Rohlin properties for $Z^d$-actions on the Cantor set

TL;DR

This paper investigates the dynamical properties of $Z^d$-actions on the Cantor set, revealing differences in generic behavior and the density of isomorphism classes for different dimensions.

Contribution

It proves that for $d \,\geq\, 2$, all isomorphism classes are meager, contrasting with the $d=1$ case, and analyzes the density and effectiveness of actions.

Findings

For $d=1$, there exists a residual isomorphism class.

For $d\geq 2$, all isomorphism classes are meager.

Effective actions are dense but not transitive for $d\geq 2$.

Abstract

We study the space of continuous $Z^d$-actions on the Cantor set, particularly questions on the existence and nature of actions whose isomorphism class is dense (Rohlin's property). Kechris and Rosendal showed that for $d=1$ there is an action on the Cantor set whose isomorphism class is residual. We prove in contrast that for $\geq 2$ every isomorphism class is meager; on the other hand, while generically an action has dense isomorphism class and the effective actions are dense, no effective action has dense isomorphism class. Thus for $d \geq 2$ conjugation on the space of actions is topologically transitive but one cannot construct a transitive point. Finally, we show that in the space of transitive and minimal actions the effective actions are nowhere dense, and in particular there are minimal actions that are not approximable by minimal SFTs.