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{\nopagenumbers
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\line{\hfill Preprint KUL-TF-95/20}
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\centerline{\BF An algebraic approach to the}
\vskip\baselineskip
\centerline{\BF Kolmogorov-Sinai entropy}
\vskip 2\baselineskip
\centerline{R.~Alicki$^{\ast\ast,1}$, J.~Andries$^{\ast,2}$,
M.~Fannes$^{\ast,3}$, and P.~Tuyls$^{\ast,4}$}
\vskip \baselineskip
\centerline{$\ast$ Instituut voor Theoretische Fysica}
\centerline{Katholieke Universiteit Leuven}
\centerline{Celestijnenlaan 200D}
\centerline{B-3001 Heverlee, Belgium} \vskip \baselineskip %
\centerline{$\ast\ast$ Institute of Theoretical Physics and Astrophysics}
\centerline{University of Gda\'nsk}
\centerline{Wita Stwosza 57}
\centerline{PL-80-952 Gda\'nsk, Poland}
\vskip 6\baselineskip plus 30pt
\centerline{\bf Abstract}
We revisit the notion of Kolmogorov-Sinai entropy for classical dynamical
systems in terms of an algebraic formalism. This is the starting point for
defining the entropy for general non-commutative systems. Hereby typical
quantum tools are introduced in the statistical description of classical
dynamical systems. We illustrate the power of these techniques by providing
a simple, self-contained proof of the entropy formula for general
automorphisms of $n$-dimensional tori.
\noindent {\bf Mathematics Subject Classification (1991):
28D20, 47A35, 46L55, 82B10}
\vfootnote1
{Email: {\tt fizra@halina.univ.gda.pl}}
\vfootnote2
{Email: {\tt johan.andries@fys.kuleuven.ac.be}}
\vfootnote3
{Onderzoeksleider NFWO Belgium, Email: {\tt
mark.fannes@fys.kuleuven.ac.be}}
\vfootnote4
{Onderzoeker IIKW, Email: {\tt pim.tuyls@fys.kuleuven.ac.be}}
\vfill\eject}
\pageno=1
\beginsection 1. Introduction
The aim of this paper is to investigate the compatibility of the
construction of the dynamical entropy for general non-commutative dynamical
systems, as proposed in~[AF1], with the well-established notion of entropy
for classical dynamical systems~[AA, CFS, P, Wal]. We follow the common
strategy that a classical notion is formulated on the level of functions on
phase space and then quantized by extending the construction to general
non-Abelian algebras. This approach is quite similar to that of introducing
quantum groups, quantum stochastic processes or non-commutative
differential geometry. As classical systems arise as classical limits of
quantum systems, one should logically reverse the approach.
In the statistical description of discrete time classical dynamical systems
one produces a coarse-grained model of the system by partitioning the phase
space $\Ex$ into a finite number of disjoint measurable subsets $(C_0,
\ldots C_{k-1})$. The measure $\mu(C_i)$ of a set $C_i$ gives the
probability that the set $C_i$ is occupied and defines, putting $\lambda_i=
\mu(C_i)$, a discrete probability space $(\Ir_k,\lambda)$ which is the
coarse-grained statistical model of the system. In the course of time, the
orbit of a point $x\in\Ex$ visits the sets $C_{i_0}, C_{i_1}, \ldots$ The
probability of the points in phase space that will visit at time $j$ the
set $C_{i_j}$, $j=0,1,\ldots m-1$ is given by
$$\mu\bigl(C_{i_0}\cap \theta^{-1}(C_{i_1})\cap \cdots\cap
\theta^{-m+1}(C_{i_{m-1}})\bigr), \eqno(1.1)
$$
where $x\mapsto\theta(x)$ is the single time step evolution. The mapping of
the orbit of a point to the ``configuration'' $(i_0, i_1, \ldots)$ defines
a symbolic dynamics. A single step in the evolution translates into a
left-shift of the configuration. Hence, a partition of the phase space
leads to a classical spin model on a half-chain with a shift invariant
measure determined by~(1.1). The dynamical entropy or Kolmogorov-Sinai
invariant is the usual statistical entropy density of the half-chain or,
more precisely, the maximal one, varying over the partitions of the phase
space. Alternatively, the half-chain model can be interpreted as a
stationary stochastic process with values in the index set $\{0,\ldots
k-1\}$.
The natural way to obtain a discrete statistical picture of a
non-commutative system is to consider density matrices $\rho$ as states on
the algebra $\Ma_k$ of $k\times k$ matrices. Such a picture can easily be
produced in terms of {\sa operational partitions of the unity}~[Lin],
which are collections $\X= (x_0, \ldots x_{k-1})$ of observables satisfying
$\sum_i x_i^*\,x_i= \idty$~. The analogue of the measure $\lambda$ of above
is the density matrix $\rho$ with matrix elements $\omega(x_j^*\,x_i)$. A
construction similar to that of above can now be performed. It yields a
state on a quantum spin half-chain which can be interpreted
either as a symbolic dynamics or as a quantum stochastic process in the
sense of~[AFL]. Unlike the classical case however, the stochastic process
is generally not stationary, due to the non-commutativity of the
quantum observables. The dynamical entropy can still be defined in terms of
the mean entropy of the state on the half-chain. A number of technical
problems limits the computations in specific quantum models to operational
partitions of a special nature~[AF1, AF2, AN, AFTA]. The problem of computing
the entropy for general partitions in the von~Neumann algebra~$\M$
associated with the dynamical system is still open but is crucial in order
to obtain a proper invariant.
One of the main motivations of this paper is to obtain a full understanding
of the algebraic construction of the dynamical entropy for classical
systems. Despite the commutativity of the observables, the statistical
models of above still produce truly non-commutative systems. It was unclear
in~[AF1] whether allowing operational partitions constructed with general
measurable functions instead of the limited class of finite linear
combinations of characteristic functions, yields the standard
Kolmogorov-Sinai invariant. A second motivation is to test how effective it
is to describe a classical system not in terms of the statistical
properties of the time evolution of phase space partitions but rather in
terms of multi-time correlation matrices and their von~Neumann entropies.
The paper is organized as follows: in Section~2 we describe the algebraic
construction of the dynamical entropy for general non-commutative dynamical
systems. In Section~3 we restrict our attention to classical systems and
prove that the algebraic construction produces the Kolmogorov-Sinai
invariant. Section~4 provides a condition, $\En$-density, on a subalgebra
$\B$ of functions which guarantees that the Kolmogorov-Sinai invariant will
be reached when restricting attention to operational partitions in $\B$. In
particular continuous and smooth functions on phase space form sufficiently
large algebras. Finally, we consider in Section~5 automorphisms of
$n$-dimensional tori. We prove that the $\En$-density condition is
satisfied for periodic functions with finite Fourier series and use this
explicit algebra to produce a self-contained computation of the entropy.
