Regularity of solutions to the measurable Livsic equation

In this note we give generalisations of Livsic's result that a priori measurable solutions to cocycle equations must in fact be more regular. We go beyond the original continuous hyperbolic examples of Livsic to consider examples of this phenomenon in the context of:

(a)

$\beta$-transformations;

(b)

rational maps; and

(c)

planar maps with indifferent periodic points.

Such examples are not immediately covered by Livsic's original approach either due to a lack of continuity or hyperbolicity.