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Banach Representations and Affine Compactifications of Dynamical Systems

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Asymptotic Geometric Analysis

Part of the book series: Fields Institute Communications ((FIC,volume 68))

Abstract

To every Banach space V we associate a compact right topological affine semigroup (V ). We show that a separable Banach space V is Asplund if and only if \(\mathcal{E}(V )\) is metrizable, and it is Rosenthal (i.e., it does not contain an isomorphic copy of l 1) if and only if \(\mathcal{E}(V )\) is a Rosenthal compactum. We study representations of compact right topological semigroups in \(\mathcal{E}(V )\). In particular, representations of tame and HNS-semigroups arise naturally as enveloping semigroups of tame and HNS (hereditarily nonsensitive) dynamical systems, respectively. As an application we obtain a generalization of a theorem of R. Ellis. A main theme of our investigation is the relationship between the enveloping semigroup of a dynamical system X and the enveloping semigroup of its various affine compactifications Q(X). When the two coincide we say that the affine compactification Q(X) is E-compatible. This is a refinement of the notion of injectivity. We show that distal non-equicontinuous systems do not admit any E-compatible compactification. We present several new examples of non-injective dynamical systems and examine the relationship between injectivity and E-compatibility.

Mathematical Subject Classifications (2010): 37Bxx, 54H20, 54H15, 46xx

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Glasner, E., Megrelishvili, M. (2013). Banach Representations and Affine Compactifications of Dynamical Systems. In: Ludwig, M., Milman, V., Pestov, V., Tomczak-Jaegermann, N. (eds) Asymptotic Geometric Analysis. Fields Institute Communications, vol 68. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6406-8_6

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