The Rokhlin dimension of topological Z^m-actions

TL;DR

This paper investigates the Rokhlin dimension for topological Z^m-actions on finite-dimensional spaces, establishing conditions under which these actions have finite Rokhlin dimension and implications for the associated C*-algebras.

Contribution

It extends the understanding of Rokhlin dimension to topological Z^m-actions on finite-dimensional spaces and links this to properties of the induced C*-algebraic actions.

Findings

Finite Rokhlin dimension implies finite nuclear dimension of the C*-algebra.

Free Z^m-actions on finite-dimensional spaces satisfy a strengthened marker property.

The results generalize previous work by Gutman on Rokhlin dimension.

Abstract

We study the topological variant of Rokhlin dimension for topological dynamical systems (X,{\alpha},Z^m) in the case where X is assumed to have finite covering dimension. Finite Rokhlin dimension in this sense is a property that implies finite Rokhlin dimension of the induced action on C*-algebraic level, as was discussed in a recent paper by Ilan Hirshberg, Wilhelm Winter and Joachim Zacharias. In particular, it implies under these conditions that the transformation group C*-algebra has finite nuclear dimension. Generalizing a result of Yonatan Gutman, we show that free Z^m-actions on finite dimensional spaces satisfy a strengthened version of the so-called marker property, which yields finite Rokhlin dimension for said actions.