TY - CHAP

T1 - Banach Representations and Affine Compactifications of Dynamical Systems

AU - Glasner, Eli

AU - Megrelishvili, Michael

PY - 2013

Y1 - 2013

N2 - To every Banach space V we associate a compact right topological affine semigroup ℰ(V). We show that a separable Banach space V is Asplund if and only if ℰ(V) is metrizable, and it is Rosenthal (i.e., it does not contain an isomorphic copy of l 1) if and only if ℰ(V) is a Rosenthal compactum. We study representations of compact right topological semigroups in 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.

AB - To every Banach space V we associate a compact right topological affine semigroup ℰ(V). We show that a separable Banach space V is Asplund if and only if ℰ(V) is metrizable, and it is Rosenthal (i.e., it does not contain an isomorphic copy of l 1) if and only if ℰ(V) is a Rosenthal compactum. We study representations of compact right topological semigroups in 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.

KW - Affine compactification

KW - Affine flow

KW - Asplund space

KW - Enveloping semigroup

KW - Nonsensitivity

KW - Right topological semigroup

KW - Semigroup compactification

KW - Tame system

KW - Weakly almost periodic

UR - http://www.scopus.com/inward/record.url?scp=84883078678&partnerID=8YFLogxK

U2 - 10.1007/978-1-4614-6406-8_6

DO - 10.1007/978-1-4614-6406-8_6

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AN - SCOPUS:84883078678

SN - 9781461464051

T3 - Fields Institute Communications

SP - 75

EP - 144

BT - Asymptotic Geometric Analysis

A2 - Ludwig, Monika

A2 - Pestov, Vladimir

A2 - Milman, Vitali

A2 - Tomczak-Jaegermann, Nicole

ER -