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Hopf’s ergodic theorem for nonlinear operators. (English) Zbl 0703.47042

We show for any order preserving, integral preserving, positively homogeneous and \(L^{\infty}\) nonexpansive mapping on \(L^ 1(E,\mu)\) with \(\mu (E)<\infty\), that \((1/n+1)S_ nf\) is a.e. convergent, where \(S_ nf\) is inductively defined by \(S_ 0f=f\) and \(S_{n+1}f=f+T(S_ nf)\). Furthermore, if \(f\in L^ 2\), then the limit function is just the best approximation in \(L^ 2\) of f with respect to the convex cone of invariant functions. Weaker results are given for mappings, which are not positively homogeneous and only \(L^{\infty}\) norm decreasing.
Reviewer: R.Wittmann

MSC:

47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47A35 Ergodic theory of linear operators
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References:

[1] Bruck, R.E.: Asymptotic behavior of nonexpansive mappings. Contemp. Math.18, 1-47 (1983) · Zbl 0528.47039
[2] Krengel, U.: Ergodic theorems. Berlin New York: deGruyter 1985 · Zbl 0575.28009
[3] Krengel, U.: An example concerning the nonlinear pointwise ergodic theorem. Isr. J. Math.58, 193-197 (1987) · Zbl 0636.47016 · doi:10.1007/BF02785676
[4] Krengel, U.: Generalized measure preserving transformations. Proceed. Conf. a.e. Convergence, Columbus, Ohio 1988 · Zbl 0688.28007
[5] Krengel, U., Lin, M.: Order preserving nonexpansive operators inL 1. Isr. J. Math.58, 170-192 (1987) · Zbl 0641.47057 · doi:10.1007/BF02785675
[6] Krengel, U., Lin, M.: An integral representation of disjointly additive order preserving operators inL 1. Stochastic Anal. Appl.6, 289-304 (1988) · Zbl 0667.60085 · doi:10.1080/07362998808809150
[7] Krengel, U., Lin, M., Wittmann, R.:A limit theorem for order preserving nonexpansive operators onL 1. Isr. J. Math.71, 181-191 (1990) · Zbl 0729.47051 · doi:10.1007/BF02811883
[8] Lin, M., Sine, R.C.: On the fixed point set of nonexpansive order preserving maps. Math. Z.203, 227-234 (1990) · Zbl 0662.47030 · doi:10.1007/BF02570732
[9] Lin, M., Wittmann, R.: Pointwise ergodic theorems for certain order preserving mappings inL 1. Preprint in: Proceedings Conference: Almost Everywhere Convergence, Evanston IL., 1989 (to appear)
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