Rohlin properties for \(\mathbb {Z}^{d}\) actions on the Cantor set (Q2880675)

\textit{P. R. Halmos} [Ann. Math. (2) 45, 786--792 (1944; Zbl 0063.01889)] initiated the study of the space of all invertible measure-preserving transformations of a Lebesgue space, showing in particular that any aperiodic transformation has a dense conjugacy class. This paper addresses a related question in a natural topological setting, namely that of continuous actions of \(\mathbb{Z}^d\) on the Cantor set. Write \(H\) for the group of homeomorphisms of the Cantor set, and \(H(d)\) for the space of continuous \(\mathbb{Z}^d\)-actions on the Cantor set. Conjugacy defines an action of \(H\) on the space \(H(d)\), and the questions studied here concern orbits or conjugacy classes of this action. The case \(d=1\) is well studied: \textit{E. Glasner} and \textit{B. Weiss} [Am. J. Math. 123, No. 6, 1055--1070 (2001; Zbl 1012.54042)] showed that there are dense conjugacy classes in \(H(1)\), and this was strengthened by \textit{A. S. Kechris} and \textit{C. Rosendal} [Proc. Lond. Math. Soc. (3) 94, No. 2, 302--350 (2007; Zbl 1118.03042)], who showed that a residual conjugacy class exists. The author also showed [Ergodic Theory Dyn. Syst. 28, No. 1, 125--165 (2008; Zbl 1171.37305)] that in the subspace of \(H(1)\) comprising topologically transitive actions (a Polish subspace of \(H(1)\)), a residual conjugacy class can also be found. In this paper, a series of striking differences between \(H(1)\) and \(H(d)\) for \(d\geq2\) are found. Assume now that \(d\geq2\). There are actions in \(H(d)\) with dense conjugacy class, but one of the main results here is that every action in \(H(d)\) has meager conjugacy class. Even in the case of dense orbits, \(H(d)\) behaves very differently than \(H(1)\). A notion of `explicit' is introduced here for actions, meaning roughly that decisions about how the actions move sets about can be decided algorithmically, and it is shown that in a precise sense it is possible to `explicitly' describe actions in \(H(1)\) with dense conjugacy class, but it is not possible to `explicitly' describe such actions in \(H(d)\). A further contrast between \(H(1)\) and \(H(d)\) is revealed by another result, that in the subspaces of minimal or of transitive actions, the `explicitly' describable actions are nowhere dense. The proofs make heavy use of the analysis of \(d\)-dimensional shifts of finite type, and along the way other surprising differences between \(\mathbb{Z}\)-actions and \(\mathbb{Z}^d\)-actions emerge. For example, a consequence of one of the main results is that the minimal (and the transitive) \(d\)-dimensional shifts of finite type are nowhere dense in the space of minimal (or transitive) topological \(\mathbb{Z}^d\)-actions.