2} -T R_+^{-2}T^*)^{-1}dTR_+^{-\frac 12}
(R_+^2 -T^*R_-^2T)^{-\frac 12}R_+^{\frac 12}.\tag{1.3}$$
Now applying Lemma 1.6 we can ``peel off'' the outside factors
using
$$
\aligned
&R_+ \geq r_1^+ I_p,\
R_+^2 \geq (r_1^+)^2 I_p, \ \ \ \
R_- \geq r_1^- I_q, \
R_-^2 \geq (r_1^-)^2 I_q,\\
&(R_+^2 -T^*R_-^2T) \geq (r_1^+)^2(I_p - T^*T), \ \ \
(R_-^{-2} -T R_+^{-2}T^* \geq (r_1^-)^{-2}(I_q - TT^*).
\endaligned
$$
We obtain that the spectrum of \thetag{1.3} does not
exceed the spectrum of
$$
\aligned
\frac 1{(r_1^+)^2}(I_p - T^*T)^{-\frac 12}dT^*(R_-^{-2} -T R_+^{-2}T^*)^{-1}
dT (I_p - T^*T)^{-\frac 12} \leq \\
\left(\frac{r_1^-}{r_1^+}\right)^2
(I_p -T^*T)^{-\frac 12} dT^*(I_q -TT^*)^{-1}dT(I_p -T^*T)^{-\frac 12}.
\endaligned
$$
To establish that the coefficient of contraction is sharp
it is enough to take $T = 0$ and calculate the infinitesimal
coefficient of contraction which is equal to
$\frac{r_1^-}{r_1^+}$.
\qed\enddemo
This theorem extends the result of Bougerol, \cite{B},
about
the contraction of the space of positive
Lagrangian subspaces under the action of
monotone symplectic operators.
In the proof we have obtained the estimate of all eigenvalues
and not just the trace. It leads to the contraction property
for other metrics, for instance the Finsler metric determined
by $\|dT^*dT\|$ at zero subspace, in parallel with \cite{L-W}.
Theorem 1.5 holds also in the pseudo unitary case
and the proof is essentially the same. In such a case the space
of complex positive subspaces of dimension $p$ is the
classical bounded domain. As shown by Suzuki, \cite{Su},
for this domain the Caratheodory and Kobayashi metrics
coincide with the Finsler metric determined
by $\|dT^*dT\|$ at zero subspace. Hence this metric must be contracted
by any holomorphic map of the domain into itself and not only
by the M\"obius maps as in Theorem 1.5.
The Kahler metric of the symmetric space is not contracted by
all holomorphic maps (cf. \cite{Ko}), but it is contracted by M\"obius maps
defined by $\Cal J$-separated operators, which is the content of
Theorem 1.5 in the pseudo unitary case.
\vskip.4cm
\subhead \S 2. The symplectic case
\endsubhead
\vskip.4cm
Let us apply these general ideas to the symplectic space
and symplectic operators.
We have $p =q $, $V = \Bbb R^p \times \Bbb R^p$ and the
symplectic form $\omega(v,w) = \langle v_1,w_2 \rangle
- \langle v_2,w_1 \rangle$. We consider a special type of
a $\Cal J$--form, which we call a $\Cal Q$--form.
It was defined geometrically in \cite{W5} .
A $\Cal Q$-- form is defined by a pair of transversal
Lagrangian subspaces $L_1,L_2$.
Using the standard splitting
$v = v_1 + v_2, v_i \in L_i, i = 1,2,$ we put
$\Cal Q (v) = \omega(v_1,v_2)$.
The positive cone $\Cal C_+$
of a $\Cal Q$--form is called a symplectic
sector between $L_1$ and $L_2$.
The boundary of a symplectic sector contains
many Lagrangian subspaces, but only two isolated ones,
$L_1$ and $L_2$. These two Lagrangian subspaces act as
the ``sides'' of the sector, which they actually
are only when $p =1$.
We can choose symplectic coordinates in such a way that
$L_1 = \Bbb R^p \times\{0\}, L_2 = \{0\}\times\Bbb R^p $
and accordingly $\Cal Q(v) = \langle v_1,v_2\rangle$.
We will call it the standard $\Cal Q$-form.
The following theorem was published by Potapov in
\cite{P2}, and independently in
\cite{W1}. The proof below is essentially taken from \cite{W1}.
\proclaim{Theorem 2.1}
If a symplectic operator is (strictly) $\Cal Q$--separated
then it is (strictly) $\Cal Q$--monotone.
\endproclaim
Before providing the proof we develop a useful factorization of
$\Cal Q$--separated symplectic operators. We will use the standard
symplectic form and the standard $\Cal Q$-form in
$V = \Bbb R^p \times \Bbb R^p$ .
Let $$ S = \left(\matrix A& B\\ C& D\endmatrix\right) $$
be a symplectic operator on $V = \Bbb R^p \times \Bbb R^p$.
Hence $A,B,C,D$ are $p\times p$ matrices such that
$C^*A = A^*C, D^*B = B^*D, \ \ \text{and} \ \ D^*A -B^*C = I$.
\proclaim{Lemma 2.2}
If a symplectic $S$ is $\Cal Q$--separated then
$A$ and $D$ are invertible.
\endproclaim
\demo{Proof}
Suppose that there is $\widehat{v}_{1} \in \Bbb R^p$ such
that $A\widehat{v}_{1}= 0 $. It follows that
$\widehat{v}_{1} = (D^*A -B^*C)\widehat{v}_{1} = -B^*C\widehat{v}_{1}$.
For $v = (\widehat{v}_1, v_2) \in \Cal{C}_+$ we must have
$Sv = (Bv_2,C\widehat{v}_1 + Dv_2 ) \in \Cal{C}_+$, i.e., if
$\langle \widehat{v}_1, v_2 \rangle > 0$ then
$$
0 < \langle Bv_2,C\widehat{v}_1 + Dv_2 \rangle =
- \langle \widehat{v}_1, v_2 \rangle + \langle v_2, B^*Dv_2 \rangle.
$$
Choosing $v_2 = \epsilon \widehat{v}_1, \epsilon > 0$ we obtain
$$
\langle \widehat{v}_1,\widehat{v}_1\rangle < \epsilon
\langle \widehat{v}_1,B^*D\widehat{v}_1 \rangle,
$$
which leads to $\widehat{v}_1 = 0$.
\qed \enddemo
The symplectic operators of the form
$$ \left(\matrix A& 0\\ 0& (A^*)^{-1}\endmatrix\right) $$
are also $\Cal{Q}$--isometries.
By Lemma 2.2 given a $\Cal{Q}$--separated symplectic $S$ we can factor
out the following
$\Cal{Q}$-isometry on the right
$$
S =
\left(\matrix A& B\\ C& D\endmatrix\right) =
\left(\matrix
I& K \\ H &\cdot \endmatrix\right)\left(\matrix A& 0\\ 0& (A^*)^{-1}
\endmatrix\right)
$$
Symplecticity of $S$ forces $H$ and $K$ to be symmetric
and allows the further unique factorization
$$
S =
\left(\matrix I& 0 \\ H &I
\endmatrix\right)\left(\matrix I& K \\ 0 &I \endmatrix\right)
\left(\matrix A& 0\\ 0& (A^*)^{-1}\endmatrix\right).
