OPERATOR ALGEBRAS AND DYNAMICAL SYSTEMS 7
entropy mp = ì ñï , where Ñ is the 0-1-matrix compatible with Ñ [Pari].
The equivalence relation
Rp
is nonsingular and ergodic with respect to ì
ñ
,
and
(À·6 ) p*r,ßp(x,x)=( Ð P(XixM))/( Ð ^ ¼ )
V -m/« Jl X-m'in' J
whenever (x,x) e
RP
satisfies (1.4). Proposition 44 in [ParTl] implies
that ì
p
is the measure of maximal entropy if and only if psp = 1. We
always assume that the space Xp and the equivalence relations R , S are
furnished with the measure ì
ñ
.
The equivalence relations
RP
and
SP
are preserved under certain kinds
of isomorphism of the shift Spaces. If Ñ and Q are stochastic matrices, the
Markov shifts óñ and aQ are metrically conjugate if there exists a measure-
preserving isomorphism ö: Xp XQ such that ö · óñ = aQ ö //p-a.e. The
isomorphism ö is finitary if there exist null sets Np c Xp , NQ c XQ , and
Borel maps á
9
m :XP—N, á -\, m -\: JTQ Í such that
(fW)o = (P(y))0
f o r
all ÷, y ^XiVp
with xn = yn for -ôçö(÷)ç áö(÷)
and
(^_1(^))o
= (P~\y))0
f o r
all x , y G ËÃâ\ËÃâ
with xn = yn for - ôçø-÷ {x)n af-x (x).
Á finitary isomoöhism ö:×ñ -+ XQ has finite expected code lengths if the
functions á , m , á -\, m -\ can all be chosen to be integrable.
Under our assumptions, óñ and aQ are metrically conjugate if and only if
they have the same (metric) entropy (óñ) = - ^ jP(i)P(i9 j) ·
log(p(i)P(i, ;))) = Çì (aQ) [FriO]. In [KeaS] M. Keane and M. Smorodinsky
prove that, if óñ and aQ are metrically conjugate, then they are also fini-
tarily conjugate. However, W. Parry [Par6] and W. Krieger [Kri5] found ob-
structions to the existence of isomorphisms with finite expected code lengths,
P P
which turn out to be connected with the equivalence relations R and S
defined above.
For every ÷ e Xp we denote by
Ws(x)
- {y G Xp'.xn = yn for all
sufficiently large « e N } and
Wu(x)
= {y e Xp:x_n = y_n for all suffi-
ciently large / I G N } the s table and uns table sets of ÷ . Á measure-preserving
isomorphism ö:×ñ XQ is hyperbolic (more precisely: preserves the hyper-
bolic structure) if there exist null sets Np c Xp and NQ c XQ such that
Q
Two nonnegative k ÷ k matrices Ñ, Q are compatible if, for all l i, j k , P(i', j) 0
if and only if Q(i, j) 0. Any two compatible matrices P, Q satisfy that Xp - XQ ,
P O P O
R = R^ , and S = S , but the measures ì
ñ
and ì
â
are in general (mutually) singular.
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