Intrinsic algebraic entropy

© 2014 Elsevier B.V. The new notion of intrinsic algebraic entropy ent~ 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 ent~ 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.