On tame dynamical systems

Tom 105 / 2006

E. Glasner Colloquium Mathematicum 105 (2006), 283-295 MSC: Primary 54H20. DOI: 10.4064/cm105-2-9


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.


  • E. GlasnerDepartment of Mathematics
    Tel Aviv University
    Tel Aviv, Israel

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