\input amstex
\documentstyle {amsppt}
\magnification \magstep1
\openup3\jot
\NoBlackBoxes
\pageno=1
\def\wnabla{\widehat{\nabla}}
\def\weta{\widehat\eta}
\topmatter
\title
W -- Flows on Weyl Manifolds and Gaussian Thermostats
\endtitle
\rightheadtext{W -- Flows in Weyl Spaces}
\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 26, 2000
\enddate
\abstract
We introduce W-flows, by modifying the geodesic flow on
a Weyl manifold, and show that they coincide with the
isokinetic dynamics. We establish some connections between
negative curvature of the Weyl structure and the
hyperbolicity of W-flows, generalizing in dimension 2
the classical result of
Anosov on Riemannian geodesic flows. In higher dimensions
we establish only weaker hyperbolic properties.
We extend the theory to billiard W-flows and introduce the
Weyl counterparts of Sinai billiards.
We obtain that the isokinetic Lorentz gas with the constant
external field $E$ and scatterers of radius $r$, studied
by Chernov, Eyink, Lebowitz and Sinai in \cite{Ch-E-L-S},
is uniformly hyperbolic, if only $r|E| < 1$, and this condition is
sharp.
\endabstract
\endtopmatter
\document
\subhead \S 1. Introduction \endsubhead
\vskip.4cm
A Weyl structure on a manifold generalizes a Riemannian metric. It is
a torsion free connection with the parallel transport preserving a
given conformal class of metrics. A presentation of Weyl manifolds in
the modern language was given by Folland, \cite{F}.
Fixing a Riemannian metric in this class we introduce the W-flow on
the unit tangent bundle whose orbits are geodesics of the Weyl
connection equipped with unit tangent vectors. Such flows turn out to
be identical with Gaussian thermostats, or isokinetic dynamics,
introduced by Hoover \cite{H}. Isokinetic dynamics provides
useful models in nonequilibrium statistical mechanics,
discussed in the papers of Gallavotti and Ruelle, \cite{G},\cite{R},
\cite{G-R}.
With a fixed Riemannian metric the Weyl structures $\wnabla$
are parametrized by 1-forms on the manifold or equivalently by
vector fields $E$
$$
\wnabla_XY = \nabla_XY +
\langle X, E \rangle Y + \langle Y, E\rangle X
- \langle X , Y\rangle E,
$$
where $X,Y$ denote arbitrary vector fields, $\langle , \rangle$
denotes the chosen Riemannian metric and $\nabla$ is its Levi-Civita
connection. If we multiply the metric by a function $e^{-2U}$, then
the vector field $E$ is replaced by $e^{2U}(E + \nabla U)$. Hence, if
the vector field $E$ has a potential i.e., $E = -\nabla U$ then the
Weyl structure coincides with the Levi-Civita connection of the
rescaled metric. This leads to the local variational formulation of
the isokinetic dynamics obtained by Choquard, \cite{Cho}. Since the
geodesic flows are Hamiltonian when considered on the cotangent bundle
we get the Hamiltonian formulation of the Gaussian thermostat obtained
by Dettmann and Morriss, \cite{D-M}. This also connects us with the
conformally symplectic formulation of the isokinetic dynamics in
\cite{W-L}. Thus the point of view presented in this paper
generalizes the previous interpretations of isokinetic dynamics.
Using the Jacobi equations for a geodesic flow we obtain the
linearized equations for W-flows in terms of the curvature tensor of
the Weyl connection. The Weyl curvature operator has a symmetric
component (distance curvature) which is a multiple of the identity.
The antisymmetric part of the curvature leads to sectional curvatures
in planar directions, which can be defined only with respect to some
fixed metric. However the sign of the sectional curvatures has
absolute character. Sectional curvatures for the Weyl structures
defined by a constant vector field on a flat two dimensional torus are
zero but it is not the case in higher dimensions. We obtain negative
sectional curvature for any plane which does not contain the vector
field $E$.
We investigate the Anosov property for the W-flows under the
assumption that the sectional curvatures are negative. In dimension 2
we obtain the counterpart of the Anosov theorem for geodesic flows,
negative sectional curvatures do imply that the W-flow is Anosov. In
dimension $n \geq 3$ we obtain only two invariant distributions of $n$
dimensional tangent subspaces, with one of them repelling and the
other contracting most of other distributions, i.e., we obtain the
dominant splitting for the flow (cf. \cite{\~M}). We establish
additionally that volumes are expanded on one invariant distribution
ond contracted on the other. We were unable to obtain any further
hyperbolic properties for the dynamics. We conjecture that there are
$3$ dimensional manifolds and vector fields $E$ such that the
sectional curvatures of the corresponding Weyl structure are negative
but the W-flow is not Anosov.
An interesting example is furnished by negatively curved surfaces with
a harmonic 1-form representing a nonzero cohomology class. The
corresponding vector field gives no contribution into the sectional
curvatures of the Weyl connection so that we can multiply the field by
a large scalar and the resulting flows are always Anosov. These
systems were studied by Bonetto, Gentile and Mastropietro,
\cite{B-G-M}, with the goal of establishing the singular nature of the
SRB measure for small fields. We obtain it for fields of arbitrary
strength. It would be interesting to find the asymptotic behavior of
the SRB measures for these flows. We conjecture that there is a limit
measure supported on the integral curves of the field.
We apply these ideas to the periodic Lorentz gas driven by
an external field with the addition of the Gaussian
thermostat, the system studied numerically by Moran and Hoover,
\cite{M-H}, and analyzed rigorously ly by Chernov, Eyink,
Lebowitz and Sinai, \cite{Ch-E-L-S}, in the case of a weak field.
It turns out that in the plane the conformal mapping by $e^z$
transforms the trajectories of the system into straight segments and
hence the character of the system depends on the convexity of the
transformed scatterers. Small disks are mapped by $e^z$ into convex
domains but beyond radius 1 they develop a dip on one side. This dip
is responsible for the stable periodic orbits observed numerically in
\cite{M-H} for sufficiently large fields. We obtain explicit and sharp
conditions for the uniform hyperbolicity of the system, which paves
the way for extending the analysis in \cite{Ch-E-L-S} beyond the case
of very small fields.
In higher dimensions the convexity of the scatterers with respect to
the Weyl structure is apparently not sufficient to guarantee the
complete hyperbolicity. More work is required to resolve this
issue.
\bigskip
\subhead \S 2. Gaussian thermostats and W -- flows \endsubhead
\bigskip
Let us consider a compact $n$-dimensional Riemannian manifold $M$
and its
tangent bundle $TM $. The metric $g$ will be also
denoted by $\langle \cdot, \cdot \rangle $.
For a smooth vector field $E$ on $M$
we introduce the flow on $TM$
defined by the equations
$$
\frac{dq}{dt} = v, \frac{Dv}{dt} = E,
$$
where $\frac{D}{dt}$ denotes the covariant derivative $\nabla_v$.
We further impose the preservation of the kinetic energy $H = \frac 12
v^2$ via the Gauss least constraint principle, \cite{H}. We obtain
{ \it the Gaussian thermostat} or {\it the isokinetic dynamics}.
$$
\frac{dq}{dt} = v, \frac{Dv}{dt} =
E- \frac{\langle E,v\rangle}{v^2}v.
$$
We fix the value of the kinetic energy, $v^2 = c$, so that the phase space
becomes the sphere bundle. By rescaling time we can assume that
$v^2 = 1$ and then the vector field $E$ is multiplied by the factor
$1 \over c$. We will be considering
Gaussian thermostats only in this restricted phase space,
i.e, in the unit tangent bundle $SM$ of $M$.
Let $\varphi$ be the 1-form associated with the vector field $E$,
i.e., $\varphi(\cdot) = \langle E, \cdot \rangle$.
Together with the Riemannian metric
it defines a Weyl structure on $M$, which is a linear torsion free
connection $\wnabla$ given by the formula (cf. \cite{F})
$$
\wnabla_XY =
\nabla_XY + \varphi(Y)X +
\varphi(X)Y - \langle X , Y\rangle E,
$$
for any vector fields $X, Y$ on $M$ (by $\nabla$ we denote the
Riemannian connection).
The Weyl structure is usually introduced on the basis
of the conformal class of $g$ rather than $g$ itself,
but in our study we fix the Riemannian metric, which plays the role
of a physical space. If we change the metric $g$ by the factor $e^{-2U}$
to $\widetilde g = e^{-2U}g $, then
the 1-form $\varphi(\cdot) = \langle E, \cdot \rangle$
is replaced by $\widetilde\varphi = \varphi + d U$.
Hence, if the vector field $E$ has a potential i.e.,
$E = -\nabla U$ then the Weyl structure coincides with
the Levi-Civita connection of the rescaled metric $\widetilde g$.
