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A Multiplicative Ergodic Theorem and Nonpositively Curved Spaces

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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 yY 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.

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Received: 27 April 1999 / Accepted: 25 May 1999

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Karlsson, A., Margulis, G. A Multiplicative Ergodic Theorem and Nonpositively Curved Spaces. Comm Math Phys 208, 107–123 (1999).

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