\beginsection 2. Preliminaries
We briefly remind the construction of the dynamical entropy for a discrete
dynamical system~[AF1]. Let $\M$ be a von~Neumann algebra of operators
acting on a Hilbert space $\H$ and let $\Om$ be a normalized vector in $\H$
which is cyclic for $\M$ \ie $\M\,\Om$ is dense in $\H$. Furthermore we are
given a unitary operator $U$ on $\H$ such that $U\,\Om= \Om$ and
$U\,\M\,U^*= \M$. The single time-step evolution is given by the
$\ast$-automorphism $x\mapsto U\,x\,U^*$ of $\M$. Mostly, a dynamical
system is given in terms of a C*-algebra $\A$, an automorphism $\Theta$ of
$\A$ and a $\Theta$-invariant state $\om$ of $\A$. By the GNS~construction,
there is a unique representation $\pi$ of such an abstract dynamical system
onto a Hilbert space $\H$ in such a way that $\om(x)= \langle\Om,
\pi(x)\,\Om\rangle$ and $\pi(\Theta(x))= U\,\pi(x)\,U^*$. The von~Neumann
algebra $\M$ of above is then the weak closure of $\pi(\A)$. In the sequel,
we will identify $\pi(x)$ with $x$, $x\in\A$ and use the notation $\Theta$
for the automorphism $x\mapsto U\,x\,U^*$ on $\M$.
An {\sa operational partition of unity} of size $k=1,2,\ldots$ is a
$k$-tuple $\X$ of elements $x_i$ of $\M$ satisfying:
$$\sum_{i=0}^{k-1} x_i^*x_i = \idty.
$$
Two partitions $\X= \bigl(x_0, x_1, \ldots, x_{k-1}\bigr)$ and $\Y=
\bigl(y_0, y_1, \ldots, y_{\ell-1}\bigr)$ can be composed to yield a new
partition $\X\circ\Y = \bigl(x_0\,y_0, x_0\,y_1, \ldots,
x_{k-1}\,y_{\ell-1}\bigr)$ of size $k\,\ell$. Also, if $\X$ is a partition,
then $\Theta(\X)= \bigl(\Theta(x_0), \Theta(x_1), \ldots,
\Theta(x_{k-1})\bigr)$ is again a partition.
To any size~$k$ partition $\X$ we associate a $k\times k$ density matrix
$\rho[\X]$ with $(i,j)$ matrix-element $\langle x_j\,\Om, x_i\,\Om\rangle$.
A partition $\X$ introduces therefore a coarse-grained description of the
system in terms of the $k\times k$ matrices $\Ma_k$ and of a density matrix
$\rho[\X]$. More explicitly, consider $\Cx^k\otimes\H$ and the algebra
$\Ma_k\otimes\M$ acting on it. Let $(e_0, \ldots, e_{k-1})$ be a fixed
orthonormal basis in $\Cx^k$, then the vector $\Psi_\X= \sum_i e_i\otimes
x_i\,\Om$ is normalized and cyclic for $\Ma_k\otimes\M$. The restriction of
the vector state $\langle\Psi_\X, \cdot\,\Psi_\X\rangle$ to $\Ma_k$ is
precisely $\rho[\X]$. Indeed,
$$\langle\Psi_\X, A\otimes\idty\, \Psi_\X\rangle= \sum_{i,j=0}^{k-1}
A_{i,j}\, \langle\Om, x_i^*\,x_j \Om\rangle= \Tr \Bigl(\rho[\X]\,A\Bigr).
$$
As $\langle\Psi_\X, \cdot\,\Psi_\X\rangle$ is a vector state, both
$\rho[\X]$ and the restriction $\sum_i |x_i\,\Om\rangle\langle x_i\,\Om|$
of the vector state to $\B(\H)$ have, up to multiplicities of zero, the
same eigenvalues~[AL]. In particular both density matrices have equal
von~Neumann entropy
$$\En[\X]= \ent(\rho[\X])= \ent\Bigl(\sum_{i=0}^{k-1} |x_i\,\Om\rangle
\langle x_i\,\Om| \Bigr). \eqno(2.1)
$$
The {\sa von~Neumann entropy} $\ent(\rho)$ of a density matrix is computed as
$\Tr \eta(\rho)$ where $\eta$ is the standard entropy function
$$\eta(x)= \cases{-x\,\log x &for $0< x\le1$ \cr
0 &for $x=0$. } \eqno(2.2)
$$
The entropy $\En[\Theta^{m-1}(\X)\circ \cdots\circ \Theta(\X)\circ \X]$
will typically grow linearly with $m$. We define the {\sa dynamical
entropy} $\en[\X]$ of the partition $\X$ as:
$$\en[\X]= \limsup_m\ {1\over m}\, \En[\Theta^{m-1}(\X)\circ \cdots\circ
\Theta(\X)\circ \X].
$$
Consider a unital $\ast$-subalgebra $\B$ of $\M$ which is globally
invariant under the dynamics. The dynamical entropy $\en_\B$ is obtained by
taking the supremum of $\en[\X]$ over all finite partitions $\X$ in
$\B$:
$$\en_\B= \sup_{\X\subset\B}\ \en[\X].
$$
Finally, we will put $\en= \en_{\M}$.
\beginsection 3. Classical systems
>From now on, we will restrict our attention to classical systems. Consider a
measurable space $\Ex$ with $\sigma$-algebra $\S$ and probability measure
$\mu$ and assume that $\S$ is complete \ie any subset $B$ of a null-set $A$
automatically belongs to $\S$ and has zero measure. The dynamics is given
in terms of an automorphism of $\Ex$ \ie a measure preserving
transformation $\theta$ of $\Ex$ with measure preserving inverse:
$$\mu(A)= \mu(\theta(A))= \mu(\theta^{-1}(A)),\quad A\in\S.
$$
The triple $(\Ex,\theta,\mu)$ is usually called a classical dynamical system,
often the space $\Ex$ carries more structure such as that of a manifold.
The {\sa Kolmogorov-Sinai invariant}~[AA, CFS, P, Wal] $\en_{\rm KS}$ of
$(\Ex,\theta,\mu)$ is constructed as follows: if $\C= (C_0, C_1, \ldots,
C_{k-1})$ is a partition of $\Ex$ into disjoint, measurable subsets, the
entropy $\En[\C]$ of $\C$ is computed as
$$\En[\C] = \sum_{i=0}^{k-1} \eta(\mu(C_i)),
$$
$\eta$ defined as in (2.2). If $\C$ and ${\cal D}$ are two partitions of
$\Ex$, then $\C\vee{\cal D}$ is the partition of $\Ex$ generated by $\C$ and
${\cal D}$. It consists of the non-empty intersections of sets $C_i$ and
$D_j$. Choosing a fixed partition $\C$, it can be shown that the following
limit exists:
$$\en[\C] = \lim_{m\to\infty} {1\over m}\ \En\left[\theta^{-m+1}(\C)\vee\cdots
\theta(\C)\vee\C\right].
$$
The Kolmogorov-Sinai entropy $\en_{\rm KS}$ is defined as:
$$\en_{\rm KS}= \sup_\C\ \en[\C].
