Topological speedups of $\mathbb{Z}^d$-actions

TL;DR

This paper investigates how speedups affect minimal $ ext{Z}^d$-actions, revealing that bounded speedups of odometers remain odometers but can differ significantly from the original in higher dimensions.

Contribution

It establishes the behavior of bounded speedups in minimal $ ext{Z}^d$-odometers, showing they are again odometers but not necessarily conjugate or isomorphic, unlike the 1D case.

Findings

Bounded speedups of $ ext{Z}^d$-odometers are again odometers.

In higher dimensions, these speedups need not be conjugate or isomorphic to the original.

The study links speedups with invariant measures, dimension groups, and orbit equivalence.

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.