\tag {2.1}
$$
Moreover $H$ and $K$ must be positive semidefinite
($H\ge 0,\,K\ge 0$). It follows from the following
\demo{Proof of Theorem 2.1}
Using the factorization \thetag{2.1} we get for $v = (v_1,\,v_2)$
$$
\Cal{Q}(Sv) = \langle v_1,\,v_2\rangle + \langle K v_2,\,v_2\rangle +
\langle H(v_1+Kv_2),\, v_1+K v_2\rangle.
$$
Taking $v_2 = 0$ we obtain $\Cal{Q}(v) = 0$ and hence
$ \Cal{Q}(Sv) = \langle Hv_1,\, v_1\rangle \geq 0$.
Taking $v_1 = -K v_2$ we obtain $\Cal{Q}(Sv) = 0$ which
implies that $\Cal{Q}(v) = - \langle Kv_2,\, v_2\rangle \leq 0$.
\qed\enddemo
As a byproduct of the proof we get the following useful observation
\proclaim{Proposition 2.3}
A $\Cal{Q}$--monotone symplectic operator $S$ maps
vectors from $L_1$ ($L_2$) either
into $\Cal{C}_+$ or into $L_1$ ($L_2$).
Moreover if $S$ maps $L_1$ and $L_2$
into $\Cal{C}_+$ then it is
strictly $\Cal{Q}$--monotone.
\endproclaim
\qed
It turns out that the modulus of a symplectic $\Cal Q$-monotone
operator is also
symplectic and hence the positive and negative singular values
are inverses of each other.
Indeed, from \thetag{2.1} we obtain the following factorization
of the
$\Cal{Q}$--monotone symplectic operator
$S \left(\matrix A^{-1}& 0\\ 0& A^{*}\endmatrix\right)$.
$$
\left(\matrix I& K \\ H & I + HK \endmatrix\right)=
\left(\matrix (I+KH)^{\frac 12}& K(I+HK)^{-\frac 12} \\
H(I+KH)^{\frac 12} & (I + HK)^{\frac 12} \endmatrix\right)
\left(\matrix (I+KH)^{-\frac 12}& 0 \\
0 & (I + HK)^{\frac 12} \endmatrix\right).
$$
Since the last matrix is a $\Cal{Q}$--isometry and
the next to last matrix is $\Cal{Q}$--symmetric we obtain that
the modulus of $S$ is equal to the next to last matrix.
We proceed to calculate its eigenvalues.
\proclaim{Proposition 2.4}
The positive singular values of a $\Cal{Q}$--monotone operator
$S = \left(\matrix A& B\\ C& D\endmatrix\right)$
are equal to $ \sqrt{1+t_i} + \sqrt{t_i} =\exp(
\sinh^{-1}\sqrt{t_i})$,
where $t_1, \dots, t_p$ are the eigenvalues of $CB^*$,
which are real and nonnegative.
\endproclaim
\demo{Proof}
To calculate the pseudo Euclidean
singular values of $S$ we will use the fact that
they are not changed if we multiply $S$ on left and on the right by
$\Cal{Q}$--isometries. In the factorization \thetag{2.1}
$CA^{-1} = H$ and $BA^* = K = AB^*$
so that $HK$ has the same eigenvalues as $CB^*$,
in particular they must be all real nonnegative (since $H \geq 0,
K \geq 0$).
Let us assume that $K > 0$, the general case will follow by
continuity. We get
$$\left(\matrix K^{-\frac{1}{2}}& 0\\ 0&K^{\frac{1}{2}}\endmatrix\right)
\left(\matrix I& K\\ H& I+HK\endmatrix\right)
\left(\matrix K^{\frac{1}{2}}& 0\\ 0& K^{-\frac{1}{2}}\endmatrix\right) =
\left(\matrix I& I\\ F& I+F\endmatrix\right)
$$
where $F= K^{\frac{1}{2}}HK^{\frac{1}{2}}$ has the same eigenvalues as $
HK$.
Finally if $Z$ is the orthogonal matrix which diagonalizes $F$, i.e.,
$Z^{-1}FZ = T$ is diagonal, then
$$
\aligned
&\left(\matrix Z^{-1}& 0\\ 0& Z^{-1}\endmatrix\right)\left(\matrix I& I\\F&
I+F\endmatrix\right)\left(\matrix Z& 0\\ 0& Z\endmatrix\right) =
\left(\matrix I& I\\ T& I+T\endmatrix\right) =\\&
\left(\matrix (I+T)^{-\frac 12}& (I+T)^{-\frac 12}\\
T(I+T)^{\frac 12}& (I+T)^{\frac 12}\endmatrix\right)
\left(\matrix (I+T)^{\frac 12}& I\\ T& (I+T)^{\frac
12}\endmatrix\right).
\endaligned
$$
It shows that the singular values of $S$ are equal to the eigenvalues
of next to last matrix and they are easy to calculate.
\qed\enddemo
It follows from the proof of Proposition 2.4 that
\proclaim{Proposition 2.5}
If a $\Cal{Q}$--monotone symplectic operator $S$ maps all
vectors in $L_1$ and $L_2$ into $\Cal{C}_+$ except for
a subspace of $L_1$ of codimension $k$ then
at least $k$ largest singular values of $S$ are strictly
larger than one.
\endproclaim
\vskip.4cm
\subhead \S 3. The Lyapunov exponents
\endsubhead
\vskip.4cm
We are going to translate the properties of $\Cal J$-separated
operators into the properties of Lyapunov exponents of
matrix cocycles with values
in the semigroup of $\Cal J$-separated matrices.
Let $X$ be a measurable space with a probabilistic
measure $\nu$ and let $\varPhi: X \to X$ be an ergodic map
preserving the measure $\nu$.
Let further $S : X \to Gl(V)$ be a measurable map
such that $\log\|S(x)\|$ and $\log\|S(x)^{-1}\|$ are
$\nu$--integrable functions. We will refer to this requirement
as {\it integrability condition}.
We define the matrix valued cocycle
$$
\aligned
S^m(x) &= S(\varPhi^{m-1}x)\dots S(\varPhi x)S(x), \ \ \text{for} \ \
m \geq 0,\\
S^m(x) &= S(\varPhi^{m}x)^{-1}\dots S(\varPhi^{-1} x)^{-1},
\ \ \text{for} \ \ m \leq -1.
\endaligned
$$
By the Oseledets Multiplicative Ergodic Theorem, \cite{Ra},\cite{Ru},
there are numbers
$\lambda_{1} < \dots < \lambda_{N}$, called
the Lyapunov exponents of the measurable
cocycle $S(x),\ x \in X$, and for almost all $x$ there is an
invariant splitting
$$
V = V_{1}(x)\oplus \dots \oplus V_{N}(x),
$$
such that
for all vectors $0 \neq v\in V_k(x)$
$$
\lim_{m \to \pm\infty} \frac 1{|m|}\log\|S^m(x)v\|
= \pm\lambda_k.
$$
The invariance of the splitting means that
$S(x)V_k(x) = V_k(\varPhi x)$.
To simplify the notation we will omit the dependence of the subspaces
on $x \in X$,
The dimensions $d_k$ of the subspaces $V_k$ are called the
multiplicities of the Lyapunov exponents.
Moreover, if we denote by $ W_k = \bigoplus_{i=1}^k V_i$,
we have that for any linear subspace $ L \subset W_k$
having the trivial intersection with $ W_{k-t-1}$ and such that
$ W_{k-t} + L = W_k$ the exponential
rate of growth of volume on $S$ is equal to the sum of
appropriate Lyapunov exponents taken with multiplicities,
$$
\aligned
&\lim_{m \to +\infty} \frac 1m\log|det \left(S^m(x)|_ L\right)| =
\widetilde d\lambda_{k-t} + d_{k-t+1}\lambda_{k-t+1}+ \dots + \dots +
d_k\lambda_k,\\
& \widetilde d = \dim W_{k-t}\cap L =
\dim L - (d_{k-t+1}+ \dots + d_k).