$$
In an algebraic description of a classical system, the algebra of classical
observables is taken as $\L^\infty(\Ex,\mu)$ acting on $\L^2(\Ex,\mu)$ as
multiplication operators: for $f\in\L^\infty(\Ex,\mu)$, $M_f$ is the operator
$\phi\mapsto (M_f\,\phi)(x)= f(x)\,\phi(x)$ on $\L^2(\Ex,\mu)$. The constant
function ${\bf 1}$ is a normalized vector in $\L^2(\Ex,\mu)$, which is cyclic
for $\L^\infty(\Ex,\mu)$. Because of the $\theta$-invariance of~$\mu$
$$\phi\in\L^2(\Ex,\mu)\mapsto (U\,\phi)(x)= \phi(\theta(x)),\quad x\in \Ex
$$
is a unitary operator and it implements the dynamics on $\L^\infty(\Ex,\mu)$:
$$U\,M_f\,U^*= M_{\Theta(f)},\quad{\rm with}\quad \Theta(f)= f\circ\theta.
$$
A partition $\F$ of size $k$ is a $k$-tuple $(f_0,\ldots,f_{k-1})$ of
$\mu$-measurable functions such that $\sum_i |f_i|^2= {\bf 1}$. The size of
a partition $\F$ will be denoted by $|\F|$. The density matrix
$\rho[\F]$ has matrix elements
$$\rho[\F]_{ij}= \mu(\overline{f_j}\, f_i)= \int_{\Ex} d\mu\,
\overline{f_j}\, f_i.
$$
We will need the following useful representation of $\rho[\F]$: let $(e_0,
\ldots, e_{k-1})$ be an orthonormal basis for $\Cx^k$. $\Psi_\F= \sum_i
f_i\,e_i$ is a measurable function from $\Ex$ to the unit sphere in $\Cx^k$
and defines a measurable function
$$\P[\F]= |\Psi_\F\rangle\langle\Psi_\F| \eqno(3.1)
$$
to the orthogonal projections of dimension~1 in the $k\times k$ matrices
$\Ma_k$. The density matrix $\rho[\F]$ can be written as
$$\rho[\F]= \int_{\ex} d\mu\, \P[\F]. \eqno(3.2)
$$
For $n$ partitions $\F_i$ it follows that
$$\rho[\F_1\circ \cdots\circ \F _n]= \int_{\ex} d\mu\, \P[\F_1]\otimes
\cdots\otimes \P[\F_n]. \eqno(3.3)
$$
\noindent
{\bf Remark 3.1}
The density matrix $\rho[\F_1\circ\F_2]$ is unitarily equivalent to
$\rho[\F_2\circ\F_1]$, furthermore, the partial trace over the middle space
of $\rho[\F_1\circ\F_2\circ\F_3]$ produces the density matrix
$\rho[\F_1\circ\F_3]$. Both these properties hold because of the
commutativity of the algebra $\L^\infty(\Ex,\mu)$ and they generally fail
for truly quantum mechanical dynamical systems.
We list some basic properties of the entropy $\En$ of a partition.
\noindent
{\bf Proposition 3.2}
{\it For any three partitions $\F_1,\, \F_2,\, \F_3$ the following properties
hold:
\item{i)}
$0\le \En[\F_1]\le \log|\F_1|$ \hfill {\rm (3.4)}
\item{ii)}
$\En[\F_1\circ\F_2\circ\F_3]= \En[\F_2\circ\F_1\circ\F_3]$ \hfill {\rm
(3.5)}
\item{iii)}
$\En[\F_1]\le \En[\F_1\circ\F_2]$ \hfill {\rm (3.6)}
\item{iv)}
$\En[\F_1\circ\F_2]\le \En[\F_1]+ \En[\F_2]$ \hfill {\rm (3.7)}%
}
\noindent
{\bf Proof:}
(3.4)~is an obvious bound, (3.5) is an immediate consequence of the
commutativity and~(3.7) follows from subadditivity. In order to
prove~(3.6), we use strong subadditivity and the representations~(3.2)
and~(3.3).
Let $\rho_{123}$ be a density matrix on a
tensor product $\H_1\otimes \H_2\otimes \H_3$ of three Hilbert spaces. The
partial trace of $\rho_{123}$ over the third space will be denoted by
$\rho_{12}$, with similar notations for the other partial traces. The
following inequality is known as the strong subadditivity of the
entropy~[LR, Weh]
$$\ent\bigl(\rho_{123}\bigr)+ \ent\bigl(\rho_2\bigr)\le
\ent\bigl(\rho_{12}\bigr)+ \ent\bigl(\rho_{23}\bigr). \eqno(3.8)
$$
First consider the case of an atomic probability measure $\mu$ supported in
$n$~points and apply strong subadditivity to the density matrix
$$\bigoplus_{j=0}^{n-1} \mu(j)\,\P[\F_1](j)\otimes \P[\F_2](j),
$$
on $\Cx^n\otimes \Cx^{|\F_1|}\otimes \Cx^{|\F_2|}$.
Then
$$\eqalign{
&\ent\Bigl(\bigoplus_{j=0}^{n-1} \mu(j)\,\P[\F_1](j)\otimes
\P[\F_2](j)\Bigr)+ \ent\Bigl(\sum_{j=0}^{n-1} \mu(j)\,\P[\F_1](j)\Bigr)\le
\cr
&\qquad \ent\Bigl(\bigoplus_{j=0}^{n-1} \mu(j)\,\P[\F_1](j)\Bigr)+
\ent\Bigl(\sum_{j=0}^{n-1} \mu(j)\,\P[\F_1](j)\otimes \P[\F_2](j)\Bigr).
}$$
As
$$\ent\Bigl(\bigoplus_{j=0}^{n-1} \mu(j)\,\P[\F_1](j)\otimes
\P[\F_2](j)\Bigr)=
\ent\Bigl(\bigoplus_{j=0}^{n-1} \mu(j)\,\P[\F_1](j) \Bigr)=
\sum_{j=0}^{n-1}
\eta(\mu(j)),
$$
it follows that
$$\ent\Bigl(\int_{\Ex} d\mu\, \P[\F_1]\Bigr)\le \ent\Bigl(\int_{\Ex} d\mu\,
\P[\F_1]\otimes \P[\F_2]\Bigr).
$$
As the $\P[\F]$ are one-dimensional projections in a Hilbert space of
fixed, finite dimension, we may approximate any measure by atomic measures
and use the continuity of the entropy to get~(3.6).
\QED
\noindent
{\bf Theorem 3.3} {\it The entropy $\en$ of a classical dynamical system
$(\L^\infty(\Ex,\mu),\Theta,\mu)$ is equal to the Kolmogorov-Sinai
invariant $\en_{\rm KS}$ of $(\Ex,\theta,\mu)$.}
\noindent
{\bf Proof:}
By restricting in the computation of $\en[\F]$ to partitions in
characteristic functions, it follows immediately that $\en_{\rm KS}\le
\en$.
\noindent
It remains to be shown that $\en_{\rm KS}$ is an upper bound for the
entropy $h[\F]$ generated by an arbitrary size $k$ partition $\F$ of the
unity. Let $\C= (C_0,\ldots, C_{\ell-1})$ be a measurable partition of $\Ex$
into disjoint sets. We will denote by $\C^{(m)}$ and $\F^{(m)}$ the
partitions which are generated from $\C$ and $\F$ by applying $m-1$~time
steps of the dynamics:
$$\eqalign{
\C^{(m)}&= \theta^{-m+1}(\C)\vee \cdots\vee\theta^{-1}(\C)\vee \C \cr
\F^{(m)}&= \Theta^{m-1}(\F)\circ \cdots\circ\Theta(\F)\circ \F.
