A Multiplicative Ergodic Theorem and Nonpositively Curved Spaces |
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Authors: | Anders Karlsson Gregory A Margulis |
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Institution: | Department of Mathematics, Yale University, New Haven, CT 06520, USA.?E-mail: andersk@math.yale.edu; margulis@math.yale.edu, US
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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.
Received: 27 April 1999 / Accepted: 25 May 1999 |
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Keywords: | |
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