OPERATOR ALGEBRAS AND DYNAMICAL SYSTEMS
9
(5) Equivalence relations ofan endomorphism. Let (X, S*
9
ì) be a mea-
sure space, and let V: X X & nonsingular, surjective Borel map such that
V~ ({·*}) is countable for every ÷ e X (i.e. V is a nonsingular, countable-
to-one endomorphism of (X, S?, ì)) . We dehne Borel equivalence relations
R = {(÷, x) e × ÷ X:
Vmx
=
Vnx
for some m, ç 0}
and
S = {(x , x ' ) E l x I : F
W
X =
Vmx
for some m 0}
and assume that the equivalence relations R and S are nonsingular with
respect to ì (this is not automatic!). Example (4) corresponds to the special
case where X = Yp and V = óñ . Another well-known example is obtained
by setting X = [0, 1]\Q with its usual Borel field SP. The measure Üì{÷) =
(1 +
x)~ldx
on SP is invariant under the Borel endomorphism Vx =
x~l
( mod 1) of X, and the equivalence relations
RF
and
SK
are non-
singular and ergodic on (X, S*, ì) . The endomorphism V is called the
continued fraction transformation, since the continued fraction expansion
÷ = [a{, a2, ... ] =
«. +
of ÷ e M+\Q is given by an =
(Vn~lx)~l
-
Vnx
for every ç 1 . The map
ö:÷ y = (a{, a2, ...) from X to F =
Mx
is a Borel isomoöhism,
and ö · V = ó ö, where ó denotes the shift {ay)k = yk+x on Õ. It is
well known (and not difficult to verify) that two points ÷, ÷ e X satisfy
that (x, x) e R (or, equivalently, that (p(x),
$?(JC;))
G
R(J)
if and only if
there exists a matrix
(acbd)
G GL(2, Z) such that x' = (AX -f b)/(cx + d).
Although equivalence relations of endomorphisms are not usually associated
with group actions in any canonical way, this shows that R = (R )
÷
, where
Ô is the action of GL(2, Z) on E\Q defined by Tgx = (ax + b)/(cx + d)
for every ÷ e R\Q and
g={ac%)e
GL(2, Z) [HardW], [CorFS].
The continued fraction transformation just described is typical in the
sense that, if V is an arbitrary, countable-to-one, measure-preserving en-
domorphism of a probability space (X, S*, ì), and if R and S are
//-nonsingular, then R and S can always be realized as in example (3)
as the equivalence relations on a one-sided shift space (possibly with infinite
aiphabet). However, if V is only assumed to be nonsingular, then R and
S may have some very unexpected properties. Here is an example where
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