}$$
We now write
$$\eqalign{
\rho[\F^{(m)}]
&=\int_{\ex} d\mu\, \P[\F^{(m)}] \cr
&=\sum_{C_\alpha\in\C^{(m)}} \int_{C_\alpha} d\mu\, \P[\F^{(m)}] \cr
&=\sum_{C_\alpha\in\C^{(m)}} \int_{C_\alpha} d\mu\,
\P[\Theta^{m-1}(\F)]\otimes \cdots\otimes P[\F].
}$$
In order to get an upper bound for $\En[\F^{(m)}]$, we will use the
estimates~[Weh]
$$\ent\Bigl(\sum_i \lambda_i\,\rho_i\Bigr)\le \sum_i
\lambda_i\,\ent(\rho_i)+ \sum_i \eta(\lambda_i) \eqno(3.9)
$$
for density matrices $\rho_i$ and $0\le\lambda_i$ with $\sum_i\lambda_i=1$
and
$$\ent(\rho_{12})\le \ent(\rho_1)+ \ent(\rho_2),
$$
where $\rho_{12}$ is a density matrix on $\H_1\otimes\H_2$ and $\rho_1$ and
$\rho_2$ denote the partial traces of $\rho_{12}$ over $\H_2$ and $\H_1$.
Then
$$\eqalign{
\En[\F^{(m)}]
&= S\Bigl(\rho[\F^{(m)}]\Bigr) \cr
&\le \sum_{C_\alpha\in\C^{(m)}} \eta(\mu(C_\alpha))+
\sum_{C_\alpha\in\C^{(m)}} \mu(C_\alpha)\, \ent\Bigl({1\over\mu(C_\alpha)}
\int_{C_\alpha} d\mu\, \P[\F^{(m)}]\Bigr) \cr
&\le\sum_{C_\alpha\in\C^{(m)}} \eta(\mu(C_\alpha))+
\sum_{C_\alpha\in\C^{(m)}} \mu(C_\alpha)\, \sum_{j=0}^{m-1}
\ent\Bigl({1\over\mu(C_\alpha)} \int_{C_\alpha} d\mu\,
\P[\Theta^j(\F)]\Bigr).
}$$
For a suitable choice of $\C$, $\P[\Theta^j(\F)]$ remains on each
$C_\alpha$ sufficiently close to a one-dimensional projection in a fixed,
finite-dimensional space of dimension at most~$k$. As the entropy $\ent$ is
a continuous function on the $k\times k$~density matrices, which vanishes on
the pure states, we can bound
$$\ent\Bigl({1\over\mu(C_\alpha)} \int_{C_\alpha} d\mu\,
\P[\Theta^j(\F)]\Bigr)
$$
by a small number which is independent of $\alpha$ and $m$. The proof is
finished by dividing by $m$ and taking the limit $m\to\infty$.
\QED
\beginsection 4. $\En$-dense subalgebras
In specific examples, one would hardly take the supremum in the definition
of the entropy over all possible measurable partitions of unity but
rather only over a limited class. The aim of this section is to state a
condition that guarantees that the Kolmogorov-Sinai entropy will indeed be
reached by considering this restricted supremum.
Let $\C= (C_0,\ldots, C_{k-1})$ be a measurable partition of $\Ex$. We can
identify such a partition with $(\chi_{C_0},\ldots, \chi_{C_{k-1}})$, which
is an operational partition of unity. For such partitions, the
composition in the sense of operational partitions agrees with that in the
sense of partitions of the measurable space into disjoint sets. As there is
therefore no danger of confusion we will keep the same notation $\C$ for
both.
\noindent
{\bf Definition 4.1}
{\it A unital subalgebra $\B$ of $\L^\infty(X,\mu)$ is called {\sa
H-dense} in $\L^\infty(X,\mu)$ if for any finite partition $\C=(C_0,\ldots,
C_{k-1})$ of $\Ex$ into measurable sets and for any $\epsilon>0$ there exists
in
$\B$ a partition $\G=(g_0,\ldots,g_{\ell-1})$ of unity such that}
$$\En[\C\circ\G]- \En[\G]\le \epsilon.
$$
\noindent
{\bf Theorem 4.2}
{\it If $\B$ is $\Theta$-invariant and $\En$-dense, then $\en_\B= \en_{\rm
KS}$.}
For the proof of this theorem we need a few auxiliary results which we
first state as a lemma.
\noindent
{\bf Definition 4.3}
{\it Let $\F$ and $\G$ be two partitions of unity. The {\sa
conditional entropy} $\En[\F\mid\G]$ of $\F$ with respect to $\G$ is
defined as}
$$\En[\F\mid \G]= \En[\F\circ \G]- \En[\G].
$$
\noindent
{\bf Lemma 4.4}
\item{i)}
$\En[\F\mid \G]\ge 0$ \hfill {\rm (4.1)}
\item{ii)}
$\En[\F_1\circ \F_2\mid \G]\le \En[\F_1\mid \G]+ \En[\F_2\mid \G]$ \hfill
{\rm (4.2)}
\item{iii)}
$\En[\F\mid \G_1\circ \G_2]\le \En[\F\mid \G_1]$ \hfill {\rm (4.3)}
\noindent
{\bf Proof:}
(4.1) is actually the same statement as (3.6). The proof of~(4.2) uses the
strong subadditivity~(3.8) of the entropy and Remark~3.1:
$$\eqalign{
\En[\F_1\circ \F_2\mid \G]
&= \En[\F_1\circ \F_2\circ \G]- \En[\G] \cr
&\le \En[\F_1\circ \G]+ \En[\F_2\circ \G]- 2\En[\G] \cr
&= \En[\F_1\mid \G]+ \En[\F_2\mid \G].
}$$
Finally~(4.3) follows from (4.2) and Remark~3.1:
$$\eqalign{
\En[\F_1\mid \G_1\circ \G_2]
&= \En[\F\circ \G_1\circ \G_2]- \En[\G_1\circ \G_2] \cr
&= \En[\F\circ \G_2\mid \G_1]+ \En[\G_1]- \En[\G_2\mid \G_1]- \En[\G_1] \cr
&\le \En[\F\mid \G_1]+ \En[\G_2\mid \G_1]- \En[\G_2\mid \G_1] \cr
&= \En[\F\mid \G_1].
}$$
\QED
\noindent
{\bf Proof of Theorem~4.2:}
Obviously, due to Theorem~3.3, $\en_\B\le \en= \en_{\rm KS}$. We use the
$\En$-density of $\B$ to obtain for each partition $\C$ of $\Ex$ and each
$\epsilon> 0$ an operational partition $\G$ in $\B$ such that
$$\En[\C\mid\G]\le \epsilon.
$$
We use the properties of lemma~4.4 to get
$$\eqalign{
&\En[\Theta^{m-1}(\C)\circ \cdots\circ \Theta(\C)\circ \C]\le \cr
&\En[\Theta^{m-1}(\G)\circ \cdots\circ \Theta(\G)\circ \G\circ
\Theta^{m-1}(\C)\circ \cdots\circ \Theta(\C)\circ \C]\le \cr
&\En[\Theta^{m-1}(\G)\circ \cdots\circ \Theta(\G)\circ \G]+
\sum_{j=0}^{m-1} \En[\Theta^j(\C)\mid \Theta^{m-1}(\G)\circ
\cdots\circ \Theta(\G)\circ \G]\le \cr
&\En[\Theta^{m-1}(\G)\circ \cdots\circ \Theta(\G)\circ \G]+
\sum_{j=0}^{m-1} \En[\Theta^j(\C)\mid \Theta^j(\G)] =\cr
&\En[\Theta^{m-1}(\G)\circ \cdots\circ \Theta(\G)\circ \G]+
m\, \En[\C\mid \G].
