Note on Cocycles in a Semidirect Product

摘要

In 2 previous papers by the author, cocyles of a nonsingular ergodic actiontaking values in a semidirect product were studied. By examining their essential range, it was proved that in the topology of convergence in measure, recurrence is a generic property. In this paper we continue studying such cocycles. We show that the essential range is an extension of certain subgroups in the essential range of each coordinate cocycle; this allows us to view the essential range as a generalized semidirect product. When the range of the cocycle is contained in R(+) varies as R, we give sufficient conditions for the projection in the second coordinate to be trivial.