On the сlassification of сocycles over dynamical systems

This talk is devoted to the problem of classifying cocycles over dynamical systems. The product of random matrices forming a stationary random process generates a cocycle over a measure-preserving transformation and is found, for example, in the hyperbolic theory of dynamical systems (Anosov systems). The asymptotic behavior of such linear cocycles is described by Oseledets's multiplicative ergodic theorem. L. Arnold, N.D. Cong, and V.I. Oseledets obtained a refinement of the behaviour of cocycles in Oseledets subspaces (random Jordan normal form). We will discuss a geometric approach to this problem using the barycenters of probability measures on the boundaries of symmetric spaces.