The defining property of the Weyl structure is (cf. \cite{F})
$$
\wnabla_X g = - 2\varphi(X) g. \tag{2.1}
$$
We consider the geodesics of the Weyl connection. They are given
by the equations in $TM$
$$
\frac{dq}{ds} = w, \ \ \ \frac{\widehat Dw}{ds} = 0,\tag{2.2}
$$
where $\frac{\widehat D}{ds}$ denotes the covariant
derivative $\wnabla_w$.
These equations provide geodesics with distinguished parameter
$s$ defined uniquely up to scale. It follows from \thetag{2.1}
and \thetag{2.2} that
$\frac{d|w|}{ds} = - \varphi(w)|w|.$
Assuming that at the initial point $q(0)$ we have $|w| = 1$ we obtain
$$
|w| = e^{-\int_{q(0)}^{q(s)}\varphi}.
$$
This formula shows that unless the form $\varphi$ is exact we should
not expect
the geodesic flow in $TM$ of a Weyl connection to preserve any
sphere bundle.
We introduce
the flow
$$\Phi^t: SM \to SM,$$
called the {\it W-flow}, by
parametrizing the geodesics of the Weyl connection with
the arc length of $g$. In other words the projection of a trajectory
of $\Phi^t$ to $M$ is a geodesic of the Weyl connection,
$t$ is the arc length parameter defined by the metric $g$
and the trajectory itself is the natural lift of the geodesic to
$SM$.
\proclaim{Theorem 2.1}
The W-flow on $SM$ coincides with the Gaussian thermostat
on $SM$
$$
\frac{dq}{dt} = v, \ \ \ \frac{Dv}{dt} =
E- \langle E,v\rangle v, \tag{2.3}
$$
\endproclaim
\demo{Proof}
Let us consider a geodesic $q(s), w(s) = \frac{dq}{ds},$ of the Weyl
connection. The parameter $s$ is related to
the arc length parameter $t$ by $\frac{dt}{ds} = |w|$. We obtain
$$
\frac{dq}{dt} = v = \frac{w}{|w|}, \ \ \
\nabla_vv = \frac 1{w^2}\nabla_ww + \varphi(v)v = E - \varphi(v)v,
$$
It shows that the reparametrized geodesic satisfies \thetag{2.3}
\qed
\enddemo
The W-flow (just like the geodesic flow)
is {\it reversible} in the following sense.
If we denote by $\Psi : SM \to SM$ the involution $\Psi(q,v) = (q,-v)$
then we have
$$
\Phi^t\Psi = \Psi\Phi^{-t}. \tag{2.4}
$$
In the case of a potential vector field $E = -\nabla U$,
the Weyl connection is the Levi-Civita connection of the
metric $\widetilde{g} = e^{-2U}g$ and hence up to reparametrization the
trajectories of the W-flow are the geodesics of this metric.
The interesting case is that of a field with only local potential,
i.e., when $\varphi$ is closed but not exact. In such a situation the
metric $\widetilde g = e^{-2U}g$ is only local. Note that the
Levi-Civita connection and the geodesics are not changed,
if we multiply the Riemannian
metric by a constant factor, so that although the metric is local the
connection and the geodesics are globally defined. To
obtain the W-flow we parametrize these geodesics by the arc length
with respect to the background metric $g$.
\proclaim{Example 2.2}\endproclaim
Let $\Bbb T^2$ be the flat torus with
coordinates $(x,y) \in \Bbb R^2$ and $E = (a,0)$ be the constant
vector field on $\Bbb T^2$ then the equations of the W-flow
$$
\ddot{x} = a\dot{y}^2 , \ \ \ddot{y} = -a\dot{x}\dot{y},
$$
can be integrated and we obtain as
trajectories translations of the curve $$ax = - \ln \cos ay$$
or the
horizontal lines. Assuming that $E$ has
irrational direction on $\Bbb T^2$ we obtain the following global
phase portrait for the W-flow. In the unit tangent bundle $S\Bbb T^2 =
\Bbb T^3$ we have two invariant tori $A$ and $R$ with the minimal
quasiperiodic motions, $A$ contains the unit vectors in the direction
of $E$ and it is a global attractor and $R$ contains the unit vectors
opposite to $E$ and it is a global repellor. We will establish
later on (proof of Proposition 5.9) that these invariant
submanifolds are normally hyperbolic so that the phase portrait is
preserved under perturbations.
This example reveals the major departure
from geodesic flows and Hamiltonian dynamics.
W-flows may contract phase volume and
they may have no absolutely continuous invariant measure.
\proclaim{Example 2.3}\endproclaim
More generally, let $N$ be an arbitrary Riemannian manifold and let
$M = N \times S^1$ with the product metric. Let $E$ be the
vector field tangent to $S^1$ of constant magnitude $|E|$.
For a velocity vector $v \in SM$ let $v = v_1 + v_0$
be the splitting into the component $v_1$ parallel to $N$ and the
component $v_0$ parallel to $S^1$. We can immediately
calculate how $v_0$ depends on time. Assuming that at $t =0, \
v_0 = 0$ we get $v_0 = \tanh |E|t$. Moreover the projection of a
trajectory of the W-flow onto $N$ is an open segment of a geodesic
of length $\frac \pi {|E|}$, with the exception of the trajectories
along $S^1$ which project to a single point. Just as in Example 2.2
we obtain the global attractor and the global repellor.
The attractor and repellor are also normally hyperbolic,
if only $|E|$ is sufficiently large.
\proclaim{Example 2.4}\endproclaim
It was calculated in \cite{W1} that the rate of dilation
of the phase volume in $SM$ (the divergence of the velocity vector
field) for isokinetic dynamics (a W-flow in our new terminology) is
equal to $-(n-1)\langle E,v \rangle$ ($n$ is the dimension of $M$).
We have
$$
\frac{d}{dt}\langle E, v \rangle =
\langle \nabla_v E, v \rangle + E^2 - \langle E, v \rangle^2,
$$
Assuming that $E$ does not vanish and multiplying it by
a sufficiently large positive scalar we obtain the flow which
crosses the submanifold $\langle E, v\rangle = 0$ in one
direction only. Moreover on one side we have volume expansion and on the
other volume contraction.
Hence in particular such flows cannot have absolutely continuous
invariant measures.
Let $E$ be a nonvanishing Killing vector field on
a Riemannian manifold $M$. Then $\langle \nabla_X E, Y\rangle $
is an antisymmetric function of $X,Y$. Thus we obtain
$$
\frac{d}{dt}\langle E, v \rangle =
E^2 - \langle E, v \rangle^2 \geq 0,
$$
and the directed crossing of the submanifold $\langle E, v \rangle$
follows, independently of the magnitude of $E$.
\vskip.4cm
\subhead \S 3. Jacobi equations and curvature
\endsubhead
\vskip.4cm
We will obtain the linearized equations of the W-flow from
the Jacobi equations for the geodesic flow of a torsion free
connection. Let us consider a family of geodesics of the Weyl
connection
$$q(s,u),w(s,u)= \frac{dq}{ds}, |u| < \epsilon.$$
We introduce the Jacobi field
$$\xi = \frac{dq}{du} \ \ \ \text{ and } \ \ \ \weta = \wnabla_\xi
w.$$
Note that
$\wnabla_\xi w = \wnabla_w\xi$ since our connection $\wnabla$
has zero torsion and the fields $w$ and $\xi$ commute.
We have the Jacobi equations
$$
\frac{\widehat D\xi}{ds} = \weta, \ \ \
\frac{\widehat D\weta}{ds} = - \widehat{R}(\xi,w)w ,
\tag{3.1}
$$
where for any tangent vector fields $X,Y$,
$$\widehat{R}(X,Y) = \wnabla_X\wnabla_Y - \wnabla_Y\wnabla_X -
\wnabla_{[X,Y]}$$
is the curvature tensor of the Weyl connection.
Let us split the vector field $\weta = \weta_0 + \weta_1$ into
the component $\weta_1$ orthogonal to $w$ and the component
$\weta_0$ parallel to $w$. The equations \thetag{3.1} can be split
accordingly
$$
\frac{\widehat D\weta_1}{ds} = - \widehat{R}_a(\xi,w)w, \ \ \
\frac{\widehat D\weta_0}{ds} = - \widehat{R}_s(\xi,w)w,
\tag{3.2}
$$
where $\widehat{R}_a(X,Y)$ is the antisymmetric and
$\widehat{R}_s(X,Y)$ the symmetric part of the
the Weyl curvature operator $\widehat{R}(X,Y)
= \widehat{R}_a(X,Y) + \widehat{R}_s(X,Y)$
($\widehat{R}_s$ is called the distance curvature and
$\widehat{R}_a$ the direction curvature, cf. \cite{F}).
To obtain \thetag{3.2} let us observe that
since $\weta_1$ is parallel to $w$ also
$ \frac{\widehat D\weta_1}{ds}$ is parallel to $w$.
