A Salad of Cocycles

We study the centraliser of locally compact group extensions of er-godic probability preserving transformations. New methods establishing ergodicity of group extensions are introduced, and new examples of squashable and non-coalescent group extensions are constructed. Smooth versions of some of the constructions are also given. §0 Introduction Let T be an ergodic probability preserving transformation of the probability space (X, B, m). Let (G, T) be a locally compact, second countable, topological group (T = T (G) denotes the family of open sets in the topological space G), and let ϕ : X → G be a measurable function. The skew product preserves the measure µ = m × m G where m G is left Haar measure on G. There is an ergodic skew product T ϕ : X × G −→ X × G iff the group G is amenable (see [G-S], references therein, and [Zim]). In this paper, we are mainly concerned with Abelian G. Recall that on any locally compact, Abelian, second countable, topological group G, there is defined a norm · G (satisfying x = − x ≥ 0 with equality iff x = 0, and x + y ≤ x + y) which generates the topology of G. The centraliser. Recall that the centraliser of a non-singular transformation R : X → X is the collection of commutors of R, that is, non-singular transformations of X which commute with R. The collection of invertible commutors (the invertible centraliser) is denoted by C(R). We study those commutors Q of T ϕ , of form (*) Q(x, y) = (Sx, f (x)w(y)) Original uncut version of A cut salad of cocycles.