On Continuity of Measurable Cocycles

It is proved that every measurable, non-vanishing cocycle defined on the product of (0,∞) and an arbitrary compact metric space is continuous. Some other sufficient conditions for continuity of a cocycle are also given. Consider functions F satisfying the translation equation F (s+ t, x) = F (t, F (s, x)) (T) and real or complex valued solutions of the equation G(s+ t, x) = G(s, x)G(t, F (s, x)). (G) The two functions F and G satisfying (T) and (G), respectively, are known as an abstract automaton (see [11]). The equation (G) occurs also (in the additive form) in ergodic theory for changing velocity in flows (see [15]). It plays a fundamental role in solving the problem of embeddability of linear functional equation (see [13], [12], and [5]) and is used for a characterization of some semigroups of operators (see [1], [7], and [17]). General solutions of 1991 Mathematics Subject Classification. Primary 39B12, 39B52; Secondary 26A18, 58F25.