On $\mathbb Z^d$-odometers associated to integer matrices

TL;DR

This paper extends the classification of $ abla^d$-odometers to higher dimensions and applies these results to odometers defined by integer matrices, enhancing understanding of their conjugacy and isomorphism properties.

Contribution

It generalizes previous results on $ abla^d$-odometers to dimensions greater than two and applies the theory to odometers defined by integer matrices.

Findings

Extended classification to higher dimensions

Characterized conjugacy and isomorphism for matrix-defined odometers

Applied theory to specific odometers from integer matrices

Abstract

We extend the results of T. Giordano, I. F. Putnam, C. F. Skau contained in ``$\mathbb Z^d$-odometers and cohomology", Groups Geom. Dyn. 13 (2019), no. 3, P. 909-938, on characterization of conjugacy, isomorphism, and continuous orbit equivalence of $\mathbb Z^d$-odometers to dimensions $d>2$. We then apply these extensions to the case of odometers defined by matrices with integer coefficients.