Further we calculate
$$
\langle \frac{\widehat D\weta_0}{ds}, w\rangle =
\frac{d}{ds} \langle \weta_0,w\rangle + 2\varphi(w)\langle
\weta_0,w\rangle = 0.$$
Now \thetag{3.2} follows from the general fact that
$\widehat{R}_a(w,\xi)w$ is
orthogonal to $w$
and from $\widehat{R}_s(w,\xi)w$ being parallel to $w$
which is a consequence
of the fact that the distance curvature is
a multiple of identity (cf. \cite{F} and Proposition 3.2 below).
We will now calculate straightforwardly the
curvature tensor of the Weyl connection.
\proclaim{Proposition 3.2} $$
\aligned
\widehat{R}_a(X,Y)Z =& R(X,Y)Z +
\langle Z,E\rangle (\langle Y, E \rangle X - \langle X, E \rangle Y)+\\
&(\langle Z, Y\rangle\langle X, E\rangle -
\langle Z, X\rangle\langle Y, E\rangle)E +
E^2(\langle Z,X\rangle Y - \langle Z,Y\rangle X)+\\
&\langle Z,\nabla_X E \rangle Y -
\langle Z ,\nabla_Y E \rangle X + \langle Z , X\rangle \nabla_Y E
- \langle Z ,Y\rangle \nabla_XE.\\
\widehat{R}_s(X,Y)Z = &(\langle Y,\nabla_XE\rangle
-\langle X,\nabla_YE\rangle) Z = -d\varphi(X,Y)Z
\endaligned
$$
\endproclaim
\qed
Together with the family of Weyl geodesics let us consider the
respective family of trajectories of the W-flow
$$q(t,u),v(t,u) = \frac{dq}{dt}, |u| < \epsilon.$$
The Jacobi field
is again $\xi = \frac{dq}{du}$. Letting $\eta = \nabla_\xi v
= \nabla_v\xi$ we can
consider $(\xi,\eta)$ as coordinates in the tangent space of $SM$,
which is described by $\langle v, \eta \rangle = 0$.
We introduce
$$\chi = \eta + \langle
E,v \rangle \xi - \langle \xi,v \rangle E$$ and use $(\xi,\chi)$
as linear coordinates in the tangent bundle of $SM$.
Note that $\langle v, \eta \rangle
= 0$ is equivalent to $\langle v, \chi \rangle = 0$ and
in these new coordinates the velocity vector field of the W-flow
\thetag{2.3} is simply $(v,0)$.
The meaning of $\chi$ is revealed in the following proposition
describing the linearization of a W-flow.
\proclaim{Proposition 3.3} We have that $\chi = \frac{\weta_1}{|w|}$,
i.e., $\chi$ is the component of
$\wnabla_\xi v$ orthogonal to $v$,
and
$$
\frac{\widehat D\xi}{dt} = \chi + \varphi(\xi)v, \ \ \
\frac{\widehat D\chi}{dt} = - \widehat{R}_a(\xi,v)v +\varphi(v)\chi.
\tag{3.3}
$$
\endproclaim
\demo{Proof}
Using the equations \thetag{3.2} we obtain
$$\frac{\widehat D\xi}{dt} = \wnabla_v\xi = \frac{\weta}{|w|} =
\frac{\weta_0}{|w|} + \frac{\weta_1}{|w|}.$$
On the other hand
$$
\wnabla_v\xi = \nabla_v\xi + \varphi(v)\xi + \varphi(\xi)v - \langle v,
\xi\rangle E = \chi + \varphi(\xi)v,
$$
which leads to $\chi = \frac{\weta_1}{|w|}$ and the first
equation of \thetag{3.3}.
Using \thetag{3.2} we get
$$
\frac{\widehat D\chi}{dt} =\wnabla_v\chi =
\frac{1}{w^2}\wnabla_w\weta_1 + \frac{d}{dt}(\frac 1{|w|}) \weta_1 =
- \widehat{R}_a(\xi,v)v +\varphi(v)\chi.
$$
\qed
\enddemo
The Weyl parallel transport along a path is a conformal
linear mapping and the coefficient of dilation is
equal to $e^{-\int \varphi}$. We choose an orthonormal
frame $v, e_1, \dots , e_{n-1}$ in an initial tangent space $T_{q_0}M$
and parallel transport it along a trajectory of
our W-flow in the direction
$v \in SM$. We obtain the orthogonal frames which we
normalize by the coefficient $e^{\int \varphi}$ and denote them
by $v(t),e_1(t),\dots,e_{n-1}(t)$. Let $(\xi_0,\xi_1,\dots,
\xi_{n-1})$ and $(0,\chi_1,\dots, \chi_{n-1})$ be the components of
$\xi$ and $\chi$ respectively in these frames. Let further
$\widetilde \xi = (\xi_1,\dots, \xi_{n-1}) \in \Bbb R^{n-1}$
and $\widetilde \chi = (\chi_1,\dots, \chi_{n-1}) \in \Bbb R^{n-1}$.
The equations \thetag{3.3} will read then
$$
\frac{d\xi_0}{dt} = \varphi(\xi -\langle \xi,v\rangle v), \ \ \
\frac{d\widetilde\xi}{dt} = - \varphi(v)\widetilde \xi +
\widetilde \chi, \ \ \
\frac{ d\widetilde\chi}{dt} = - \widehat{R}_a(\xi,v)v,
\tag{3.4}
$$
where the vector $\widehat{R}_a(\xi,v)v$,
being orthogonal to $v$, is considered
as an element in $\Bbb R^{n-1}$ by the expansion in
the basis $e_1(t),\dots,e_{n-1}(t)$.
Note that $\varphi(\xi -\langle \xi,v\rangle v)$
and $\widehat{R}_a(\xi,v )v$ depend only on
$\widetilde \xi \in\Bbb R^{n-1}$.
The operator
$\widetilde \xi \to \widehat{R}_a(\xi,v )v \in \Bbb R^{n-1}$,
in contrast to the Riemannian case, is not in general symmetric.
We introduce the sectional curvatures $\widehat K(\Pi)$
of the Weyl structure in the direction of a plane $\Pi$ as
$$
\widehat K(\Pi) = \langle \widehat{R}_a(X,Y)Y,X\rangle,
$$
for any orthonormal basis $\{X,Y\}$ of $\Pi$. The sectional
curvatures depend on the choice of a Riemannian metric $g$
in the conformal class and not on
the Weyl structure alone. However the sign of sectional curvatures
is well defined.
Denoting by $K(\Pi)$ the Riemannian sectional
curvature in the direction of $\Pi$ and by $E_\Pi$
the orthogonal projection of $E$ on $\Pi$ we obtain from Proposition 3.2
$$
\widehat K(\Pi) = K(\Pi)-
\left(E^2 - E_\Pi^2\right)
- \langle \nabla_{X} E, X \rangle -
\langle \nabla_{Y} E, Y \rangle.
$$
Note that the term in the brackets vanishes automatically,
if $M$ is 2-dimensional.
In dimension 2 we have further that $\langle \nabla_{X} E, X \rangle
+\langle \nabla_{Y} E, Y \rangle$ is the divergence of the vector
field $E$. By the theorem of Gauduchon, \cite{Ga}, we can always
multiply the metric by a positive function
and make the new
vector field $E$ divergence free (in dimension 2 this theorem
follows from the theory of harmonic forms).
Such a special choice of the metric is called the Gauduchon gauge.
We obtain that in dimension 2 the
curvature of a Weyl structure
with respect to the Gauduchon gauge is equal to the Gaussian
curvature.
Let us inspect the Examples 2.2 and 2.3 for the presence of negative
sectional curvatures.
Taking a constant vector field $E$ on a flat
torus in dimension 2 we obtain the Weyl structure with zero
curvature tensor. Moreover by the previous observation,
any Weyl structure on the 2-dimensional torus with
nonpositive sectional curvature must have zero
sectional curvature.
In dimension 3 and higher the Weyl structure has negative
sectional curvatures everywhere except
for the planes containing the
vector $E$, in contrast to the Riemannian case.
More generally in the Example 2.3 for any Riemannian metric
on $N$ with $dim N \geq 2$,
if the magnitude of the field $E$ exceeds the maximal
positive sectional curvature of $N$ then the Weyl sectional curvatures
of $M$ are negative except for the planes containing $E$
on which the Weyl sectional curvature is zero.
Using the normally hyperbolic invariant submanifolds
described in Examples 2.2, 2.3 we will arrive at the conclusion
(Proposition 5.9)
that small perturbations of these Weyl manifolds cannot
have strictly negative sectional curvatures everywhere.
Clearly one can obtain manifolds with negative Weyl sectional curvatures
by starting with the Riemannian metric of negative sectional
curvature and allowing sufficiently small fields $E$.