\endaligned
\tag{3.1}
$$
We will be assuming that the matrices $S(x), x \in X,$ are all
$\Cal J$--separated.
Let us denote by
$r_{p}^+(x) \geq \dots \geq r_1^+ (x) \geq r_{1}^-(x)\geq \dots
\geq r_{q}^-(x) >0$ the pseudo Euclidean singular
values of $S(x)$.
\proclaim{Lemma 3.1} All the functions $\log r^\pm_k(x)$ are
$\nu$-integrable.
\endproclaim
\demo{Proof} It is enough to check integrability for
$\log r_{p}^+(x)$ and $\log r_{q}^-(x)$. We have
$$
(r_{p}^+)^2 = \|R^2\| = \|SJS^*J\| \leq \|S\|^2, \ \
(r_{q}^-)^{-2} = \|R^{-2}\| = \|J(S^*)^{-1}JS^{-1}\| \leq \|S^{-1}\|^2,
$$
which proves the claim due to the assumed integrability of
$\log\|S(x)\|$ and $\log\|S(x)^{-1}\|$.
\qed\enddemo
The Lyapunov exponents $\lambda^{\pm}(v)$ (forward and backward
in time) are defined for almost every
initial point $x \in X$ and for all nonzero vectors $v \in V$.
Let us denote by
$l_\pm = \int \log r_1^\pm(x) d\nu(x)$.
\proclaim{Proposition 3.2}
For any vector $v \in \Cal C_\pm, v \neq 0$ we have
$\lambda^\pm(v) \geq \pm l_\pm$.
\endproclaim
\demo{Proof}
Since $\frac 1{r_1^+(x)}S$ is $\Cal J$-monotone
we have
$\Cal J(S(x)v) \geq \left(r_1^+(x)\right)^2 \Cal J(v)$ for all vectors $v$.
In the positive cone the square of norm is not smaller than
$\Cal J$. Hence the exponential growth of $\Cal J$
implies the exponential growth of the norm.
More precisely we have for $v \in \Cal C_+,$
$$
\|S^m(x)v\|^2 \geq \Cal J(S^m(x)v) \geq
\prod_{i=0}^{m-1}\left(r_1^+(\varPhi^ix)\right)^2
\Cal J(v).
$$
It follows from Lemma 3.1 and the Birkhoff Ergodic Theorem that
$$
\lambda^+(v) \geq \int \log r_1^+(x) d\nu(x) = l_+.
$$
To obtain the inequality $\lambda^-(v) \geq -l_-$.
we need to use the fact that the
operators $S(x)^{-1}$ are $(-\Cal J)$--separated
and apply the argument above to $S(x)^{-1}$.
\qed\enddemo
\proclaim{Proposition 3.3}
If $l_+ > l_-$ then for almost every $x\in X$ there
is $m \geq 1$ such that $S^m(x)$ is strictly $\Cal J$-separated.
\endproclaim
A matrix cocycle with the last property will be called
{\it eventually strictly $\Cal J$-separated}.
\demo{Proof}
It follows immediately from Corollary 1.4
since a $\Cal J$-separated operator $S$ is strictly $\Cal J$-separated
if and only if $r_1^+(S) > r_1^-(S)$.
\qed\enddemo
We introduce another labeling of the Lyapunov exponents and
the subspaces of the splitting.
If $d_1 > q$ then we define $s_- =0$. Similarly if $d_N > p$ we
define $s_+ =0$. Otherwise we let
$$
s_- = \max\{k | \sum_{i=1}^{k}d_i \leq q\}\ \ \ \text{and}\ \ \
s_+ = \max\{k | \sum_{i=N-k+1}^{N}d_i \leq p\}.
$$
When $s_- + s_+ =N$ we relabel the subspaces as
$$
V_{-s_-}\oplus \dots \oplus V_{-1} \oplus
V_{1} \oplus \dots \oplus V_{s_+},
$$
and when $s_- + s_+ =N-1$ we use also the zero label
$$
V_{-s_-}\oplus \dots \oplus V_{-1} \oplus V_0 \oplus
V_{1} \oplus \dots \oplus V_{s_+}.
$$
Hence the subspace $V_0$ may be present or absent.
The interval of separation $[r_1^-(x),r_1^+(x)]$ varies with the
operator $S(x)$. We get a gap in the Lyapunov spectrum if
$l_+ = \int \log r_1^+d\nu > \int \log r_1^-d\nu = l_-$ and in such a case
$ V_0$ is absent.
\proclaim{Proposition 3.4}
If $V_0$ is present in the new labeling, then $l_- = l_+ =\lambda_0$.
If $l_+ > l_-$, then $V_0$ is absent in the new labeling,
$\lambda_{1} \geq l_+ > l_- \geq \lambda_{-1}$
and
$$
V_{-s_-}\oplus \dots \oplus V_{-1} \subset \Cal C_- , \ \
V_{1} \oplus \dots \oplus V_{s_+} \subset \Cal C_+.\tag{3.2}
$$
\endproclaim
\demo{Proof}
If $V_0$ is present then the dimensions of
$V_{-s_-}\oplus \dots \oplus V_{-1} \oplus V_0$ and
$V_0 \oplus V_{1} \oplus \dots \oplus V_{s_+}$ exceed $q$ and $p$
respectively. It follows that the former subspace must contain
positive vectors and the latter negative vectors.
Using Proposition 3.2 we conclude that
$\lambda_0 \geq l_+$ and $-\lambda_0 \geq -l_-$.
In the same way we obtain that if $V_0$ is absent then
$\lambda_{1} \geq l_+$ and $ l_- \geq \lambda_{-1}$.
It remains to show the inclusions \thetag{3.2}. First of all we
have the weaker inclusions
$$
V_{-s_-}\oplus \dots \oplus V_{-1} \subset \Cal C_- \cup \Cal C_0, \ \
V_{1} \oplus \dots \oplus V_{s_+} \subset \Cal C_+\cup \Cal C_0.
\tag{3.3}
$$
Indeed, if for example
$V_{-s_-}\oplus \dots \oplus V_{-1}$ contains any positive vectors
then by Proposition 3.2 $\lambda_{-1} \geq l_+$, which violates the
assumption that $l_+ > l_-$.
Further by Proposition 3.3 the assumption $l_+ > l_-$ implies
that the matrix cocycle is eventually strictly $\Cal J$-separated.
By the invariance of the splitting we get \thetag{3.2}
almost everywhere.
\qed\enddemo
It seems plausible that the weaker inclusions \thetag{3.3}
must hold without any further assumptions, but we were unable to prove
it.
\proclaim{Proposition 3.5}
If the matrix cocyle is eventually strictly $\Cal J$-separated
then $V_0$ is absent, $\lambda_1 > \lambda_{-1}$ and
the inclusions \thetag{3.2} hold.
\endproclaim
\demo{Proof}
By Proposition 3.4 the claims must hold for
strictly $\Cal J$-separated cocycles. The general case of the
eventually strictly $\Cal J$-separated cocycles reduces to
the special case by considering the subset
$X_m = \{ x\in X | S^m(x) \ \ \ \text{is strictly} \ \ \ \Cal
J\text{-separated}\}$. For large $m \geq 1$ the set $X_m$ must
have measure close to 1(the full measure).
We further consider the return map
to such a set $X_m$ and the induced matrix cocycle
which is automatically strictly $\Cal J$-separated. But the splitting
for the induced cocycle coincides with the original splitting
restricted to the subset $X_m$ and the Lyapunov exponents
are changed by the factor equal to the average return time
(cf. \cite{K-H}, Lemma S.2.8, page 665).
