30 I. MEASURE PRESERVING AUTOMORPHISMS
F such that for every countable Borel equivalence relation E there is E ⊆ F
= 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
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
π (z) = (π(z), 0).
Then π witnesses E
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
= E. Now E
= E ⊕ R and
clearly there is an E-invariant Borel set A ⊆ Z such that E|A
(we are using here that E is not smooth). So
E ⊕ R
(E|(Z \ A) ⊕ R ⊕ . . . ) ⊕ R
(E|(Z \ A)) ⊕ (R ⊕ R ⊕ . . . )
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
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] = ∅.