Endomorphisms of abelian groups with small algebraic entropy

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