\qed\enddemo
Let us introduce $d_0^- = q - d_{-1} - \dots - d_{s_-}$ and
$d_0^+ = p - d_{1} - \dots - d_{s_+}$. Clearly
$d_0^- + d_0^+ = \dim V_0$.
\proclaim{Theorem 3.6}
If a measurable cocycle $S(x), \ x \in X$, satisfies the integrability
condition and
it has values in the semigroup of $\Cal J$--separated matrices then
for $ 0 \leq k \leq s_+$
$$
d_0^+\lambda_0 + d_1\lambda_1 + \dots + d_k\lambda_k \geq
\sum_{i=1}^{D}\int \log r_i^+ d\nu,
$$
where $D = d_0^+ + d_1 + \dots + d_k$
and for $ 0 \leq k \leq s_-$
$$
d_0^-\lambda_0 + d_{-1}\lambda_{-1} + \dots + d_{-k}\lambda_{-k} \leq
\sum_{i=1}^{D}\int \log r_i^- d\nu.
$$
where $D = d_0^-+ d_{-1} + \dots + d_{-k}$.
\endproclaim
\demo{Proof of Theorem 3.6}
We will use the formula \thetag{3.1} for the rate of volume growth on
subspaces.
We proceed as in the proof of Proposition 3.2.
Let $ L \subset W_k$ be a subspace transversal to
$ W_{-1}$ and such that $ W_0 + L = W_k$.
An important point is that we can choose such a subspace in the
positive cone $\Cal C_+ $,
if its dimension $D$ is equal to
$D = \dim W_k -q = d_0^+ +d_1 +\dots + d_k$.
Indeed, the family of subspaces of $W_k$ of dimension
$D$ (or equivalently of codimension $q$) which
are transversal to $W_{-1}$, is open and dense in the
respective Grassmanian. At the same time the intersection of $W_k$
with any maximal positive subspace (of dimension $p$)
must be of dimension at least $D$. Hence there are positive subspaces
in $W_k$ of dimension $D$. Moreover the family of such subspaces is
open in the respective Grassmanian. It follows that there must be
positive subspaces of $W_k$ of dimension $D$ transversal to $W_{-1} $
(actually there must be an open family of such subspaces in the
respective Grassmanian).
If the subspace $ L$ is in the positive cone
then the Euclidean volume on it is not smaller than
the $\Cal J$--volume. Using Proposition 1.3 we get
$$
\aligned
\log|det \left(S^m(x)|_ L\right)| &\geq
\log \alpha_D(L,S^m(x) ) + c =
\sum_{i = 0}^{m-1} \log \alpha_D(L , S(\varPhi^i x) ) +c \\
&\geq \sum_{i = 0}^{m-1} \log \sigma_D(S(\varPhi^i x) ) +c
= \sum_{i = 0}^{m-1} (\log r_1^+(\varPhi^i x) + \dots +
\log r_D^+(\varPhi^i x)) + c ,
\endaligned
$$
where the constant $c$ is the logarithm of the ratio of
the $\Cal J$-volume and the Euclidean volume on the subspace $L$.
Hence in view of \thetag{3.1} we obtain the first inequality.
The other inequality is obtained by observing that
$S(x)^{-1}$ are $(-\Cal J)$--separated and applying the same
argument.
\qed\enddemo
If we introduce the $n$ Lyapunov exponents repeated according to
the multiplicities and labeled as
$\mu_{-q} \leq \mu_{-q+1} \leq \dots \leq \mu_{-1} \leq \mu_1 \leq
\dots \leq \mu_{p}$ we can formulate the following
\proclaim{Corollary 3.7}
For $1 \leq k_1 \leq p$ and $1 \leq k_2 \leq q$
$$
\mu_1 + \dots + \mu_{k_1} \geq
\sum_{i=1}^{k_1}\int \log r_i^+ d\nu \ \ \ \text{and} \ \ \
\sum_{i=1}^{k_2}\int \log r_i^- d\nu \geq
\mu_{-1} + \dots + \mu_{-k_2}.
$$
\endproclaim
\demo{Proof}
It follows formally from all the inequalities of Theorem 3.7.
One can also repeat the proof of this theorem
starting from a full noninvariant flag of subspaces
as introduced in the proof of Theorem 3.9 below.
\qed\enddemo
Let us now formulate the symplectic version of the estimates.
Hence we assume that the matrix cocycle $S(x), x \in X,$ has values
in the symplectic group.
In such a case the Lyapunov exponents come in pairs of opposite numbers.
The following proof of this fact is taken from \cite{W-L}.
\proclaim{Lemma 3.8} For an integrable cocycle $S(x), \ x \in X$,
with values in the symplectic group
$Sp(\Bbb R^{2p})$, for any pair of non skew-orthogonal vectors
$u,v \in \Bbb R^{2p}$, i.e., $\omega(u,v) \neq 0$, we have
$$
\lambda(u)+\lambda(v)\geq 0 .
$$
\endproclaim
\demo{Proof}
For the standard Euclidean norm $\|\cdot\|$
we have $|\omega(u,v)| \leq \|u\| \|v\|$. By symplecticity
$$
\log|\omega(S^m(x)u, S^m(x)v)| = \log|\omega(u,v)|,
$$
and
$$
\frac 1m \log|\omega(S^m(x)u, S^m(x)v)| \leq
\frac 1m \log\|S^m(x)u\| + \frac 1m \log\|S^m(x)v\|.
$$
\qed\enddemo
\proclaim{Theorem 3.9}
If an integrable cocycle $S(x), \ x \in X$,
has values in the symplectic group $Sp(\Bbb R^{2p})$
then we have the following symmetry of the Lyapunov spectrum
$$
\lambda_0 = 0, \
\ \ \text{if} \ \ V_0 \ \ \text {is present}, \ \ \
s_+ = s_{-} = s, \ \lambda_{-k} + \lambda_{k} = 0,
$$
and the multiplicities of $\lambda_{-k}$ and $ \lambda_{k}$
are equal, for $k = 1,2, \dots, s$.
Moreover the subspace $W_k = \oplus_{i= -s}^kV_i$ is the skew-orthogonal
complement of $W_{-(k+1)}=\oplus_{i= -s}^{-(k+1)}V_i$.
\endproclaim
\demo{Proof}
Let $\mu_1 \leq \mu_2 \leq \dots \leq \mu_{2p}$ be the Lyapunov
exponents listed with repetitions according to their multiplicities.
We have $\mu_1 + \mu_2 + \dots + \mu_{2p} = 0$.
We can choose a flag of subspaces
$$
\{0\} = Z_0 \subset Z_1(x) \subset \dots \subset Z_{2p-1}(x)
\subset Z_{2p} = \Bbb R^{2p},$$
such that $\dim Z_l =l$ and
for all vectors $v\in Z_l(x) \setminus Z_{l-1}(x)$ the
Lyapunov exponent $\lambda(v) = \mu_l$, for $l =1,2,\dots,2p$.
(Note that except in the case of all multiplicities equal to $1$
there is a continuum of such flags.)
Since for any $l \leq p$,
$\dim Z_l + \dim Z_{2p-l+1} = 2p+1$,
there are vectors $u \in Z_l$ and $v \in Z_{2p-l+1}$
such that $\omega(u,v)\neq 0$. By continuity there must be also
vectors $\tilde u \in Z_l \setminus Z_{l-1}$ and
$\tilde v \in Z_{2p-l+1} \setminus
Z_{2p-l}$ such that $\omega(\tilde u,\,\tilde v)\neq 0$.
