Algebraic actions of the discrete Heisenberg group: Expansiveness and homoclinic points

TL;DR

This paper explores algebraic actions of the discrete Heisenberg group, introducing a new approach to expansiveness criteria and presenting the first example of an absolutely summable homoclinic point for a nonexpansive action, with applications to symbolic covers.

Contribution

It proposes a novel method based on Allan’s local principle for analyzing expansiveness in the Heisenberg group and constructs an explicit homoclinic point for nonexpansive actions.

Findings

New approach to expansiveness criteria using Allan’s local principle

First example of an absolutely summable homoclinic point in this context

Construction of an equal-entropy symbolic cover of the system

Abstract

We survey some of the known criteria for expansiveness of principal algebraic actions of countably infinite discrete groups. In the special case of the discrete Heisenberg group we propose a new approach to this problem based on Allan's local principle. Furthermore, we present a first example of an absolutely summable homoclinic point for a nonexpansive action of the discrete Heisenberg group and use it to construct an equal-entropy symbolic cover of the system.