The structure of tame minimal dynamical systems

By Eli Glasner. Ergodic Theory and Dynamical Systems, 2006

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Abstract 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 $\beta \mathbb {N}$, or it is a ‘tame’ topological space whose topology is determined by the convergence of sequences. In the latter case, the dynamical system is said to be tame. We use the structure theory of minimal dynamical systems to show that, when the acting group is Abelian, a tame metric minimal dynamical system (i) is almost automorphic (i.e. it is an almost one-to-one extension of an equicontinuous system), and (ii) admits a unique invariant probability measure such that the corresponding measure-preserving system is measure-theoretically isomorphic to the Haar measure system on the maximal equicontinuous factor.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1464996756469507fd36df46213e82ba7ec6f849

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Citations in the corpus

Isomorphic extensions and applications (2015, background): “details see [4], [1] and [11, Example 8.31(6)]). However, none of the minimal systems (X;T G) = (Z Y;T G) given by Theorem3.1is tame. In fact, a theorem of Huang [15], Kerr & Li [17], and Glasner [10] asserts that a tame minimal dynamical system is necessarily almost automorphic, i.e., an almost”
Definably amenable NIP groups (2015, background): “ Theorem 5.3. Remark 1.6. We remark that the study of NIP deﬁnably amenable groups can be thought of as a model-theoretic version of tame dynamics as studied by Glasner, Megrelishvili and others, see [Gla07b] (in fact, we discovered the connection only after having essentially completed this work). ”
Representations of dynamical systems on Banach spaces not containing $l_1$ (2008, result): “Further results concerning tame systems can be found in [28], [12], [ 13 ], [22], [27].”
Local entropy theory (2009, background): “In §§10 and 11 we survey some of the results obtained by Glasner [33, 34], Huang [47], and Kerr–Li [74], concerning tame dynamical systems and various independence notions, including the basic combinatorial lemma of Sauer–Perles– Shelah, several generalizations of this lemma, and their applications ”
Infinite-step nilsystems, independence and complexity (2011, background): “It was shown in [10, 20, 24] that a minimal null (respectively tame) system is an almost 1–1 extension of an equicontinuous TDS and is uniquely ergodic.”
Combinatorial lemmas and applications to dynamics (2009, background): “Huang, Li, Shao and Ye showed that if a minimal system is null then it is an almost one-to-one extension of an equicontinuous system [16] (see [24,15,10] for some recent results concerning tame systems).”