2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/58998Suppose G is a connected, simple, real Lie group with real rank at least two, M is an ergodic G-space with invariant probability measure, and f is a Homeo(T)-valued Borel cocycle, where Homeo(T) denotes the group of homeomorphisms of the circle T. We use an argument of E.Ghys to show that there is a G-invariant probability measure on the skew product of M and T. Furthermore, if the image of f consists of diffeomorphisms, then there is an invariant measure that is equivalent to the product measure; therefore, f is cohomologous to a cocycle with values in the isometry group of T.29 pages. Latex2e file requires style files from Kluwer Academic Publishers: http://www.wkap.nl/kapis/stylefiles/kluwer.zipDynamical SystemsRepresentation Theory22F10 (Primary) 28D15, 37A20 (Secondary)text