30 I. MEASURE PRESERVING AUTOMORPHISMS

F such that for every countable Borel equivalence relation E there is E ⊆ F

with E

∼B

= E . His argument goes as follows.

Let E∞ be a countable Borel equivalence relation on X such that for any

countable Borel equivalence relation E we have E

B

E∞ (see Dougherty-

Jackson-Kechris [DJK], 1.8). Let I(N) = N × N and put F = E∞ × I(N),

an equivalence relation on Y = X × N. We will show that F works. Take

first a non-smooth E (on some space Z). Let π : Z → X be Borel and 1-1

that witnesses E

B

E∞. Put

π (z) = (π(z), 0).

Then π witnesses E

B

F . Let P = π (Z). Clearly F |(Y \ P ) is compress-

ible, via (x, n) → (x, n + 1), so there is an aperiodic smooth subequivalence

relation R ⊆ F |(Y \ P ) (see [DJK], 2.5). Let E = (F |P ) ∪ R. Clearly

E ⊆ F , so it is enough to show that E

∼B

= E. Now E

∼B

= E ⊕ R and

clearly there is an E-invariant Borel set A ⊆ Z such that E|A

∼B

= R⊕R⊕...

(we are using here that E is not smooth). So

E

∼

=

B

E ⊕ R

∼

=

B

(E|(Z \ A) ⊕ R ⊕ . . . ) ⊕ R

∼

=

B

(E|(Z \ A)) ⊕ (R ⊕ R ⊕ . . . )

∼

=

B

E.

Finally, when E is smooth, the proof is easy as F contains an aperiodic

smooth subequivalence relation.

Comments. The method of proof and most of the results about [E]

up to 4.11 come (with some modifications) from the papers Eigen [Ei1, Ei2]

and Fathi [Fa]. The numbers 5 and 10 in 4.2 and 4.5 can be replaced by 1,

3 resp., see Miller [Mi].

5. Turbulence of conjugacy

(A) Foreman and Weiss [FW] have shown that the conjugacy action of

the group (Aut(X, µ),w) on (ERG, w) is turbulent. We will provide below

(see 5.3) a different proof of (a somewhat stronger version of) this result. Our

argument also shows that the conjugacy action of ([E],u) on (APER∩[E],u)

is turbulent for any ergodic but not E0-ergodic E. We will first give the proof

of this for hyperfinite E in order to explain in a somewhat simpler context

the main idea. In this section, we use Hjorth [Hj2] and Kechris [Kec3] as

references for the basic concepts and results of Hjorth’s theory of turbulence.

Theorem 5.1 (A special case of 5.2). Let E be a hyperfinite ergodic equiv-

alence relation. Then the conjugacy action of ([E],u) on (APER ∩ [E],u) is

turbulent.

Proof. We have already seen in 3.4 that if T ∈ APER ∩ [E], then its

conjugacy class (in [E]) is dense in (APER∩[E],u). We next verify that if

T ∈ APER ∩ [E], then its conjugacy class (in [E]) is meager in (APER ∩

[E],u). First, by 2.5, the conjugacy class of T in (Aut(X, µ),w) is meager,

so it is disjoint from a conjugacy-invariant dense Gδ in (Aut(X, µ),w), say

C. We can also assume that C ⊆ APER by 2.3. By 3.14, D = C ∩ [E] = ∅.