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\begin{document}
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\begin{titlepage}
\begin{flushright}
UWThPh-1994-10 \\
ESI 103 (1994) \\
\end{flushright}
\vspace{0.5cm}
\begin{center}
{\Large \bf Anosov Actions on Non--Commutative Algebras}\\[24pt]
G. G. Emch$^{*,**}$, H. Narnhofer$^{**}$, W. Thirring$^{**}$ \\
Institut f\"ur Theoretische Physik \\
Universit\"at Wien \\
Boltzmanngasse 5, A-1090 Vienna, Austria\\[7pt]
and \\[7pt]
G. L. Sewell$^{***}$ \\
Department of Physics \\
Queen Mary and Westfield College \\
Mile End Road, London E1 4NS, England
\vfill
{\bf Abstract} \\
\end{center}

We construct an axiomatic framework for a quantum mechanical
extension to the theory of Anosov systems, and show that this
retains some of the characteristic features of its classical
counterpart, e.g. positive Lyapunov exponents,  a vectorial
K--property, and exponential clustering. We then investigate 
the effects of quantisation
on two prototype examples of Anosov systems, namely the
iterations of an automorphism of the torus (the `Arnold Cat'
model) and the free dynamics of a particle on a surface of
negative curvature. It emerges that the
Anosov property survives quantisation in the case of the former
model, but not of the latter one. Finally, we show that the
modular dynamics of a relativistic quantum field on the Rindler
wedge of Minkowski space is that of an Anosov system.
\vfill
\small
\begin{enumerate}
\item[*] Permanent address: Department of Mathematics, University
of Florida, Gainesville, Florida 32611, USA.
\item[**] International Erwin Schr\"odinger Institute for Mathematical
Physics, Pasteurgasse 6/7, A-1090 Vienna, Austria
\item[***] Partially supported by European Capital and
Mobility Contract No. CHRX-Ct. 92-0007.
\end{enumerate}
\end{titlepage}
\normalsize
\section{Introduction} 
In classical mechanics, the theory of
Anosov systems [An,AA] provides a paradigm for the unstable
dynamics governing the mixing process, which is the basis of
statistical mechanics. In particular, this theory was a
forerunner to Sinai's [Si] classical works on the ergodic
properties of systems of colliding particles.

In quantum mechanics, on the other hand, there is as yet no
corresponding general theory of a mechanism that gives rise to
similary unstable dynamics, though there is at least one quantum
model that appears, {\em prima facie\/}, to be a good candidate for
Anosov--type properties: this is the quantum version of the
`Arnold Cat' [BNS,Nar].  In view of this situation, it seems
worthwhile to investigate the feasability of a general theory of
unstable quantum dynamical systems, that is at least roughly
parallel to the classical Anosov theory.

Our aim in this article is to provide a framework for a
non--commutative extension of the theory of Anosov systems, that
is applicable to unstable, or chaotic, quantum dynamics. This
evidently requires some non--commutative generalisation of the
differential structure, that is at the centre of the classical
theory. We provide this in the form of derivations of the 
(non--abelian) algebra of observables, the natural generalisation of
classical vector fields on a manifold (cf. [Co]).

In this way, we are able to construct an axiomatic formulation
of quantum Anosov systems, and to show that, like their classical
counterparts, these have an unstable dynamics, characterised by
non--zero Lyapunov exponents and a certain vectorial, as distinct
from algebraic, K--mixing property. Furthermore, we show that
there are models of physical interest that satisfy our axioms,
e.g. the quantised Arnold Cat and the modular dynamics of a
relativistic quantum field on the Rindler wedge of Minkowski
space. On the other hand, the canonical quantisation of one of
the prototype classical Anosov systems, namely the free motion
of a particle on a surface of constant negative curvature,
destroys its Anosov property. We shall discuss this point
further in section 7.

We shall organise our material as follows. In section 2, we shall
present a brief review of the classical theory of Anosov systems.
In section 3, we shall set out our axioms, which generalise this
to the non--commutative regime, and we shall show there that the
resultant generalised Anosov conditions imply an unstable
dynamics, with positive Lyapunov exponents
and vectorial K--mixing properties. In section 4, we shall provide
a treatment of the quantum Arnold Cat, showing that it is indeed
an Anosov system, and inferring therefrom certain cluster
properties, stronger than those obtained in [BNS]. In section 5,
we shall formulate the quantum dynamics of a free
particle on a surface of constant negative curvature, and 
shall show that this is neither Anosov nor quasi--free, in the
standard sense of algebraic quantum theory. In section 6, we shall
show that the modular dynamics of a relativistic quantum field
on a Rindler wedge is an Anosov system. Finally, in section 7, we
shall briefly summarise our conclusions.

\section{Classical Anosov Systems}
Classical Anosov systems are characterized by the so--called 
``condition C'' specified in [AA]. The phase--space of these 
systems is assumed to be a compact, connected Riemann manifold $M$. 
The dynamics is an action
$$
T : (t,m) \in {\bf R} \times M \mapsto T(t)[m] \in M \eqno(2.1)
$$
which admits two foliations $\F^+$ and $\F^-$, stable with respect to
$T$, and such that for each $m \in M$ the following two conditions are
satisfied:
\begin{enumerate}
\item the two leaves $F^\pm_m$ of $\F^\pm$ through $m$ intersect
transversally; with $E_m^\pm$ denoting the tangent space of $F^\pm_m$ at
$m$, $\dim E^\pm_m \equiv k^\pm > 0$ is independent of $m$, and
$k^+ + k^- + 1 = n +1 \equiv \dim M$; one also assumes that the orbit
$T(\,\cdot\,)[m]$ intersects both $F^\pm_m$ transversally and that
its tangent vector never vanishes;
\item there exist strictly positive constants $a,b,\lambda$ such that
for every $X^\pm_m \in E^\pm_m$
$$
\| T(t)^* [X^\pm_m]\| \leq a \; e^{\mp \lambda t} \; \|X^\pm_m\|
\quad \forall \; t \in {\bf R}^\pm \eqno(2.2.a)
$$
$$
\| T(t)^* [X^\pm_m]\| \geq b \; e^{\pm \lambda t} \; \|X^\pm_m\|
\quad \forall \; t \in {\bf R}^\mp . \eqno(2.2.b)
$$
\end{enumerate}

The archetype [Had] for this structure is the geodesic flow on a compact,
connected, two--dimensional, Riemannian manifold 
$M_c = \Gamma \setminus SL(2,{\bf R})/K$ of constant negative curvature,
where $SL(2,{\bf R})$ acts on the Poincar\'e half--plane 
$(\ol{M}_c,g)$; specifically
$$
\ol{M}_c = \{ z \in {\bf C}\,\mid\, \frac{z - z^*}{i} > 0\}; \qquad
g = - 4 (z - z^*)^{-2} \; dz \; dz^* \eqno(2.3)
$$
and $SL(2,{\bf R})$ acts by fractional transformations
$$
(m,z) \in SL(2,{\bf R}) \times \ol{M}_c \mapsto m[z] =
\frac{az + b}{cz + d} \in \ol{M}_c \; ; \eqno(2.4)
$$
$K = S^1 \equiv \{ m \in SL(2,{\bf R})\,\mid\, m[i] = i\}$; 
$\ol{M}_c$ can be identified with $SL(2,{\bf R})/R : SL(2,{\bf R}) \ni
m \ra (m(i) \in \ol{M}_c$ and
$\Gamma$ is a discrete, co--compact subgroup of $SL(2,{\bf R})$. The
``phase--space'' of the model is the unit tangent bundle $M = T_1 \;
M_c$ which can thus be identified with $\Gamma \setminus SL(2,{\bf R})$.
The geodesic flow on $M$ is the action
$$
T : (t,m) \in {\bf R} \times M \mapsto T(t) [m] \in M \eqno(2.5)
$$
given by
$$
T(t) [m] = m \cdot \gamma(t); \qquad
\gamma(t) = \left( \ba{cc} e^{-t/2} & 0 \\ 0 & e^{t/2} \ea \right).
\eqno(2.6)
$$
Two horocyclic actions are defined
$$
S^\pm : (s,m) \in {\bf R} \times M \mapsto S^\pm (s) [m] \in M
\eqno(2.7)
$$
and 
$$
S^\pm(s)[m] = m \cdot \gamma^\pm(s) ; \qquad
\gamma^+(s) = \left( \ba{cc} 1 & s \\ 0 & 1 \ea \right); \qquad
\gamma^-(s) = \left( \ba{cc} 1 & 0 \\ s & 1 \ea \right).
\eqno(2.8)
$$

One now verifies straightforwardly that the orbits of the horocyclic
actions $S^\pm$ are the leaves $F^\pm$ of two foliations $\F^\pm$ that 
equip the dynamical system $(M,\mu,T)$ --- where $\mu$ is obtained
from the right Haar 
measure on $SL(2,{\bf R})$ --- with the structure of an  Anosov 
system. Note, moreover, that
$$
T(t) \; S^\pm(s) \; T(-t) = S^\pm (e^{-\lambda_\pm t} \; s) \quad
\forall \; (s,t) \in {\bf R}^2 \eqno(2.9)
$$
and $\lambda_\pm = \pm 1$.
This is a stronger version of (2.2), and, in higher
dimensions, would subsume the Frobenius theorem on involutive distributions.

