A map maintaining the orbits of a given $\mathbb {Z}^d$-action
Volume 143 / 2016
Colloquium Mathematicum 143 (2016), 1-15
MSC: 37B05, 37B10, 37A20.
DOI: 10.4064/cm6361-12-2015
Published online: 2 December 2015
Abstract
Giordano et al. (2010) showed that every minimal free $\mathbb {Z}^d$-action of a Cantor space $X$ is orbit equivalent to some $\mathbb {Z}$-action. Trying to avoid the K-theory used there and modifying Forrest's (2000) construction of a Bratteli diagram, we show how to define a (one-dimensional) continuous and injective map $F$ on $X\setminus \{\textrm {one point}\}$ such that for a residual subset of $X$ the orbits of $F$ are the same as the orbits of a given minimal free $\mathbb {Z}^d$-action.
