On the disjointness property of groups and a conjecture of Furstenberg

Abstract

In his seminal 1967 paper "Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation" Furstenberg introduced the notion of disjointness of dynamical systems, both topological and measure preserving. In this paper he showed that for actions of the integers the Bernoulli system $\Omega = \{0, 1\}^\mathbb{Z}$, is disjoint from every minimal system, and that the subring $R_0$, over the field $\mathbb{Z}_2 =\{0, 1\}$, generated by the minimal functions in $\Omega$, is a proper subset of $\Omega$. He conjectured that a similar result holds in general and in our 1983 work "Interpolation sets for subalgebras of $l^\infty(\mathbb{Z})$" we confirmed this by showing that the closed subalgebra $\mathfrak{A}$ of $l^\infty(\mathbb{Z})$, generated by the minimal functions, is a proper subalgebra of $l^\infty(\mathbb{Z})$. In this work we generalize these results to a large class of groups. We call a countable group $G$ a DJ group if for every metrizable minimal action of $G$ there exists an essentially free minimal action disjoint from it. We show that amenable groups are DJ and that the DJ property is preserved under direct products. We define a simple dynamical condition DDJ on minimal systems, which is a strengthening of the Gottchalk-Hedlund property, and we say that a group $G$ is DDJ if every minimal $G$-system has this property. The DJ property implies DDJ and by means of an intricate construction we show that every finitely generated DDJ group is also DJ. Residually finite, maximally almost periodic and $C^*$-simple groups are all DDJ. Finally we show that Furstenberg's conjecture holds for every DDJ group.