Topological speedups of ℤ^d-actions

Abstract

We study minimal ${\mathbb{Z}}^d$-Cantor systems and the relationship between their speedups, their collections of invariant Borel measures, their associated unital dimension groups, and their orbit equivalence classes. In the particular case of minimal ${\mathbb{Z}}^d$-odometers, we show that their bounded speedups must again be odometers but, contrary to the 1-dimensional case, they need not be conjugate, or even isomorphic, to the original.  Furthermore, we give examples of speedups of ${\mathbb{Z}}^d$-odometers which show the significant role played by a choice of ‘cone’ associated to the speedup.

Keywords:

• Orbit equivalence
• minimal Cantor systems
• topological ${\mathbb{Z}}^d$-speedups
• ${\mathbb{Z}}^d$-odometers
• Kakutani-Rohklin partitions

2020 Mathematics Subject Classification:

• 37B05

Disclosure statement

No potential conflict of interest was reported by the author(s).