}$$
Dividing by $m$ and taking the limit for large $m$, it follows that
$\en_\B\ge \en_{\rm KS}$.
\QED
\noindent
{\bf Corollary 4.5}
{\it Let $(\Ex,\theta,\mu)$ be a dynamical system, then the algebra of
finite linear combinations of characteristic functions of measurable sets
is $\En$-dense in $\L^\infty(\Ex,\mu)$.}
Consider a dynamical system $(\Ex,\theta,\mu)$ where $\Ex$ is a compact
Hausdorff space equipped with its Borel $\sigma$-algebra $\B(\Ex)$,
$\theta$ is an automorphism of $\Ex$ and $\mu$ a $\theta$-invariant regular
measure on $\Ex$. We prove that the unital algebra $\C(\Ex)$ of continuous,
complex-valued functions on $\Ex$ is $\En$-dense in $\L^\infty(\Ex,\mu)$.
\noindent
{\bf Theorem 4.6:}
{\it Let $(\Ex,\theta,\mu)$ be a dynamical system on a compact Hausdorff
space $\Ex$ with regular, $\theta$-invariant measure $\mu$, then
$\en_{\C(\hbox{{\sevensanserif X}})}= \en_{\rm KS}$.}
\noindent
{\bf Proof:}
Fix a partition $\C= (C_0, \ldots, C_{k-1})$ of $\Ex$ into disjoint Borel
sets
$C_i\in \B(\Ex)$, $i=0, \ldots, k-1$. By the regularity of $\mu$ and
Urysohn's lemma there exist for any $\epsilon> 0$ open sets $V_i$, compact
sets $K_i$ and continuous functions $f_i$ such that
$$K_i\subset C_i\subset V_i,\quad \chi_{K_i}\le f_i\le \chi_{V_i},\quad
{\rm and}\quad \mu(V_i\setminus K_i)< \epsilon.
$$
One readily verifies that $\G= (g_0,\ldots,g_{k-1})$, defined by
$$g_i= \sqrt{f_i\over{\sum_j f_j}}\quad i=0, \ldots, k-1,
$$
is a partition of unity in $\C(\Ex)$.
\noindent
The density matrix associated to $\F\circ \G$ can be written as
$$\rho[\F\circ \G]= \bigoplus_{j=0}^{k-1} V^j
$$
with
$$\eqalignno{
[V^j]_{j,j}
&= \mu(K_j)+ \int_{C_j\setminus K_j} d\mu\, g_j^2 &(4.4)\cr
[V^j]_{i,i}
&= \int_{C_j\cap(V_i\setminus C_i)} d\mu\, g_i^2 \quad
\hbox{for}\quad i\neq j. &(4.5)\cr
[V^j]_{p,q}
&= \int_{C_j \cap V_p\cap V_q } d\mu\, g_p\, g_q \quad \hbox{for}
\quad p\neq q. &(4.6)
}$$
By construction, it is possible to choose the $g_i$ such that~(4.4) is
arbitrary close to $\mu(C_j)$ and such that~(4.5) and~(4.6) are arbitrary
small. Hence, for any $\epsilon> 0$, there is a partition $\G$ in $\C(\Ex)$
for which $\En[\F\circ \G]- \En[\F]< \epsilon$. Similarly $|\En[\G]-
\En[\F]|< \epsilon$.
\QED
If the compact Hausdorff space has more structure, such as that of a
differentiable manifold, then $C^{\infty}(\Ex)$, the unital algebra of
differentiable functions on $\Ex$, is $\En$-dense in $\L^\infty(\Ex,\mu)$.
The proof of this assertion follows easily since there is an extension of
Urysohn's lemma on a compact differentiable manifold in terms of
differentiable functions~[BC].
\noindent
{\bf Corollary 4.7:} {\it Let $(\Ex,\theta,\mu)$ be a dynamical system on
an infinitely differentiable compact Hausdorff manifold $\Ex$ with
$\theta$-invariant, regular measure $\mu$. Then $\en_{\C^{\infty}(\Ex)}=
\en_{\rm KS}$.}
\beginsection 5. Automorphisms on tori
The aim of this section is to show that the entropy of a simple dynamical
system can be computed using only the algebraic description. We will not
rely on any standard computation of the entropy or of the Lyapunov
exponents on the level of the manifold which supports the dynamics~[Ledr].
The proof that we obtain is even shorter than the usual proofs such as
given in~[P, Wal].
We will denote by $\Ts$ the one-dimensional torus $\Rl/2\pi$, which can be
thought of as the unit circle in the complex plane. The algebra of
continuous, complex-valued functions on $\Ts$ is the uniform closure of the
subalgebra of finite linear combinations of functions
$$W(k): \Ts\to\Cx: x\mapsto \exp ikx,\quad k\in\Ir.
$$
$\mu$ will denote the normalized Lebesgue measure on $\Ts$. The algebra of
continuous functions on the \hbox{$n$-dimensional} torus $\Ts^n$ is the
uniform closure of the algebra $\D$ of the finite linear combinations of
functions $W(\k)$ with $\k\in\Ir^n$ where $W(\k)$ is the function
$\x\in\Ts^n\mapsto \exp(i\k\x)$. The dynamics $\theta$ on $\Ts^n$ is
determined by a matrix $T\in\hbox{{\sa SL}}(n,\Ir)$, the group of $n\times
n$ matrices with integer entries and determinant 1:
$$\theta\,\x= T\,\x\ {\rm mod}\ 2\pi.
$$
As $T\in\hbox{{\sa SL}}(n,\Ir)$, $\theta$ is a continuous transformation of
$\Ts^n$ with continuous inverse and $\mu\circ\theta= \mu$. On the level of
the von~Neumann algebra $\L^\infty(\Ts^n,\mu)$, the evolution becomes
$\Theta(f)(\x)= f(\theta(\x))$. From the definition of $\theta$ it follows
that $\Theta(W(\k))= W(T^\dagger\,\k)$ where $T^\dagger$ is the adjoint of
$T$.
The matrix $T$ has $n$ linearly independent eigenvectors $\phi_i$
corresponding to eigenvalues $t_i$. The vectors $\phi_i$ are generally not
mutually orthogonal. The spectrum of $T$ is self-adjoint in the sense that
if $t$ is eigenvalue then also $\overline t$ is an eigenvalue. Furthermore
the product of the $t_i$ is equal to 1.
