Running list of HW2 Problems:
1. Show that the following are equivalent for a measure preserving Zd+ action i 7→ T i on
a probability space (M, A, µ):
(a) whenever A ∈ A and µ(A) > 0, then µ((∪i∈Zd+ T −i (A)) = 1
(b) if A, B ∈ A and µ(A), µ(B) > 0, there exists n ∈ Zd+ \{0} s.t. µ(T −n (A)∩B) > 0.
(c) i 7→ T i is ergodic, i.e. if A ∈ A and T −i (A) = A for all i ∈ Zd+ , then µ(A) = 0 or
1.
(d) if A ∈ A and µ(T −i (A)∆A) = 0 for all i ∈ Zd+ , then µ(A) = 0 or 1.
(e) If f ∈ Lp and f ◦ T i = f a.e. for all i ∈ Zd+ , then f is constant a.e. (here,
0 ≤ p ≤ ∞)
(f) if A, B ∈ A, then
lim (1/nd )
X
µ(T −i (A) ∩ B) = µ(A)µ(B)
n→∞
i∈Zd+ ,|i|≤n
(here, |i| is the L∞ norm)
2. (a) An Zd+ action is totally ergodic if T i is ergodic for all nonzero i ∈ Zd+ . Show that
the action is totally ergodic ⇒ Some element is ergodic ⇒ the action is ergodic
(b) A Zd+ action is totally mixing if T i is mixing for all nonzero i ∈ Zd+ . Show that
the action is mixing ⇒ the action is totally mixing ⇒ Some element is mixing.
3. (a) Show that a Zd action is measure-preserving as a Zd action iff it is measurepreserving as a Zd+ action.
(b) Show that a measure-preserving Zd action is ergodic as a Zd action iff it is ergodic
as a Zd+ action.
4. Given that γ α and H(β|α) = H(β|γ), what can you conclude about α, β and γ?
5. Let α, β, γ and δ be finite measurable partitions of a probability space. Prove:
(a) H(α ∨ β|γ) = H(α|γ) + H(β|α, γ)
(b) H(β|γ, α) ≤ H(β|γ) with equality iff α ⊥γ β, i.e., for each element C of γ, the
restrictions of α to C and β to C are independent.
(c) If α ⊥γ (β, δ), then H(β|δ, γ, α) = H(β|δ, γ).
(d) If γ α, then H(γ|β) ≤ H(α|β)
(e) If T is an MPT, then for all i, j, k,
H(T −i α ∨ . . . ∨ T −j α) = H(T −i−k α ∨ . . . ∨ T −j−k α)
6. Prove the entropy inequalities (conditioning on sub-σ-algebras): see page 5 of Lectures
18-20 on course website).
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7. True or False:
Iα|B = E(Iα |B)
8. Prove the two-sided generator theorem by modifying the proof of the one-sided generator theorem.
9. For an MPT T and positive integer n show that h(T n ) = nh(T ).
10. Let T be a Zd action and let T −1 be the Zd action defined by i 7→ T −i . Show that
h(T −1 ) = h(T )
11. Show that if an MP Zd action T has a one-sided generator, then h(T ) = 0.
12. Compute the entropy of a rotation of the circle.
13. For each of the following, give one-sided generators and two-sided generators (whenever
they make sense and exist):
(a) The doubling map
(b) The Baker’s transformation
14. Let T be an MP Zd+ action with d ≥ 2. Suppose that for some i ∈ Zd+ \{0}, h(T i ) < ∞.
Show that h(T ) = 0.
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