Endomorphism rings of toroidal solenoids

2024/11/18 by Sabitova, Maria
#Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2411.11634

Abstract

We study the endomorphism ring End(G_A) of a subgroup G_A of ℚⁿ defined by a non-singular n× 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 ℤⁿ-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 ℤⁿ-odometer is computable.

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