Rohlin properties for $\mathbb{Z}^{d}$ actions on the Cantor set

Abstract

We study the space $\mathcal {H}(d)$ of continuous $\mathbb {Z}^{d}$-actions on the Cantor set, particularly questions related to density of isomorphism classes. For $d=1$, Kechris and Rosendal showed that there is a residual conjugacy class. We show, in contrast, that for $d\geq 2$ every conjugacy class in $\mathcal {H}(d)$ is meager, and that while there are actions with dense conjugacy class and the effective actions are dense, no effective action has dense conjugacy class. Thus, the action by the group homeomorphisms on the space of actions is topologically transitive but one cannot construct a transitive point. Finally, we show that in the spaces of transitive and minimal actions the effective actions are nowhere dense, and in particular there are minimal actions that are not approximable by minimal shifts of finite type.