[SOLVED] Math154 homework 8-transformation p0

MATH154
Homework 8
Transformation
Problem 8.1: a) Check that the automorphisms of a probability space form a group. There is a subset of ergodic automorphisms. Investigate whether (i) ergodic, (ii) weakly mixing, (iii) mixing automorphisms form a subgroup.
b) For every T ∈ Aut(Ω,A,P) we have a unitary transformation U : L2 → L2 given by Uf = f(T). Check the orthogonality condition ⟨Uf,Ug⟩ = ⟨f,g⟩.
c) Classical mechanics is the theory of automorphisms of probabilityspaces, where the unitary evolution is given by a dynamics Uf = f(T). Quantum mechanics allows for a larger automorphism group consisting of all unitary operator Uf = eitAf with a self-adjoint operator A on the Hilbert space L2(Ω). Assume our probability space is finite. What is its classical automorphism group? What is its quantum automorphism group?
Problem 8.2: Show that if a measure-preserving transformation T has the property that for any A,B ∈ A there is m such that P[A∩T−n(B)] = P[A]P[B] for all n ≥ m, then A is a trivial algebra.
Ergodicity
Problem 8.3: Let (Ω,A,P) be a probability space, and let T : Ω → Ω be a measure-preserving transformation. Verify that the following conditions are equivalent:
(i) T is ergodic
(ii) If A ∈ A and P[T−1(A)∆A] = 0, then P[A] = 0 or P[A] = 1.
(iii) If A ∈ A satisfies P[A] > 0 then P[Sn T−n(A)] = 1.
(iv) If A,B ∈ A satisfy P[A] > 0,P[B] > 0 then there is n such that P[T−n(A) ∩ B] > 0.
Instead of checking all 12 possible ordered pairs, use the Merry-Go-Round proof technique: (i) → (ii) → (iii) → (iv) → (i).
Probability Theory
Proof. (i) → (ii) P[T−1(A)∆A] = 0 means that T(A) = A up to a measure zero. By definition A has measure 0 or 1. (ii) → (iii) The set B = Sn T−n(A) is invariant and so has measure 0 or 1. Since it contains A which has positive measure, it has measure
1.
(iii) → (iv) If there existed a set B which never can be reached, then B would be disjoint of T−n(A). But P[Sn T−n(A)] = 1. Assume T−1(A) = A and A has measure different from one. Then take
□
Weak mixing
Problem 8.4: a) In the proof showing that T is mixing implies T2 is mixing, we use the following Lemma from calculus or real analysis: the following two things are equivalent:
(i) cn ≥ 0 is a bounded sequence with
(ii) There exists a set J of density 1 in N on which limj∈J |ck| → 0. b) Use a) to verify that if cn ≥ 0 is a
bounded sequence is equivalent to
c) Conclude that weakly mixing can be rephrased as the property = 0 for all A,B ∈ A.
Mixing
Problem 8.5: a) Prove the following result of R´enyi: A dynamical system T is mixing if and only if µ(A ∩ T−nA) → µ(A)2 for n → ∞.
b) State and give a proof of the Riemann-Lebesgue lemma. Why does this lemma imply that T has only absolutely continuous spectrum, then T is mixing? (Use a).