We do not know any nonperturbative examples.
\proclaim{Problem 3.4} Construct a Weyl structure with negative
sectional curvatures on a manifold which does not carry
a Riemannian metric of negative sectional curvature.
\endproclaim
\bigskip
\subheading{4. Monotonicity,
$\Cal J$--algebra of Potapov and dominated splittings }
\bigskip
To study the hyperbolic properties of the W-flows we
will be using general criteria developed in \cite{W2}.
They are a combination of
the ideas of Lewowicz, \cite{L} and our,
\cite{W3}, \cite{W4}, \cite{W5}, with
the $\Cal J$-algebra of Potapov,
\cite{P1},\cite{P2},\cite{P3}.
For the convenience of the reader we will formulate them briefly.
The detailed exposition can be found in \cite{W2}
Let us consider an $n$-dimensional linear space
$V$ 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, 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 $q$.
\proclaim{Definition 4.1}
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
$v\in V$,
\item
$\Cal J$--symmetric if $\langle JS \cdot, \cdot \rangle$ is a symmetric
bilinear form on $V$.
\endroster
\endproclaim
There are natural infinitesimal versions
of properties 1 -- 5 for one parameter subgroups of linear operators
(elements of the Lie algebra).
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. He developed the theory
for a pseudo unitary form $\Cal J$ rather than pseudo Euclidean,
but in our applications we need only the real version.
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 $S$.
Let us denote the eigenvalues of the modulus of $S$ by
$$ r_p^+ \geq \dots \geq r_2^+\geq r_1^+
\geq r_{1}^- \geq r_{2}^- \geq \dots \geq r_q^- >0.
$$
$S$ is strictly $\Cal J$--separated iff $r_1^+
> r_{1}^-$. It is strictly $\Cal J$--monotone iff $r_1^+
> 1 > r_{1}^-$.
The eigenvalues of the modulus
can be interpreted as pseudo Euclidean singular values.
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
of 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 $L$. We have
\proclaim{Proposition 4.1\cite{W2}}
$\sigma_d(S) = r_1^+r_2^+ \dots r_d^+$.
\endproclaim
Similarly we can obtain the other eigenvalues of the modulus
by considering the negative subspaces. Note that
if $S$ is $\Cal J$-separated then $S^{-1}$ is $(-\Cal J)$-separated.
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},
$$
(dT)^2 = tr\left( (I_{p} - T^*T)^{-1} dT^* (I_{q} - TT^*)^{-1}dT\right).
$$
A $\Cal J$--separated operator $S$ maps naturally $\Cal L_+$
into itself. This mapping is a contraction.
More precisely we have
\proclaim{Theorem 4.2\cite{W2} }
For a $\Cal J$--separated operator $S$ and $T_1,T_2 \in \Cal L_+$
$$
dist(ST_1,ST_2) \leq \frac {r_1^-}{r_1^+} dist(T_1,T_2).
$$
\endproclaim
Let us move the discussion to a compact manifold $M$ of dimension $n$
and a diffeomorphism
\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 field 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.
\proclaim{Theorem 4.3\cite{W2}} 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
$$ T_xM = \Cal E^-(x) \oplus \Cal E^+(x),
$$
such that for some $0 < \lambda < 1, c > 0$, and all $k \geq 1$
$$
\|D{\varPhi^k}_{|\Cal E^-}\|\ \|D{\varPhi^{-k}}_{|\Cal E^+}\| < c\lambda^k.
$$
\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 4.2 that the diameter of
$ D\varPhi^k \Cal L_+(D\varPhi^{-k} x) $
does not exceed $c_0
\prod_{i= 1}^{k-1} \frac{r_1^-(\varPhi^{-i}x)}{r_1^+(\varPhi^{-i}x)}
$. Since the subsets
$$ D\varPhi^{k+1} \Cal L_+(D\varPhi^{-k-1} x)
\subset D\varPhi^k \Cal L_+(D\varPhi^{-k} x) $$
are nested and their
diameter decays exponentially we get a unique point of intersection
$$
\Cal E^+(x) =
\bigcap_{k =1}^\infty D\varPhi^k \Cal L_+(D\varPhi^{-k}x).\tag{4.1}
$$
It is clear that the subspace depends continuously on $x$.
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 the separation property with $\lambda =
\sup_x \frac{r_1^-(x)}{r_1^+(x)}$.
\enddemo\qed
As a corollary we obtain the theorem of Lewowicz, \cite{L},
\proclaim{Theorem 4.5} If a diffeomorphism is
strictly $\Cal J$--monotone then it is Anosov.
\endproclaim
\demo{Proof} By Theorem 4.3 we obtain the continuous
invariant splitting. On the unstable subspace the $\Cal J$--form
increases exponentially and it can be estimated by any
norm.
\enddemo\qed
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 $D \varPhi$, i.e., the eigenvalues
of the modulus of $D \varPhi$.
Having in mind the applications to billiard W-flows we
allow piecewise differentiable mappings.
\proclaim{Theorem 4.6\cite{W2}}
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
$$
\mu_{-p} \leq \mu_{-{p-1}} \leq \dots \leq \mu_{-1}\leq
\mu_1 \leq \mu_2 \leq \dots \leq \mu_{q},
$$
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
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 $D\varPhi^t$ factors 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 Theorem 4.3
we get a dominated splitting of the quotient space, but it does not
necessarily lift to the splitting of the tangent space.
We can lift the splitting in the case of strict $\Cal J$--monotonicity
(cf. \cite{W1}) and we obtain the counterpart of the
Lewowicz's Theorem 4.5.
\proclaim{Theorem 4.7}
If the derivative cocycle of a smooth flow is strictly
$\Cal J$--monotone then the flow is Anosov.
\endproclaim
\bigskip
\subheading{5. Hyperbolic properties of W-flows}
\bigskip
We introduce a quadratic form $\Cal J$ in the tangent spaces of the
phase space of $SM$
by $\Cal J(\xi,\chi) = \langle \xi ,\, \chi\rangle$.
The form $\Cal J$ factors naturally to the quotient bundle
(the quotient by the span of the vector field \thetag{2.3}, i.e.,
in the $(\xi,\chi)$ coordinates the quotient by the span of $(v,0)$),
(note that it is so only for the restricted phase space $SM$).
The quotient space can be represented by the subspace
$\langle \xi, v \rangle = 0$, but this subspace is not invariant
under the linearization of the flow \thetag{3.3} (or \thetag{3.4}).
The form $\Cal J$ in the quotient space is clearly
nondegenerate and it has equal positive and
negative indices of inertia.
We are going to calculate the Lie derivative of $\Cal J$.
The result is by necessity a form that factors onto the
quotient bundle, like the $\Cal J$ itself. Hence after
(but not before) the
differentiation we can consider $\xi$ to be orthogonal to
$v$.
Using \thetag{2.1} and \thetag{3.3} we obtain
$$
\frac{d}{dt} \Cal J(\xi,\chi) = \frac{d}{dt} \langle \xi,\chi\rangle =
\chi^2\ - \varphi(v)\langle \xi,\chi\rangle -
\langle \widehat{R}_a(\xi,v)v,\xi\rangle.\tag{5.1}$$
\proclaim{Theorem 5.1}
If the sectional curvatures of the Weyl structure
are negative everywhere in $M$
then the W -- flow is strictly $\Cal J$-separated
and hence it has the dominated splitting.
If for every plane $\Pi$ the sectional curvatures
$$- \widehat K(\Pi) > \frac 14 E_\Pi^2,$$
where $E_\Pi$ is the orthogonal projection of $E$ on $\Pi$,
then the W -- flow is Anosov.
\endproclaim
\demo{Proof}
It follows immediately from \thetag{5.1} and Theorems 4.3 and 4.7.
Indeed we get
$$
\frac{d}{dt} \Cal J(\xi,\chi) =
\left(\chi - \frac 12\langle E,v\rangle \xi\right)^2
+ \left(-\frac 14\langle E,v\rangle^2 - \widehat K(\Pi)\right)\xi^2,
$$
where the plane $\Pi$ is spanned by $v$ and $\xi$, ($\xi$ is assumed
to be orthogonal to $v$). We obtain that the W-flow is
strictly $\Cal J$-monotone if
only $- \widehat K(\Pi) > \frac 14 E_\Pi^2$.
\enddemo
\qed
In the case of negative sectional curvature let us denote the
invariant subspaces of the dominant splitting in the quotient
spaces by $\Cal E^{+}(q,v)$ -- the positive subspace and by
$\Cal E^{-}(q,v)$ --
the negative subspace. We do not claim in general (nor do we expect)
to have exponential growth in $\Cal E^{+}(q,v)$ or exponential decay in
$\Cal E^{-}(q,v)$.
