5. TURBULENCE OF CONJUGACY 39
This concludes the proof of (a).
(b) By the proof in part (a), it is clearly enough to show that if σ, τ are
symmetric measures on T, and σ τ, then Tσ,Tτ are isomorphic transfor-
mations, i.e., there is an isomorphism S :
(RZ,µϕ(σ))

(RZ,µϕ(τ))
sending
to .
Suppose σ τ. Then the map
f
L2(T,σ)



f
L2(T,τ)
is an isomorphism of the Hilbert space
L2(T,σ)
to
L2(T,τ)
that sends to
. Since σ, τ are both symmetric, i.e., invariant under conjugation, so is


,
i.e.,


(z) =


(¯) z (τ−a.e.). Let
S2(T,σ)
be the closed subset of
L2(T,σ)
consisting of all f
L2(T,σ)
satisfying f(¯) z = f(z). This is a real subspace
of
L2(T,σ)
invariant under the operator and
L2(T,σ)
is the complexifi-
cation of
S2(T,σ),
since any f
L2(T,σ)
is uniquely written as f1 + if2,
with f1,f2
S2(T,σ),
namely f1(z) =
1
2
[f(z) + f(¯)],f2(z) z =
1
2i
[f(z)
f(¯)]. z Moreover, on
L2(T,σ)
is the complexification of
Vσ|S2(T,σ).
Now the spectral theory of the Gaussian shift asserts that there is an
isomorphism between the real Hilbert space =
H:1:
(the first chaos of
L2(RZ,µϕ(σ),
R) associated with the Gaussian space
(RZ,µϕ(σ)))
and the real
Hilbert space
S2(T,σ),
which sends Uσ|Hσ to
Vσ|S2(T,σ)
(see Cornfeld-
Fomin-Sinai [CFS], p. 368). Similarly for τ. Since


is invariant under
conjugation, the map f L2(T,σ)


f L2(T,τ) sends S2(T,σ) to
S2(T,τ)
and
Vσ|S2(T,σ)
to
|S2(T,σ).
Thus there is an isomorphism T
between and that sends Uσ|Hσ to |Hτ . By Appendix D, there is
an isomorphism S :
(RZ,µϕ(σ))

(RZ,µϕ(τ))
such that if OS(f) = f
S−1
is the corresponding isomorphism of
L2(RZ,µϕ(σ),
R) with
L2(RZ,µϕ(τ),
R),
then OS|Hσ = T. Since T sends Uσ|Hσ to |Hτ , S sends to and the
proof is complete.
(C) Very recently Foreman, Rudolph and Weiss [FRW] have shown that
the conjugacy equivalence relation on ERG is not Borel (in fact it is a
complete Σ1
1
subset of
ERG2).
This in particular shows that conjugacy on
ERG cannot be Borel reduced to unitary equivalence on ERG and thus, in
combination with 5.7, it shows that, in terms of Borel reducibility, conjugacy
on ERG is strictly more complicated that unitary equivalence on ERG.
Although the place of unitary equivalence on ERG in the Borel reducibil-
ity hierarchy of complexity of equivalence relations is completely understood,
it is still open to determine this for conjugacy on ERG. For example, how
does it compare with the universal equivalence relation induced by a Borel
action of (Aut(X, µ),w) (or equivalently of U(H), which is isomorphic to a
closed subgroup of (Aut(X, µ),w), by Appendix E)?
Another question (motivated by a discussion with Yehuda Shalom) on
which not much seems to be known is the following. It has been a classical
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