Bounded orbit injections and suspension equivalence for minimal $\mathbb{Z}^2$ actions

Abstract

In this paper we prove that there exist bounded orbit injections from minimal $\mathbb{Z}^{2}$ actions of a Cantor set $T$ and $S$ into a common action $R$ if and only if the suspension spaces associated to $T$ and $S$ are homeomorphic. In this way we prove a two-dimensional analog of a result of Parry and Sullivan on flow equivalence and discrete cross-sections for minimal systems. At the same time the result is a topological analog of a result of del Junco and Rudolph on Kakutani equivalence for ergodic $\mathbb{Z}^{d}$ actions. We also prove a structural result about such suspension spaces. Namely, that they are a finite union of products of Cantor sets with polygons, $C_{i}\times P_{i}$, after an identification on the boundary, $C_{i}\times \partial P_{i}$, with the action given by ${\mathbb{R}}^{2}$ on the polygon. The polygons $P_{i}$ can be chosen to have properties associated with Voronoi or Delaunay tilings corresponding to a set of points located uniformly throughout the plane.