We can represent these subspaces as graphs of operators defined
on the tangent subspace orthogonal to $v$, i.e.,
$$\Cal E^{\pm}(q,v) = \{(\xi,\chi) | \chi = U_\pm(q,v) \xi\}.$$
The reversibility of the flow (\thetag{2.4}) leads us to the conclusion that
$$
U_\mp(q,v) = - U_\pm(q, - v). \tag{5.2}.
$$
For 2-dimensional surfaces the negative curvature alone implies
Anosov property.
\proclaim{Theorem 5.2}
For a 2-dimensional surface $M$ if the curvature of the
Weyl structure is negative,
i.e, $\widehat K = K - div E < 0 $ on $M$,
then the W-flow is a transitive Anosov flow.
\endproclaim
\demo{Proof}
Let us calculate the Lie derivative of $\chi^2$. We get using
\thetag{2.1} and \thetag{3.3}
$$
\frac{d}{dt}\chi^2 = -2 \langle \widehat{R}_a(v, \xi)v,\chi \rangle
\tag{5.3}
$$
In general we cannot claim that this derivative is positive in the
positive cone of $\Cal J$. In the case of a surface
both $\chi$ and $\xi$ can be assumed to be
orthogonal to $v$ and hence they must be
colinear, i.e., $\chi = \frac{\langle \xi, \chi \rangle}
{\xi^2} \xi$. Substituting this into \thetag{5.3} we obtain
$$
\frac{d}{dt}\chi^2 = 2(-K + div E)\langle \xi, \chi \rangle,
\tag{5.4}
$$
Since our condition ensures that the W-flow is strictly $\Cal
J$-separated we have a continuous bundle of positive lines in the
quotient spaces, invariant under the flow. Hence there is a positive
continuous function $u$ on $SM$ such that the positive invariant line
is described as $\chi = u \xi$. On the invariant line \thetag{5.4}
yields
$$
\frac{d}{dt}\chi^2 = \frac 2u(-K + div E)\chi^2.
$$
Since $u$ is positive and bounded the
last formula implies that there is exponential growth
on the positive invariant lines.
By the reversibility \thetag{5.2} we conclude immediately
that there is exponential
decay on the negative invariant lines which proves that
the system is Anosov.
The transitivity of the system follows from the observation
made in Section 3 that
we can always consider the field $E$ to be divergence free and then
the sectional curvature of the Weyl structure coincides with the Gaussian
curvature. By assumption it is negative everywhere. Hence multiplying
the vector field $E$ by the parameter $a, 0\leq a \leq 1$ we obtain
a continuous family of Anosov flows ending with the geodesic flow
on a surface of negative Gaussian curvature
which is transitive. By the Anosov
structural stability theorem we obtain that
all these systems are topologically transitive.
\qed
\enddemo
In higher dimensions under the assumption of negative
sectional curvatures we obtain some hyperbolicity
which is weaker than the Anosov
property.
\proclaim{Theorem 5.3} If the sectional curvatures of a Weyl structure
are negative everywhere in $M$
then there is uniform exponential growth of volume on the
positive invariant subspace $\Cal E^+$ and uniform exponential decay of
volume on the negative invariant subspace $\Cal E^-$.
\endproclaim
\demo{Proof}
Since the W-flow is reversible (\thetag{2.4},\thetag{5.2})
it is enough to prove the
exponential growth on $\Cal E^+$.
The crucial element in the proof is the choice of
the ``Lyapunov volume element'' on the invariant subspaces
which is increased under the dynamics.
For convenience we employ the coordinate form
\thetag{3.4} of the linearized equations.
We can represent $\Cal E^+$ as the graph of an operator $U: \Bbb R^{n-1}
\to \Bbb R^{n-1}$ with positive definite symmetric part,
$$\Cal E^{+} = \{(\xi,\chi) | \widetilde\chi = U^{+} \widetilde\xi\}.$$
The operator $U$ satisfies the Riccatti equation
$$
\frac{d}{dt}U = \varphi(v)U - U^2 - R , \ \ \ \text{where} \ \ \
R\widetilde\xi = \widehat{R}_a(\xi,v)v.\tag{5.5}
$$
Note that in contrast to the Riemannian case the operator $R$
is not symmetric and hence the operator $U$ cannot be expected
to be symmetric.
We split the operators $U =U_s + U_a$ and $R = R_s + R_a$
into the symmetric and antisymmetric parts. For the symmetric
part $U_s$ we obtain from \thetag{5.5}
$$
\frac{d}{dt}U_s = \varphi(v)U_s - U^2_s - U^2_a - R_s.\tag{5.6}
$$
By the assumption of negative sectional curvatures
the operator $-R_s$ is positive definite. Moreover
since $U_a$ is antisymmetric $-U^2_a$ is
positive semidefinite.
We introduce new linear coordinates $\kappa \in \Bbb R^{n-1}$ on
$\Cal E^+$ by the formula $\kappa = U_s\widetilde\xi$.
Using \thetag{3.4} and \thetag{5.6} we obtain
$$
\frac{d}{dt}\kappa = \left(-U_s +U_s U U_s^{-1} +
(- U_a^2 - R_s)U_s^{-1}\right)\kappa.
$$
Since $tr (U_sUU_s^{-1}) = tr U_s$
the trace of the operator in right hand side of this linear equation
is equal to $tr [(- U_a^2 - R_s)U_s^{-1}] > 0$ and
we obtain that
the standard volume element in the coordinates $\kappa$ is uniformly
exponentially expanded.
\enddemo\qed
\proclaim{Corollary 5.4}
If the sectional curvatures of the Weyl structure
are negative everywhere in $M$
then for any ergodic invariant measure of the W-flow
the largest Lyapunov exponent is positive
and the smallest Lyapunov exponent is negative.
\endproclaim
We can apply this corollary to an individual
periodic orbit and we obtain linear instability.
Moreover there are also no linearly repelling or
linearly neutral periodic orbits.
In the case of a potential field $E = -\nabla U$ the
Weyl connection is the Levi-Civita connection of the
Riemannian metric $\widetilde g = e^{-2U}g$.
For such flows we can rewrite \thetag{5.1} as
$$
\frac{d}{dt} \left( e^{-U}\Cal J(\xi,\chi)\right) =
e^{-U}\left(\chi^2\ -
\langle \widehat{R}_a(\xi,v)v,\xi\rangle\right).$$
This allows us to make a stronger claim for some periodic orbits
than Corollary 5.4.
\proclaim{Proposition 5.5}
If the potential $U$ is single valued on a periodic orbit
of a W-flow
and the sectional curvatures are negative then the periodic orbit
is hyperbolic and it has
half of the Floquet multipliers outside of the unit circle
and half of them inside.
\endproclaim
In the 2-dimensional case we have $div E = -\triangle U$
and Theorem 5.1 leads directly to
\proclaim{Corollary 5.6}
If $K < -\triangle U $ on a 2-dimensional surface $M$
then the W -- flow is a transitive Anosov flow.
\endproclaim
\proclaim{Corollary 5.7}
If the vector field $E$ has a local potential $E = -\nabla U$
which is harmonic
and the Gaussian curvature $K < 0$ on $M$
then the W -- flow is a transitive Anosov flow.
\endproclaim
We can conclude that in the case of fields given by automorphic forms
on surfaces of constant negative curvature, which were studied by
Bonetto, Gentile and Mastropietro, \cite{B-G-M}, the flow is always
Anosov. It was shown in \cite{W1} that once such a flow is Anosov it
is also automatically dissipative, i.e., the SRB measure is singular.
Note that in this situation we can multiply the vector field $E$ by an
arbitrary scalar and we still get a transitive Anosov flow. It would
be interesting to understand the asymptotics of the SRB measure as
$\lambda \to \infty$. Is the limit supported on the union of integral
curves of $E$ ? Let us stress that this scenario differs from the
perturbative conditions in \cite{Go}, \cite{Gr}, \cite{W1}, where the
geodesic curvature of the trajectories cannot be too large. Our
trajectories may have arbitrarily large geodesic curvatures and yet
give a transitive Anosov flow.
The following conjecture is of considerable interest
\proclaim{Conjecture 5.8}
There are manifolds of dimension $\geq 3$ and 1-forms $\varphi$
such that the Weyl sectional curvatures are negative
everywhere but the W-flow is not Anosov.
\endproclaim
It is also plausible that under the assumption of negative sectional
curvatures we can obtain W-flows which are nontransitive
Anosov flows, as in the examples of Franks and Williams, \cite{F-W}.
To construct such examples we need to find nice examples of
Weyl manifolds with negative sectional curvatures. In that direction
there are some notable obstructions.
\proclaim{Proposition 5.9} There are no Weyl structures with negative
sectional curvatures in a small neighborhood of the structure
defined by the constant vector field $E$ on a flat torus
(Example 2.2) or in a small neighborhood of the structure
defined in Example 2.3.
