16 I. MEASURE PRESERVING AUTOMORPHISMS

The proof is similar to that of 3.2, so we omit it. Notice that this

condition is also necessary if E is ergodic.

This has the following consequence; below E0 denotes the following

equivalence relation on

2N:

xE0y ⇔ ∃m∀n ≥ m(xn = yn).

Proposition 3.8. The group of dyadic permutations of

2N

is uniformly

dense in [E0].

Proof. We use 3.7. Take Γ = {dyadic permutations}, A = the algebra

of clopen sets in

2N.

Thus A consists of all sets of the form

s∈I

Ns, I ⊆

2n

for some n.

Then

EΓN

2

= E0 and A is closed under Γ. To verify 3.7 for ∆ = Γ take

{Ai}in ⊆ A, {γi}in ⊆ Γ with {Ai}in, {γi(Ai)}in both pairwise disjoint.

We can assume that for some large enough N,

Ai =

s∈Ii

Ns, Ii ⊆

2N

,

γi(Ai) =

s∈Ji

Ns, Ji ⊆

2N

,

and γi is a dyadic permutation of rank N, i = 0,...,n − 1. Thus each of

{Ii}in, {Ji}in is pairwise disjoint and if γi(sˆx) = πi(s)ˆx, πi a permutation

of

2N

, then πi(Ii) = Ji,i = 0,...,n − 1. So there is a permutation π of

2N

with π|Ii = πi|Ii. Thus if γ ∈ Γ is defined by γ(sˆx) = π(s)ˆx, s ∈

2N

,

clearly γ|Ai = γi|Ai,i = 0,...,n − 1 and we are done. ✷

Remark. In connection with 3.3 and 3.8, Ben Miller has shown that if X

is a zero-dimensional Polish space, µ a non-atomic Borel probability measure

on X, Γ a countable group acting by homeomorphisms on X, E = EΓ

X

the

equivalence relation generated by this action, and [E]C the group of all

homeomorphisms f of X such that ∀x ∈ X(f(x)Ex), then the periodic (i.e.,

having finite orbits) homeomorphisms in [E]C are uniformly dense in [E].

(C) We next verify that a generic pair in APER∩[E] generates a free

group.

Theorem 3.9. The set

{(S, T) ∈ (APER ∩

[E])2

: S, T generate a free group}

is dense Gδ in

(APER∩[E],u)2.

Proof. It is clearly Gδ. To prove density it is enough, by the Baire

Category Theorem, to verify that for each reduced word w(x, y),

{(S, T) ∈ (APER ∩

[E])2

: w(S, T) = 1}

is uniformly dense in

(APER∩[E])2.