On tame dynamical systems

E. Glasner. "On tame dynamical systems." Colloquium Mathematicae 105.2 (2006): 283-295. <http://eudml.org/doc/284021>.

@article{E2006,
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.},
author = {E. Glasner},
journal = {Colloquium Mathematicae},
keywords = {enveloping semigroup; enveloping operator semigroup; tame; injective dynamical system; minimal system; equicontinuous system; Rosenthal compact; Fréchet compact; Helly compact; angelic topological space; distal metric},
language = {eng},
number = {2},
pages = {283-295},
title = {On tame dynamical systems},
url = {http://eudml.org/doc/284021},
volume = {105},
year = {2006},
}

TY - JOUR
AU - E. Glasner
TI - On tame dynamical systems
JO - Colloquium Mathematicae
PY - 2006
VL - 105
IS - 2
SP - 283
EP - 295
AB - 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.
LA - eng
KW - enveloping semigroup; enveloping operator semigroup; tame; injective dynamical system; minimal system; equicontinuous system; Rosenthal compact; Fréchet compact; Helly compact; angelic topological space; distal metric
UR - http://eudml.org/doc/284021
ER -