6 I. MEASURE PRESERVING AUTOMORPHISMS

∑

j∈B

µ(T(Dj)) =

M

2N

+

∑

j∈B

µ(Dj)

M

2N

+

2N

−M

2N

= 1, a contradiction.

Similarly reversing the roles of i, j. ✷

In particular, 2.1 implies that (Aut(X, µ),w) is topologically locally finite,

i.e., has a locally finite countable dense subgroup.

We also have the following result which is a consequence of the Rokhlin

Lemma.

Theorem 2.2 (Uniform Approximation Theorem, Rokhlin, Halmos

[Ha]). If T ∈ Aut(X, µ) is aperiodic, then for each N ≥ 1, 0 there is a

periodic S ∈ Aut(X, µ) of period N such that δu(S, T) ≤

1

N

+ .

Consider now the set APER of all aperiodic elements of Aut(X, µ). Then

as

T ∈ APER ⇔ ∀n (δu(T

n,

1) = 1) ⇔ ∀n∀m δu(T

n,

1) 1 −

1

m

,

APER is Gδ in (Aut(X, µ),w). (It is also clearly closed in (Aut(X, µ),u).)

In fact the following holds.

Proposition 2.3. APER is dense Gδ in (Aut(X, µ),w).

Proof. Take X =

2N

with the usual measure µ. Then the sets of the

form

k

i=1

{T : dµ(T(Di),T0(Di)) },

Di finite unions of basic nbhds, T0 ∈ Aut(X, µ), 0 form a nbhd basis

for T0 in w. Thus it is enough for each dyadic permutation π, of rank say

n, to find T ∈ APER such that for each s ∈

2n,T(Ns)

= Tπ(Ns) = Nπ(s).

Let ϕ :

2N

→

2N

be any aperiodic element of Aut(X, µ), e.g., the odometer:

ϕ(1nˆ0ˆx)

=

0nˆ1ˆx, ϕ(1∞)

=

0∞.

Define then T(sˆx) = π(s)ˆϕ(x). Clearly

this works. ✷

Moreover, using the Uniform Approximation Theorem, one obtains the

following result.

Theorem 2.4 (Conjugacy Lemma, Halmos [Ha]). Let T ∈ APER.

Then its conjugacy class

{STS−1

: S ∈ Aut(X, µ)} is uniformly dense in

APER. Therefore its conjugacy class is weakly dense in Aut(X, µ). Con-

versely, if the conjugacy class of T is weakly dense, T ∈ APER.

Proof. Observe that if U, V are periodic of period N, then they are

conjugate. Indeed let A ⊆ X be a Borel set that meets (almost) every orbit

of U in exactly one point and let B ⊆ X be defined similarly for V . Then

µ(A) = µ(B) =

1

N

. Thus there is S0 ∈ Aut(X, µ) with S0(A) = B. Define

now S ∈ Aut(X, µ) by

S(x) = S0(x), if x ∈ A,

S(U

n(x))

= V

n(S0(x)),

if x ∈ A, 0 ≤ n N.