2. SOME BASIC FACTS ABOUT Aut(X, µ) 11
space of probability measures on T with the
weak∗-topology.
Then we have
T ERG σT
0
({1}) = 0,
T WMIX σT
0
is non-atomic,
T MMIX σT
0

D⊥,
T MIX σT
0
R,
where R is the class of Rajchman measures in P (T), i.e., those measures
µ P (T) such that ˆ(n) µ 0 as |n| ∞, D is the class of closed Dirichlet
sets in T, i.e., the class of closed E T such that there is nk with
znk
1 uniformly for z E, and finally
D⊥
is the class of all µ P (T)
that annihilate all Dirichlet sets. All the preceding characterizations in
terms of UT , σT
0
can be found, for example, in Queff´ elec [Qu]. The reader
should note that in Queff´ elec [Qu], III.21 one finds the characterization:
T MMIX σT
0

D⊥
in the form T MMIX σT
0
LI, where LI is a
certain class of measures defined there. A proof that LI =
D⊥
can be found
in Proposition 3, p. 212 and Proposition 9, p. 243 of Host-M´ela-Parreau
[HMP]. For more about Dirichlet sets, see Lindahl–Poulsen [LP], Ch.1.
For each σ P (T) which is symmetric (i.e., invariant under conjuga-
tion), ˆ(n) σ =
z−ndσ(z)
is a positive-definite real function and let be
the Gaussian shift on
RZ
corresponding to ˆ σ (see Appendix C). It is known
(see Cornfeld-Fomin-Sinai [CFS], 14.3, Theorem 1) that the maximal spec-
tral type of UTσ
|C⊥
is σ∞ =
∑∞
n=1
1
2n
σ∗n,
where
σ∗n
= σ∗···∗σ is the n-fold
convolution of σ. We also have (see Cornfeld-Fomin-Sinai [CFS], 14.2) and
Queff´ elec [Qu], III. 21):
σ is non-atomic ERG WMIX,
σ
D⊥
MMIX,
σ R MIX.
For the equivalence σ
D⊥
MMIX one needs to notice that
σ
D⊥
σ∞
D⊥,
which follows from the fact that
D⊥
is closed un-
der convolution by any measure, since D is closed under translations (see
Lindahl–Poulsen [LP], p.4). We already noted that ERG, WMIX are in
(Aut(X, µ),w) and it is easy to see that MIX is Π3.
0
However we have the
following result.
Theorem 2.7 (Kaufman, Kechris). The set of mild mixing transfor-
mations in Aut(X, µ) is co-analytic but not Borel in the weak topology of
Aut(X, µ).
Proof. It is clear that MMIX is co-analytic. The map σ
Aut(X, µ) is Borel and
D⊥
restricted to symmetric measures is known to be
non-Borel (Kechris-Lyons [KLy], Host-Louveau-Parreau).
Remark. Recall that a subset A of N \ {0} is an IP-set if there are
p1,p2, · · · N \ {0} such that A = {pi1 + · · · + pik : i1 · · · ik}. The filter
Previous Page Next Page