EXERCISE SHEET #2
Exercise 1 (Rigidity of the Dyadic odometer):
Let X = {0, 1}N be the dyadic odometer and for 0 < p < 1, µp is the product measure on X
with µp [xj = 0] = p = 1 − µp [xj = 1] . In this exercise you will prove the rigidity property
along 2n for the dyadic odometer on X. That is for any f : X → R measurable and > 0,
n
µp f ◦ τ 2 − f > −−−→ 0.
n→∞
Steps: (a) Show that if f = g (x1 , .., xn ) depends only on the first n coordinates then for all
k ≥ n, f ◦ τ 2k ≡ f. Hint: Odometer property.
ψ(x)
dµp ◦τ −1
1−p
(b) Let ψ(x) := min {n ∈ N : xn = 1} − 2 so that dµ
(x)
=
. Show that if q > 1
p
p
q
< 1 then,
so that p1 1−p
p
1 − p ψ q
M := < ∞.
p
Lq(X,µp )
(c) Show that for all n ∈ N,
the chain rule to show that
2−n
dµp ◦τ
dµp
dµp ◦τ 2
dµp
=
−n
=
1−p
p
1−p
p
ψ◦S n
P2n −1
k=0
with S : X → X the shift map. Hint: Use
ψ◦τ −k
.
(d) Use steps (b) and (c) to show that for any A ∈ BX with q > 1 as in step (b),
−n ˆ dµ ◦ τ 2−n
q−1
p
µp τ 2 A =
dµp ≤ M µp (A) q .
dµp
A
Hints: S preserves µp and Hoelder’s inequality.
(e) Finish the proof. Hint: Since functions which depend only on finitely many coordinates are
dense, it follows that for general f : X → R measurable and > 0 there exists g = g (x1 , .., xn )
s.t. µp |f − g| > 2 < . It then follows that
n
n
n
µp f ◦ τ 2 − f > ≤ µp f ◦ τ 2 − g ◦ τ 2 > /2 + µp (|f − g| > /2) .
Exercise #2 (Ergodic Maharam extension for the dyadic odometer):
Let (X, B, µp , τ ) be the dyadic odometer and
τ̃ (x, n) := (τ (x), y + φ(x)) : X × Z → X × Z,
with φ(x) := min {n ∈ N : xn = 0} − 2. Let mp be the measure on X × Z defined by
n
p
mp (A × {n}) = µp (A) 1−p
.
(a) Show that τ̃ preserves mp (mp ◦ τ̃ = mp ). Hint: For any z ∈ Z and A ∈ B,
A × {z} := ∪∞
l=−1 (A ∩ [φ = l]) × {z}.
1
EXERCISE SHEET #2
2
(b) Prove ergodicity of (X × Z, B × BZ , mp , τ̃ ).
P n −1
φ ◦ τ k (x) =: φ2n (x) = φ ◦ S n (x) and thus for any
Steps: (b.1) Recall that 2k=0
n
n
F ∈ L∞ (X × Z, mp ) and n ∈ N, F ◦ τ̃ 2 (x, y) = F τ 2 x, n + φ ◦ S n (x)
(b.2) Show that by the rigidity proposition (exercise 1) for all F ∈ L∞ (X × Z, mp ) there exists
nk → ∞ s.t.
n
F τ 2 k x, z −−−→ F (x, z), ∀z ∈ Z and a.e. x ∈ X.
k→∞
(b.3) Show that for any nk → ∞ there exists a subseqence nkl → ∞ s.t. for a.e. x ∈ X
φ ◦ S nkl (x) = −1, ∀l ≥ 1. Hint: the events An = [φ ◦ S n = −1] are independent with
µp (An ) = p, Borel Cantelli Lemma.
(b.4) Assume now that F ∈ L∞ (X, mp ) is τ̃ invariant. Show that there exists a susequence
nkl → ∞ s.t for a.e. x ∈ X and for all z ∈ Z,
nk
nk
F (x, z) = F ◦ τ̃ 2 l (x, z) = F τ 2 l x, z − 1 −−−→ F (x, z − 1)
l→∞
(b.5) Show that if F ∈ L∞ (X, mp ) is τ̃ invariant, then ∃f : X → R such that f (x) = F (x, z).
Conclude that F is a.e. constant by ergodicity of (X, B, µp , τ ).
Exercise #3 (Maharm’s Theorem):
In this exercise you will prove Maharam’s Theorem which says that if (X, B, m, T ) is an
invertible conservative non singular transformation then it’s Maharam extension T̃ : X × R is a conservative measure preserving transformation
of (X ×
R, B × BR , µ) with
dµ◦T
y
dµ(x, y) := dm(x)e dy and T̃ (x, y) := T x, y − log dµ (x) .
(a) Show that T̃ preserves µ.
(b) Show that if T is conservative then for any f ∈ L1 (X, m), if T̂ f ≤ f then T̂ f = f a.e.. The
second property is called “dual incompressibility”. Here T̂ : L1 (X, m) is the dual operator
´
´
defined by X T̂ f gdm = X f g ◦ T dm for all g ∈ L∞ (X, m). Hint: Hopf’s reccurence theorem
(see exercise sheet #1).
n
(c) Writing (T n )0 (x) := dµ◦T
and F (x) = supn≥0 (T n )0 (x), show that F ◦ T ≤ F/T 0 hence
dµ
since T is dual incompressible (by part (b)) F ◦ T = F/T 0 .
(d) For a ∈ R, define Xa = X × (−∞, a). Show that µ (Xa ) = ea < ∞ and the set
−n X satisfies T̃ −1 K = K .
Ka := ∪∞
a
a
a
n=0 T̃
(e) Deduce that T̃ |Ka is conservative hence X × R = ∪a∈R Ka ⊂ C T̃ where C T̃ is the
conservative part of T̃ .
Exercise 4 (Lin’s Theorem):
n Prove that a non singular transformation (X, B, m, T ) is exact if and only if T̂ f −−−→ 0
1 n→∞
´
for every f ∈ L1 (X, m) with X f dm = 0.
´
(a)
Show
that
if
T
is
not
exact
then
∃f
∈
L
(X,
m)
with
1
X f dm = 0 and α > 0 such that
n −n B choose f = 1 − m(A) 1 c . Show
T̂ f ≥ α for all n. Hint: for non trivial A ∈ ∩∞
A
n=1 T
m(Ac ) A
´1 n
´
that A T̂ f dm = A f dm > 0.
EXERCISE SHEET #2
(b) From now on assume T is exact.
(b.1) Show that ∃gn ∈ L∞ (X, m) such that
3
´
n dm = T̂ n f .
f
g
◦
T
n
X
1
(b.2) Show that by exactness of T , every weak ∗ limit in L∞ of gn ◦ T n is constant.
(b.3) Show that for every subsequence nj → ∞, there exists a subsubsequence njk → ∞ such
that
njk T̂
f
−−−→ 0
1 k→∞
and deduce the final part of Lin’s Theorem.
Exercise 5:
Let X = [0, 1] and T x = 2x mod 1. Calculate Tˆ :L1 (X, m) and show that
ˆ
n
T̂ f −−−→
f dm uniformly on X.
n→∞
X
(b) Deduce that T is exact.
Exercise 6:
An invertible measure preserving transformation (X, B, m, T ) is a K-system (K for
Kolmogorov) if there exists F ⊂ B such that T −1 F ⊂ F (F- is a factor), ∩n∈N T −n F (exactness
n
of F) and ∨∞
n=0 T F = B. Show that if (X, B, m, T ) is a conservative K- system then T is
ergodic. Hint: Similar to Parry’s Theorem.