Endomorphisms of abelian groups with small algebraic entropy

Abstract

We study the endomorphisms ϕ of abelian groups G having a “small” algebraic entropy h (where “small” usually means h(ϕ)<log2). Using essentially elementary tools from linear algebra, we show that this study can be carried out in the group Qd, where an automorphism ϕ with h(ϕ)<log2 must have all eigenvalues in the open circle of radius 2, centered at 0 and ϕ must leave invariant a lattice in Qd, i.e., be essentially an automorphism of Zd. In particular, all eigenvalues of an automorphism ϕ with h(ϕ)=0 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.