We now seek to generalize the relation (2.9) in a way that lifts the actions 
from the points $m$ of the manifold $M$ to a (sufficiently large) algebra
$\A$ of functions $f$ on $M$, constituating the observables of the 
classical system. Once that is achieved we extend the formalism to the
non--commutative algebras of quantum systems. The first step is immediately 
achieved by defining the actions
$$
\tau : (t,f) \in {\bf R} \times \A \mapsto f \circ T(-t) \in \A
\eqno(2.10)
$$
$$
\sigma^\pm : (s,f) \in {\bf R} \times \A \mapsto f \circ S^\pm(-s) \in
\A \eqno(2.11)
$$
satisfying
$$
\tau(t) \; \sigma^\pm(s) \; \tau(-t) = \sigma^\pm 
(e^{-\lambda_\pm t}\; s). \eqno(2.12)
$$

\section{Non--commutative Anosov Systems}

In this section we introduce first one of the most convenient extensions
of the Anosov structures to non--commutative systems. Some refinements of
our initial definitions are discussed at the end of the section (Remarks
3.7).

\paragraph{Definition 3.1}  Let $\cal A$ be a von Neumann algebra, $\phi $ 
be a faithful normal state on $\cal A\, , $ $\tau $ an action --- the
dynamics --- of ${\bf R}$ on $ \cal A $ leaving $ \phi $ invariant, i.e.
a map
$$ 
\tau : (t,A) \in {\bf R} \times {\cal A} \mapsto \tau ( t ) [A] \in {\cal A}
\eqno(3.1) 
$$
such that for all $t$ and $t'$ in ${\bf R} $ : 
$\tau(t) \in \mbox{Aut }({\cal A}) \, ; $
$\tau(t)\tau(t') = \tau(t+t') \, ; $
$\tau $  continuous in $t \, ; $
and $\phi \circ \tau(t) = \phi \, . $

We say that the dynamical system $ ({\cal A},\phi,\tau) $ admits an 
`integrable Anosov structure' if there exists a collection
$$ 
\{\,\sigma_j \mid j=1,\ldots,k;k+1,\ldots,n\,\}
\eqno(3.2) 
$$
of `horocyclic' actions
$$ 
\sigma_j : (s,A) \in {\bf R} \times {\cal A} \mapsto {\sigma}_j (s)[A] 
\in {\cal A}
\eqno(3.3)
$$
leaving $\phi$ invariant and satisfying
$$ 
\tau(t) \sigma_j(s) \tau(-t) = \sigma_j(e^{-\lambda_j t}s)
\quad \forall \quad (s,t) \in {\bf R}^2 
\eqno(3.4)
$$
with $ \lambda_j \in {\bf R} $ and
$$   
\lambda_1 \, \leq \, \ldots \, \leq \, \lambda_k < 0 < 
\lambda_{k+1} \, \leq \, \ldots \, \leq \, \lambda_n .
\eqno(3.5)
$$

Without loss of generality, we can assume (by GNS construction) that: 
$\cal A $ acts on a Hilbert space $ \cal H \, ; $
$ \phi $ is a vector state, i.e. 
$ \langle \phi ; A \rangle = (\Phi , A \Phi) \, \forall \, A 
\in {\cal A} \, , $
with $ \Phi \in \cal H $  cyclic and separating for ${\cal A} \, ; $
$\tau $ [resp. $\sigma_j $] is implemented by a 
strongly continuous one-parameter 
unitary group $ V $ [resp. $ U_j $] , i.e. $ \forall \, (s,t)\in {\bf R}^2 $ 
and $\forall \, A \in {\cal A} $
$$  
\tau(t)[A] = V(t) \, A \, V(-t) \qquad
\sigma_j(s)[A] = U_j(s) \, A \, U_j(-s) .
\eqno(3.6)
$$
We have then, as a consequence of (3.4)
$$ 
V(t) \, U_j (s) \,  V(-t) = U_j (e^{-\lambda_j t} s)
\quad \forall \quad (s,t) \in {\bf R}^2 .
\eqno(3.7)
$$

\paragraph{Theorem 3.2 (Lyapunov exponents)}
Let $ \delta_j $ be the derivation generating the horocyclic action
    $\sigma_j ({\bf R})$
and $ {\cal D} ({\delta}_j ) $ be its domain. Then:
\begin{enumerate}
\item[(1)] $\tau(t) \, \delta_j \, \tau (-t) = e^{-\lambda_j t } \delta_j 
\quad $  on $ {\cal D} ({\delta}_j ) \quad \forall \quad t \in {\bf R} $

\item[(2)] for every $ A \in {\cal D} (\delta_j) $ with $ \delta_j[A] \neq 0 $
$$ 
\lambda_j = \lim_{t\to\infty} {1\over t} \, 
\ln \| \, {\delta}_j \tau(t)[A] \, \| 
$$
\item[(3)] with $ \cal D $ denoting the space of derivations of ${\cal A}\, , $
    $ g $ any positive bilinear map
$$ 
g : (\delta , \delta ') \in {\cal D} \times {\cal D} \mapsto
    g(\delta ,\delta ') \in {\cal A} \, , 
$$ 
and, for every $ \delta \in {\cal D} \, : $ 
    $ \| \delta \|_g \equiv g(\delta,\delta)^{1/2}  \, ;$
    if we assume that for every horocyclic $ \delta_j $:
      $ \| \delta_j \|_g \neq 0 \, , $  we have:
 $$ 
\lambda_j  = \lim_{t\to\infty} {1\over t} \,
       \ln \| \, \tau(-t) \delta_j \tau(t) \|_g 
$$
\item[(4)] for every $ A \in {\cal D} ({\delta}_j) $ with $ \delta_j[A] 
\neq 0 $,  $ f(A,s) \equiv \| \, \sigma_j(s)[A] - A \, \|  $ satisfies
$$ 
{\lambda_j} = {{1\over t}} \ln \, \lim_{s\to 0 }
       \left\{ {{f(\tau(t)[A],s)}\over {f(A,s)}} \right\} .  
$$
\end{enumerate}

\paragraph{Proof:} Eqn. (3.4) implies that ${\cal D} ({\delta}_j)$ 
is stable under $\tau({\bf R})\, , $
and in fact, for any $ A \in {\cal D}({\delta}_j)$
$$ 
\delta_j \, \tau(t)[A] = e^{\lambda_j t} \tau(t)[{\delta}_j \, A] 
$$
which is (1). We have then
$$
\ln \| \, \delta_j \,\tau(t)[A] \, \| = 
  \lambda_j t + \ln \| \delta_j[A] \| 
$$
from which (2) follows; (3) follows similarly from (1). 
Finally:
$$ 
\lim _{s\to 0} \,
{ {f(\tau(t)[A],s)}\over {f(A,s)} }
 \, = \, 
 e^{\lambda_j t} \, \lim_{s\to 0} \,
{{(s e^{\lambda_j t})^{-1} f(A, e^{\lambda_j t}s)}\over { s^{-1} f(A,s) } }
 \, = \, e^{\lambda_j t} 
$$
which proves (4). \hfill\ q.e.d.

\paragraph{Comments on Thm 3.2} 
\begin{enumerate}
\item  Eqn. (3.7) implies that $ \delta_j$ are unbounded derivations,
hence 
$ \| \tau(-t) \delta_j \tau(t) \| $
would be of no use in the above computations.

\item The conclusion (3) involves a non--commutative extension of the 
classical definition of a metric
$$ 
g : (X,X') \in {\cal X}^{\infty}(M) \times {\cal X}^{\infty}(M) \mapsto
   g (X,X') \in {\cal C}^{\infty}(M) 
$$
where $ {\cal X}^{\infty}(M) $ denotes the space of smooth vector fields on
$ M \, . $ Hence (3) generalizes the classical result, according to which
the Lyapunov exponents do not depend on the metric (see e.g. [Wal]).

\item  $ f(A,s) $ provides a measure of how far $ A $ moves along 
an orbit of the horocyclic action $ \sigma_j \, , $ so that (4) compares
this to the similar expression for the time--translate $ \tau(t) [A] $ of
$ A \, . $

\item  $ f(A,s) $ can be replaced in 3.2.4 by
$ f(\Phi,A,s) = \| ( \sigma_j(s) [A] - A ) \Phi \| \, . $
\end{enumerate}

\paragraph{Theorem 3.3 (spectral properties)}  Let $ V({\bf R}) $ and 
$ U({\bf R}) $ be two strongly continuous one--parameter groups of 
unitary operators acting on some Hilbert
space $ \cal H $ and satisfying for some $ \lambda \neq 0 $
$$ 
V(t) \, U(s) \, V(-t) = U(e^{-\lambda t } s ) 
\quad \forall \quad (s,t) \in {\bf R}^2 .
\eqno (3.8)
$$
With $ \{ E(-\infty,k] \, \mid \, k \in {\bf R} \} $
denoting the spectral family of the generator $K$ of $ U({\bf R}) $, let
$$ 
{\cal H}^o \equiv \{ \Psi \in {\cal H }\, \mid \, U(s) \Psi = \Psi
   \quad \forall \quad s \in {\bf R} \}
\eqno (3.9)
$$
$$ 
\Ha^{\perp} \equiv \{ \Psi \in {\cal H} \, \mid \, (\Psi,\Psi') = 0
    \quad \forall \quad  \Psi' \in {\cal H}^o \}
\eqno (3.10)
$$
$$ 
\Ha^- \equiv E(-\infty,0] \, {\cal H}^{\perp}
\quad ; \quad 
\Ha^+ \equiv E [0,\infty) \, {\cal H}^{\perp} .
\eqno (3.11)
$$
Then:
\begin{enumerate}
\item[(1)] the decomposition 
${\cal H}^- \oplus {\cal H}^o \oplus {\cal H}^+ $ is stable
under $V({\bf R})$ and  $U({\bf R})$;

