6 OPERATOR ALGEBRAS AND DYNAMICAL SYSTEMS
Ð Â: Á e 9") and ì
Â
is the restriction of ì to 5?B . If R is transitive or
ergodic, then the same is true for RB .
Let V be a nonsingular automorphism of (ËÃ, ^ , ì) , and let R =
{(÷, Ê ;c):k e Z} be the equivalence relation induced by the Z-action ç —•
F" on I . The automorphism V is conservative if ì(Â) = 0 for every
à Å ^ with ì(Â Ð F*5) = 0 for all 0 ö k e AE . If V is conservative then
(RV)B
=
RVß
( mod ì) for every  e S* with ì(£ ) 0, where VB is the
automorphism induced by V on Â:
,m(x) ..,.. .„,__, I 0 if 0: Tkx e B} = 0 ,
VBx = Ê ÷ì-a.e . with m(x)
•{
r * min{/c 0: Ã JC G 5} otherwise
(3) Equivalence relations on Markov shifts. Let Ñ = (P(i, j), 1 /, j k)
7 Ø
be a nonnegative, irreducible matrix , and let Xp = {(xn) G {1, ... , k} :
Ñ(÷
Ë
,÷/2+1) 0 for all ç e Z} be the Markov shift space (or simply
Markov shift) defined by Ñ. Then Xp is a closed, shift invariant subset of
,z
{1, ... , k} , and we write óñ for the restriction of the shift {ox)n = xn+l
W Ñ
on {1, ... , k} to Xp . We define a Borel equivalence relation R on Xp
by setting (x, x) e
RP
if and only if there exist m, m , n9 ri 0 with
(!·4)
*_m_5 = *l
m
'_
5
and xn+s = xn*+s
ñ ñ
for every s 0 and denote by S c R the subrelation consisting of all pairs
(x, x) e
RP
satisfying (1.4) with m = m and ç = ç . The equivalence
ñ ñ
relation S was defined in [Kri5], but was already implicit in [Hed], and R
is taken from [Sch7]. All these equivalence relations are shift invariant.
The matrix Ñ can be converted into a stochastic matrix as follows:
let ë be the unique maximal eigenvalue of Ñ, and let í be a nonzero
right eigenvector for Ñ with eigenvalue ë. Then the matrix Ñ defined by
P(/
?
j) =
ë~é
· P(i, j) · v(j)/v(i) for /, j = 1, ... , k , is stochastic, and we
define the Markov measure ì
ñ
on the Borel field 5? of Xp by setting
(1.5) M
c
)=P(
,
o)
p
(
i

/
i)---
p
(
/
n-M
i
n )
for every cylinder set C = [/0, ... , in]r = {x e Xp'xr+k = ik for k =
0 , . . . , « } , where /? = (/?(1), ... , p(k)) is the unique vector satisfying pP =
ñ and J2i=l fc?(0 = 1. Of particular interest is the measure of maximal
Á k ÷ k matrix Ñ is nonnegative if all its entries are nonnegative, and a nonnegative
matrix Ñ is irreducible if there exists, for every (/', j) with 1 /, 7 /c, an Ë 1 with
Pn(i, 7) 0, where Pw is the nih power of Ñ under matrix multiplication. If there exists
an ç 1 with Pn(i,j) 0 for all i,j then Ñ is aperiodic. Although we shall always
assume aperiodicity, this assumption is inessential and can be removed at the expense of a slight
technical complication.
Á nonnegative k ÷ k matrix Ñ is stochastic if its row sums are all equal to 1.
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