On the bounded cohomology of ergodic group actions
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JON AARONSON∗ AND BENJAMIN WEISS Dedicated, with admiration, to Larry Zalcman upon concluding thirty years as editor of the Journal d’Analyse Abstract. In this note we show existence of bounded, continuous, transitive cocycles over a transitive action by homeomorphisms of any finitely generated group on a Polish space, and bounded, measurable, ergodic cocycles over any ergodic, probability-preserving action of Zd .
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Introduction
Cocycles and skew product actions. Let be a countable group and let X be a space. In the sequel, X will represent either a Polish, metric space X = (X, d) or a standard probability space X = (X, B, m). A -action on X is a homomorphism T : → Aut(X). In the topological case, Aut(X) = Homeo(X) in and Aut(X) = PPT(X, B, m) the group of probability-preserving transformations of (X, B, m) in the probabilistic case. Let G be an abelian topological group equipped with a norm · G . To define a -skew product action on X × G, we need a T-cocycle, that is, a function F : × X → G satisfying (L)
F(nk, x) = F(k, x) + F(n, Tk x)
(n, k ∈ ).
The cocycle F : × X → G is assumed to be continuous in the topological case and measurable in the probabilistic case. The F-skew product transformations are then defined on X × G by Tn(F) (x, z) := (Tn (x), z + F(n, x)) ∗
(n ∈ ).
Aaronson’s research was partially supported by ISF grant No. 1289/17.
1 ´ JOURNAL D’ANALYSE MATHEMATIQUE, Vol. 141 (2020) DOI 10.1007/s11854-020-0123-6
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J. AARONSON AND B. WEISS
The assumptions and (L) ensure that T (F) : × (X × G) → Aut(X × G) is a -action, called the skew product action. We will also consider cocycles which are bounded in the sense that sup F(γ, x)G < ∞ x∈X
∀γ ∈ .
In case = Z, it is easy to exhibit cocycles. F = F (ϕ) : Z × X → G by ⎧ n−1 k ⎪ ⎪ ⎨ k=0 ϕ(T x), F(n, x) = 0, ⎪ ⎪ ⎩ |n| − k=1 ϕ(T −k x),
Let ϕ : X → G and define
n ≥ 1, n = 0, n ≤ −1.
This is a cocycle and indeed any cocycle is of this form. We sometimes write Tn(F) = Tϕn where Tϕ : X × G → X × G is the skew product transformation defined by Tϕ (x, z) := (T(x), z + ϕ(x)). Construction of cocycles for the actions of multidimensional groups (e.g., = Z2 ) is more difficult. Note that a constant cocycle for an action T : → Aut(X) is given by a homomorphism h : → G an in this case; the skew product action T (h) : → Aut(X × G) is given by the (direct) product action T ×h where (T ×h)γ(x, y) := (Tγ (x), y+h(γ)). The simplest G-valued, non-constant T-cocycles for an action T : → Aut(X) are given by a coboundary, that is, a function h : × X → G defined by h(n, x) = c(x) − c(Tn x), where c : X → G (the transfer function) is measurable or continuous in the probabilistic and topological cases respectively. It is not hard to see that a coboundary is a cocycle. For full shifts of Zd , the only H¨older continuous Rκ -valued cocycles are sums of a coboundary and a constant cocycle (homomorphism). See §4. The dynamical properties of such cocycles are somewhat limited (see §4). However, for certain infinitely generated groups, co
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