\item[(2)] with $ H^{\pm} $ and $ K^{\pm} $ denoting the self--adjoint 
generators of the restrictions $V^{\pm}({\bf R})$ 
and $U^{\pm}({\bf R})$ of  $V({\bf R})$ and $U({\bf R})$ to 
${\cal H}^{\pm} \, , $ the four operators 
$ H^{\pm} $ and $\Lambda^{\pm} \equiv \ln \{ \pm K^{\pm} \} $
have homogeneous, absolutely continuous Lebesgue spectrum covering 
${\bf R} \, . $
\end{enumerate}

\paragraph{Proof:} Since $ {\cal H}^i \, (i = -,o,+) $
are defined in terms of the spectral family of $U({\bf R})\, ,$
these spaces are stable under $ U(s) $ for all $ s \in {\bf R} \, .$
Moreover Eqn. (3.4.1) is equivalent to the relation
$$ 
V(t) \, E( \Delta ) \, V(-t) = E( e^{\lambda t} \Delta )
\eqno (3.12)
$$
holding for every $ s \in {\bf R} $ and every Borel subset
$ \Delta \subset {\bf R} \, .$ This relation  implies that 
$ {\cal H}^i $ are also stable under $ V(s) $ for all
$ s \in {\bf R} \, . $ This proves (1).

On $ {\cal H}^{\pm} $ the operators $ \pm \, K^{\pm} $
are strictly positive so that 
$ \Lambda ^{\pm} = \ln ( \pm \, K^{\pm} ) $
are well defined self--adjoint operators with spectral families
$ F^{\pm} $ such that, for every $ \Delta \subset {\bf R} $ 
$ F^{\pm} ( \Delta ) \equiv E^{\perp} ( \exp[\pm \Delta] ) $
where $ E^{\perp} $
denotes the restriction of $ E $ to $ {\cal H}^{\pm} \, .$

With 
$ W^{\pm} (t) \equiv V^{\pm} ( { t\over \lambda } ) $
Eqn. (3.4.2) reads 
$$ 
W^{\pm} (t) \, F^{\pm} ( \Delta ) \, W^{\pm} (-t) \, = \, 
  F^{\pm} ( \Delta \, + \, t ) 
\eqno (3.13)
$$
for every $ t \in {\bf R} $ and every Borel set
$ \Delta \subset {\bf R} \, .$
Since Eqns. (3.13) are systems of imprimitivity on $ {\bf R} \, , $ the
conclusion (2) of the theorem is a straightforward consequence of
the Mackey--von Neumann uniqueness theorem. \hfill\ q.e.d.\\

As a consequence of Eqn. (3.4) [see (3.6) -- (3.7)] the above spectral 
properties are satisfied for every horocyclic action 
$ \sigma_j$ $( j = 1, \ldots , n) $ of a non--commutative Anosov system.

In the sequel, we shall need to control the behaviour of 
$E(-\epsilon,\epsilon)A\Phi$ for small $\epsilon$. With this in mind, 
we introduce the following notation, in which the index $j$ is omitted
whenever it is not explicitly required.  For every $ A \in {\cal A} $
and every $ f \in L^1 ({\bf R}, ds) $ we introduce
$$ 
A(f) \equiv \int ds \, f(s) \sigma (s) [A] \, .
\eqno (3.14)
$$
We denote with $ \hat f $ the Fourier transform of $ f \, , $
and we introduce
$$ 
{\cal F}^o \equiv \{ f \in L^1 ({\bf R},dx) \, \mid \, 
   \hat f (0) = 0 \, ; \, \hat f \, \mbox{ continuous around the origin} \} 
\eqno (3.15)
$$
and for every Borel set $ \Delta \subset {\bf R} $
$$ 
{\cal F}^o (\Delta ) \equiv \{ f \in {\cal F}^o \, \mid \, 
  \mbox{ess--supp } \hat f \cap \Delta = \emptyset \}.
\eqno (3.16)
$$
We define
$$ 
{\cal A}^o \equiv \{ A(f) \, \mid \, A \in {\cal A} \, ; 
  \, f \in {\cal F}^o \}
\eqno (3.17)
$$
and
$$ 
{\cal A}^o ( \Delta ) \equiv \{ A \in \A^o \mid A(f) = 0 
  \quad \forall \; f \in {\cal F}^o (\Delta ) \}\, 
\eqno (3.18)
$$
such that $\Delta \subset \ol{\Delta}$ implies $\A^o(\Delta) 
\subset \A^o(\ol{\Delta})$.

\paragraph{Corollary 3.4 (vectorial K--filtering)} Let
$ ( {\cal A} , \phi , \tau ) $ be a non--commutative Anosov system, 
$ \sigma $ be one of its horocyclic actions, and
$ {\cal A}^o( \Delta ) $ be defined as in Eqn. (3.18) with 
$\Delta = (-\infty,-a) \cup (a,\infty)$, $a \in {\bf R}^+$. Then
\begin{enumerate}
\item[(1)] $ {\cal A}^o ( \Delta ) $ is a subspace of $ {\cal A}^o $

\item[(2)] $ \tau (t) [{\cal A}^o ( \Delta )] = 
                {\cal A}^o ( e^{-\lambda t} \Delta ) $

\item[(3)] for all $ \Delta = (-\infty,-a] \cup [b,\infty)$,  
$a,b \in {\bf R}^+$  and for all $ t $ such that $ \lambda t > 0 $:\\
$ {\cal A}^o ( \Delta ) \subset \tau(t)\, [ {\cal A}^o ( \Delta ) ] $.
\end{enumerate}

\paragraph{Proof:} (1) is straightforward. For (2), note that for every
$ A \in {\cal A}\, :$ $ (\tau(t) [A])(f) = \tau(t)[A(f_t)] $
with $ f_t(s) = e^{-\lambda t} f(e^{-\lambda t} s) $
and thus $ \hat f_t (k) = \hat f ( e^{ \lambda t } k ) \, ; $
together with (3.18) this indeed implies (2). 
Finally, (3) follows straightforwardly from (2).\hfill\ q.e.d.

\paragraph{Lemma 3.5} Let $ V({\bf R}) $ and $ U({\bf R}) $
be as in Theorem 3.3; $ K $ be the infinitesinal generator of
$ U({\bf R}) \, , $ and $ E $ be its spectral family. Let further
$ {\Psi}_1 $ and $ {\Psi}_2 $ be two vectors in
$ \cal H \, , $ with $ E(-a,a) {\Psi}_1 = 0 $ for some 
$ a > 0 \, , $ and $ {\Psi}_2 \in {\cal D} (K^r) $ for some
$ r > 0 \, . $ Then for every $ t \in {\bf R} \, : $
$$ 
\vert ( {\Psi}_2 , V(t) {\Psi}_1 ) \vert \leq
 e^{-\lambda t r } \, a^{-r} \, \| {\Psi}_1 \| \, \| K^r \, {\Psi}_2 \|.
\eqno (3.19)
$$

\paragraph{Proof:} With 
$ \Delta = ( -\infty ,- a] \cup [a,\infty) $
we have as a consequence of Eqn. (3.12)
$$ 
\vert ( {\Psi}_2 , V(t) {\Psi}_1 ) \vert =
   \vert ( E(e^{\lambda t} \Delta ) {\Psi}_2 , V(t) {\Psi}_1 ) \vert \leq
   \| {\Psi}_1 \| \, ( {\Psi}_2 , E(e^{\lambda t }\Delta ) {\Psi}_2 )^{1/2}
\eqno (3.20)
$$
from which the conclusion of the lemma follows upon using the
classical inequality, holding for every $ r > 0 \, : $
$$ 
\chi_{(-\infty,-1] \cup [1, \infty)} (x) \leq x^{2r} 
   \quad \forall \quad x \in {\bf R }
\eqno (3.21)
$$
where $\chi_\Delta$ denotes the characteristic function of the subset
$\Delta$; for  $ \Delta $ as above, we have:
$$ 
\chi_{ \exp (\lambda t ) \Delta } (x) \leq (a \exp (\lambda t))^{-2r} x^{2r}
   \quad \forall \quad (x,t) \in {\bf R}^2 
\eqno (3.22)
$$
so that
$$ 
( {\Psi}_2 , E( e^{\lambda t} \Delta ) {\Psi}_2 ) ^{1/2} \leq
   (a \exp (\lambda t))^{-r} \, ( {\Psi}_2 , K^{2r} {\Psi}_2 )^{1/2} \, .
\eqno (3.23)
$$

\hfill\ q.e.d.

