6 OPERATOR ALGEBRAS AND DYNAMICAL SYSTEMS

{Á 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 Â 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

o»

/

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.