It follows from Lemma 3.8 that
$$
\mu_l + \mu_{2p-l+1} \geq 0,
$$
for $l =1,2,\dots, p$.
Adding these inequalities together, we get
$$
0 = \sum_{l=1}^p(\mu_l + \mu_{2p-l+1}) \geq 0,
$$
which shows that all the inequalities must be actually
equalities. It follows immediately that for any $k = 1,\dots, s,$
the multiplicities of $\lambda_k$ and $\lambda_{-k}$ are equal
and $\lambda_k + \lambda_{-k} = 0$.
To show that the subspace
$W_k$ is the skew-orthogonal complement
of the subspace $W_{-(k+1)}$ (denoted by $W_{-(k+1)}^\sk$)
we observe that $\omega(u,\,v) = 0$ for any $u\in W_k$
and $ v \in W_{-(k+1)}$. Indeed, if this is not the case we
could use Lemma 3.8 to claim that
$ \lambda_k + \lambda_{-(k+1)} \geq 0$, which leads to
the contradiction
$$
0 = \lambda_k + \lambda_{-k} > \lambda_k + \lambda_{-(k+1)} \geq 0.
$$
We obtain $ W_k \subset W_{-(k+1)}^\sk$.
Since the dimensions of these subspaces are equal they must coincide.
\qed\enddemo
If all the symplectic matrices
$S(x) = \left(\matrix A(x)& B(x)\\ C(x)& D(x)\endmatrix\right) ,
x \in X,$ are also $\Cal Q$--monotone (with the standard form $\Cal Q$)
then using Proposition 2.4 we can reformulate Theorem 3.6 as
\proclaim{Theorem 3.10}
For any $1 \leq k \leq p$
$$
\mu_1 + \dots + \mu_{k} \geq \sum_{i =1}^k \int \sinh^{-1} \sqrt{t_i}
d\nu,
$$
where $0 \leq t_1(x) \leq \dots \leq t_p(x)$ are the eigenvalues of
$CB^*$.
\endproclaim
\vskip.4cm
\subhead \S 4. The $\Cal J$--separated diffeomorphisms
and flows
\endsubhead
\vskip.4cm
Let us move the discussion to a manifold of dimension $n$
and a diffeomorhism
\hbox{$\varPhi : M^n \to M^n$.} We assume that the manifold is
equipped with a continuous pseudo Riemannian metric of type
$(p,q), p+q = n$, which we will denote again by $\Cal J$.
We obtain the fields of positive and negative cones
$\Cal C_\pm(x), x \in M$.
A diffeomorphism $\varPhi$
is called $\Cal J$--separated if $D\varPhi \Cal C_+(x) \subset
\Cal C_+(\varPhi x)$.
Similarly we define diffeomorphisms which are strictly
$\Cal J$--separated, $\Cal J$--monotone and
strictly $\Cal J$--monotone.
If a diffeomorphism is $\Cal J$--separated then the
pseudo Riemanian singular values are well defined for
the derivative $D\varPhi$ at any point $x \in M$.
We will denote them by
$0 < r_q^-(x) \leq \dots \leq
r_1^-(x) \leq r_1^+(x) \leq \dots \leq r_p^+(x)$.
\proclaim{Proposition 4.1} If a diffeomorphism $\varPhi$ is
strictly $\Cal J$--separated then it has a dominated splitting,
i.e,, it has a continuous
invariant splitting of the tangent bundle
$$ T_xM = E^-(x) \oplus E^+(x),
$$
such that for some $0 < \lambda < 1, c > 0$, and all $m \geq 1$
$$
\|D{\varPhi^m}_{|E^-(x)}\|
\|D{\varPhi^{-m}}_{|E^+(\varPhi^m x)}\| \leq c\lambda^m.
\tag{4.1}
$$
\endproclaim
\demo{Proof}
At every point $x \in M$
we have the manifold of positive subspaces $\Cal L_+(x)$.
We will use the contraction property of the action of $D\varPhi$
on $\Cal L_+(x)$. The diameter of $D\varPhi \Cal L_+(x)$
is finite because $D\varPhi$ is strictly
$\Cal J$--separated, and hence it is uniformly bounded by a constant
$c_0$. It follows from Theorem 1.5 that the diameter of
$ D\varPhi^m \Cal L_+(D\varPhi^{-m} x) $
does not exceed $c_0
\prod_{i= 1}^{m-1} \frac{r_1^-(\varPhi^{-i}x)}{r_1^+(\varPhi^{-i}x)}
$. Since the subsets
$$ D\varPhi^{m+1} \Cal L_+(D\varPhi^{-m-1} x)
\subset D\varPhi^m \Cal L_+(D\varPhi^{-m} x) $$
are nested and their
diameter decays exponentially we get a unique point of intersection
$$
E^+(x) = \bigcap_{m =1}^\infty D\varPhi^m \Cal L_+(D\varPhi^{-m} x).
$$
It is clear that the subspace depends continuously on $x$
(by the standard argument it is actually H\"older continuous).
To obtain the other subspace of the splitting we need to
observe that $\varPhi^{-1}$ is strictly $(-\Cal J)$--separated
and repeat the above argument.
Finally we get \thetag{4.1} with $\lambda =
\sup_x \frac{r_1^-(x)}{r_1^+(x)} < 1$. Indeed by the definition of
the pseudo Euclidean singular values we have for every
vector $v \in T_{\varPhi x}M$ such that $D\varPhi^{-1}v$ is positive
$$
\frac {\|v\|} {\|D\varPhi^{-1}v\|} \geq r_1^+(x).
$$
Since the subspaces $E^+(x)$ are by construction positive
and invariant
we conclude that
$$
\|D{\varPhi^{-1}}_{|E^+(\varPhi x)}\| \leq \frac 1{r_1^+(x) }.
$$
Similarly we can estimate $\|D{\varPhi}_{|E^-(x)}\| $.
\qed\enddemo
As a corollary we obtain the theorem of Lewowicz, \cite{L},
\proclaim{Theorem 4.2} If a diffeomorphism is
strictly $\Cal J$--monotone then it is Anosov.
\endproclaim
\demo{Proof} By Proposition 4.1 we obtain the continuous
invariant splitting. On the unstable subspace the $\Cal J$--form
increases exponentially and it can be compared with any
norm.
\enddemo
Establishing that a dynamical system is
$\Cal J$--separated is an important step,
but this property alone allows varied and complicated
behavior different from an Anosov system. For example
the DA attractors can be made $\Cal J$--separated, \cite{Ro}.
Also the examples studied by Hu and Young, \cite{H-Y},
are $\Cal J$--separated without being Anosov.
In the piecewise differentiable setup with
a chosen invariant measure $\nu$ we get
for a strictly $\Cal J$--separated mapping $\varPhi$
a measurable invariant
splitting $T_xM = E^-(x) \oplus E^+(x)$ defined $\nu$ almost everywhere
and $E^+(x) \subset \Cal C_+ , \
E^-(x) \subset \Cal C_-$.
Now the contraction in the manifold of positive subspaces is
not uniform but by the Birkhoff Ergodic Theorem it is
pointwise exponential almost everywhere.
For an ergodic invariant measure $\nu$ we can obtain
the estimates of the Lyapunov exponents in terms of the
pseudo-Riemannian singular values of the
$\Cal J$--separated $D \varPhi$. We need to assume only
that $D \varPhi$ is $\Cal J$--separated almost everywhere
with respect to the measure.
