12 I. MEASURE PRESERVING AUTOMORPHISMS

IP∗

consists of all A ⊆ N \ {0} that intersect every IP-set (see Furstenberg

[Fur]). It is shown in [Fur], 9.22, that T ∈ Aut(X, µ) is mild mixing iff for

any two Borel sets A, B ⊆ X, µ(A ∩ T

−n(B))

→ µ(A)µ(B) in the filter IP*.

It is clear that A, B can be restricted to any countable dense subset of the

measure algebra. It follows that IP*, which is clearly a co-analytic set in

the compact metric space of subsets of N \ {0}, is not Borel.

In Kechris-Lyons [KLy] a canonical Π1-rank

1

on

D⊥

is defined and used

to prove, by a boundedness argument, the non-Borelness of

D⊥.

It might be

interesting to define a canonical Π1-rank

1

on MMIX, as this would probably

give an interesting transfinite hierarchy of progressively “milder” mild mix-

ing notions. One could even wonder whether mixing would occupy exactly

the lowest level of this hierarchy.

(C) We next show that Aut(X, µ) is contractible.

Theorem 2.8 (Keane [Kea]). The group Aut(X, µ) is contractible in both

the weak and the uniform topology.

Proof. For each A ∈ MALGµ, T ∈ Aut(X, µ), define the induced

transformation TA ∈ Aut(X, µ) as follows: Let B0 = X \ A, B1 = A ∩

T

−1(A),...,Bn

= A ∩ T

−1(B0)

∩ · · · ∩ T

−(n−1)(B0)

∩ T

−n(A),n

≥ 2. The

Poincar´ e Recurrence Theorem guarantees that {Bn} is a partition of X

(a.e.). Let TA(x) = T

n(x),

if x ∈ Bn. Then (T, A) → TA is continuous from

(Aut(X, µ),u) × MALGµ → (Aut(X, µ),u) and also from (Aut(X, µ),w) ×

MALGµ → (Aut(X, µ),w). Assume now X = [0, 1], µ = Lebesgue measure,

and define ϕ : [0, 1] × Aut(X, µ) → Aut(X, µ) by ϕ(λ, T) = T[λ,1]. Then this

is a contraction to the identity of Aut(X, µ). ✷

For the weak topology a complete topological characterization has been

obtained.

Theorem 2.9 (Nhu [Nh]). The space (Aut(X, µ),w) is homeomorphic to

2.

(D) Finally we mention a number of algebraic properties of Aut(X, µ).

Theorem 2.10 (Fathi [Fa], Eigen [Ei2], Ryzhikov [Ry1]). The group

Aut(X, µ) has the following properties:

(i) Every element is a commutator and the product of 3 involutions.

(ii) It is a simple group.

(iii) Every automorphism is inner.

Proofs of (ii), (iii) and a somewhat weaker version of (i) are contained

in the arguments given in the course of the proof of Theorem 4.1 below

Comments. For 2.5 see Nadkarni [Na], §8 and references contained

therein. For 2.10, see Choksi-Prasad [CP] for some generalizations.