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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.
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