For any measurable subset $C$ of $\Ts^n$, let $Q_C$ be the
orthogonal projection operator from $\L^2(\Ts^n,\mu)$ on $\L^2(C,\mu)$. It
is the multiplication operator on $\L^2(\Ts^n,\mu)$ by the characteristic
function of $C$. Similarly, for any subset $\Lambda$ of $\Ir^n$,
$P_\Lambda$ is, in the Fourier space of $\Ts^n$, the projection operator on
$\ell^2(\Lambda)\subset \ell^2(\Ir^n)$. $P_\Lambda$ can be written as
$$P_\Lambda= \sum_{\k\in\Lambda} |W(\k)\rangle \langle W(\k)|.
$$
\noindent
{\bf Lemma 5.1}
{\it
\item{i)}
For any finite $\Lambda\subset\Ir^n$ and any measurable $C\subset\Ts^n$,
$P_\Lambda\, Q_C\, P_\Lambda$ is unitarily equivalent to $Q_C\,
P_\Lambda\, Q_C$. Both operators have range of dimension at most
$\#(\Lambda)$ and trace $\#(\Lambda)\, \mu(C)$.
\item{ii)}
Let $\Lambda_n$ be a sequence of finite subsets of $\Ir^n$, increasing to
$\Ir^n$ and such that for any $\k\in\Ir^n$
$$\lim_{n\to\infty} {\#(\Lambda_n\cap (\Lambda_n+\k))\over
\#(\Lambda_n)}= 1,
$$
then
$$\lim_{n\to\infty} {1\over \#(\Lambda_n)\,\mu(C)}\; \exp
\ent\Bigl({1\over\#(\Lambda_n)\,\mu(C)}\,P_{\Lambda_n}\, Q_C\,
P_{\Lambda_n}\Bigr)=1.
$$
}
\noindent
{\bf Proof:}
The unitary equivalence of $P_\Lambda\, Q_C\, P_\Lambda$ and $Q_C\,
P_\Lambda\, Q_C$ follows from the polar decomposition of $P_\Lambda\, Q_C$
and implies immediately that the dimension of the range of both operators
is at most $\#(\Lambda)$ because $P_\Lambda$ has range of dimension
$\#(\Lambda)$. The trace can easily be computed:
$$\Tr P_\Lambda\, Q_C\, P_\Lambda= \sum_{\k\in\Lambda} \langle W(\k),Q_C\,
W(\k)\rangle= \sum_{\k\in\Lambda} \int_C d\mu= \#(\Lambda)\,\mu(C).
$$
\noindent
>From the definition of the entropy we obtain
$$\ent\Bigl({1\over\#(\Lambda)\,\mu(C)}\,P_\Lambda\, Q_C\, P_\Lambda\Bigr)=
\log\#(\Lambda)+ \log\mu(C)+ {1\over\#(\Lambda)\,\mu(C)}\, \Tr
\eta(P_\Lambda\, Q_C\, P_\Lambda).
$$
The lemma will be proven if we show that
$$\lim_{n\to\infty} {1\over\#(\Lambda_n)}\, \Tr
\eta(P_{\Lambda_n}\, Q_C\, P_{\Lambda_n})= 0.
$$
For any $\epsilon>0$ there is a $C(\epsilon)>0$ such that
$$0\le \eta(x)\le \epsilon+ C(\epsilon)\,x\,(1-x),\quad 0\le x\le1.
$$
As $0\le P_\Lambda\, Q_C\, P_\Lambda\le \idty$ and as the dimension of the
range of $P_\Lambda\, Q_C\, P_\Lambda$ is equal to $\#(\Lambda)$ we can
make the following estimate
$${1\over\#(\Lambda)}\, \Tr \eta(P_\Lambda\, Q_C\, P_\Lambda) \le
\epsilon+ C(\epsilon)\, {1\over\#(\Lambda)}\, \Bigl(\Tr
\bigl(P_\Lambda\, Q_C\, P_\Lambda)- \Tr \bigl(P_\Lambda\, Q_C\,
P_\Lambda)^2 \Bigr).
$$
We rewrite the last term:
$$\eqalign{
&{1\over\#(\Lambda)}\, \Bigl(\Tr \bigl(P_\Lambda\, Q_C\, P_\Lambda)- \Tr
\bigl(P_\Lambda\, Q_C\, P_\Lambda)^2 \Bigr)=\cr
&\qquad\qquad = \mu(C)- {1\over\#(\Lambda)}\, \sum_{\k,\k'\in\Lambda}
|\langle W(\k),Q_C\,W(\k')\rangle|^2= \cr
&\qquad\qquad =\mu(C)- \sum_{\k\in\Lambda}
{\#(\Lambda\cap(\Lambda+\k))\over\#(\Lambda)}\, \left|\int_C d\mu(\x)\,
\exp i\k\x\right|^2.
}$$
As $n\to\infty$, the functions
$$\k\mapsto{\#(\Lambda_n\cap(\Lambda_n+\k))\over\#(\Lambda_n)}
$$
tend to 1 and are evidently bounded by 1. Because $\k\mapsto\left|\int_C
d\mu(\x)\, \exp i\k\x\right|^2$ is absolutely summable with sum $\mu(C)$,
we can apply the dominated convergence theorem and conclude that
$$\lim_{n\to\infty}\ \sum_{\k\in\Lambda_n}
{\#(\Lambda_n\cap(\Lambda_n+\k))\over\#(\Lambda_n)}\, \left|\int_C
d\mu(\x)\, \exp i\k\x\right|^2= \mu(C).
$$
\QED
\noindent
{\bf Remark 5.2}
Lemma~5.1 has an obvious quantum mechanical interpretation. Both
unnormalized density matrices $P_\Lambda\, Q_C\, P_\Lambda$ and $Q_C\,
P_\Lambda\, Q_C$ are possible quantum candidates for a microcanonical
ensemble for a free particle on the torus localized in $C$ and with
momentum in $\Lambda$. Statement~{\sl ii)} of Lemma~5.1 claims that
the standard picture of the entropy as the logarithm of the phase space
volume occupied by the system is asymptotically correct.
\noindent
{\bf Theorem 5.3}
{\it The algebra $\D$ of finite linear combinations of functions $W(\k)$,
$\k\in\Ir^n$ is $\En$-dense in $\L^\infty(\Ts^n,\mu)$.}
\noindent
{\bf Proof:}
Fix an arbitrary partition $\C= (C_0, \ldots, C_{k-1})$ of $X$ into disjoint
measurable subsets. We prove that the partition $\G$ of unity with
$$\G= \Bigl\{\Bigl.{1\over\sqrt{\#(\Lambda)}}\,W(\k)\Bigr| \k\in\Lambda\Bigr\}
$$
satisfies $\En[\C\circ\G]- \En[\G]\le \epsilon$ for $\Lambda$ sufficiently
large. Using the same notation as above for the projection operators
$P_\Lambda$ and $Q_C$, we compute
$$\eqalign{
\En[\C\circ\G]
&= \ent\Bigl({1\over\#(\Lambda)} \sum_{j=0}^{k-1} Q_{C_j}\, P_\Lambda\,
Q_{C_j}\Bigr)
\cr
&= \sum_{j=0}^{k-1} \mu(C_j)\, \ent\Bigl({1\over\#(\Lambda)\, \mu(C_j)}
Q_{C_j}\,
P_\Lambda\, Q_{C_j}\Bigr)+ \sum_{j=0}^{k-1} \eta\bigl(\mu(C_j)\bigr).
