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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.11634 |
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| _version_ | 1866917840623239168 |
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| author | Sabitova, Maria |
| author_facet | Sabitova, Maria |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_11634 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Endomorphism rings of toroidal solenoids Sabitova, Maria Number Theory Dynamical Systems 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. |
| title | Endomorphism rings of toroidal solenoids |
| topic | Number Theory Dynamical Systems |
| url | https://arxiv.org/abs/2411.11634 |