Note that the corollary can be strengthened in case $ {\Psi}_2 $
satisfies $ E( {\Delta }_2) {\Psi}_2 = {\Psi}_2 $ for 
$ {\Delta}_2 $ compact; indeed there exists then $ T  \geq 0 $
such that for all $ t > T \, : $
$ e^{\lambda t} \Delta \cap {\Delta}_2 = \emptyset $ and thus
$$ 
( {\Psi}_2 , V(t) {\Psi}_1 ) = 0 \quad \forall \quad t > T .
\eqno (3.24)
$$

\paragraph{Theorem 3.6 (exponential clustering)}
Let $ ({\cal A},\phi,\tau) $ be a non--commutative Anosov system;
$ \sigma $ be one of its horocyclic actions,  for which 
$ {\cal A}^o $ is defined as in (3.17);
$ \delta $ be the derivation generating $ \sigma \, ; $
$ {\cal D} (\delta^r) $ be the domain of $ \delta^r \, , $ 
with $ r > 0 \, .$

Then for every $A \in {\cal A}^o $ and $ \epsilon > 0 \, , $
there exists $ a > 0 $ such that for all $ t \in {\bf R} \, , $
all $ r > 0 $ and all $ B \in {\cal D}(\delta^r) \, : $
$$ 
\ba{rcl}
\vert  \langle  \phi ; A^* \tau(t)[B] \rangle \vert  &\leq &
e^{-\lambda \, t \, r} \, a^{-r}
           \langle \phi ; A^*A\rangle^{1/2} 
    \langle \phi ; \delta^r [B]^* \delta^r [B]\rangle^{1/2} \\[12pt]
&& +\; \epsilon \langle \phi ; A^*A\rangle^{1/2} \, 
           \langle \phi ; B^*B\rangle^{1/2}   \ea
\eqno (3.25)
$$
with $\langle \phi ; A \rangle \; \langle \phi ; B \rangle = 0$, since by
(3.15) and (3.17) $\langle \phi;A\rangle = 0$.

\paragraph{Proof:} Given $ A \in {\cal A}^o $ we have
$ A = A_o(f) $ for some $ A_o \in {\cal A} $ and some 
$ f \in {\cal F}^o \, , $ hence $\langle \phi ; A \rangle = 0$.
Without loss of generality, we can assume 
$ \| A \Phi \| \neq 0 \, ; $ for every $ a > 0 \, , $ we have then:
$$ 
\| E(-a,a)\, A_o(f) \Phi \| =
\left( \int_{-a}^a dk \vert \hat f (k) \Phi_{A_o}(k) \vert^2 \right)^{1/2} 
\leq \sup_{[-a,a]} \vert \hat f(k) \vert 
{ {\| A_o \Phi \|}\over {\| A \Phi \|} } \, \| A \Phi \|
\eqno (3.26)
$$
where $ \{ \Phi_{A_o} (k) \, \mid \, k \in {\bf R} \} $
denotes the representation of $ A_o \Phi $ with respect to the 
spectral family of $ K $ --- the generator of the unitary group
$ U({\bf R}) $ implementing $ \sigma ({\bf R}) \, . $ 
Since $ f \in {\cal F}^o \, , $ given $ \epsilon > 0 $ there exists
$ a > 0 $ such that
$$ 
\| E(-a,a) A \Phi \| \leq \epsilon \| A \Phi \| .
\eqno (3.27)
$$

We now write, for $ A $ and $ B $ as stated in the theorem:
$$ 
\vert \langle \phi ; A^*\tau[B]\rangle \vert \leq
   \vert (A \Phi , V(t) B \Phi   ) \vert \leq
   \vert ( \Psi_2 , V(t) \Psi_1  ) \vert \, + \,
   \vert ( \Psi_2^c , V(t) \Psi_1 ) \vert
\eqno (3.28)
$$
with
$ \Psi_2   \equiv E(\Delta)   A \Phi \, , $
$ \Psi_2^c \equiv E(\Delta^c) A \Phi \, , $
$ \Delta   \equiv (-\infty, -a] \cup [a,\infty) \, , $
$ \Delta^c \equiv (-a,a) $
and
$ \Psi_1   \equiv B \Phi \, . $

The theorem then follows from (3.28) upon applying Lemma 3.5 to
$ \vert (\Psi_2 , V(t) \Psi_1) \vert \, , $
and using Schwartz inequality for
$ \vert (\Psi_2^c , V(t) \Psi_1 )\vert $
together with the inequality (3.27).\hfill\ q.e.d.

\paragraph{Remarks 3.7}
\begin{enumerate}
\item In the definition of $ \tau \, , $
one may want to allow for ${\bf R} $ to be replaced by $ Z \, ; $ 
this modification requires some adjustments that will be illustrated in
section 4.
Although we set as a separate assumption the $\sigma$--invariance of $\phi$, 
this property alternatively can be viewed as a consequence of the
following three assumptions: (i) $\sigma$ is unitarily implemented;
(ii) Eqn. (3.7) holds with $(s,t) \in {\bf R} \times {\bf R}$ or 
${\bf R} \times {\bf Z}$; and (iii) $\phi$ is $\tau$--invariant.
\item To emphasize the fact that classical Anosov systems are structures
appearing in differential geometry 
(i.e. with $ {\cal A}_o = C^{\infty}(M) $ ),
rather than probability theory
(i.e.  with ${\cal A} = L^{\infty}(M,d\mu) $ ),
and the fact that derivations provide a natural non--commutative
extension of the classical concept of vector field, one may want to
consider the conclusion (1) in Theorem 3.2 as the starting point of the 
theory rather than the  relation (3.4) used in definition 3.1.

\item Although it has played no role in this section, one might ultimately 
want to impose that: (i) there exists a dense domain
$ {\cal D} \subset {\cal A} $ 
(compare with $ C^{\infty}(M) \subset  L^{\infty}(M) $ for compact $M$),
stable under every $ \delta_j (j=1,\ldots,n) \, ; $
and (ii) that these derivations form a $n$--dimensional Lie algebra over
$ {\cal D} \, . $

\item In the same vein, it may be interesting to pursue the fact that 
there exists non--commutative Anosov systems
(see e.g. section 4) for which the generators of the horocyclic
actions form a basis in the space of outer derivations.

\item We did {\em not\/} assume, in the present section, that the dynamics
$\tau({\bf R})$ was the modular group for $\phi$, as this would have 
precluded the existence of horocyclic actions $\sigma_j({\bf R}) 
\subset \mbox{Aut }(\A)$. Nevertheless, see section 6 (e.g. Remark 
6.3.2, Lemma 6.6 and Theorem 6.7) for a possibility to have $\phi$ KMS
for $\tau({\bf R})$ and still keep the main results of the present
section.

\item The consequence (3.7) of the defining relation (3.4) rather than
this relation itself was the operative condition for the present section.
Hence, once (3.7) is obtained, 
W*--character of $\A$ is inessential. After that, $\A$ could equally
well be C*--algebra, a normed *--algbra or just a *--algebra.

\item It is also possible to weaken the assumptions without damaging the 
essential structures and not require that $ \phi $ be a faithful state on 
${\cal A} \, , $ or even dispense entirely with the introduction of a state.

\item In addition to the fact that the subspace $ {\cal H}^o $
in theorem 3.3 is stable under $ V({\bf R}) $
and contains the cyclic and separating vector $ \Phi \, , $
one might wish to demand that $ \dim ({\cal H}^o_j) = 1 $
i.e. that $ {\cal H}^o_j = C \Phi $ for each $ j $ separately; this would
imply that the state $ \phi $ is  
{\em extremal  $\sigma_j$--invariant\/}  for each $ j \, . $
This might be too much to ask. Nevertheless, a less stringent assumption
will ensure an ergodic behaviour under the dynamics $ \tau({\bf R}) \, .$
Specifically, to obtain that
$ {\cal H}^o_V \equiv \{ \Psi \, \mid \, V(t) \Psi = \Psi \quad 
\forall \quad t \in {\bf R} \} $ is one--dimensional, and thus (since we
assumed $\A$ to be in standard form with respect to $\phi$)
$ \phi $ is {\em extremal  $ \tau$--invariant\/},
it is sufficient, by virtue of the absolute continuity of
$ H $ over $ {\cal H}^{\perp} \, , $ to assume that 
$$ 
\dim \left( \bigcap_{j \in S} {\cal H}^o_j\right) = 1 
\eqno(3.30)
$$
where $ S $ is any subset of the index set $ j = 1, \ldots, n \, ; $
the cases where  $ S = \{ 1,\ldots,k \} $ 
[resp. $ S = \{ k+1,\ldots,n \} $]
 could be interpreted as a non--commutative analog of the standard
feature of classical Anosov flows, namely that the stable [resp.
unstable] manifolds are dense in $ M \, . $
\end{enumerate}

\section{The Quantized Arnold Cat as Anosov System}
Recall (see e.g. [AA]) that the phase space $(M,\omega)$ of the
classical Arnold cat map is the torus $M = \Gamma \setminus \ol{M}$,
with $\Gamma = \{ \zeta = (\xi,\eta) \in {\bf Z}^2\}$ and
$\ol{M} = \{ z = (p,q) \in {\bf R}^2\}$, equipped with the symplectic
form $\omega = dp \wedge dq$. The {\bf discrete} ``dynamics'' of the
model is given by the iterates $\{ T^n \,\mid\, n \in {\bf Z}\}$ of
the natural action on $M$ of $T \in SL(2,{\bf Z})$ with tr~$T > 2$.
Note that the eigenvalues $\ve_j$ $(j = 1,2)$ of $T$ satisfy
$0 < \ve_1 < 1 < \ve_2 < \infty$; and that the principal directions
$X_j = (u_j,v_j)$ $(j = 1,2)$ of $T$, defined by
$$
T \; X_j = \ve_j \; X_j \quad \mbox{and} \quad \|X_j\| = 1,
\eqno(4.1)
$$
have irrational slope $(v_j/u_j)$, so that each of the integral curves of
$X_j$ is dense in $M$.

