Nearly approximate transitivity (AT) for circulant matrices

TL;DR

This paper explores how ergodic actions derived from circulant matrices relate to the approximately transitive (AT) property, showing that under certain conditions, these actions tend to be AT, with numerous examples illustrating the results.

Contribution

It demonstrates that ergodic actions from circulant matrices are generally close to AT, and establishes conditions under which ATness naturally arises, extending understanding of their measure-theoretic classification.

Findings

Non-AT actions can be constructed from circulant matrices.

Under modest conditions, these actions are AT.

Dropping positivity leads to all such actions satisfying an AT-like property.

Abstract

By previous work of Giordano and the author, ergodic actions of $\Z$ (and other discrete groups) are completely classified measure-theoretically by their dimension space, a construction analogous to the dimension group used in C*-algebras and topological dynamics. Here we investigate how far from AT (approximately transitive) can actions be which derive from circulant (and related) matrices. It turns out not very: although non-AT actions can arise from this method of construction, under very modest additional conditions, ATness arises; in addition, if we drop the positivity requirement in the isomorphism of dimension spaces, then all these ergodic actions satisfy an analogue of AT. Many examples are provided