A multiplicative ergodic theorem and nonpositively curved spaces

We study integrable cocycles u(n, x) over an ergodic measure preserving tra nsformation that take values in a semigroup of nonexpanding maps of a nonpo sitively curved space Y, e.g. a Cartan-Hadamard space or a uniformly convex Banach space. It is proved that for any y is an element of Y and almost al l x, there exist A greater than or equal to 0 and a unique geodesic ray gam ma(t, x) in Y starting at y such that lim(n-->infinity)1/nd(gamma(An, x),u(n, x)y) = 0. In the case where Y is the symmetric space GL(N)(R)/O-N(R) and the cocycles take values in GL(N)(R), this is equivalent to the multiplicative ergodic theorem of Oseledec. Two applications are also described. The first concerns the determination o f Poisson boundaries and the second concerns Hilbert-Schmidt operators.