\proclaim{Theorem 4.3}
For a piecewise differentiable $\Cal J$--separated $\varPhi$
and an an ergodic invariant measure $\nu$, if
$\log \|D\varPhi^{\pm 1}\|$ are integrable functions
then the Lyapunov exponents
$$
\mu_{-q} \leq \mu_{-{q-1}} \leq \dots \leq \mu_{-1}\leq
\mu_1 \leq \mu_2 \leq \dots \leq \mu_{p},
$$
satisfy
$$
\mu_1 + \dots + \mu_{k_1} \geq
\sum_{i=1}^{k_1}\int \log r_i^+ d\nu \ \ \ \text{and} \ \ \
\sum_{i=1}^{k_2}\int \log r_i^- d\nu \geq
\mu_{-1} + \dots + \mu_{-k_2},
$$
for any $k_1 \leq p, k_2 \leq q$, where
the coefficients \hbox{$0 < r_q^-(x) \leq \dots \leq
r_1^-(x) \leq r_1^+(x) \leq \dots \leq r_p^+(x)$} are the
singular values of the
$\Cal J$--separated operators $D_x\varPhi$.
\endproclaim
\demo{Proof}
We have an auxiliary Riemannian metric on $M$
which defines the norm $\| \cdot \|$
in the integrability condition (in $\log \|D\varPhi^{\pm 1}\|$).
We introduce a measurable
Riemannian metric associated to the pseudo Riemannian
metric $\Cal J$ by
choosing a measurable field of positive subspaces $P(x), x\in M$.
and considering their $\Cal J$--orthogonal complements. which
we denote by $Q(x), x \in M,$. The Riemannian metric is equal
to $\Cal J$ on $P$ and $-\Cal J$ on $Q$, and the two subspaces
are orthogonal. Let $\|\cdot \|_J$ denote the respective norm.
If we had the integrability of $\log \|D\varPhi^{\pm 1}\|_J$,
we could apply Theorem 3.6 directly.
It turns out that we do not need to impose any
further integrability conditions.
Indeed, there is a measurable function $c(x) > 0$
such that
$$ \frac 1{c(x)} \| \cdot \| \leq \| \cdot \|_J \leq c(x) \| \cdot \|.$$
For every $\epsilon >0$ consider the subset
$M_\epsilon = \{ x\in M\ | \ c(x) \geq \epsilon\}$ and
the return map \hbox{$\varPhi_\epsilon : M_\epsilon \to M_\epsilon$.}
We clearly have the integrability of
$\log \|D\varPhi_\epsilon^{\pm 1}\|$ which leads to the
integrability of $\log \|D\varPhi_\epsilon^{\pm 1}\|_J$.
Hence we can apply to $D\varPhi_\epsilon$ the estimates
in Theorem 3.6. The Lyapunov exponents
for $D\varPhi_\epsilon$ and $D\varPhi$ differ by the average return
time to $M_\epsilon$ which is close to $1$ for small $\epsilon$.
The final observation is that the average of
the coefficient $\sigma_d(D\varPhi_\epsilon)$
is not smaller than the average of the coefficient
$\sigma_d(D\varPhi)$.
It follows from the
supermultiplicativity \thetag{1.1} of $\sigma_d$.
Hence in the limit of $\epsilon \to 0$ we obtain the desired estimate.
\qed\enddemo
To develop the theory of $\Cal J$--separated flows
$\varPhi^t: M^n \to M^n, t\in \Bbb R,$
we need the velocity vector field $F(x)$
of the flow, $F(x) = \frac{d}{dt}\varPhi^t(x)_{|t=0}$,
to be nonzero everywhere and the form $\Cal J$
to be degenerate. More precisely
the positive index of inertia of $\Cal J$ is equal to $p\geq 1$,
the negative index of inertia is equal to $q \geq 1$ and $p+q = n-1$,
and
for the bilinear form $\Cal J$ the linear form
$\Cal J (F,\cdot)$ vanishes identically.
Equivalently we consider the quotient tangent spaces
$\widetilde T_xM^n = T_xM^n/span\{F(x)\}$ and the nondegenerate
forms $\Cal J$ on them. The derivative $D\varPhi^t$ projects naturally
to the quotient spaces and we can apply the theory developed
previously to the derivative cocycles on the quotient spaces.
In particular if for a smooth flow the derivative cocycle
is strictly $\Cal J$--separated then by Proposition 4.1
we get a splitting of the quotient space, but in general
it does not lift to the splitting of the tangent space.
We can lift the splitting in the case of strict $\Cal J$--monotonicity
by the classical proof of Anosov \cite{A} (see also \cite{W8})
and we obtain the counterpart of the
Lewowicz's Theorem 4.2 for flows.
\proclaim{Theorem 4.4}
If the derivative cocycle of a smooth flow is strictly
$\Cal J$--monotone then the flow is Anosov.
\endproclaim
\vskip.4cm
\subhead \S Appendix. Boltzmann -- Sinai Gas of Hard Spheres
\endsubhead
\vskip.4cm
In our discussion we will follow the approach developed in
\cite{W2} and \cite{W5}.
We consider the system of $N$ spherical particles
in a $d$-dimensional vessel, either a $d$-dimensional
torus or a box.
The particles have arbitrary masses: $m_1,m_2,\dots, m_N$,
and they collide elastically with each other (and the walls of the
box). The Hamiltonian of the system
is $$ H = \sum_{i=1}^N\frac{p_i^2}{2m_i} $$ where $p_i\in
\Bbb{R}^d$ is the momentum of $i$-th particle,
$i= 1,\dots ,N, d \ge 2$.
The Hamiltonian
differential equations are linear
$$ \dot{q_i}= \frac {p_i}{m_i},\ \ \ \ \dot {p_i}= 0, \ \
i=1,...,N.
$$
Our phase space is a linear symplectic space. All its tangent
spaces are naturally identified with it but for the sake of clarity we
will distinguish between a point in the phase space and a tangent
vector by placing $\delta$ in front of $q$ and $p$. We choose the
constant Lagrangian subspaces in the $\left(\qq,\pp\right)$-space
$\Bbb{R}^{dN}\times \Bbb{R}^{dN}$
$$ L_1 \equiv \Bbb{R}^{dN}\times\{0\} = \{ dp_1=\dots =dp_N =0\} $$
and
$$ L_2 \equiv\{0\}\times\Bbb{R}^{dN}= \{ dq_1=\dots =dq_N =0\}.$$
This is the choice which makes the integrable system of noninteracting
particles $\Cal Q$--monotone.
We see that the velocity vector field is in $L_1$. The
quadratic form $\Cal{Q}$ is equal to
$$
\Cal Q = \sum_{k=1}^N\langle dq_k, dp_k\rangle,
$$
where
$\langle dq_k, dp_k \rangle = \sum_{i=1}^ddq_{k,i}dp_{k,i}$.
The linearized equations are
$$
\frac{d}{dt}\qq_i = \frac{\pp_i}{m_i}, \ \ \ \
\frac{d}{dt}\pp_i = 0.
$$
We obtain immediately that between collisions
$$
\frac{d}{dt} Q = \sum_{i=1}^N\frac{(dp_i)^2}{m_i} \geq 0,
$$
which means that the system is $\Cal Q$--monotone.
Moreover it follows that $L_2$ enters $\Cal C_+$ immediately and completely.
We proceed to investigate collisions of two particles in dimension $d
\geq 2$.
Mathematically speaking the collisions are described by a
symplectomorphism defined on the boundary of the phase space. More precisely
not all positions of the balls are allowed since they cannot overlap.
The configurations with at least two particles touching each other form
the boundary of the phase space. The intersection of this
boundary with a given total energy level, say\hbox{ $\{H= \frac{1}{2}\}$,}
has a canonical symplectic structure (its tangent can be identified
with the factor space by the line spanned by the velocity vector).
