A Multiplicative Ergodic Theorem and Nonpositively Curved Spaces

Abstract:

We study integrable cocycles u(n,x) over an ergodic measure preserving transformation that take values in a semigroup of nonexpanding maps of a nonpositively curved space Y, e.g. a Cartan–Hadamard space or a uniformly convex Banach space. It is proved that for any y∈Y and almost all x, there exist A≥ 0 and a unique geodesic ray γ (t,x) in Y starting at y such that

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 of Poisson boundaries and the second concerns Hilbert-Schmidt operators.