Tame systems and scrambled pairs under an Abelian group action

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, Glasner (On tame enveloping semigroups. Colloq. Math.105 (2006), 283–395) calls the system tame. In this paper, we study the tame system under an Abelian group action. We introduce the notion of scrambled pairs for a dynamical system under an Abelian group action and show that a tame system has no scrambled pair. At the same time, we give some sufficient conditions such that a pair is a scrambled one. Moreover, using these sufficient conditions we prove that a minimal tame system under an Abelian group action is almost automorphic and uniquely ergodic. This gives a positive answer to Problem 2.5 in the paper cited above. Finally, we prove that for ${\mathbb Z}$-actions a tame system is $M$-null, that is, the metric sequence entropy is zero for any invariant measure and any increasing sequence of natural numbers. We also give examples to show that tameness is strictly weaker than nullness for ${\mathbb Z}$-actions.