\endproclaim
\demo{Proof}
As noted in the discussion of Examples 2.2, 2.3,
the W-flow has two invariant
submanifolds. Let us consider the attracting
submanifold. It is defined by the
equation $ |E|v = E$. Its tangent bundle is given by
$\chi = |E|(\xi - \langle \xi ,v \rangle v)$. On the invariant
manifold the linearized equations \thetag{3.4} read
$$
\frac{d\xi_0}{dt} = 0, \ \ \
\frac{d\widetilde\xi}{dt} = - |E|\widetilde \xi +
\widetilde \chi, \ \ \
\frac{ d\widetilde\chi}{dt} = 0.
$$
The bundle defined by $\chi = 0$ is an invariant normal bundle.
On this bundle we have exponential decay while on the invariant
submanifold the dynamics is by isometries. Hence the W-flow is
normally contracting at the invariant
submanifold. It follows that the invariant manifold
persists under small perturbations
(cf. \cite{Ro}, p.445). The invariant submanifold of the perturbed
system is $C^1$ close to the original one.
The tangent subspaces of the unperturbed
invariant submanifold are in the positive cone of the form $\Cal J$
and so are the tangent subspaces of
the perturbed invariant submanifold,
at least when the perturbation is sufficiently small.
Hence for such perturbations,
if the sectional curvatures are negative then the invariant
submanifold must be tangent to
$\Cal E^+\oplus span\{(v,0)\}$ (cf. \thetag {4.1}).
But on this bundle we have by Theorem 5.4 uniform exponential
growth of the volume element. It is incompatible with the compactness
of the invariant manifold.
\qed\enddemo
We do not know, if the d-dimensional, $d \geq 3$,
torus carries a Weyl structure with negative sectional curvatures.
\proclaim{Example 5.10}\endproclaim
Let us consider a vector field $E$ on the
two dimensional torus $\Bbb R^2/\Bbb Z^2$ with finite number of
points $\{s_1,s_2,\dots,s_N\}$ removed. We allow the field to have
singularities at these points. We assume that $E$ has a local
potential $U$ which is superharmonic on the punctured torus,
$\triangle U < 0$. We can represent $U = U_0 + U_1$ where
$U_0$ is a linear function and $U_1$ is a doubly periodic function
with singularities which is superharmonic away from the singularities.
Such a function with only one singularity
can be produced by taking a superharmonic
function in $\Bbb R^2$ with one singularity and with fast decay at
infinity, e.g., the function $f$
given in polar coordinates $(r,\alpha)$ by
$f = \ln r - \ln (e^{-r} + r)$. Then we take the sum over the integer
translates
$$
U_1(q) = \sum_{a\in \Bbb Z^2} f(q+a).
$$
This function has the desired properties.
Let us consider the W-flow with such a potential
$U$. Locally, up to the time change,
it is the geodesic flow of the metric $e^{-2U}ds^2$
where $ds^2$ denotes the flat metric of the torus. Such a flow
is in general incomplete, moreover
open sets of orbits may end up at the singularities. This
can be excluded by allowing only logarithmic singularities.
\proclaim{Proposition 5.11} With the above assumptions on the local
potential $U$, if at every singularity point $s$ we have
$U_1(q) =\beta \ln r +B(q)$ for $0< \beta <1$ and
some smooth function $B(q)$ defined in a neighborhood of $s$,
with $r = |q-s|$,
then for every point on the punctured torus
the trajectories of the W-flow
are defined globally except for countably many directions
which result in the trajectories
reaching one of the singularities in finite
time.
\endproclaim
\demo{Proof} Let us assume that $s = 0$ and $q \in \Bbb C$.
We lift the metric $e^{-2U}ds^2$ to the Riemann surface of the function
$w = q^\gamma, \ \gamma >0$ . We obtain the metric
$$
\frac{1}{\gamma} |w|^{\frac{2(1-\gamma -\beta )}{\gamma}}
e^{-2U_0(w^{\frac 1{\gamma}})-2B(w^{\frac 1\gamma})}dwd\bar{w}.
$$
Choosing $\gamma = 1-\beta$ we obtain a finite metric on the Riemann
surface with the boundary $w = 0$. We do not have differentiability at
the boundary but we know that inside the curvature is nonpositive.
This allows us to conclude that through every point on the surface
there is at most one geodesic that reaches the boundary $w = 0$.
\qed\enddemo
For such special singularities we obtain a measurable flow
with hyperbolic properties. It seems that the results of
Chernov \cite{Ch} could be extended to cover these flows and then we
would obtain a unique SRB measure.
It is definitely so for special values of $\beta_i = 1
-\frac 1{k_i}$ with $k_i$ a positive integer. For these
values of $\beta$-s we obtain a smooth system on a branched
covering of the torus. This construction was used before by
Knauf, \cite{Kn}, in his study of motion in the fields with Coulombic
singularities.
\proclaim{Theorem 5.12} For the W-flow $\Psi^t: S\Bbb T^2 \to
S\Bbb T^2 $
described in Proposition 5.11, if $\beta_i = 1
-\frac 1{k_i}$ with $k_i$ equal to positive integers, i = 1,\dots,N,
then there
exists a
unique SRB measure $\nu_0$ on $S\Bbb T^2$, i.e., an invariant
measure such that
a. for every measure $\nu$
with positive density with respect to the Lebesgue measure
$$
\Psi_*^t(\nu) \to \nu_0
$$
weakly as $t \to +\infty$.
b. for every continuous function $f$ on $S\Bbb T^2$ we have
for Lebesgue almost all $x \in S\Bbb T^2$
$$
\frac 1T \int_0^T f(\Psi^tx) dt \to \int_{S\Bbb T^2} f(x) d\nu_0(x).
$$
Moreover, if the linear part $U_0$ of the potential $U$ is nonzero
then the measure $\nu_0$ is singular with respect to the Lebesgue
measure.
\endproclaim
\demo{Proof}
We split the potential $U$ into the linear
and the periodic parts, $U = U_0+ U_1$. We remove the
periodic part by changing the metric to $e^{-2U_1}ds^2$.
By the construction of Knauf \cite{Kn} we can lift this metric to
a smooth metric of negative Gaussian curvature on a
branched covering of $\Bbb T^2$. On this branched covering we
get the smooth W-flow with the local potential equal to the lift of $U_0$.
The lift of $U_0$ is a harmonic function on the surface and hence
by Corollary 5.8 the W-flow is a transitive Anosov flow.
Now the theorem follows from the general theory of SRB measures for
transitive Anosov flows, \cite{S}\cite{B-R}.
(Note that the conclusions of our theorem are not effected
by the time change in passing from one metric to the other.)
It was shown in \cite{W1} that the SRB
measure has to be singular, if the potential is not global.
\enddemo \qed
Another way to obtain hyperbolic W-flow is to enclose
the singularities of a
superharmonic potential on a torus in round scatterers
and assume elastic reflections from the scatterers.
We will discuss these models in the next section.
\bigskip
\subhead \S 6. Billiard W -- flows
\endsubhead
\bigskip
Let us now assume that the Riemannian manifold $M$ has a boundary.
Given a vector field $E$ we define the W -- flow with
collisions by the condition that
when a trajectory of our flow reaches the boundary the velocity is
reflected by the rules of elastic collisions.
We can for instance consider a closed manifold without a boundary
and remove certain number of open sets with smooth boundaries
(``obstacles'' or ``scatterers''). It is clear that we obtain in this way
a flow with collisions, as defined in \cite{W6}.
Such systems have discontinuities which are associated with
tangencies of the trajectories of the W-flow to the boundary
and any singularities (corners) of the boundary.
The orbits that do not experience any discontinuity will be called
smooth.
We will restrict our attention to billiard W-flows such that
on all smooth orbits there are infinitely many collisions with the
boundary and the time between collisions is uniformly bounded
away from zero and infinity (finite horizon).
In particular we assume that
the boundaries have no corners.
For such flows we can introduce also the Poincare section of the
billiard W-flow, {\it billiard map} for short, defined on the
manifold of ``outward'' unit tangent vectors over the
boundary.
The description of the linearized dynamics is similar to the
case of Riemannian billiards. Let us consider a scatterer $O$
and a point $q_0 \in O$. We denote by $N$ the field of unit
vectors orthogonal to the boundary $O$ and pointing inwards
(of the manifold $M$).
The collision map is described by the formula
$$
v^+ = v^- - 2\langle v^-,N\rangle N = Zv^-, \tag{6.1}
$$
where $v^+$ denotes the velocity immediately after,
$v^-$ the velocity immediately before the collision and
$Z$ denotes the reflection in the tangent subspace to $O$.
To obtain the linearized dynamics let us consider a
family of trajectories of the W-flow
$q(t,u),v(t,u) = \frac{dq}{dt}, |u| < \epsilon$ coming to a collision
with $O$ in the vicinity of $q_0 = q(t_0,0)$. At the moment of
collision $t_0 = t_0(u)$ the velocity is changed by the reflection
in the tangent plane of $O$ described by \thetag{6.1}.