To quantize this model, we introduce first the Weyl algebra
$\{ W(\zeta)\, \mid \, \zeta = (\xi,\eta) \in {\bf R}^2\}$ for $\ol{M}$,
satisfying the defining relations
$$
\ba{l}
W(\zeta)^* = W(-\zeta); \qquad \|W(\zeta)\| = 1 \\[10pt]
W(\zeta_1) \; W(\zeta_2) = e^{i2\pi \theta\sigma(\zeta_1,\zeta_2)}\;
W(\zeta_1+\zeta_2) \ea
\eqno(4.2)
$$
where $\sigma(\zeta_1,\zeta_2) = \frac{1}{2} (\xi_1 \; \eta_2 - \xi_2 \; 
\eta_1)$ and the ``deformation parameter'' $\theta \neq 0$ plays here the
role of the Planck constant or the magnetic field, see [Bel].

The Weyl algebra for $M = \Gamma \setminus \ol{M}$ is then obtained
by restricting the domain of $W$ to those $\zeta$ that satisfy the 
``periodic boundary conditions''
$$
W(\eta) \; W(\zeta) \; W(\eta)^* = W(\zeta) \quad \forall \; \eta 
\in \frac{1}{\theta}\; {\bf Z}^2. \eqno(4.3)
$$
Hence $W$ will now be restricted to $\zeta \in {\bf Z}^2$. We denote by
$\C$ the C*--algebra obtained as the norm--closure of the linear span
$\D$ of $W = \{W(\zeta)\,\mid \, \zeta \in {\bf Z}^2\}$. Note that
$$
\phi : W(\zeta) \in W \mapsto \delta_{0,\zeta} \eqno(4.4)
$$
extends to a faithful tracial state $\phi$ over $\C$. Let $\pi$ be the 
faithful representation of $\C$ obtained by GNS from $\phi$, and let
$\A = \pi (\C)''$. Clearly $\A$ is a non--abelian von Neumann algebra;
in particular, for irrational values of $\theta$, $\A$ is the 
hyperfinite type II$_1$--factor. We henceforth identify $W(\zeta)$ 
and $\pi(W(\zeta)$); similarly $\D$ and $\pi(\D)$. Note finally that
$$
\tau : W(\zeta) \in W \mapsto W(\wh T \zeta) \in W \eqno(4.5)
$$
(where $\wh T$ denotes the transposed of $T$) extends to a 
continuous action
$$
\tau : (n,A) \in {\bf Z} \times \A \mapsto \tau(n) [A] \in \A
\eqno(4.6)
$$
satisfying
$$
\phi \circ \tau(n) = \phi \quad \forall \; n \in {\bf Z}.
\eqno(4.7)
$$
Eqn. (4.5) expresses the fact that the dynamics $\tau$ is quasi--free.

\paragraph{Definition 4.1} [BNS,Nar]: The Quantized Arnold Cat (for
$\theta \neq 0$) is the non--commutative dynamical system $(\A,\phi,
\tau)$ constructed above.\\

In the sense of definition 3.1, {\em except\/} that the time runs here over
{\bf Z} rather than over {\bf R}, we immediately obtain the following
result.

\paragraph{Proposition 4.2} The Quantized Arnold Cat is a non--commutative 
Anosov system.

\paragraph{Proof:} For $j = 1,2$ and $X_j$ as in (4.1), define for
every $s \in {\bf R}$:
$$
\sigma_j(s) : W(\zeta) \in W \mapsto e^{-is(X_j,\zeta)} \; W(\zeta) 
\in \D \eqno(4.8)
$$
where $(\;\cdot\;,\;\cdot\;)$ denotes the usual scalar product in ${\bf R}^2$.
Then $\sigma_j$ extends to a weakly--continuous action of {\bf R} on $\A$:
$$
\sigma_j : (s,A) \in {\bf R} \times \A \mapsto \sigma_j(s)[A] \in \A
\eqno(4.9)
$$
satisfying
$$
\phi \circ \sigma_j(s) = \phi \quad \forall \; s \in {\bf R} \eqno(4.10)
$$
and the discrete--time version of (3.4), namely
$$
\tau(n) \; \sigma_j(s) \; \tau(-n) = \sigma_j(e^{-\lambda_j n} \; s)
\quad \forall \; (s,n) \in {\bf R} \times {\bf Z} \eqno(4.11)
$$
where $\lambda_j = \ln \ve_j$ and $\ve_j$ as in (4.1), and thus
$- \infty < \lambda_1 < 0 < \lambda_2 < \infty$. \hfill\ q.e.d.\\

As indicated in Rem. 3.7.1, the fact that the domain of the 
time--parameter is {\bf Z} rather than {\bf R} requires some minor
modifications to the theory presented in section 3. The following result 
is at the root of most of these modifications.

\paragraph{Theorem 4.3} With $(\A,\phi,\tau)$ and $\sigma_j({\bf R})$ as 
in Proposition 4.2
\begin{enumerate}
\item[(i)]
$$
U_j(s) : W(\zeta) \Phi \in \Ha \mapsto \sigma_j(s) [W(\zeta)] \Phi \in
\Ha \eqno(4.12)
$$
extends to a weakly--continuous, one--parameter unitary group 
$U_j({\bf R})$; the spectrum of the generator $K_j$ of $U_j({\bf R})$
is a discrete, but dense, subgroup of {\bf R}, and it is simple.
\item[(ii)]
$$
V : W(\zeta) \Phi \in \Ha \mapsto \tau(1) [W(\zeta)] \Phi \in \Ha
\eqno(4.13)
$$
extends to a unitary operator $V$ on $\Ha$; and $V$ has homogeneous 
Lebesgue spectrum (in the sense of Kolmogorov).
\end{enumerate}

\paragraph{Proof:} Since $\C$ is a deformation $(\theta \neq 0$) of the
classical ($\theta = 0$) algebra of functions on $T^2$, and since
the quantum dynamics is quasi--free (4.5), the classical proof, which depends
effectively only of the vector space structure of the algebra, can be
extended straightforwardly to the present case. We first notice 
(see (4.4)) that
$$
\{ \Phi_\zeta \equiv W(\zeta) \Phi \, \mid \, \zeta \in {\bf Z}^2\}
\eqno(4.14)
$$
is an orthonormal basis in $\Ha$. From Eqns. (4.12) and (4.8):
$$
K_j \; \Phi_\zeta = (X_j,\zeta) \Phi_\zeta \quad \forall \, \zeta \in
{\bf Z}^2 \eqno(4.15)
$$
and thus, with $X_j = (u_j,v_j)$ and $\zeta = (\xi,\eta)$:
$$
\mbox{Sp }(K_j) = \{ u_j \; \xi + v_j \; \eta \, \mid \, (\xi,\eta)
\in {\bf Z}^2\} . \eqno(4.16)
$$
Hence Sp~$(K_j)$ is a discrete subgroup of {\bf R}; since $m \equiv
v_j/u_j$ is irrational, Sp~$(K_j)$ is dense in {\bf R}, and is simple.
This proves (i). Note in particular that
$$
\Ha^0 \equiv \{ \psi \in \Ha \,\mid\, U_j(s)\; \psi = \psi \; \forall
\; s \in {\bf R}\} = {\bf C} \Phi. \eqno(4.17)
$$
To prove (ii), we parametrize ${\bf Z}^2 \setminus \{0\}$, and hence
$$
\Ha^\perp \equiv \{ \psi \in \Ha \, \mid\, (\psi,\Phi) = 0 \}
\eqno(4.18)
$$
as follows. Since the principal directions $\wh X_j$ of $\wh T$ (also)
have irrational slopes: $({\bf R} X_j) \, \cap \, {\bf Z}^2 = \{0\}$,
and the points of ${\bf Z}^2 \setminus \{0\}$ can be re--indexed as
$\zeta = (\wh \xi,\wh \eta)$ with $\wh \xi \in {\bf Z} \setminus \{0\}$
indexing the orbits of $\wh T$ (except the trivial orbit $\{0\}$), 
and $\wh \eta \in {\bf Z}$ indexing the successive points of the orbit
$\wh \xi$. We thus have:
$$
\Ha^\perp = \overline{\mbox{span}}\; \{ \Phi_{\wh \xi,\wh \eta} \,\mid\,
\wh \xi \in {\bf Z} \setminus \{0\}, \wh \eta \in {\bf Z} \}
\eqno(4.19)
$$
$$
V \; \Phi_{\wh \xi,\wh \eta} = \Phi_{\wh \xi,\wh \eta +1} \quad
\forall \quad \wh \xi \in {\bf Z} \setminus \{0\}, \; \wh \eta \in
{\bf Z} \eqno(4.20)
$$
$$
\{ \psi \in \Ha \,\mid\, V \; \psi = \psi \} = \Ha^0 = {\bf C} \; \Phi.
\eqno(4.21)
$$
Eqns. (4.20 -- 4.21), together with (4.17 -- 4.19), are the conclusion (ii) 
of the theorem. 

