8 OPERATOR ALGEBRASAND DYNAMICAL SYSTEMS

9(Wu(x)\Np)

=

Wu(p{x))\NQ

and

p{Ws{x)\Np)

=

Ws((p(x))\NQ

for ev-

ery ÷ e *

p

. Since

SP(x)

=

Wu{x)nWs(x)

and

RP(x)

= U

m

,„

€Z

^"(V*)

Ð

Ws(apnx)

for every X G I ^ , every hyperbolic isomorphism ö:×ñ - XQ

ñ ñ Ï Ï

is an isomorphism of the equivalence relations S c R and S c R .

By [Kri5] every shift-commuting, finitary isomorphism ö:×ñ —• XQ with

finite-expected code lengths is hyperbolic. If ö is finitary, but only the func-

tions m and á are integrable, then (ö ÷ ö)(¸ß

Ñ)

C R

ö

and (ö ÷ ^(S^) c

S

ö

. According to [Sch8] a hyperbolic isomorphism ö:×ñ —• XQ is finitary

if and only if there exist null sets Np c Xp, NQ c XQ, and Borel maps

V ¹*:ËÃ Ñ-Í , á*-é, m*-i:ATß-*N, such that

(ö(÷))ç = (^(x'))^ for all ç 0 (Ë 0) whenever ÷, ÷' e XP\Np

and xm = x^ for all m -m*(x) (m á*(÷)),

and

(1.8)

(^ , (x))n = (ö (x'))„ for all ç 0 (ç 0) whenever ÷, ÷ e X^\NQ

and xm — x'm for all m -ra*-i(x) (m a*-i(x)).

(4) Equivalence relations on one-sided Markov shifts. In the notation of the

example (3), let Yp = {x = ( x j e {1, ... ,

k}N:P(xn,

xn+l) 0 for every

ç e N} be the one-sided Markov shift space (or Markov shift) defined by Ñ,

and denote the shift on Yp again by óñ . We set

R' = {(x, x) eYpX Yp\ ó!Ã(÷ ) =

onp(x)

for some m, ç 0}

and

v/

{(x, x) eYpX Yp:

onp(x)

=

óçñ(÷)

for some ç 0},

• ñ «ñ

and we furnish Yp, R , and S with the probability measure vp = ì

ñ

·

ð

- 1

, where ð:×ñ —• 7

ñ

is the projection obtained by forgetting negative

10

//* iP iP iP

coordinates. Then S c R , the equivalence relations R and S are

both nonsingular and ergodic with respect to vp , and

(1.9)

p*,Vp(x÷) = (p(x0)• Ð

ñ(÷ßé ÷

+é))/(ñ(¼)·

÷

Ð

ñ(÷ç÷Ìç

V

0;« '*

V

oin'

Fl Fl / Ñ / Ñ

whenever ó^÷) = ó^ (÷ ). Finally we note that n(R (x)) = R (ð(÷)) and

7T(SP(JC))

=

S,P(n(x))

for every

× Å É

Ñ

,

and that

R/P

and

S,P

are invariant

under the shift óñ .

If mp is the measure of maximal entropy on Xp then we denote the measure mp · ð~

on Õ ñ again by mp .