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 hß (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.