On tame dynamical systems

E. Glasner¹

¹ Department of Mathematics Tel Aviv University Tel Aviv, Israel

Colloquium Mathematicum, Tome 105 (2006) no. 2, pp. 283-295.

Résumé

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 $\beta {\cal N}$, 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.

Détail

DOI : 10.4064/cm105-2-9

Keywords: dynamical version bourgain fremlin talagrand dichotomy shows enveloping semigroup dynamical system either large contains topological copy beta cal tame topological space whose topology determined convergence sequences latter say dynamical system tame metric distal minimal system tame equicontinuous abelian acting group tame metric minimal system hence weakly mixing minimal system never tame iii tame minimal cascade has zero topological entropy minimal distal but not equicontinuous systems canonical map enveloping operator semigroup ellis semigroup never isomorphism answers long standing question complete characterization minimal systems whose enveloping semigroup metrizable particular follows abelian acting group system equicontinuous