Endomorphisms of Abelian Groups with Small Algebraic Entropy

Authors

D. Dikranjan, Department of Mathematics and Computer Science, University of Udine,Viadelle Scienze,206,Loc.Rizzi,33100 Udine, Italy
Ketao Gong, Dublin Institute of TechnologyFollow
P. Zanardo, Department of Pure and Applied Mathematics,University of Padua,Via Trieste 63,35121 Padua,Italy

Document Type

Article

Rights

This item is available under a Creative Commons License for non-commercial use only

Disciplines

1.1 MATHEMATICS

Publication Details

https://www.sciencedirect.com/science/article/pii/S0024379513003637

Abstract

We study the endomorphisms ϕ of abelian groups G having a “small” algebraic entropy h (where “small” usually means ). Using essentially elementary tools from linear algebra, we show that this study can be carried out in the group , where an automorphism ϕ with must have all eigenvalues in the open circle of radius 2, centered at 0 and ϕ must leave invariant a lattice in , i.e., be essentially an automorphism of . In particular, all eigenvalues of an automorphism ϕ with must be roots of unity. This is a particular case of a more general fact known as Algebraic Yuzvinskii Theorem. We discuss other particular cases of this fact and we give some applications of our main results.

DOI

https://doi.org/10.1016/j.laa.2013.05.021

Recommended Citation

Dikranjan, D., Gong, K. and Zanardo, P. (2013). Endomorphisms of abelian groups with small algebraic entropy. Linear Algebra and its Applications, vol. 439, no. 7, pg. 1894–1904 doi.org/10.1016/j.laa.2013.05.021