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.
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