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)(¸ß
Ñ)
C R
ö
and ÷ ^(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 0) whenever ÷, ÷' e XP\Np
and xm = x^ for all m -m*(x) (m á*(÷)),
and
(1.8)
(^ , (x))n = (x'))„ for all ç 0 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 .
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