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On tame dynamical systems

100%
Glasner E.
Colloquium Mathematicum
|
2006
|
tom 105
|
nr 2
283-295
EN
A dynamical version of the Bourgain-Fremlin-Talagrand dichotomy shows that the enveloping semigroup of a dynamical system is either very large and contains a topological copy of β𝓝, or it is a "tame" topological space whose topology is determined by the convergence of sequences. In the latter case we say that the dynamical system is tame. We show that (i) a metric distal minimal system is tame iff it is equicontinuous, (ii) for an abelian acting group a tame metric minimal system is PI (hence a weakly mixing minimal system is never tame), and (iii) a tame minimal cascade has zero topological entropy. We also show that for minimal distal-but-not-equicontinuous systems the canonical map from the enveloping operator semigroup onto the Ellis semigroup is never an isomorphism. This answers a long standing open question. We give a complete characterization of minimal systems whose enveloping semigroup is metrizable. In particular it follows that for an abelian acting group such a system is equicontinuous.
2
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Hereditarily non-sensitive dynamical systems and linear representations

63%
Glasner E., Megrelishvili M.
Colloquium Mathematicum
|
2006
|
tom 104
|
nr 2
223-283
EN
For an arbitrary topological group G any compact G-dynamical system (G,X) can be linearly G-represented as a weak*-compact subset of a dual Banach space V*. As was shown in [45] the Banach space V can be chosen to be reflexive iff the metric system (G,X) is weakly almost periodic (WAP). In the present paper we study the wider class of compact G-systems which can be linearly represented as a weak*-compact subset of a dual Banach space with the Radon-Nikodým property. We call such a system a Radon-Nikodým (RN) system. One of our main results is to show that for metrizable compact G-systems the three classes: RN, HNS (hereditarily non-sensitive) and HAE (hereditarily almost equicontinuous) coincide. We investigate these classes and their relation to previously studied classes of G-systems such as WAP and LE (locally equicontinuous). We show that the Glasner-Weiss examples of recurrent-transitive locally equicontinuous but not weakly almost periodic cascades are actually RN. Using fragmentability and Namioka's theorem we give an enveloping semigroup characterization of HNS systems and show that the enveloping semigroup E(X) of a compact metrizable HNS G-system is a separable Rosenthal compact, hence of cardinality $ ≤ 2^{ℵ₀}$. We investigate a dynamical version of the Bourgain-Fremlin-Talagrand dichotomy and a dynamical version of the Todorčević dichotomy concerning Rosenthal compacts.
3
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The universal minimal system for the group of homeomorphisms of the Cantor set

63%
Glasner E., Weiss B.
Fundamenta Mathematicae
|
2003
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tom 176
|
nr 3
277-289
EN
Each topological group G admits a unique universal minimal dynamical system (M(G),G). For a locally compact noncompact group this is a nonmetrizable system with a rich structure, on which G acts effectively. However there are topological groups for which M(G) is the trivial one-point system (extremely amenable groups), as well as topological groups G for which M(G) is a metrizable space and for which one has an explicit description. We show that for the topological group G = Homeo(E) of self-homeomorphisms of the Cantor set E, with the topology of uniform convergence, the universal minimal system (M(G),G) is isomorphic to Uspenskij's "maximal chains" dynamical system (Φ,G) in $2^{2^E}$. In particular it follows that M(G) is homeomorphic to the Cantor set. Our main tool is the "dual Ramsey theorem", a corollary of Graham and Rothschild's Ramsey's theorem for n-parameter sets. This theorem is used to show that every minimal symbolic G-system is a factor of (Φ,G), and then a general procedure for analyzing G-actions of zero-dimensional topological groups is applied to show that (M(G),G) is isomorphic to (Φ,G).
4
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New algebras of functions on topological groups arising from G-spaces

63%
Glasner E., Megrelishvili M.
Fundamenta Mathematicae
|
2008
|
tom 201
|
nr 1
1-51
EN
For a topological group G we introduce the algebra SUC(G) of strongly uniformly continuous functions. We show that SUC(G) contains the algebra WAP(G) of weakly almost periodic functions as well as the algebras LE(G) and Asp(G) of locally equicontinuous and Asplund functions respectively. For the Polish groups of order preserving homeomorphisms of the unit interval and of isometries of the Urysohn space of diameter 1, we show that SUC(G) is trivial. We introduce the notion of fixed point on a class P of flows (P - fpp) and study in particular groups with the SUC-fpp. We study the Roelcke algebra (= UC(G) = right and left uniformly continuous functions) and SUC compactifications of the groups S(ℕ), of permutations of a countable set, and H(C), of homeomorphisms of the Cantor set. For the first group we show that WAP(G) = SUC(G) = UC(G) and also provide a concrete description of the corresponding metrizable (in fact Cantor) semitopological semigroup compactification. For the second group, in contrast, we show that SUC(G) is properly contained in UC(G). We then deduce that for this group UC(G) does not yield a right topological semigroup compactification.
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