Geodesic
Lyapunov exponents, symmetric spaces, and a multiplicative ergodic theorem for semisimple Lie groups
Zapiski Nauchnykh Seminarov POMI, Differential geometry, Lie groups and mechanics. Part IX, Tome 164 (1987), pp. 30-46

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The classical theory of dyapunov characteristic exponents is reformulated in invariant geometric terms and developed for arbitrary non-compact semi-simple Lie groups with finite centre. A multiplicative ergodic theorem and a global law of large numbers for semi-simple Lie groups, and criteria for the Lyapunov regularity of linear systems of ordinary differential equations with subexponential growth of coefficients are proved. Bibl. – 21. 
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     author = {V. A. Kaimanovich},
     title = {Lyapunov exponents, symmetric spaces, and a multiplicative ergodic theorem for semisimple {Lie} groups},
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V. A. Kaimanovich. Lyapunov exponents, symmetric spaces, and a multiplicative ergodic theorem for semisimple Lie groups. Zapiski Nauchnykh Seminarov POMI, Differential geometry, Lie groups and mechanics. Part IX, Tome 164 (1987), pp. 30-46. http://geodesic.mathdoc.fr/item/ZNSL_1987_164_a3/