Mixing actions of zero entropy for countable amenable groups (Q2833637)

The property of mixing (asymptotic independence of measurable sets when moved apart in time) makes sense for a measure-preserving action of any group, where `moving apart in time' can be expressed as a sequence of group elements that eventually leaves any finite subset of the group. The measure-theoretic entropy of a measure-preserving action also has a natural definition for countable amenable group actions, where it may be defined by averaging information over suitable Følner sequences. The latter also has been extended to non-amenable group actions.NEWLINENEWLINEThis paper is a further contribution to our knowledge of amenable group actions, in this case showing that they behave in one regard much like the classical case of a \(\mathbb{Z}\)-action in that any discrete countable infinite amenable group has a free action by measure-preserving transformations on a standard probability space that is on the one hand mixing (indicative of the action chopping up the space in a complicated way) and on the other has zero entropy. While well known this is not entirely obvious even for a \(\mathbb{Z}\)-action, where the simplest examples come from a Gaussian measure space construction or from combinatorial constructions, and for \(\mathbb{Z}^d\)-actions with \(d\geqslant2\) the dynamical systems of algebraic origin provide a host of natural examples with zero entropy known to be mixing of all orders by work of \textit{K. Schmidt} and the reviewer [Invent. Math. 111, No. 1, 69--76 (1993; Zbl 0824.28012)].NEWLINENEWLINEThe general case dealt with here uses methods from the Poisson suspension of infinite-measure preserving actions and builds on an earlier study of actions of the Heisenberg group by the author [Ergodic Theory Dyn. Syst. 34, No. 4, 1142--1167 (2014; Zbl 1315.37004)].