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Topological groups with Rokhlin properties

100%
Glasner E., Weiss B.
Colloquium Mathematicum
|
2008
|
tom 110
|
nr 1
51-80
EN
In his classical paper [Ann. of Math. 45 (1944)] P. R. Halmos shows that weak mixing is generic in the measure preserving transformations. Later, in his book, Lectures on Ergodic Theory, he gave a more streamlined proof of this fact based on a fundamental lemma due to V. A. Rokhlin. For this reason the name of Rokhlin has been attached to a variety of results, old and new, relating to the density of conjugacy classes in topological groups. In this paper we will survey some of the new developments in this area.
2
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Locally equicontinuous dynamical systems

100%
Glasner E., Weiss B.
Colloquium Mathematicum
|
2000
|
tom 84/85
|
nr 2
345-361
EN
A new class of dynamical systems is defined, the class of "locally equicontinuous systems" (LE). We show that the property LE is inherited by factors as well as subsystems, and is closed under the operations of pointed products and inverse limits. In other words, the locally equicontinuous functions in $l_{∞}(ℤ)$ form a uniformly closed translation invariant subalgebra. We show that WAP ⊂ LE ⊂ AE, where WAP is the class of weakly almost periodic systems and AE the class of almost equicontinuous systems. Both of these inclusions are proper. The main result of the paper is to produce a family of examples of LE dynamical systems which are not WAP.
3
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On two recurrence problems

100%
Boshernitzan M., Glasner E.
Fundamenta Mathematicae
|
2009
|
tom 206
|
nr 1
113-138
EN
We review some aspects of recurrence in topological dynamics and focus on two open problems. The first is an old one concerning the relation between Poincaré and Birkhoff recurrence; the second, due to the first author, is about moving recurrence. We provide a partial answer to a topological version of the moving recurrence problem.
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