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Subadditive and multiplicative ergodic theorems. (English) Zbl 1440.37006
The asymptotic behaviour of products of random matrices is governed by the multiplicative ergodic theorem of V. I. Oseledets [Trans. Mosc. Math. Soc. 19, 197–231 (1968; Zbl 0236.93034); translation from Tr. Mosk. Mat. Obshch. 19, 179–210 (1968)]. In many related situations (composition of random noncommuting operations) a metric which is either invariant or not expanded by the operations exists, giving rise to associated functions exhibiting a form of sub-additivity. This allows the authors to use the J. F. C. Kingman’s subadditive ergodic theorem [J. R. Stat. Soc., Ser. B 30, 499–510 (1968; Zbl 0182.22802)] to generalize the pointwise ergodic theorem to subadditive cocycles.
This, for example, shows that random products of group elements exhibit a well-defined asymptotic growth rate. Here more refined questions about whether random products grow in specific directions are studied, using the notion of horofunctions. To do this a refined version of Kingman’s subadditive ergodic theorem is proved and then used to prove a general multiplicative ergodic theorem and, in particular, results on the asymptotic behaviour of random products of \(1\)-Lipschitz maps on metric spaces.
Applications of these ideas include ergodic theorems for cocycles of bounded linear operators.

MSC:
37A30 Ergodic theorems, spectral theory, Markov operators
37H15 Random dynamical systems aspects of multiplicative ergodic theory, Lyapunov exponents
47A35 Ergodic theory of linear operators
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