Endomorphism rings of toroidal solenoids

TL;DR

This paper investigates the structure of endomorphism rings of certain subgroups of rational vector spaces, revealing conditions under which these rings are commutative and linking them to number fields, with applications to toroidal solenoids.

Contribution

It establishes the commutativity of endomorphism rings under specific conditions and connects these rings to number fields, advancing understanding of toroidal solenoids and odometers.

Findings

Endomorphism rings are commutative under irreducibility conditions.

Provides a formula for counting periodic points of toroidal solenoid endomorphisms.

Shows the linear representation group of a $\

Abstract

We study the endomorphism ring $End(G_A)$ of a subgroup $G_A$ of $\mathbb{Q}^n$ defined by a non-singular $n\times n$-matrix $A$ with integer entries. In the case when the characteristic polynomial of $A$ is irreducible and an extra assumption holds if $n$ is not prime, we show that $End(G_A)$ is commutative and can be identified with a subring of the number field generated by an eigenvalue of $A$. The obtained results can be applied to studying endomorphisms of associated toroidal solenoids and $\mathbb{Z}^n$-odometers. In particular, we build a connection between toroidal solenoids and $S$-integer dynamical systems, provide a formula for the number of periodic points of a toroidal solenoid endomorphism, and show that the linear representation group of a $\mathbb{Z}^n$-odometer is computable.