40 I. MEASURE PRESERVING AUTOMORPHISMS

problem of ergodic theory to distinguish up to conjugacy ergodic transfor-

mations which are unitarily equivalent. The concept of entropy provided

a powerful tool for attacking this problem. In this vein one can raise the

general problem of understanding the complexity of conjugacy within each

unitary equivalence class. More precisely, let C ⊆ ERG denote a given uni-

tary equivalence class in ERG. How complicated is conjugacy restricted to

C? The answer will depend of course on C. If C corresponds to a dis-

crete spectrum ergodic transformation T (i.e., one for which UT has discrete

spectrum, which means that the corresponding maximal spectral type con-

centrates on a countable set), then, by the classical Halmos-von Neumann

Theorem, C consists of a single conjugacy class. On the other hand not

much seems to be known about the complexity of conjugacy for C whose

maximal spectral type is continuous (i.e., non-atomic). A particular case of

interest is the class C which has countable homogeneous Lebesgue spectrum

(i.e., the unitary equivalence class of the shift on

2Z

with the usual product

measure), which of course contains continuum many conjugacy classes.

Remark. Concerning the spectral theory of ergodic measure preserving

transformations, it is a very hard and still unsolved problem to determine

what are the possible spectral invariants, i.e., pairs of measure classes in

P (T) and multiplicity functions, that correspond to (the unitary operator

associated to) an ergodic measure preserving transformation. Equivalently,

this can be viewed as the question of characterizing the unitary operators

that are realized by ergodic, measure preserving transformations up to iso-

morphism (i.e., conjugacy in the unitary group) - see here also Appendix

H, (F). Fox example, one can ask if the set of such of operators is Borel in

U(H). (It is obviously Σ1.)

1

It also appears to be unknown what measure

classes of probability measures on T can appear as maximal spectral types

of ergodic measure preserving transformations. (See, e.g., Katok-Thouvenot

[KTh].) For example, one can ask whether the set of maximal spectral types

of such transformations is a Borel set in P (T). (It is clearly

Σ1.)1

6. Automorphism groups of equivalence relations

(A) For each measure preserving countable Borel equivalence relation

E on (X, µ) we denote by N[E] the group of all T ∈ Aut(X, µ) such that

xEy ⇔ T(x)ET(y),

for all x, y in a conull set. Note that N[E] is the normalizer of [E] in

Aut(X, µ). If E is not smooth, then T.-J. Wei [We] showed that N[E] is

Π3-complete

0

in (Aut(X, µ),w). Clearly N[E] is closed in (Aut(X, µ),u) but

it may not be separable. For example, if E = E0 and for A ⊆ N we let

fA(x) = x + χA, where addition is pointwise modulo 2, then fA ∈ N[E]

and δu(fA,fB) = 1 if A = B. Next we will see that N[E] is a Polishable

subgroup of (Aut(X, µ),w), if E is aperiodic.