}$$
Remark that the formula of above is a special case of~(3.9), for density
matrices with mutually orthogonal ranges~(3.9) indeed becomes an
equality~[Weh]. We can now write the difference between $\En[\C\circ \G]$
and $\En[\G]$ as:
$$\eqalign{
\En[\C\circ \G]- \En[\G]
&= \sum_{j=0}^{k-1} \mu(C_j)\, \ent\Bigl({1\over\#(\Lambda)\, \mu(C_j)}
Q_{C_j}\,
P_\Lambda\, Q_{C_j}\Bigr)+ \sum_{j=0}^{k-1} \eta\bigl(\mu(C_j)\bigr)-
\En[\G] \cr
&= \sum_{j=0}^{k-1} \mu(C_j)\, \ent\Bigl({1\over\#(\Lambda)\, \mu(C_j)}
Q_{C_j}\,
P_\Lambda\, Q_{C_j}\Bigr)+ \sum_{j=0}^{k-1} \eta\bigl(\mu(C_j)\bigr)-
\log\#(\Lambda) \cr
&= \sum_{j=0}^{k-1} \mu(C_j)\,\Bigl(\ent\Bigl({1\over\#(\Lambda)\, \mu(C_j)}
Q_{C_j}\,
P_\Lambda\, Q_{C_j}\Bigr)- \log\mu(C_j)- \log\#(\Lambda)\Bigr).
}$$
Using Lemma~5.1, we can find a $\Lambda\subset\Ir^n$ such that this
difference becomes arbitrarily small.
\QED
Let us introduce as in~[AFTA] the following notation. For
$f,g\in\ell^{\,0}(\Ir^n)$, the complex sequences indexed by $\Ir^n$ with
only a finite number of elements different from 0, we put
$$W(f) =\sum_{\k\in\Ir^n} f(\k)\,W(\k).
$$
Clearly $\D= \{W(f)\mid f\in\ell^{\,0}(\Ir^n)\}$. Furthermore,
$\overline{W(f)}= W(f^\dagger)$ with $f^\dagger(\k)= \overline{f(-\k)}$,
$W(f)\,W(g)= W(f\ast g)$ with
$$f\ast g (\k) = \sum_{\k'\in\Ir^n} f(\k-\k' )\, g(\k'),
$$
$\mu(W(f))= f({\bf 0})$ and $\mu(\overline{W(f)}\,W(g))= \langle
f,g\rangle$. For any partition $\F= \bigl(W(f_0),\ldots,W(f_{k-1})\bigr)$
in $\D$, we will denote
$$\Sup(\F)= \bigcup_{j=0}^{k-1} \Sup(f_j).
$$
The density matrix $\rho[\F]$ has matrix elements $\rho[\F]_{ij}= \langle
f_j\mid f_i\rangle$ and therefore by~(2.1)
$$\En[\F]= \ent\bigl(\rho[\F]\bigr) = \ent\bigl(\bigl[\langle f_j\mid
f_i\rangle\bigr]_{ij}\bigr) = \ent\Bigl(\sum_j\mid f_j\rangle\langle
f_j\mid\Bigr)\le \log \#\bigl(\Sup(\F)\bigr). \eqno(5.1)
$$
\noindent
{\bf Lemma 5.4}
{\it For any partition $\F\in\D$ we have $\en[\F]\le {\displaystyle
\sum_{j\atop1\le|t_j|}} \log |t_j|$}
\noindent
{\bf Proof:}
Let $\F^{(m)}$ be the partition generated by applying $m-1$~times the
dynamics to $\F$. As $\F$ is a partition in $\D$ it can be written as $\F=
\bigl(W(f_0),\ldots,W(f_{k-1})\bigr)$ with $f_i\in\ell^{\,0}(\Ir^n)$. The
support of the partition $\F^{(m)}$, which also belongs to $\D$, satisfies:
$$\Sup(\F^{(m)})\subset \sum_{j=0}^{m-1} (T^\dagger)^j\,
\Sup(\F).
$$
From~(5.1) it follows that
$$\en[\F]\le \lim_{m\to\infty} {1\over m}\, \log\#(\Sup(\F^{(m)})).
$$
It therefore remains to estimate the number of multi-integers contained in
$\Sup(\F^{(m)})$.
Let $\k\in\Sup(\F)$ and write $\k= \sum_i k_i\,\phi_i$, where
$\phi_i$ is the eigenvector of $T^\dagger$ corresponding to the
eigenvalue $t_i$. Clearly
$$(T^\dagger)^j\,\k= \sum_i (t_i)^j\, k_i\, \phi_i,
$$
but this implies that for large $m$
$$\eqalign{
\#\Bigl(\Sup(\F^{(m)})\Bigr)
&= \#\Bigl(\sum_{j=0}^{m-1} \sum_{\k\in\hbox{{\sevensanserif
Supp}}(\F)}(T^\dagger)^j\, \k\Bigr) \cr
&\le {\rm Constant}\, \prod_{j\atop1<|t_j|} |t_j|^m.
}$$
Taking the logarithm, dividing by $m$, and taking the limit $m\to\infty$
finishes the proof.
\QED
\noindent
{\bf Lemma 5.5}
{\it $\en_\D\ge \sum_{1\le|t_j|} \log |t_j|$ or equivalently, for any
$\epsilon>0$ there is a partition $\F$ in $\D$ such that $\en[\F]\ge
\sum_{1\le|t_j|} \log |t_j|- \epsilon$.}
\noindent
{\bf Proof:}
We will consider partitions $\F$ generated by $f_j\in \ell^{\,0}(\Ir^n)$
that live on distinct single points of $\Ir^n$. For such a partition the
density matrix $\rho[\F]$ is diagonal with equal weights $1/|\F|$ on the
diagonal and therefore its entropy is precisely equal to $\log|\F|$. As we
wish to compare for $r= 1,2,\ldots$ the entropies of the dynamical systems
$(\D,\Theta,\mu)$ and $(\D,\Theta^r,\mu)$ we will explicitly include a
subscript referring to the dynamics. So, $\en_\Theta(\F)$ will denote the
entropy of the partition $\F$ generated by the dynamics $\Theta$. Note that
$\Theta^r$ is generated by the matrix $T^r$ in $\hbox{{\sa SL}}(n,\Ir)$.
>From the monotonicity~(3.6) it follows that
$$\En\bigl(\Theta^{r(m-1)}(\F)\circ \Theta^{r(m-2)}(\F)\circ \cdots\circ
\F\bigr)\le \En\bigl(\Theta^{r(m-1)}(\F)\circ \Theta^{r(m-1)-1}(\F)\circ
\cdots\circ \Theta(\F)\circ
\F\bigr)
$$
and hence:
$$\en_{\Theta^r}(\F)\le r\,\en_\Theta(\F). \eqno(5.2)
$$
Let $T^\dagger$ be a matrix in $\hbox{{\sa SL}}(n,\Ir)$ with eigenvectors
$\phi_i$ and corresponding eigenvalues $t_i$. If $t_i$ is real, then the
corresponding $\phi_i$ may be chosen real. If $t_i$ is complex then
$\overline{t_i}$ is also an eigenvalue and its corresponding eigenvector is
the complex conjugate of $\phi_i$. We have to distinguish between the case
of a real and that of a pair of adjoint, complex eigenvalues of
$T^\dagger$.
