5. TURBULENCE OF CONJUGACY 31

Clearly D ⊆ APER ∩ [E] is conjugacy invariant in [E] and, being non-

empty, it is dense in (APER ∩ [E],u) by 3.4. Finally as w ⊆ u, C is Gδ in

(Aut(X, µ),u) and so D is Gδ in (APER ∩ [E],u). As D is disjoint from the

conjugacy class of T in [E], we are done.

We finally verify that if T ∈ APER ∩ [E], then T is turbulent (for the

conjugacy action of ([E],u) on (APER ∩ [E],u)).

Fix 0 and let

U = {S ∈ APER ∩ [E] : δu(S, T) }.

Fix also a nbhd V of 1 in ([E],u). It is enough to show that the local orbit

O(T, U , V )

is dense in U .

Since

{gTg−1

: g ∈ [E]} is dense in (APER ∩ [E],u), it is clearly dense

in (U , u). So fix any g ∈ [E] with

gTg−1

∈ U . It is enough to show that

gTg−1

∈ O(T, U , V )

u

.

Let E =

∞

n=1

En, where E1 ⊆ E2 ⊆ . . . are Borel with card([x]En ) ≤

n, ∀x. Then

n

[En] is dense in ([E],u). Indeed, given S ∈ [E], if Xn = {x :

S(x) ∈ [x]En }, then X1 ⊆ X2 ⊆ . . . and

n

Xn = X. So for any ρ 0,

choose n large enough so that µ(Xn) 1 − ρ. Then, as x ∈ Xn ⇒ S(x) ∈

[x]En , we can find S ∈ [En] such that S|Xn = S |Xn and so δu(S, S ) ρ.

So we can clearly assume that g ∈ [En], for some large enough n.

Notice that it is enough to find a continuous path λ → gλ in ([E],u),λ ∈

[0, 1], such that g0 = 1,g1 = g and gλTgλ

−1

∈ U , ∀λ. Because then we

can find λ0 = 0 λ1 · · · λk = 1 with gλi+1

gλi1 −

∈ V, ∀i k. If

T1 = gλ1 , T2 = gλ2

gλ11,...,Tk −

= gλk

gλk1 −

−1

, then Ti ∈ V and

TiTi−1 . . . T1TT1

−1

. . .

Ti−1

−1

Ti−1

= gλi

Tgλi1 −

∈ U , ∀i ≤ k.

Choose δ0 small enough so that if α =

δu(gTg−1,T),

then α + 4δ0 .

Find then N ≥ n large enough, so that there is S ∈ [EN ] with δu(S, T) δ0.

Then δu(S,

gSg−1)

δ0 + δu(T,

gTg−1)

+ δ0 = α + 2δ0.

Claim. There is a continuous path λ → gλ in ([E],u) with g0 = 1,g1 = g

and δu(S, gλSgλ

−1)

α + 2δ0.

Then δu(T, gλTgλ

−1)

δ0 + α + 2δ0 + δ0 = α + 4δ0 , i.e., gλTgλ

−1

∈

U , ∀λ.

Proof of the claim. Let Z be a Borel transversal for EN and let

λ → Zλ be continuous from [0, 1] into MALGµ, so that Z0 = ∅ ⊆ Zλ ⊆ Zν ⊆

Z1 = Z, µ(Zν \ Zλ) ≤ (ν − λ), for 0 ≤ λ ≤ ν ≤ 1. Let Xλ = [Zλ]EN , so that

µ(Xν \ Xλ) ≤ N · (ν − λ). Finally, define

gλ =

g on Xλ,

id on X \ Xλ.