10 I. MEASURE PRESERVING AUTOMORPHISMS

to check that for each finite {n1,...,nk} ⊆ Z and clopen sets {A1,...,Ak}

the function

T →

k

i=1

χAi (xni )dνT ((xn))

=

k

i=1

χAi (T

−ni

(x))dµ(x)

=

k

i=1

χT

ni

(Ai)(x)dµ(x)

= χ

k

i=1

T

ni

(Ai)

dµ

= µ(

k

i=1

T

ni

(Ai))

is continuous, which is clear. ✷

Concerning stronger notions of ergodicity, the set of weak mixing trans-

formations, WMIX, is also dense Gδ (Halmos [Ha]; see 12.1 for a more

general statement) but the set of mild mixing transformations, MMIX, and

the set of (strong) mixing transformations, MIX, are meager (Rokhlin for

MIX; see Katok-Thouvenot [KTh], 5.48 and Nadkarni [Na], Chapter 8).

(B) Recall that if T ∈ Aut(X, µ) and UT ∈

U(L2(X,

µ)) is the cor-

responding unitary operator, then ERG, MMIX, WMIX, MIX are charac-

terized as follows in terms of UT , where we put below L0(X,

2

µ) = {f ∈

L2(X,

µ) : fdµ = 0} =

C⊥

(the orthogonal of the constant functions):

T ∈ ERG ⇔ UT (f) = f, ∀f ∈ L0(X,

2

µ) \ {0},

T ∈ WMIX ⇔ UT (f) = λf, ∀λ ∈ T∀f ∈ L0(X,

2

µ) \ {0},

T ∈ MMIX ⇔ UT

nk

(f) → f, ∀nk → ∞, ∀f ∈ L0(X,

2

µ) \ {0},

T ∈ MIX ⇔ U

n(f),f

→ 0, ∀f ∈ L0(X,

2

µ).

We have MIX ⊆ MMIX ⊆ WMIX ⊆ ERG and these are proper inclusions.

(Note that if for any T ∈ Aut(X, µ) we denote by κT

0

the Koopman repre-

sentation of Z on L0(X,

2

µ) induced by UT , then T is in ERG, etc., iff κT

0

is

ergodic, etc., according to the definition in Appendix H.)

Also denote by σT

0

the maximal spectral type of UT |L0(X,

2

µ) (thus σT

0

is

uniquely defined up to measure equivalence); see Appendix F (where we take

∆ = Z,H = L0(X,

2

µ),π(n) = UT

n|L0(X, 2

µ)). Note that σT

0

can be chosen

so that the map T → σT

0

is continuous from (Aut(X, µ),w) into P (T), the