Let us denote by $P_{\pm}$ the projection onto the tangent
subspace of $O$
along the velocity vector $v^{\pm}$. We introduce
$\xi_0 = \frac{d}{du} q(t_0(u),u)_{|u=0} = P_{-}\xi^- = P_+\xi^+$,
where $\xi^{\pm}$ are the values of the Jacobi field
$\xi = \frac{dq}{du}$
immediately before and immediately after the collision. We have
clearly $\xi^+ = Z\xi^-$. To calculate the change in the $\chi$
coordinate let us recall (Proposition 3.3) that
$\chi^{\pm}$ is the component of
$\wnabla_{\xi^{\pm}} v$ orthogonal to $v$ at time $t_0$.
Hence we can replace $\xi^{\pm}$ in the derivative by $\xi_0$, i.e.,
$\chi^{\pm}= \wnabla_{\xi_0}v^{\pm} -
\langle \wnabla_{\xi_0}v^{\pm}, v^{\pm}\rangle v^{\pm}$.
\proclaim{Proposition 6.1}
$$
\chi^+ = Z\chi^- - 2\langle v^-,N\rangle P^*_+\widehat K\xi_0,
\tag{6.2}
$$
where
$\widehat K\xi_0 = \wnabla_{\xi_0}N - \langle\xi_0,E\rangle N
= \nabla_{\xi_0}N + \langle N,E\rangle \xi_0$.
\endproclaim
Note that the operator $\widehat K$ is the orthogonal projection of
the '' Weyl shape operator'' $ \wnabla_{\xi_0}N$ to the boundary $O$.
\demo{Proof}
Differentiating \thetag{6.1} we obtain
$$
\aligned
\wnabla_{\xi_0}v^+ =& \wnabla_{\xi_0}v^- -
2\langle \wnabla_{\xi_0}v^-,N\rangle N -\\
&- 2\langle v^-,\wnabla_{\xi_0}N
\rangle N -2\langle v^-,N\rangle \wnabla_{\xi_0} N
+ 4\langle \xi_0,E\rangle\langle v^-,N\rangle N =\\
=& Z\wnabla_{\xi_0}v^- - 2\langle v^-, \widehat K\xi_0
\rangle N -2\langle v^-,N\rangle \widehat K\xi_0,
\endaligned
$$
\enddemo\qed
If we denote by $K\xi_0 = \nabla_{\xi_0}N$ the
Riemannian shape operator
of the boundary $O$ we have
$$\widehat K\xi_0 = K\xi_0 +
\langle N,E\rangle \xi_0.
\tag{6.3}
$$
We say that the submanifold $O$ is (strictly) Weyl concave if
$\widehat K$ is negative (definite) semidefinite.
We want to establish what happens at collisions to the
the quadratic form $\Cal J(\xi,\chi) = \langle \xi,\chi \rangle$,
introduced in the previous section.
\proclaim{Proposition 6.2}
The billiard W-flow on a manifold $M$ of nonpositive
sectional Weyl curvature and (strictly) Weyl concave boundary $O$ is
(strictly) $\Cal J$-separated.
\endproclaim
\demo{Proof}
At the collision we have by Proposition 6.1
$$
\Cal J (\xi^+,\chi^+) = \Cal J (\xi^-,\chi^-) +
\langle \widehat K\xi_0 ,\xi_0\rangle.\tag{6.4}
$$
\enddemo
\qed
The fact that the flow is strictly $\Cal J$-separated does not allow
to construct the dominated splitting on every smooth orbit,
as in Theorem 4.3. The problem is that the coefficients of
contraction on $\Cal L_+$ from Theorem 4.2 are not bounded away
from $1$, except in dimension 2.
We saw in Section 5 the special
status of the 2-dimensional case. Similarly to Theorem 5.1
we obtain.
\proclaim{Theorem 6.3}
If the Weyl curvature of
a 2-dimensional surface $M$ with the boundary $O$
is nonpositive and
the boundary $O$ is strictly Weyl concave then the
the billiard map has the dominated splitting
on every smooth orbit and there is uniform expansion on
the positive invariant line and uniform contraction
on the negative invariant line.
\endproclaim
\demo{Proof}
To construct the dominated splitting
we need to estimate the coefficients of contraction on $\Cal L_+$.
The derivative of the billiard map is the composition of the
derivative \thetag{6.2} of the collision map and the derivative
of the W-flow between collisions.
We will be using
the orthonormal frames $e_0(t) = v(t), e_1(t),$ which were used in
\thetag{3.4}. We continue these frames across collisions by
the reflection $Z$ in the boundary.
In the 2-dimensional case
the relation \thetag{6.2} is scalar. Let us denote by $k$ and
$\widehat k = k + \langle E,N\rangle$ the Riemannian
geodesic curvature and the ``Weyl geodesic curvature'' of $O$
respectively. By the assumption of strict concavity
$\widehat k$ is bounded away from zero.
When
$\xi^-$ is in the plane spanned by $v^-$ and $N$ then
$$P^*_+\xi_0 = \frac{1}{\langle v^-,N\rangle^2} Z\xi^-. $$
Hence we obtain $\chi^+ =Z(\chi^- + h \xi^-)$ where
$h = -2\frac{\widehat k }{\langle v^-,N\rangle} > 0$.
Taking into account that $\xi^+ =Z\xi^-$
we conclude that the derivative of the collision map is described
by the matrix
$
\left(\matrix 1 & 0 \\ h &1 \endmatrix \right),
$
with the coefficient $h$ bounded away from zero, and unbounded.
Between collisions we get from \thetag{3.4}
$$
\frac{d \xi}{dt} = - g(t)\xi + \chi, \ \ \
\frac{ d \chi}{dt} = - r(t)\xi,
\tag{6.5}
$$
where $\xi,\chi$ are scalars, and
$g(t) = \langle E(t), v(t) \rangle, r(t) = \langle
\widehat{R}_a(\xi,v)v, e_1 \rangle$.
The contribution into the derivative of the billiard
map is described by the fundamental
matrix solution of these equations.
Let us denote this contribution by
$
\left(\matrix a & b \\ c &d \endmatrix \right).
$
Since the dynamics is $\Cal J$-separated the matrix
has nonnegative entries positive entries.
By compactness the determinant of this matrix is bounded away from
zero and from infinity, and all the entries are bounded.
It also follows from the form of the equations \thetag{6.5}
that $b$ is bounded away from zero.
The derivative of the billiard map is given by the product of
these matrices
$$
\left(\matrix a & b \\ c &d \endmatrix \right)
\left(\matrix 1 & 0 \\ h &1 \endmatrix \right) =
\left(\matrix a +hb & b \\ c+hd &d \endmatrix \right).
$$
The coefficient of contraction
on $\Cal L_+$ appearing in Theorem 4.2 is equal to
$$
\frac{r_1^-}{r_1^+} = \frac{\sqrt{(a+hb)d} - \sqrt{(c+hd)b}}
{\sqrt{(a+hb)d} + \sqrt{(c+hd)b}} =
\frac{\sqrt A - 1}{\sqrt A +1},
$$
where $A = \frac{(a+hb)d}{(c+hd)b} \leq \frac a{hb} +1$.
Since $a$ is bounded and $h,b$ are bounded away from zero
we conclude that the coefficient of contraction is bounded away from
$1$. Consequently on every smooth orbit we obtain the two invariant
line bundles. Let us describe the positive line bundle by
$\chi = u \xi$. Clearly
$$
\frac{c+hd}{a+hb} < u < \frac db,
$$
so that $u$ is bounded away from zero and infinity.
(Let us note that $u$ is bounded because in our billiard map
the reflection in the boundary goes first. It is not so if we
consider the billiard map in which we start with the flow.)
As in the proof of Theorems 5.1 and 5.2 we calculate now the change in
$\chi^2$ at the collision.
On the positive invariant line $\chi^- = u \xi^-$ we obtain
$$
(\chi^+)^2 = (\chi^-)^2(1+\frac hu)^2.
$$
Since $u$ was established to be bounded and
$\chi^2$ does not decrease between collisions we obtain the
desired uniform expansion.
\enddemo
\qed
In dimension 3 and higher
we do not obtain the dominated splitting on all orbits
since we do not get the required uniform estimates at the collision
with the boundary. We do get the dominated splitting
on individual periodic orbits or more generally
for almost all orbits of some ergodic invariant measure
as guaranteed by Theorem 4.6. Moreover parallelly to Theorem 5.3
we obtain exponential growth and decay of volumes on
the invariant subspaces. This is the content of the
next theorem. We will consider only the ergodic invariant measures
for which the derivative of the billiard map
satisfies the integrability condition which
guarantees the existence of Lyapunov exponents. In special
examples it may be difficult to establish such integrability
for singular invariant measures,
since the collisions lead to unbounded derivatives.