\hfill\ q.e.d.

\paragraph{Remarks 4.4}
\begin{enumerate}
\item The other results of section 3 (Lyapunov exponents and exponential
clustering) hold for the present model. Note also that
 conditions (3.17) and (3.18) in Cor. 3.4 and Thm 3.6 can be replaced 
by the following
(perhaps more visualizable) condition: for all $a > 0$, let
$$
\D_a \equiv \{  \zeta \in {\bf Z}^2 \,\mid \, (X,\zeta)^2 \geq a\}
\eqno(4.22)
$$
where $X$ is one of the principal directions of $T$; and
$$
\A_a \equiv \mbox{Span } \{W(\zeta)\,\mid \, \zeta \in \D_a\} ;
\eqno(4.23)
$$
in this case, we can even write $\ve = 0$ in (3.25).
\item In connection with Rem. 3.7.4, note that in the present model,
the differential version of (4.12), namely
$$
\tau(n) \; \delta_j \; \tau(- n) = e^{-\lambda_j n}\; \delta_j \quad
\forall \; n \in {\bf Z}
\eqno(4.24)
$$
holds on the dense domain $\D = \mbox{Span } \{ W(\zeta)\,\mid\,
\zeta \in {\bf Z}^2\}$, where
$$
\delta_j [W(\zeta)] = (X_j,\zeta)\; W(\zeta) \quad \forall \; 
\zeta \in {\bf Z}^2. \eqno(4.25)
$$
Hence [Nar,BEJ,Bre,CR] the generators $\delta_j$ of the horocyclic
actions (4.9) form a basis in the space of outer derivations for the
differential structure attached to $\D$, i.e. every derivation
$\delta : \D \ra \D$ can be written as
$$
\delta = c_1 \; \delta_1 + c_2 \; \delta_2 + \delta_3 \eqno(4.26)
$$
with $\delta_3$ (approximatively) inner.
\end{enumerate}

\section{Free Quantum Particle on the Poincar\'e Half--Plane}

The free motion of a classical particle on a compact manifold of
constant negative curvature, such as a principal domain of the
Poincar\'e half--plane, provides a prototype example of an Anosov
flow (see section 2 above). The question therefore naturally 
arises as to whether its
Anosov property survives quantization. We shall now show that,
by contrast with the dynamics of the Arnold Cat, it does not.

\subsection*{The Model} 
As in section 2, let $\ol{M}_c \equiv \{z = (x,y) \mid x \in {\bf R},
y \in {\bf R}^+\}$ be
the Poincar\'e half--plane; the measure 
$d  \mu_c = y^{-2} dx dy$ is invariant under the free and
transitive action on $\ol{M}_c$ obtained by restricting (2.4) to the
subgroup $H$ of  upper--diagonal matrices in $SL(2,{\bf R})$.
We parametrize $H$ with its two subgroups
$$
H_1 \equiv \{ \gamma^+(v) \mid v \in {\bf R} \} \quad ; \quad
H_2 \equiv \{ \gamma(u) \mid u \in {\bf R}^+\} \eqno(5.1a)
$$
with
$$
\gamma^+(v) = \left( \ba{cc} 1 & v \\ 0 & 1 \ea \right) \quad ; \quad
\gamma(u) = \left( \ba{cc} u^{-1/2} & 0 \\ 0 & u^{1/2} \ea \right).
\eqno(5.1b)
$$

We define
${\cal H}$ to be the Hilbert space $L^{2}(M,d{\mu})$ and $U$ to
be the unitary representation of $H$ in ${\cal H},$ given by
$$
(U(h)f)(z) \equiv f(h^{-1}[z]) \quad \forall \; (z,h) \in \ol{M}_c \times H . 
\eqno(5.2)
$$
Thus, the infinitesimal generators of $U(H_{1}), \
U(H_{2})$ are $-iw_{1}, \ -2iw_{2},$ respectively, where
$$
w_{1} \equiv {{\partial}\over {\partial}x}\; ; \quad
w_{2} \equiv x{{\partial}\over {\partial}x}+
y{{\partial}\over {\partial}y}. 
\eqno(5.3)
$$

We assume the algebra of observables, ${\cal A},$ for a quantum
particle on $\ol{M}_c$ to be (cf. [Ma;Em3])
${\lbrace}U(H),C(\ol{M}_c){\rbrace}^{{\prime}{\prime}},$ the
elements of $C(\ol{M}_c)$ acting multiplicatively on ${\cal H}.$ Thus,
${\cal A}={\cal B}({\cal H}).$ Further, [Em2,3], ${\cal A}$ is
the $W^{\star}-$algebra generated by an extension of $U$ to a
faithful unitary representation in ${\cal H}$ of the Weyl group,
$W,$ of the CCR for a particle in $\ol{M}_c.$ To be specific, the Lie
algebra ${\cal W}$ of $U(W)$ has as basis the operators
$(w_{1},w_{2},w_{3},w_{4},w_{5}),$ where $w_{1},w_{2}$ are as
defined by (5.3) and
$$
w_{3}=xy^{-1}\; ; \quad w_{4}=-(1+y^{-1}) \; ; \quad w_{5}=1
\eqno(5.4)
$$
all acting multiplicatively on ${\cal H}.$ Thus, the structure
of ${\cal W}$ is given by
$$
[w_{1},w_{2}]=w_{1}\; ; \quad [w_{1},w_{3}]=
w_{4}+w_{5} \; ;\quad  [w_{2},w_{4}]=w_{4}+w_{5}
\eqno(5.5)
$$
all other commutators between the $w_{k}$'s vanishing. The group
$W$ is then [Em3] a central extension by ${\bf R}$ of a group
$G,$ which is itself a central extension of $H$ by ${\bf R}^{2}.$

In a standard way, we term an automorphism, ${\alpha},$ of ${\cal
A},$
quasi--free if $U(W)$ is stable under ${\alpha},$ i.e., if there
is an
automorphism, ${\wt {\alpha}},$ of $W,$ such that
${\alpha}U(F) \equiv U({\wt {\alpha}}F).$

The free dynamics of a particle on $\ol{M}_c$ is governed by the
one--parameter group ${\tau}({\bf R})$ of automorphisms of ${\cal
A},$ given by
$$
{\tau}(t)[A] \equiv V(t)AV(-t)\eqno(5.6)
$$
where
$$
V(t)={\exp}(-i{\Delta}t)\eqno(5.7)
$$
and ${\Delta}$ is the Laplace--Beltrami operator for $\ol{M}_c,$ i.e.
$$
{\Delta} \equiv y^{2}({{\partial}^{2}\over {\partial}x^{2}}+
{{\partial}^{2}\over {\partial}y^{2}}) . \eqno(5.8)
$$
The model of a quantum particle on $\ol{M}_c$ is thus given by
$({\cal A},{\tau}).$
One defines similarly the dynamics on a fundamental domain 
$M_c = \Gamma \setminus \ol{M}_c$ (see section 2 and [Em3]), with
$\Ha_c = L^2(M_c, d\mu_c)$, 
$\A_c = \B(\Ha_c)$ and $\Delta_c$ the Laplace--Beltrami operator for
$M_c$.

\subsection*{Violation of Anosov and Quasi--Free Conditions} 
In the case of
the Arnold Cat, the Anosov property of ${\tau}$ stems from the fact
that it is quasi--free and implemented by classical Anosov
automorphisms
of its Weyl group. Since, the dynamics of
the present models are also obtained by quantisation of free
Anosov dynamics, it is reasonable to ask whether the same
situation also prevails here. However, the following Propositions
show that it does not.

\paragraph{Proposition 5.1}  For any normal, ${\tau}_{\Gamma}-
$invariant state, ${\phi}_{\Gamma},$ on ${\cal A}_{\Gamma},$ the
dynamical system $({\cal A}_{\Gamma},{\tau}_{\Gamma},{\phi}_{\Gamma})$
is not Anosov.

\paragraph{Proof:} Since [BGM] the spectrum of  ${\Delta}_{\Gamma}$ is
discrete, it follows from Theorem 3.3 that the model cannot be
Anosov. \hfill\ q.e.d.

\paragraph{Proposition 5.2}  The model $({\cal A},{\tau})$ is
neither quasi--free nor Anosov.

\paragraph{Proof:} 
 Since the spectrum of  ${\Delta}$ is positive,
 it follows again from Theorem 3.3 that the model cannot satisfy the
Anosov condition.

The proof that ${\tau}$ is not quasi--free, i.e., that
$U(W)$ is not stable under ${\tau}({\bf R}),$ is an
immediate consequence of Lemmas 5.3 and 5.4 below. 
We include them for completeness but the result is to be expected
since here even classically the Lie--algebra does not close if one 
adds the $\Delta$ to the $w$'s. \hfill\ q.e.d.

