We introduce the algebraic entropy for endomorphisms of arbitrary abelian groups, appropriately modifying existing notions of entropy. The basic properties of the algebraic entropy are given, as well as various examples. The main result of this paper is the Addition Theorem showing that the algebraic entropy is additive in appropriate sense with respect to invariant subgroups. We give several applications of the Addition Theorem, among them the Uniqueness Theorem for the algebraic entropy in the category of all abelian groups and their endomorphisms. Furthermore, we point out the delicate connection of the algebraic entropy with the Mahler measure and Lehmer Problem in Number Theory

The new notion of adjoint algebraic entropy of endomorphisms of Abelian groups is introduced. Various examples and basic properties are provided. It is proved that the adjoint algebraic entropy of an endomorphism equals the algebraic entropy of the adjoint endomorphism of the Pontryagin dual. As applications, we compute the adjoint algebraic entropy of the shift endomorphisms of direct sums, and we prove the Addition Theorem for the adjoint algebraic entropy of bounded Abelian groups. A dichotomy is established, stating that the adjoint algebraic entropy of any endomorphism can take only values zero or infinity. As a consequence, we obtain the following surprising discontinuity criterion for endomorphisms: every endomorphism of a compact Abelian group, having finite positive algebraic entropy, is discontinuous

The algebraic entropy h defined for endomorphisms f of abelian groups G measures the growth of the trajectories of non-empty finite subsets F of G with respect to f. We show that this growth can be either polynomial or exponential. The greatest f-invariant subgroup of G where this growth is \ud polynomial coincides with the greatest f-invariant subgroup P(G,f) of G (named Pinsker subgroup of f) such that h(f|_P(G,f))=0. \ud We obtain also an alternative characterization of P(G,f) from the point of view of the quasi-periodic points of f

The new notion of intrinsic algebraic entropy (ent) over tilde of endomorphisms of Abelian groups is introduced and investigated. The intrinsic algebraic entropy is compared with the algebraic entropy, a well-known numerical invariant introduced in the sixties and recently deeply studied also in its relations to other fields of Mathematics. In particular, it is shown that the intrinsic algebraic entropy and the algebraic entropy coincide on endomorphisms of torsion Abelian groups, and their precise relation is clarified in the torsion-free case. The Addition Theorem and the Uniqueness Theorem are also proved for ent, in analogy with similar results for the algebraic entropy. Furthermore, a relevant connection of eat to the algebraic entropy of a continuous endomorphism of a locally compact Abelian group G is pointed out; this allows for the computation of the algebraic entropy in case G is totally disconnected

We show that the topological entropy of a continuous endomorphism of a compact abelian group coincides with the algebraic entropy of the dual endomorphism of the (discrete) Pontryagin dual group. As an application a relation is given between the topological Pinsker factor and the algebraic Pinsker subgroup