\proclaim{Theorem 6.4} If the Weyl sectional curvatures of a
manifold $M$ are nonpositive everywhere
and the boundary $S$ is strictly Weyl concave,
then for any ergodic invariant measure
of the billiard
flow the largest Lyapunov exponent is positive
and the smallest Lyapunov exponent is negative.
\endproclaim
\demo{Proof} We follow the scheme of the proof of Theorem 5.3.
We obtain the invariant bundles of the dominated splitting
defined almost everywhere. We need to study the growth at collisions
of the volume element on the positive invariant subspaces
introduced in the proof of Theorem 5.3.
On the positive subspace
$\Cal E^+$ given by $\chi^\pm = U^\pm\xi^\pm $ we can rewrite \thetag{6.2}
as
$$
\chi^+ = Z\chi^- + H\xi^+, \ \ \ \text{where} \ \ \
H = - 2\langle v^-,N\rangle P^*_+\widehat K P_+.
$$
The operator $H$ is symmetric and positive definite.
It follows from the above formula that $U^+ = ZU^-Z^{-1} +H$.
As in Theorem 5.3 the operators
$U$ are not in general symmetric but their symmetric parts
$U_s$ are positive definite and $U^+_s = ZU^-_sZ^{-1} +H$.
For $\kappa = U_s \xi$ we get
$$\kappa^+ = (ZU^-_sZ^{-1} +H)Z(U^-_s)^{-1}\kappa^-.$$
We observe that
$$
|\det(ZU^-_sZ^{-1} +H)Z(U^-_s)^{-1}| =
\frac{\det(U^-_s + Z^{-1}HZ)}{\det U^-_s} > 1.
$$
We cannot guarantee that the determinant is bounded away from $1$
but using the ergodic theorem we obtain the exponential growth
of volume on almost all orbits.
Note that we used the fact established in the proof of
Theorem 5.3 that between collisions the standard volume
element in $\kappa$ coordinates is not decreased.
\qed\enddemo
As in the case of W-flows we can apply this theorem to
individual periodic orbits and we obtain that all such orbits
are linearly unstable and there are no repelling orbits.
In the case of the Weyl structure with a local potential
we can formulate a counterpart of Proposition 5.5
because by \thetag{6.4} the form $\Cal J$ is not decreased
at the collisions.
\proclaim{Proposition 6.5}
For a locally potential Weyl structure with
nonpositive sectional curvatures and strictly Weyl concave boundary,
if the potential is single valued on a periodic orbit of the
billiard W-flow then the periodic orbit is hyperbolic and it has
half of the Floquet multipliers outside of the unit circle
and half of them inside.
\endproclaim
Let us consider the case of a a 2-dimensional flat
torus and the harmonic local potential $U$,
i.e., $\triangle U = 0$. We have calculated in Section 3
that in such a case the Weyl curvature is zero.
It turns out that we can also integrate the equations
of the W-flow. We take the torus $\Bbb T^2 = \Bbb C/{(\Bbb Z+ i\Bbb
Z)}$. Let $f(z), z= x+iy,$ be the holomorphic function
such that $U = Re(f)$
\proclaim{Proposition 6.5}
The conformal mapping defined by
$$F(z) = \int e^{-f(z)}dz$$
takes the trajectories of the W-flow into straight lines.
In particular, if $z(t)$ is a trajectory (parametrized by
the arc length of the flat background metric) then
$e^{-iIm(f)}z' = const$.
\endproclaim
\demo{Proof}
$$
dFd\bar F = e^{-f}dz e^{-\bar f}d\bar z = e^{-2U}(dx^2+ dy^2).
$$
We know that the trajectories of the W-flow are geodesics of
the last metric which proves the first claim.
To prove the second claim we observe that $F(z(t))$ is a straight
line and hence $\frac{d}{dt}F(z(t))$ has a constant direction.
\enddemo\qed
Now we can easily analyze the linear stability of periodic orbits
of the W-flow by mapping them onto periodic orbits of
a planar billiard. For planar billiards the linear stability
of an orbit is determined by the lengths of the segments of
the orbit, the angles at which they reflect from the boundaries
and the curvatures of the boundaries. The angles do not change
under the conformal map and
the length of the image of a trajectory from $z_0 = z(t_0)$ to
$z_1 = z(t_1)$ is equal to $|F(z_1) -F(z_0)|$.
It remains to express the geodesic curvature of the
image of the boundary. Let $z(s)$ be a parametrized curve
oriented by the field of normals $N = i \frac{z'}{|z'|}$.
The signed curvature of the curve is equal to $k = -\frac 1{|z'|}
Im(\frac {z''}{z'})$. By straightforward calculation we obtain that
the signed curvature $\widetilde k$ of the image curve
$ F(z(s))$ is equal to
$$
\widetilde k = \frac {1}{|F'|}\left( k +
Im\left(\frac {F''z'}{F'|z'|} \right)\right).\tag{6.6}
$$
Finally let us consider the system with round scatterers
and constant external field,
studied by Moran and Hoover , \cite{M-H}, and
Chernov, Eyink, Lebowitz and Sinai, \cite{Ch-E-L-S}.
We put $U = -ax$ and then we get symmetric periodic orbits
shown in Fig.1. It turns out that the ``flat'' orbit is never
linearly stable and the ``curved'' orbit is linearly stable
as soon as the Weyl curvature of the scatterers at the
points of reflections becomes positive. Hoover and Moran
observed numerically these stable orbits and the
KAM ``bottles'' enclosing them.
The stability of an orbit depends on its length $l$ and the
signed curvatures at the points of reflections $k_0,k_1$
($k$ is negative for a strictly concave boundary).
The orbit is linearly stable if and only if (cf. \cite{W7})
$$
lk_0k_1 - k_0 - k_1 < 0, \ \ \ \text{and} \ \ \ (lk_0 - 1)(lk_1 -1)
>0.
\tag{6.7}
$$
We have $F(z) = \frac 1a e^{az}$. Let us assume that the points of
reflection are $z_0 = x_0 + iy_0$ and $z_1 = x_1 + iy_1$
Using Proposition 6.5 and \thetag{6.6} we obtain for the flat orbit
in Fig 1 ($y_0 = y_1, x_0 < x_1$)
$$
l = \frac 1a(e^{ax_1} - e^{ax_0}),\ \ k_0 = e^{-ax_0}(-\frac 1{r_0} -a),
\ \ k_1 = e^{-ax_1}(-\frac 1{r_1} +a),
$$
where $r_0$ and $r_1$ are the radii of the scatterers (which we allow
to be different).
It follows that
$$
lk_0k_1 - k_0 - k_1 = \frac 1{ar_0r_1} (e^{-ax_0} - e^{-ax_1})
+\frac 1{r_0}e^{-ax_1} + \frac 1{r_1} e^{-ax_0} > 0.
$$
Hence the orbit is always unstable, even if the scatterers
have different radii.
On the contrary for the curved symmetric orbit, $ax = -\ln \cos ay, \ \
y_1 = -y_0 > 0, r_0 = r_1 =r, k_0 = k_1 =k,
$
in Fig.1 we have
$$
l = \frac 1a|\tan ay_1 - \tan a y_0|,\ \ k = \cos ay_1 (-\frac 1{r}
+ a \sin ay_1).
$$
The second condition in \thetag{6.7} is satisfied automatically,
while the first reads $k(lk - 2) < 0$.
But $ lk = 2 \sin ay_1(-\frac 1{ar}+ \sin ay_1) < 2$,
if only the curvature $k$ is positive
($ ar \sin ay_1 > 1$).
In dimension 3 and higher the sectional curvatures
are nonpositive and we still get the types of periodic
orbits considered above. Since the potential on these periodic
orbits is singlevalued we can apply Proposition 6.5 and we obtain
hyperbolicity of these orbits under the assumption that
at the collision points the
boundary is Weyl strictly concave.
By \thetag{6.3} the boundary is Weyl strictly concave
if $r|E| < 1$ where $r$ is the radius of
a scatterer. Under this assumption and the assumption
of finite horizon we get complete hyperbolicity in
dimension 2 by Theorem 6.3.
It sets the stage for the study of existence and
properties of SRB measures under
such assumption. It seems that the results of Chernov \cite{Ch}
could be generalized to include this case.
Further we expect that in \cite{Ch-E-L-S} the assumption of
weak field can be replaced by $r|E| < 1$,
at least for some parts of the theory.
In dimension 3 and higher with this assumption we obtain
by Theorem 6.4 at least one positive Lyapunov exponent and one negative
Lyapunov exponent for
``nice'' invariant measures. The issue of complete hyperbolicity
requires further study.
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\enddocument
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%%EndProlog
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% Polyline
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