Rigidity of Continuous Coboundaries

We consider the functional equation F_oT–F=f, where T isa measure-preserving transformation and f is a continuous function.We show that if there is an L function F which satisfies thisequation, then F is constrained to satisfy a number of regularityconditions, and, in particular, if T is a one-sided Bernoullishift, then we show that there is a continuous function F satisfyingthis equation. We show that this is not the case for the two-sidedshift. 1991 Mathematics Subject Classification 28D05, 58F11.