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Nonlinear ergodic theorems. (English) Zbl 0649.47042
Ergodic theory and related topics II, Proc. Conf., Georgenthal/GDR 1986, Teubner-Texte Math. 94, 99-107 (1987).
[For the entire collection see Zbl 0627.00018.]
The paper surveys the main results concerning asymptotic properties of nonlinear operators T in $$L_ 1$$, which forms a developing branch of the ergodic theory.
First a nonexpansive operator in $$L_ p$$ of a measure space ($$\Omega$$,$${\mathfrak a},\mu)$$ is defined, and then results are mentioned for order preserving nonexpansive aperiodic operators. Some speed limit operators are discussed next. One theorem is then mentioned on the problem of a.e. convergence. New results regarding disjointly additive operators are also discussed. The paper concludes by mentioning several papers where nonlinear ergodic theorems are discussed.
Reviewer: P.C.Sinha

##### MSC:
 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47A35 Ergodic theory of linear operators 47H20 Semigroups of nonlinear operators