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

Tσ to Tτ .

Suppose σ ∼ τ. Then the map

f ∈

L2(T,σ)

→

dσ

dτ

f ∈

L2(T,τ)

is an isomorphism of the Hilbert space

L2(T,σ)

to

L2(T,τ)

that sends Vσ to

Vτ . Since σ, τ are both symmetric, i.e., invariant under conjugation, so is

dσ

dτ

,

i.e.,

dσ

dτ

(z) =

dσ

dτ

(¯) 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 Vσ 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, Vσ on

L2(T,σ)

is the complexification of

Vσ|S2(T,σ).

Now the spectral theory of the Gaussian shift Tσ asserts that there is an

isomorphism between the real Hilbert space Hσ =

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

dσ

dτ

is invariant under

conjugation, the map f ∈ L2(T,σ) →

dσ

dτ

f ∈ L2(T,τ) sends S2(T,σ) to

S2(T,τ)

and

Vσ|S2(T,σ)

to Vτ

|S2(T,σ).

Thus there is an isomorphism T

between Hσ and Hτ that sends Uσ|Hσ to Uτ |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 Uτ |Hτ , S sends Tσ to Tτ 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