5. TURBULENCE OF CONJUGACY 35
Theorem 5.5. Let E be an ergodic equivalence relation. Then the conjugacy
action of ([E],u) on (Aut(X, µ),w) is generically turbulent.
Proof. Use the argument in 5.1 to show that every ergodic T ∈ [E]
is turbulent for this action (since the conjugacy class of T in Aut(X, µ) is
weakly dense, this shows that the action is generically turbulent). ✷
(B) Given equivalence relations E, F on standard Borel spaces X, Y , we
say that E can be Borel reduced to F , in symbols
E ≤B F,
if there is a Borel map f : X → Y such that
xEy ⇔ f(x)Ff(y).
We say that E is Borel bireducible to F , in symbols
E ∼B F,
if E ≤B F and F ≤B E. An equivalence relation E on a standard Borel
space X can be classified by countable structures if there is a countable
language L and a Borel map f : X → XL, where XL is the standard Borel
space of countable structures for L (with universe N), such that xEy ⇔
= f(y), where
= is the isomorphism relation for structures, i.e, E is
Borel reducible to the isomorphism relation of the countable structures of
some countable language. Hjorth [Hj2] has shown that if E is induced by a
(generically) turbulent action, then E restricted to any dense Gδ set cannot
be classified by countable structures.
From 5.3 and the fact that WMIX is dense Gδ one can of course de-
rive that conjugacy in WMIX cannot be classified by countable structures.
Earlier such a result for ERG was proved in Hjorth [Hj1]. Moreover, by
also using 2.5, it also follows that unitary equivalence in WMIX cannot be
classified by countable structures.
Theorem 5.6 (Hjorth [Hj1] for ERG, Foreman-Weiss [FW]). Con-
jugacy and unitary (spectral) equivalence in WMIX cannot be classified by
In fact one can prove stronger results which apply as well to MIX (which
is meager in Aut(X, µ)).
Fix an uncountable standard Borel space Y and let P (Y ) be the standard
Borel space of Borel probability measures on Y . As usual call µ, ν ∈ P (Y )
equivalent measures (or mutually absolutely continuous) if they have the
same null sets. Thus denoting equivalence of µ, ν by µ ∼ ν and absolute
continuity by µ ν, we have
µ ∼ ν ⇔ µ ν and ν µ.
Clearly, up to Borel isomorphism, ∼ is independent of the choice of Y . It is
known that ∼ is a Borel equivalence relation which cannot be classified by
countable structures, see Kechris-Sofronidis [KS]. The spectral theorem for