On extension of cocycles to normalizer elements, outer conjugacy, and related problems

Abstract:

Let $T$ be an ergodic automorphism of a Lebesgue space and $\alpha$ a cocycle of $T$ with values in an Abelian locally compact group $G$. An automorphism $\theta$ from the normalizer $N[T]$ of the full group $[T]$ is said to be compatible with $\alpha$ if there is a measurable function $\varphi : X \to G$ such that $\alpha (\theta x, \theta T\theta ^{-1}) = - \varphi (x) + \alpha (x, T) + \varphi (Tx)$ at a.e. $x$. The topology on the set $D(T, \alpha )$ of all automorphisms compatible with $\alpha$ is introduced in such a way that $D(T , \alpha )$ becomes a Polish group. A complete system of invariants for the $\alpha$-outer conjugacy (i.e. the conjugacy in the quotient group $D(T, \alpha )/[T])$ is found. Structure of the cocycles compatible with every element of $N[T]$ is described.