\paragraph{Lemma 5.3}  The restriction of ${\cal W}$ to ${\cal D}$
is not stable under $Ad({\Delta}).$

\paragraph{Lemma 5.4}  If $i{\xi}$ is the infinitesimal generator
of a one--parameter subgroup, $S({\bf R}),$ of $U(W),$ such that
${\tau}({\bf R}):S({\bf R}){\rightarrow}U(W),$ then
$Ad({\Delta})[{\xi}],$ as defined on ${\cal D},$ belongs to
${\cal W}.$

\paragraph{Proof of Lemma 5.3} It follows from our definitions that
$$
[{\Delta},w_{3}]=2(y{{\partial}\over {\partial}x}-
x{{\partial}\over {\partial}y}-w_{3}) \quad \mbox{on} \; {\cal D}. 
$$
The required result follows from the fact that, by eqns. (5.3)
and (5.4), the r.h.s. of this equation manifestly does not belong
to ${\cal W}.$ \hfill\ q.e.d.

\paragraph{Proof of Lemma 5.4} Let
$S_{t}(s) \equiv {\tau}(t)[S(s)] \equiv V(t)S(s)V(-t), \
{\forall}s,t \in {\bf R},$ and let $i{\xi}(t)$ be the
infinitesimal generator of $S_{t}({\bf R}).$ Then the assumption
that ${\tau}({\bf R}):S({\bf R}){\rightarrow}U(W)$ implies that
${\xi}(t)$ lies in ${\cal W}.$ Hence, ${\xi}(t)$ is a complex
linear
combination of the $w_{j}'s,$ i.e.,
$$
{\xi}(t)={\sum}_{k=1}^{5}c_{k}(t)w_{k}.\eqno(5.9)
$$
Further, by the above definition of $S_{t},$
$$
(V(t)f,S_{t}(s)g)=(S(-s)f,V(-t)g) \quad {\forall}f,g \in
{\cal H}. \eqno(5.10)
$$
Hence, as ${\cal D}$ lies in the domains of all elements of
${\cal W},$
$$
(V(t)f,{\xi}(t)g)=({\xi}f,V(-t)g) \quad {\forall}f,g \in
{\cal D}\eqno(5.11)
$$
i.e., by (5.11),
$$
{\sum}_{k}c_{k}(t)(V(t)f,w_{k}g)=({\xi}f,V(-t)g) \quad
{\forall}f,g \in {\cal D} . \eqno(5.12)
$$
Consequently, if $f_{1},.. \ ,f_{5},g \in {\cal D}$ and
$$
L_{jk}(t) \equiv (V(t)f_{j},w_{k}g) \quad \mbox{and} \quad
d_{j}(t) \equiv ({\xi}f_{j},V(-t)g);
\eqno(5.13)
$$
then
$$
{\sum}_{k}L_{jk}(t)c_{k}(t)=d_{j}(t) . \eqno(5.14)
$$
We note here that, since ${\cal D}$ lies in the domains of the
polynomials in the $w_{j}$'s and ${\Delta},$ it follows that
$L_{jk}$ and $d_{j}$ are $C^{\infty}$ functions on ${\bf R}.$

In order to ensure that the matrix
$[L_{jk}(t)]$ is invertible, at least for $t$ in a neighbourhood
of the origin, we choose $g$ so that the vectors $\{ w_j g \mid
j = 1,\ldots 5\}$ are linearly independent; and then we choose
each $f_j$ to be orthogonal to $\{w_k g\mid k \neq j\}$, but not
to $w_j g$.
Thus,  $[L_{jk}(0)]$ is a diagonal matrix,
whose determinant is non--zero. Hence, it follows from (5.13) and
the $C^{\infty}$ character of $[L_{jk}(t)]$ that, for $t$ in a
neighbourhood of ${\lbrace}0{\rbrace},$ this matrix is invertible
and its inverse, $[L^{-1}_{jk}(t)]$ is $C^{\infty}.$
Consequently, by (5.14) and the $C^{\infty}$ property of the
$d_{j}$'s, the functions $c_{j}(t)$ are $C^{\infty},$ for $t$ in
this neighbourhood of ${\lbrace}0{\rbrace}.$

Reverting now to the situation where $f,g$ are arbitrary elements
of ${\cal D},$ we find, on differentiating eqn. (5.14) w.r.t.
$t$ at $t=0,$ that
$$
{\sum}_{k}[{\dot
c}_{k}(0)(f,w_{k}g)+ic_{k}(0)({\Delta}f,w_{k}g)]=
i({\xi}f,{\Delta}g) \quad {\forall}f,g \in {\cal D},
$$
i.e, as ${\xi}={\sum}_{k}c_{k}(0)w_{k},$ by (5.11),
$$
[{\Delta},{\xi}]=i{\sum}_{k}{\dot c}_{k}(0)w_{k},
 \quad \mbox{on }\; {\cal D}. 
$$
Since the r.h.s. of this equation lies in ${\cal W},$ this
establishes the required result. \hfill\ q.e.d.
 
\section{Quantized Field on the Rindler Wedge}
Let $(M^{1,1},g)$ denote the 2--dimensional Minkowski space, with null,
future--directed, coordinates $(u,v)$; and with metric $g = 4 \; du dv$.
In the Poincar\'e group $P^{1,1}$, we consider the following
one--parameter subgroups: the Lorentz boosts
$$
\Lambda : (t;(u,v)) \in {\bf R} \times M^{1,1} \mapsto (e^t\; u,
e^{-t} \; v) \in M^{1,1}
\eqno(6.1)
$$
and the two null--translations
$$
N_1 : (s,(u,v)) \in {\bf R} \times M^{1,1} \mapsto (u + s,v) \in
M^{1,1} \qquad \forall \; s \in {\bf R} \eqno(6.2)
$$
$$
N_2 : (s,(u,v)) \in {\bf R} \times M^{1,1} \mapsto (u,v + s) \in
M^{1,1} \qquad \forall \; s \in {\bf R}. \eqno(6.3)
$$
For any weakly continuous, unitary representation $U$ of $P^{1,1}$,
we write (with $j = 1,2)$:
$$
V(t) = U(\Lambda(t)) \quad \mbox{and} \quad 
U_j(s) = U(N_j(s)) \quad \forall \; s,t \in {\bf R}. \eqno(6.4)
$$
Note that these operators satisfy the Anosov property:
$$
V(t) \; U_j(s) \; V(-t) = U_j(e^{-\lambda_j t}\; s) \quad \forall \;
(s,t) \in {\bf R}^2 \eqno(6.5)
$$
with $\lambda_1 = -1$ and $\lambda_2 = +1$.

This relation is therefore satisfied, in particular, for every
relativistic QFT on $M^{1,1}$ with $U(P^{1,1})$--invariant vacuum $\Phi$.
>From the Reeh--Schlieder theorem [RS], $\Phi$ is cyclic and separating 
for the von Neumann algebra $\cal A$ of the observables relative to
the wedge
$$
W = \{ (u,v) \in M^{1,1} \, \mid \, u \geq 0, v \leq 0\}
\eqno(6.6)
$$
which is, moreover, stable under the action (6.1). Let finally
$$
\phi : A \in {\cal A} \mapsto (\Phi,A \; \Phi) \in {\bf C} \eqno(6.7)
$$
$$
\tau : (t,A) \in {\bf R} \times {\cal A} \mapsto V(t) \; A \; V(-t) 
\in {\cal A}. \eqno(6.8)
$$
\paragraph{Definition 6.1} [Rin]: The quantized Rindler wedge (in
Minkowski space) is the
non--commutative dynamical system $({\cal A},\phi,\tau)$ constructed 
above.
\paragraph{Proposition 6.2} On the quantized Rindler wedge 
$({\cal A},\phi,\tau)$ 
the semi--groups of endomorphisms, defined for $j = 1,1$, by
$$
\sigma_j : (s,A) \in {\bf R}_j \times {\cal A} \mapsto U_j(s) \; A \; 
U_j(-s) \in {\cal A} \eqno(6.9)
$$
(where ${\bf R}_1 \equiv {\bf R}^+$, and ${\bf R}_2 \equiv {\bf R}^-$)
satisfy the Anosov condition
$$
\tau(t) \; \sigma_j(s) \; \tau(-t) = \sigma_j (e^{-\lambda_j t}\; s)
\quad \forall\; (s,t) \in {\bf R}_j \times {\bf R}
\eqno(6.10)
$$
with $\lambda_1 = -1$ and $\lambda_2 = +1$.

\paragraph{Proof:} Immediate from (6.5, 6.8, 6.9) and from the fact
that the wedge (6.6) is stable under $N_j({\bf R}_j)$. \hfill\ q.e.d.

