Entropy on abelian groups

Abstract

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

Intrinsic algebraic entropy

“…The detailed description of this case allows us to prove the Addition Theorem for the intrinsic algebraic entropy, stating that ent is an additive invariant of the category Mod(Z[X]). The proof closely follows the proof of the Addition Theorem for the algebraic entropy given in [6]. This result, together with the property of being upper continuous, makes the intrinsic algebraic entropy a length function on Mod(Z[X]), similarly to the algebraic entropy h (see [6]).…”

“…In Section 6 we prove a Uniqueness Theorem, characterizing the intrinsic algebraic entropy in a similar fashion, mutatis mutandis, to the analogous theorems proved for the two above mentioned entropies ent and h in [9] and [6], respectively. Indeed, we see that ent is the unique length function on Mod(Z[X]) coinciding with ent on torsion Abelian groups and satisfying (1.5).…”

“…Peters gave a different definition of algebraic entropy in [15] for automorphisms of Abelian groups using finite subsets instead of finite subgroups; he obtained in this way the new invariant h which is non-trivial also for torsion-free Abelian groups. This notion was extended to endomorphisms of Abelian groups in [6], where it was proved that h coincides with the algebraic entropy ent for endomorphisms of torsion Abelian groups. Imitating [16] and [6], in [23] the definition of algebraic entropy ✩ This research was partially supported by "Progetti di Eccellenza 2011/12" of Fondazione CARIPARO.…”

“…This notion was extended to endomorphisms of Abelian groups in [6], where it was proved that h coincides with the algebraic entropy ent for endomorphisms of torsion Abelian groups. Imitating [16] and [6], in [23] the definition of algebraic entropy ✩ This research was partially supported by "Progetti di Eccellenza 2011/12" of Fondazione CARIPARO. The fourth named author was partially supported by DGI MINECO MTM2011-28992-C02-01, and by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya.…”

“…Following [6], the definition of algebraic entropy h(φ) of an endomorphism φ : G → G of an Abelian group G makes use of finite subsets F of G and their partial φ-trajectories T n (φ, F ) = F + φ(F ) + φ 2 (F ) + . .…”