On tame dynamical systems

Abstract

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 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.

Representations of dynamical systems on Banach spaces not containing $l_{1}$

“…Extending Köhler's definition to arbitrary topological groups G, we say that a compact G-space X is regular if, for any f ∈ C(X), the orbit fG does not contain an l 1 -sequence (in other words, the second alternative is ruled out in the Rosenthal dichotomy). As we will see later, in Corollary 5.8, a G-system is regular if and only if it is tame (for metrizable X this fact was established in [10]). …”

“…In [10] a compact metric dynamical system is called tame if the first alternative occurs, i.e. if E(X) is a Rosenthal compactum.…”

“…A compact metric dynamical G-system X is called tame [10] if in the dynamical BFT-dichotomy (Fact 1.2) the first alternative occurs; i.e., E(X) is Rosenthal compact (see also Proposition 6.15 below).…”

The dynamical hierarchy for Roelcke precompact Polish groups

“…3. Tame ∩ UC = NIP = Null ∩ UC Tame functions have been studied by Glasner and Megrelishvili in [GM12], after the introduction of tame dynamical systems by Köhler [Köh95] (who called them regular systems) and later by Glasner in [Gla06]. If the translation of Ben Yaacov and Tsankov for Roelcke precompact Polish groups identifies WAP functions with stable formulas, we remark in this section that tame functions correspond to NIP (or dependent) formulas.…”

On metrizable enveloping semigroups

“…(2) Theorem 6.2 answers negatively Problem 3.3 in [19]. (3) In his paper [13] Ellis, following Furstenberg's classical work, investigates the projective action of GL(n, R) on the projective space P n−1 .…”