Ergodic Averages and Integrals of Cocycles

Abstract. This paper concerns the structure of the space C of real valued cocycles for a flow (X,Zm). We show that C is always larger than the set of cocycles cohomologous to the linear maps if the flow has a free dense orbit. By considering appropriate dual spaces for C, we obtain the concept of an invariant cocycle integral. The extreme points of the set of invariant cocycle integrals parallel the role of ergodic measures and enable us to investigate different ergodic averages for cocycles and the uniform convergence of such averages. The cocycle integrals also enable us to characterize the subspace of the closure of the coboundaries in C, and to show that C is the direct sum of this space with the linear maps exactly when the invariant cocycle integral is unique.