MATH 663—ERGODIC THEORY
Problems
Fall 2014
1. Give a proof of the mean ergodic theorem using the spectral theorem for unitary operators.
2. Prove Khintchine’s recurrence theorem: If G is a countable amenable group and G y (X, µ)
a p.m.p. action then for every measurable set A ⊆ X and ε > 0 the set
{s ∈ G : µ(sA ∩ A) ≥ µ(A)2 − ε}
is syndetic.
3. Let G be a countable group and let G y (Y G , ν G ) be a Bernoulli action. Determine the
associated Koopman representation of G on L2 (Y G , ν G ).
4. Let G be a countably infinite amenable discrete group. Show that there are infinitely many
left invariant means on `∞ (G).
5. Give examples of unitary representations π and ρ of Z such that π ⊗ ρ is ergodic but neither
π nor ρ is weakly mixing.
6. Give an example of a unitary representation of Z on an infinite-dimensional Hilbert space
which is weakly mixing but not mixing.
7. Show that a countable discrete group G is amenable if and only if every continuous action
of G on a compact Hausdorff space has an invariant Borel probability measure.
8. Show that a subgroup of an amenable countable discrete group is amenable.
9. Show that every compact p.m.p. action of a countable amenable group has zero entropy.
10. Let G be a countable sofic group and Σ = {σi : G → Sym(di )} a sofic approximation
sequence. Let G y (X, µ) be the trivial action on a probability space, i.e., sx = x for all
x ∈ X and s ∈ G. Show that hΣ,µ (G, X) = 0.
11. Let G y (X, µ) be a p.m.p. action of a countable group, Show that the set N of functions
in L∞ (X, µ) which are compact as elements of L2 (X, µ) under the Koopman representation
is a G-invariant von Neumann subalgebra, and that the L2 -closure of N in L2 (X, µ) is the
subspace of compact vectors.
12. Let G be a counatble amenable group and (X, µ) a standard probability space. Let
Act(G, X, µ) be the space of p.m.p. actions G y (X, µ) with the topology which has as
basic open sets
Uβ,Ω,F,δ = {α ∈ Act(G, X, µ) : µ(α(A)∆β(A)) < ε for all A ∈ Ω}
where β ∈ Act(G, X, µ), Ω is a finite collection of measurable subsets of X, F is a finite
subset of G, and δ > 0. Show that the set of actions in Act(G, X, µ) with zero entropy is a
dense Gδ .
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13. Let Act(G, X, µ) be as in Question 12. Show that the set of actions in Act(G, X, µ) which
are weakly mixing is a dense Gδ .
14. Give an example of a p.m.p. action G y (X, µ) of a sofic group and two sofic approximation
sequences Σ1 and Σ2 for G such that hΣ1 ,µ (G, X) 6= hΣ2 ,µ (G, X).