ON EXTENSION OF COCYCLES TO NORMALIZER ELEMENTS, OUTER CONJUGACY, ANDRELATED PROBLEMS

Let T be an ergodic automorphism of a Lebesgue space and alpha a cocyc le of T with values in an Abelian locally compact group G. An automorp hism theta from the normalizer N[T] of the full group [T] is said to b e compatible with alpha if there is a measurable function phi : X --> G such that alpha(theta x, theta T theta(-1)) = -phi(x) + alpha(x, T) + phi(Tx) at a.e. x. The topology on the set D(T, alpha) of all automo rphisms compatible with alpha is introduced in such a way that D(T, al pha) becomes a Polish group. A complete system of invariants for the a lpha-outer conjugacy (i.e. the conjugacy in the quotient group D(T, al pha)[T]) is found. Structure of the cocycles compatible with every ele ment of N[T] is described.