Lau, Anthony To-Ming; Takahashi, Wataru Nonlinear ergodic theorems for amenable semigroups. (English) Zbl 1152.47045 Takahashi, Wataru (ed.) et al., Nonlinear analysis and convex analysis. Proceedings of the 4th international conference (NACA 2005), Okinawa, Japan, June 30–July 4, 2005. Yokohama: Yokohama Publishers (ISBN 978-4-946552-27-4/hbk). 317-328 (2007). Let \(S\) be a semigroup and let \(\ell^{\infty}(S)\) be the space of all bounded real-valued functions on \(S\) endowed with the supremum norm. The authors present a survey of the most representative recent results concerning the relationship between invariant means on a subspace of \(\ell^{\infty}(S)\) and ergodic properties of \(S\) when the latter is represented as a semigroup of nonexpansive mappings on a nonempty, closed and convex subset of a Banach space.For the entire collection see [Zbl 1104.47002]. Reviewer: Ioan I. Vrabie (Iaşi) MSC: 47H20 Semigroups of nonlinear operators 47H10 Fixed-point theorems 28D15 General groups of measure-preserving transformations 47-02 Research exposition (monographs, survey articles) pertaining to operator theory Keywords:semigroup; invariant mean; translation invariant subspace; semigroup of nonexpansive mappings PDFBibTeX XMLCite \textit{A. T. M. Lau} and \textit{W. Takahashi}, in: Nonlinear analysis and convex analysis. Proceedings of the 4th international conference (NACA 2005), Okinawa, Japan, June 30--July 4, 2005. Yokohama: Yokohama Publishers. 317--328 (2007; Zbl 1152.47045)