Assume first that $t$ is a real eigenvalue of $T^\dagger$. Let $\Delta(t)$
be the centered line segment in $\Rl^n$ of length 1 in the direction of
$\phi$:
$$\Delta_t= \{a\,\phi\mid a\in\Rl,\ \|a\,\phi\|\le1/2\}.
$$
Under $T^\dagger$, $\Delta(t)$ is mapped into $|t|\,\Delta_t$. \hfill\break
Assume next that $t$ is complex with non-zero imaginary part. Let
$\Delta_t$ be the non-degenerate, centered ellipse with major axis of
length~1 in the two-dimensional subspace of $\Rl^n$ spanned by the vectors
$z\,\phi+ \overline{z\,\phi}$, $z\in\Cx$:
$$\Delta_t= \Bigl\{z\,\phi+ \overline{z\,\phi}\Bigm| z\in\Cx,\
|z|\le1/\sqrt{2(\|\phi\|^2+ |\langle\phi,
\overline{\phi}\rangle|)}\Bigr\}.
$$
Again $\Delta_t$ is mapped under $T^\dagger$ into $|t|\,\Delta_t$.
Let $\k$ be an arbitrary vector in $\Ir^n$. We write $\k= \sum_t P_t\,\k$,
where $P_t$ is either the one-dimensional, generally not orthogonal,
projection operator onto the subspace spanned by an eigenvector $\phi$
corresponding to a real eigenvalue $t$, or a projection operator on the
two-dimensional real subspace spanned by $z\,\phi+ \overline{z\,\phi}$,
$z\in\Cx$ corresponding to a complex eigenvalue. In the summation we
implicitly assume that, for complex eigenvalues, only one of the eigenvalues
$t$ or $\overline t$ appears.
Consider for $r\in\Nl$ large enough, the solid in $\Rl^n$
$$\Lambda(r)= \sum_{1<|t|} (|t|^r-1)\, (\Delta_t/2\sqrt n)+
\Bigl\{\psi=\sum_{1\ge|t|} c_t\,\phi_t\,\Bigm| \|\psi\|\le 1/2\Bigr\}.
$$
For $r\in\Nl$ sufficiently large, a large number of points of $\Ir^n$ will
be contained in the interior of $\Lambda(r)$: there exists a constant $C$,
independent of $r$ such that
$$\#(\Lambda(r))\ge C\, \prod_{1<|t|} |t|^r. \eqno(5.3)
$$
Assume for the time being, that for all $m\in\Nl$, the maps
$$(\k_0, \ldots, \k_m)\in\Ir^n\cap\Lambda(r)\mapsto \k_0+
(T^\dagger)^r\,\k_1+
\cdots+ (T^\dagger)^{rm}\,\k_m \eqno(5.4)
$$
are injective. The density matrix $\rho[\F^{(m)}]$ corresponding to the
partition generated by applying $m-1$ steps of the dynamics $(T^\dagger)^r$
to the partition
$$\F= \{W(\k)/\sqrt{|\Lambda(r)|}\mid \k\in\Lambda(r)\cap\Ir^n\}
$$
is diagonal. Therefore
$$\en_{\Theta^r}[\F]\ge \log \#(\Lambda(r)).
$$
It now suffices to use~(5.2) and~(5.3) and to take the limit $r\to\infty$
in order to get the result.
It still remains to show the injectivity of the maps~(5.4). Consider two
sequences $(\k_0, \ldots, \k_m)$ and $(\k'_0, \ldots, \k'_m)$ in
$\Lambda(r)\cap\Ir^n$ and assume, without loss of generality, that
$\k_m\neq \k'_m$. At least one of the $P_t\,\k$ with $|t|>1$ has a length
not less than $\|\k\|/2\sqrt n$. We now estimate the difference of the
projections of the first $m$~terms. By the construction of $\Delta_t$ the
maximal distance between points in $P_t\,\Lambda$ will not exceed
$(|t|^r-1)/2\sqrt n$ and, after $j$~applications of $(T^\dagger)^r$, it
will not exceed $|t|^{jr}/2\sqrt n$. Therefore
$$\eqalign{
\Bigl\|\sum_{j=0}^{m-1} P_t\,\bigl({T^\dagger}^{jr}\,(\k_j- \k'_j)\bigr)
\Bigr\|
&\le \sum_{j=0}^{m-1} \left\|P_t\,\bigl((T^\dagger)^{jr}\,(\k_j-
\k'_j)\bigr)\right\| \cr
&\le \sum_{j=0}^{m-1} (|t|^r-1)\, |t|^{jr}/2\sqrt n \cr
&< |t|^{mr}/2\sqrt n.
}$$
Clearly the first $m$~terms can never compensate for the minimal difference
due to the $(m+1)$-th term.
\QED
\noindent
{\bf Theorem 5.6}
{\it For the dynamical system $(\Ts^n,\Theta,\mu)$ described above}
$$\en_{\rm KS}= \en_\D= \en= \sum_{i\atop|t_i|>1} \log |t_i|$$
\noindent
{\bf Proof:}
In Theorem~5.3 it was shown that $\D$ is $\En$-dense in
$\L^\infty(\Ts^n,\mu)$. The proof follows immediatly from Lemma~5.4
and Lemma~5.5.
\QED
\noindent
{\bf Remark 5.7}
There is a natural quantization of the dynamical systems on the
$n$-dimensional tori that we have considered. The commutative algebra
generated by the $W(\k)$ is deformed by introducing a symplectic form
$\sigma$ and imposing the relations
$$W(\k)\,W(\k)^*= \idty \quad\hbox{and}\quad W(\k)\,W(\k')=
\exp(i\sigma(\k,\k'))\, W(\k+\k')\quad \k,\k'\in\Ir^n.
$$
On the level of the C*-algebra generated by such $W(\k)$, the dynamics is
given by the automorphism $W(\k)\mapsto W(T^\dagger\,\k)$ and the invariant
state $\omega$ by $\omega(W(\k))= 0$ for $\k\neq {\bf 0}$. The computation
of the quantum dynamical entropy $\en_\D$ with respect to partitions in the
algebra $\D$ of finite linear combinations of $W(\k)$ can be performed
along the same lines as that in Theorem~5.6 and it leads to the natural
generalization of the result obtained in~[AFTA]. We conjecture that
$\en_\D= \en$, where $\en$ is the entropy computed allowing general
partitions in the von~Neumann algebra $\M$ of the representation of
$\omega$. The ergodic properties and Lyapunov exponents of dynamical
systems on non-commutative tori have been considered in~[ENTS]. It is
obvious that there is for these models a connection between these exponents
and the dynamical entropy. The intrinsic connection between Lyapunov
exponents and entropy for quantum dynamical systems should certainly be
investigated.
\noindent
{\bf Acknowledgements}
The paper was completed while R.A. visited the K~U~Leuven. He is grateful
to A.~Verbeure for the hospitality and acknowledges financial support from
KUL~grant OT/92--09 and KBN~grant 2~PO3B~144~09.
\vfill\eject
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\parskip=6pt
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\bye