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 =
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 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.
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