Collisions are then described by a symplectomorphism of this symplectic
manifold (a gross simplification -- this boundary has many
discontinuities and so we only get piecewise differentiability,
but it is still possible to talk about Lyapunov exponents). So we have
here a combination of a Hamiltonian flow and a symplectomorphism.
Such systems were discussed abstractly in \cite{W3} and they are
called there flows with collisions, the boundary being called the
collision manifold.
Let the colliding particles have positions $q_1,q_2$, momenta
$p_1,p_2$, masses $m_1,m_2$ and radii $r_1, r_2$. The collision
manifold is given locally by $\|q_2-q_1\|=r = r_1+r_2$ and
its tangent space by
$$ \langle e, dq_2-dq_1 \rangle = 0 \ \ \ \ \text{where} \ \ \ \
e=\frac1r (q_2-q_1), \|e\|=1.$$
To facilitate the calculations we introduce locally new
symplectic coordinates $(x,y)$, the center of mass, relative position
and the respective momenta
$$
\aligned
x_1 = \frac{m_1}{m_1+m_2}q_1+\frac{m_2}{m_1+m_2}q_2, \ \ &
p_1 = \frac{m_1}{m_1+m_2}y_1 -y _2\\
x_2 = q_2-q_1,
\ \ \ \ \ \ \ \ \ \ \ \ \ \ & p_2 = \frac{m_2}{m_1+m_2}y_1 +y_2.
\endaligned
$$
In these coordinates the collision manifold is $\|x_2\| =r$, its
tangent space $\langle x_2,dx_2\rangle = 0$ and the collision map
is given by the formulas
$$
y_1^+ = y_1^-, \ \ \ y_2^+ = y_2^-
- \frac2{r^2}\langle y_2^-,x_2\rangle x_2.
$$
Differentiating we obtain
$$
dy_2^+ = dy_2^- - \frac2{r^2}\langle dy_2^-,x_2\rangle x_2
- \frac2{r^2}\langle y_2^-,dx_2\rangle x_2
- \frac2{r^2}\langle y_2^-,x_2\rangle dx_2
\tag{A.1}
$$
Since we have that
$$
Q = \langle dq_1,dp_1 \rangle + \langle dq_2,dp_2 \rangle + \dots =
\langle dx_1,dy_1 \rangle + \langle dx_2,dy_2 \rangle + \dots,
$$
we obtain on the collision manifold
$$
\aligned
Q^+ - Q^- &=
\langle dx_2,dy_2^+ \rangle -\langle dx_2,dy_2^-
\rangle =
- \frac2{r^2}\langle y_2^-,x_2\rangle \langle dx_2,dx_2\rangle \\
&= - \frac{2m_1m_2}{(m_1+m_2)r^2}
\langle \frac{p_2^-}{m_2} - \frac{p_1^-}{m_1} ,
q_2-q_1 \rangle \langle dq_2-dq_1 , dq_2-dq_1\rangle \geq 0.
\endaligned
\tag{A.2}
$$
Thus monotonicity is verified also for the collisions. (The inequality
\hbox{$\langle \frac{p_2^-}{m_2} - \frac{p_1^-}{m_1} ,
q_2-q_1 \rangle > 1$} holds automatically for all nondegenerate collisions.
It simply means that the particles are indeed colliding and not flying
away from each other.)
Let us note that when approaching
collision before we can apply \thetag{A.1} and \thetag{A.2}
to a vector
$\left(\qq,\pp\right)$ we have to project it onto the tangent space of
the collision manifold, $\langle e, dq_2- dq_1 \rangle = 0,$ along the
velocity vector of the flow.
$$ \left(\qq_i,\pp_i\right) \mapsto
\left(\qq_i - \lambda \frac{p_i}{m_i},\pp_i\right), \ \ \ \lambda
=\frac{\langle \qq_2 -\qq_1,e \rangle }{\langle \frac{p_2}{m_2} -
\frac{p_1}{m_1},e \rangle}.$$
This projection reflects the fact that nearby trajectories do not
arrive at the collision at the same time. It does not change the value
of the form $\Cal{Q}$ but it does change the tangent vector.
The formula for the increase of the form $\Cal{Q}$ shows
that if a vector in $L_1$ does not enter $\Cal{C}_+$ as a result of a
nondegenerate collision of the first two particles then
$$\qq_2^--\qq_1^- = 0.$$ If we take into account the necessary
projection preceding the application of the derivative of the collision map it means
that
$\qq_2-\qq_1 $ is parallel to $\frac{p_2}{m_2}-\frac{p_1}{m_1}$ (then the projection
makes the two components equal). Thus in every nondegenerate collision
of two particles all vectors from $L_1$ enter $\Cal C_+$ except for
the subspace of codimension $d-1$ which stays in $L_1$.
It is clear that there are many
special orbits and vectors in $L_1$ which do not enter $\Cal{C}$ for a
long time (ever).
Establishing strict monotonicity for almost all orbits
in the system is difficult. This result was obtained by
Sim\'anyi and Sz\'asz, \cite{S-Sz}.
One obstacle to strict monotonicity is the possibility that
the particles are split into two families with all the collisions
occuring inside one of the two families.
Let us denote the union of orbits on which no such splitting
is possible by $Z$.
Sim\'anyi and Sz\'asz, \cite{S-Sz}, Theorem 5.1, discovered a short and
transparent proof of the fact
that for the system on the torus the set $Z$ has
full measure.
Using Proposition 2.5 and Theorem 3.10 we can easily prove the
following
\proclaim{Proposition A.1}
There are at least $(N-1)(d-1)$ positive Lyapunov exponents
almost everywhere on the set $Z$.
\endproclaim
\demo{Proof}
We have established that each nondegenerate collision
pushes vectors from $L_1$ into $\Cal C_+$ except for a
codimension $d-1$ subspace which stays in $L_1$.
This exceptional subspace evolves then inside $L_1$
until the next collision. At that time some vectors
from this evolved exceptional subspace are pushed into $\Cal C_+$
unless the new exceptional subspace coincides with it.
In general we will obtain a new exceptional subspace
of higher codimension. The question is what is the guaranteed
codimension of this exceptional subspace after many collisions.
Each collision contributes $d-1$ linear conditions (equations)
only in variables corresponding to the particles involved in the
collision. During the evolution between collisions these
equations change but no new variables may be introduced into them.
On orbits from $Z$ after some time all the particles are
connected by a chain of collisions. Let us record the
equations determining the evolved exceptional subspaces
generated by a minimal chain of $N-1$ collisions connecting all the
particles. We obtain $(N-1)(d-1)$ equations. We claim that they
are linearly independent. Indeed, suppose $k$ particles
have collided generating $(k-1)(d-1)$ equations. These equations
contain only the variables associated with the $k$ particles.
Note that in the forward evolution these equations are changed
but no new variables may be added into them until a new particle collides
with one of these $k$ particles.
A collision with a new particle generates
$d-1$ equations linearly independent in the $d$ variables
associated with the new particle. Hence these new $d-1$ equations
are linearly independent of the previous equations (because they
do not depend on these new variables). We conclude that
after the $N-1$ collisions in the minimal chain of collisions
we will have $(N-1)(d-1)$ linearly independent equations.
It follows from the above discussion that on orbits from
$Z$ after sufficiently long time all the vectors from $L_1$
are pushed into $\Cal C_+$ except possibly for a subspace
of codimension at least $(N-1)(d-1)$.
Our Propositions follows now from Proposition 2.5 and Theorem 3.10.
