On tame dynamical systems

By Eli Glasner. 2006

84 citations recorded by the source corpus.

According to a dynamical version of the BourgainFremlin-Talagrand dichotomy an enveloping semigroup of a dynamical system is either tame: has cardinality ≤ 2א0 , or it is topologically wield and contains a copy of βN, the Cech-Stone compactificaltion of a discrete countable set. As a general principle one can measure the usefulness of a new mathematical notion by the number of seemingly unrelated ways by which it can be characterized. According to this principle the notion of tameness stands rather high. Tameness can also be characterized by the lack of a certain independence property — where combinatorial Ramsey type arguments take a leading role — by the fact that the elements of the enveloping semigroup of a tame system are Baire class 1 maps, and, using results of Talagrand, as systems in which the topological Glivenko-Cantelli property is satisfied. Finally a dynamical system is tame iff it can be represented on a Banach space which does not contain an isomorphic copy of `1. I will review some of these results and indicate some applications. Department of Mathematics, Tel-Aviv University, Tel Aviv, Israel E-mail address: glasner@math.tau.ac.il

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d51124cbc10ebee40368fc81d5b76ba43118368b

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Citations in the corpus

Weak forms of topological and measure-theoretical equicontinuity: relationships with discrete spectrum and sequence entropy (2014, background): “Another class of rigid TDSs are the tame systems introduced in [16, 28].”
Representations of dynamical systems on Banach spaces not containing $l_1$ (2008, background): “In fact the notions of regularity and tameness coincide (see [10] (for metrizable systems) and Corollary 5.”
Local entropy theory (2009, background): “It is known that there exists a transitive non-minimal tame TDS [33].”
The structure of tame minimal dynamical systems (2006, methodology): “here as well as in [13], is a proposition about diffused measures (Proposition 3.”