Cocycles and Almost Periodicity

We discuss different types of almost periodicity of trajectories of cocycles on compact solenoidal groups in relation to various notions of triviality of these cocycles. Introduction Let G be a compact solenoidal group, that is, a compact Abelian group containing, as a dense subgroup, the image of IR by a continuous homomorphism, say a. We shall assume that G is not a circle by requiring a to be injective. The dual group Y(g,t) (geG) will be referred to as a trajectory of Y. In this paper we exhibit various relations between different types of almost periodicity of trajectories of cocycles and different notions of triviality of cocycles. In Section 1 we prove that c-trivial cocycles are the only trivial cocycles all of whose trajectories are Hartman almost periodic. Next, generalizing results of D. Lowdenslager and C. G. R. Carlson, we characterize trivial cocycles as those which have all trajectories almost periodic in a non-classical generalized sense; half of this result appears in Section 2, the other half in Section 3. Section 3 presents, inter alia, a theorem to the effect that the trajectories of a cocycle are Ryll-Nardzewski almost periodic if and only if the cocycle is either c-trivial or non-trivial. 1. Besicovitch and Hartman almost periodicity of trajectories of trivial cocycles Given //eIR and a complex locally integrable function/on U, the limit