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 Π0-complete 3 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.
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