\qed\enddemo
Note that in the above proof no distinction is made between
the cases of the torus and the cube as vessels.
In the case of the torus the total momentum is the first integral
of the system which leads to $2d$ zero Lyapunov exponents.
Complete hyperbolicity has not yet been established for the case of
particles in a box, although there is no doubt that it does
hold. The special case of two particles was studied by
Sim\'anyi, \cite{S2}.
\Refs
\widestnumber\key{XXXX}
\ref \key{A} \by D.V. Anosov
\paper Geodesic flows
on Riemannian manifolds with negative
curvature
\jour Proc. Steklov Inst. Math. \vol 90 \yr 1967
\endref
\ref \key{B} \by P. Bougerol \paper Kalman filtering with random
coefficients and contractions
\jour Siam J. Control Optim.
\pages 942 -- 959
\vol 31
\yr 1993
\endref
\ref \key{Ch-S} \by N.I.Chernov, Ya.G.Sinai \paper Ergodic
properties of some systems of 2-dimensional discs and 3-dimensional
spheres \jour Russ.Math.Surv. \vol 42 \yr 1987 \pages 181-207
\endref
\ref \key{C} \by W.A. Coppel
\paper Dichotomies in stability theory
\jour Lecture Notes in Math. 629,
Springer-Verlag
\yr 1978
\endref
\ref \key{H-Y} \by H.Y. Hu, L.-S. Young
\paper Nonexistence of SBR measures for some diffeomorphisms
that are ``almost Anosov''
\pages 67 -- 76
\yr 1995
\jour Erg. Th. Dyn. Sys.
\vol 15
\endref
\ref\key{K-H}\by A. Katok, B. Hasselblatt \book
Introduction to the modern theory of dynamical systems
\publ Cambridge UP \yr 1995
\endref
\ref \key{Ko} \by A. Kor\'anyi
\paper A Schwarz lemma for bounded symmetric domains
\pages 210 - 213
\yr 1966
\jour Proc. AMS
\vol 17
\endref
\ref\key{L}\by J. Lewowicz \paper Lyapunov functions and topological
stability
\jour J.Diff.Eq. \vol 38 \yr 1980 \pages 192 -- 209
\endref
\ref\key{L-W}\by C. Liverani, M.P. Wojtkowski
\paper Generalization of the Hilbert metric to the
space of positive definite matrices
\jour Pacific Journal of Math.
\vol 166
\yr 1994 \pages 339 -355
\endref
\ref\key{M}\by R. Markarian \paper Billiards with Pesin
region of measure one
\jour Commun.Math.Phys. \vol 118 \yr 1988 \pages 87 -- 97
\endref
\ref\key{P1} \by V.P. Potapov
\paper The multiplicative structure
of J--contractive matrix functions
\jour AMS Translations (2)
\vol 15
\yr 1960
\pages 131 -- 243
\endref
\ref\key{P2} \by V.P. Potapov
\paper Linear fractional transformation of matrices
\jour AMS Translations (2)
\vol 138
\yr 1988
\pages 21 -- 35
\endref
\ref\key{P3} \by V.P. Potapov
\paper A theorem on the modulus. I
\jour AMS Translations (2)
\vol 138
\yr 1988
\pages 55 -- 65
\endref
\ref\key{Py} \by I.I. Pyatetskii-Shapiro
\book Automorphic functions and the geometry of
classical domains
\publ Gordon \& Breach
\yr 1969
\endref
\smallskip\noindent
\ref\key{Ra}\by M.-S. Raghunathan,
\paper A proof of Oseledec's multiplicative
ergodic theorem
\jour Israel J. Math.
\vol 32
\yr 1979
\pages 356 - 362
\endref
\ref\key{Ro}\by C. Robinson
\book Dynamical Systems: Stability, Symbolic Dynamics and Chaos
\publ CRC Press \yr 1995
\endref
\ref\key{Ru}\by D. Ruelle,
\paper Ergodic theory of differential dynamical
systems
\jour Publ. Math. IHES
\vol 50
\yr 1979 \pages 275 - 306
\endref
\ref\key{S1}\by N. Sim\'anyi \paper The Characteristic Exponents
of the Falling Ball Model
\jour Commun.Math.Phys. \vol 182 \yr 1996 \pages 457 -- 468
\endref
\ref\key {S2}
\by N. Sim\'anyi
\paper Ergodicity of Hard Spheres in a Box
\jour Ergodic theory and dynamical systems
\vol 19
\yr 1999
\pages 741-766
\endref
\ref\key{Su}\by M. Suzuki
\paper The intrinsic metrics on the circular domains in
$\Bbb C^n$
\jour Pacific Journal of Math.
\vol 112
\yr 1984 \pages 249 -256
\endref
\ref\key {Sa-K}
\by A.M. Samoilenko, V.L. Kulik
\paper Exponential dichotomy of an invariant torus of dynamical
systems
\jour Differential Equations
\vol 15
\yr 1979
\pages 1019 - 1025
\endref
\ref\key {S-Sz}
\by N. Sim\'anyi, D. Sz\'asz
\paper Hard Ball Systems Are Completely Hyperbolic
\jour Annals of Math.
\vol 149
\yr 1999
\pages 35-96
\endref
\ref\key{W1}\by M.P. Wojtkowski
\paper Invariant families of cones and Lyapunov exponents
\jour Ergodic Theory and Dynamical Systems
\vol 5
\yr 1985\pages 145--161
\endref
\ref\key{W2}\by M.P. Wojtkowski\paper Measure Theoretic Entropy
of the system of hard spheres
\jour Ergodic Theory and Dynamical Systems\vol 8
\yr 1988\pages 133--153
\endref
\ref \key{W3} \by
M.P. Wojtkowski \paper A system of one dimensional balls with gravity
\jour Commun.Math.Phys. \vol 126 \yr 1990 \pages 507 -- 533 \endref
\ref \key{W4} \by
M.P. Wojtkowski \paper The system of one dimensional balls in an external
field. II
\jour Commun.Math.Phys. \vol 127 \yr 1990 \pages 425 -- 432 \endref
\ref \key{W5} \by
M.P. Wojtkowski \paper Systems of classical interacting particles
with nonvanishing Lyapunov exponents \pages 243 -- 262
\yr 1991 \jour Lecture Notes in Math. 1486,
Springer-Verlag \paperinfo Lyapunov Exponents, Proceedings,
Oberwolfach 1990, L. Arnold, H. Crauel, J.-P. Eckmann (Eds)
\endref
\ref \key{W6} \by
M.P. Wojtkowski \paper Hamiltonian systems with linear potential
and elastic constraints
\jour Fundamenta Mathematicae \vol 157 \yr 1998 \pages 305 -- 341 \endref
\ref \key{W7} \by
M.P. Wojtkowski \paper Complete hyperbolicity in Hamiltonian systems
with linear potential and elastic constraints
\jour Reports on Mathematical Physics
\vol 44 \yr 1999 \endref
\ref \key{W8} \by
M.P. Wojtkowski \paper Magnetic flows and Gaussian thermostats
\paperinfo{to appear in Fund. Math.} \yr 2000\endref
\ref \key{W9} \by
M.P. Wojtkowski \paper W-flows on Weyl manifolds and Gaussian thermostats
\paperinfo{preprint} \yr 2000\endref
\ref\key{W-L}\by M.P. Wojtkowski, C. Liverani
\paper Conformally symplectic dynamics
and symmetry of the Lyapunov spectrum
\jour Commun. Math. Phys.
\vol 194
\yr 1998 \pages 47 -- 60
\endref
\endRefs
\enddocument