\paragraph{Remarks 6.3}
\begin{enumerate}
\item The inclusion $\sigma_j(s) \; [{\cal A}] \subset {\cal A}$ for
$s \in {\bf R}_j \setminus {0}$ is strict; this follows from the fact
that $\sigma_j(s) \; [{\cal A}]$ is the algebra of observables w.r.t.
the wedge $N_j(s) \; [{\cal W}] \subset {\cal W}$.
\item The fact that the $\sigma_j(s)$ for $s \in {\bf R}_j \setminus
{0}$ are *--algebraic maps that are injective but not surjective
reflects the fact, first pointed out by Bisognano and Wichmann [BW]
that $\tau({\bf R})$, as defined by (6.8), (6.4) and (6.1), is the 
modular group for $\phi$, so that (6.10) is incompatible with
$\phi \circ \sigma_j(s) = \phi$ and $\sigma_j(s) \in \mbox{Aut }
({\cal A})$, since the latter would imply [Tak] that $\sigma_j(s)$
would commute with $\tau({\bf R})$.
\item The modular automorphisms, $\tau$, represent the timme--translations 
of relativistic quantum fields, as percieved by a uniformly accelerated 
observer, and are directly relevant to the Unruh and Hawking effects 
[Se].
\end{enumerate}

\paragraph{Corollary 6.4} For every $s \in {\bf R}$ (not just 
${\bf R}_j$), let
$$
\sigma_j(s) : A \in {\cal A} \mapsto U_j(s) \; A \: U_j(s)^* \in
{\cal B}({\cal H}). \eqno(6.11)
$$
Then
\begin{enumerate}
\item[(1)] $\sigma_j (s) [{\cal A}] \subset {\cal A} \quad \forall \,
s \in {\bf R}_j$
\item[(2)] $\bigcap_{s \in {\bf R}} \sigma_j(s) [{\cal A}] = {\bf C}\; I$
\item[(3)] $\bigvee_{s \in {\bf R}} \sigma_j(s) [{\cal A}] = {\cal B}
({\cal H})$
\item[(4)] For each $s \in {\bf R}\;$ $\sigma_j(s) [{\cal A}]\Phi$ is dense 
in ${\cal H}$.
\item[(5)] With $K_j$ denoting the generator of the unitary group
$U_j({\bf R})$ implementing $\sigma_j({\bf R}): \mbox{Sp } (K_j) =
{\bf R}^+$ and the eigenvalue 0 of $K_j$ is simple.
\end{enumerate}

\paragraph{Proof:} (1) -- (4) are immediate consequences of the argument 
in Remark 6.3.1. From $\phi \circ \sigma_j(s) = \phi$ and (6.5) and by
Theorem 3.3, we know that Sp~$(K_j)$ contains at least ${\bf R}^+$
or ${\bf R}^-$; ${\bf R}^- \setminus {0}$ however is ruled out by: the
spectral condition of the energy--momentum $P = (P^0,P^1)$ of a
relativistic QFT; and $K_j = P_0 + (-1)^{j+1} \; P^1$. $\{ 0 \}$ simple
follows similarly. \hfill\ q.e.d.

\paragraph{Remark 6.5} Thus, the quantum field on the Rindler wedge 
induces an {\em algebraic\/} K--structure [Em3,NT] on the
dynamical systems $({\cal B}({\cal H}), \phi,\sigma_j)$, attached to
the whole Minkowski space $M^{1,1}$, 
 {\em in addition\/} to the fact that the dynamical system
$({\cal A},\phi,\tau)$, attached to the wedge ${\cal W}$, satisfies
the {\em vectorial\/} K--filtering of Corollary 3.4. Notice moreover
that the generator $H$ of $V({\bf R})$ has homogeneous Lebesgue spectrum
over the full real line {\bf R}, in contrast to the above conclusion
(5), namely Sp~$(K_j) = {\bf R}^+$ only.\\

The following result, essentially due to Borchers [Bor], shows the sense 
in which the above properties of the wedge are generic.

\paragraph{Lemma 6.6} Let $\phi$ be a faithful, non--tracial, normal
state on a von Neumann algebra $\cal A$ (presented in the standard
form w.r.t. $\phi$ on $\cal H$). Let $\sigma_j$ ($j = 1,2)$ be a
weakly continuous action
$$
\sigma_j : (s,A) \in {\bf R}_j \times {\cal A} \mapsto 
\sigma_j(s) [A] \in {\cal A} \eqno(6.12)
$$
of the additive semi--group ${\bf R}_j = {\bf R}^+$ [resp. ${\bf R}^-$]
for $j = 1$ [resp. $j = 2$]; and assume that for each $s \in {\bf R}_j$
$$
\sigma_j (s) \in \mbox{End }( {\cal A}) \quad ; \quad \sigma_j(s)[{\cal A}] 
\Phi \; \mbox{ dense in } {\cal H}. \eqno(6.13)
$$
Then, there exists a weakly continuous, one--parameter group of unitaries
$U_j({\bf R})$
$(j = 1,2)$, acting on $\cal H$, such that
$$
\sigma_j(s) [A] = U_j(s) \; A \; U_j(-s) \quad \forall \; (s,A) \in
{\bf R}_j \times {\cal A}. \eqno(6.14)
$$
Moreover, if the generator $K_j$ of $U_j({\bf R})$ has positive spectrum,
then $\sigma_j({\bf R})$ satisfy the Anosov condition
$$
\tau(t) \; \sigma_j(s) \; \tau(-t) = \sigma_j (e^{-\lambda_j t}\; s)
\quad \forall \; (s,t) \in {\bf R} \times {\bf R} \eqno(6.15)
$$
for $\lambda_1 = -1$, $\lambda_2 = +1$, and for the modular action
$$
\tau : (t,A) \in {\bf R} \times \A \mapsto \Delta^{it/2\pi} \; A \;
e^{-it/2\pi} \in \A \eqno(6.16)
$$
relative to $\phi\;$; and 
$$
\phi \circ \sigma_j(s) = \phi. \eqno(6.17)
$$
\paragraph{Proof:} (6.13) imply that for every $s \in {\bf R}_j$
$$
U_j(s) : A \Phi \in \D_o \mapsto \sigma_j(s) [A] \Phi \in \D_s
\eqno(6.18)
$$
are isometries, with dense domain $\D_o = \A \Phi$ and dense range
$\D_s = \sigma_j(s) [\A] \Phi$; they can therefore be extended
uniquely to unitaries acting on $\Ha$; $U_j(\cdot)$ is then extended
from $s \in {\bf R}_j$ to $s \in {\bf R}$ by $U_j(-s) \equiv U_j(s)^*$.
This proves (6.14). From [Bor], we know that $V(t) = \Delta^{it/2\pi}$
and $U_j(s)$ satisfy the Anosov condition (6.5), from which (6.15 --
6.16) follow immediately; so does (6.17) by virtue of the second part 
of Remark 3.7.1.

\paragraph{Theorem 6.7} The quantum field on the Rindler wedge, 
and more generally any 
dynamical system described as in Lemma 6.6, satisfy the essential 
properties (Lyapunov exponents, and exponential clustering) ascribed 
to the non--commutative Anosov systems of section 3.

\paragraph{Proof:} These properties are consequence of (3.7) $\equiv$
(6.5) which is satisfied for the Rindler wedge by construction, and
in general from the assumption of Lemma 6.6. \hfill\ q.e.d.

\paragraph{Remarks 6.8}
\begin{enumerate}
\item For the quantum field on the Rindler wedge, the exponential 
clustering w.r.t. Lorentz
boosts could have been obtained directly from the Lehmann--K\"allen
representation [Le,Ka].
\item Theorem 6.7  opens applications beyond the Rindler wedge
since Lemma 6.6 holds independently of whether the actions $\sigma_j$
commute with one another (see e.g. the discussion of the de Sitter
universe in [Thi]).
\end{enumerate}

\section{Concluding Remarks}
We have provided a framework for a generalisation of the
classical theory of Anosov systems to the quantum regime, where
the algebra of observables is non--commutative. Here, as in
Connes's non--commutative geometry [Co], the required differential
structure is provided by derivations of this algebra, which, in
the classical case, correspond to vector fields on a Riemannian
manifold. The dynamical system obtained by imposing the Anosov
hyperbolicity conditions onto this structure then has the {\it
vectorial} K--property.

Our principal results on concrete models are the following.
\begin{enumerate}
\item[(1)] The quantum version of the 'Arnold Cat', as given by
automorphisms of the non--commutative torus, is a quasi--free,
exponentially clustering, Anosov system, whose hyperbolic
dynamics is inherited from that of its Weyl group.

\item[(2)] On the other hand, the dynamics of a quantum particle, moving
freely, i.e. without external forces, on the Poincar\'e half--plane,
is neither quasi--free nor Anosov. Thus, in this case,
quantisation destroys the Anosov property.

\item[(3)] The modular dynamics of an arbitrary relativistic quantum
field on the Rindler wedge of Minkowski space possesses the Anosov
property.
\end{enumerate}

Thus, of the above examples, (2) is the only one that lacks the
Anosov property. It is also the only one whose algebra
of observables is a type I factor. For the algebra of (1) is
(cf. [BNS]) either a type II$_{1}$ factor or a tensor product of
a classical algebra $L^{\infty}(T^{2})$ and a finite type I
factor I$_{n},$ depending on whether the non--commutativity
parameter of the model is rational or irrational; while the
algebra of (3) is always of type III. Hence, our
results are in line with the conventional wisdom that the only
systems, apart from the (essentially) classical ones, that enjoy
good ergodic properties are those whose algebras of observables are
of type II or III.

\section*{Acknowledgements} 
This work was partially carried out during
visits of G.G.E. to London and Vienna. G.G.E. and G.L.S. would
like to express their thanks to the SERC for financially supporting
their collaboration by enabling G.G.E. to visit with the
Department of Physics, Queen Mary and Westfield College,
London. G.G.E. also wants to acknowledge the stimulating hospitality
and the support of the Institut f\"ur Theoretische Physik der
Universit\"at Wien and of the International Erwin Schr\"odinger
Institute for Mathematical Physics.

\newpage
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\end{document}