Algebraic entropy for Abelian groups

Abstract

Abstract. The theory of endomorphism rings of algebraic structures allows, in a natural way, a systematic approach based on the notion of entropy borrowed from dynamical systems. Here we study the algebraic entropy of the endomorphisms of Abelian groups, introduced in 1965 by Adler, Konheim and McAndrew. The so-called Addition Theorem is proved; this expresses the algebraic entropy of an endomorphism φ of a torsion group as the sum of the algebraic entropies of the restriction to a φ-invariant subgroup and of the endomorphism induced on the quotient group. Particular attention is paid to endomorphisms with zero algebraic entropy as well as to groups all of whose endomorphisms have zero algebraic entropy. The significance of this class arises from the fact that any group not in this class can be shown to have endomorphisms of infinite algebraic entropy, and we also investigate such groups. A uniqueness theorem for the algebraic entropy of endomorphisms of torsion Abelian groups is proved.

A soft introduction to algebraic entropy

“…The Addition Theorem was one of the main achievements in the study of the algebraic entropy of endomorphisms of Abelian groups in [6]. In that paper, the theorem was proved to hold for the subcategory of torsion groups, and it was proved to fail for the whole category of Abelian groups.…”

“…The attempt to give a precise answer to the above questions, not in the setting of vector spaces, but in the less elementary setting of Abelian groups, originated the theory of the algebraic entropy (see [1,6,14,20]). Nowadays, this theory is extended to R-modules over arbitrary rings R (see [2,15,16,18,21]), and also to topological groups (see [9]).…”

“…As motivation for our goal, it is worthwhile to remark that the general study of the algebraic entropy reduces in many different cases to linear transformations of vector spaces: for instance, if φ : G → G is an endomorphism of an Abelian group, the study of ent(φ), the algebraic entropy of φ, reduces to that of ent(φ), where φ : G/ pG → G/ pG ( p a prime number) is the linear transformation induced by φ on the vector space G/ pG over the field with p elements (see [6]). Furthermore, in the investigation of the rank entropy, which is associated with the rank of modules over an integral domain R, one can reduce to linear transformations of vector spaces over the field of quotients K of R (see [16]).…”

Algebraic Entropy of Shift Endomorphisms on Abelian Groups

“…The algebraic entropy of φ with respect to F is Since the definition is based on finite subgroups F , and in particular F is contained in the torsion part t(G) of G, the algebraic entropy depends only on the restriction of φ on t(G), that is ent(φ) = ent(φ ↾ t(G) ). The basic properties of the algebraic entropy can be found in [4,6]. The most relevant of them, known also as Addition Theorem, can be found in §2 (Theorem 2.3), which collects all relevant properties of the algebraic entropy used in this paper.…”

“…More precisely, for a non-trivial finite abelian group K the restriction β K ↾ K (N 0 ) has entropy log |K| [4, Example 1.9] and one can show that every function f defined on all endomorphisms of torsion abelian groups with values in the extended non-negative reals and satisfying f (β Z(p) ↾ Z(p) (N 0 ) ) = log |p|, the Addition Theorem and a few other natural properties (namely, Lemmas 2.1, 2.2 and Remark 2.4 (b)) must necessarily coincide with the algebraic entropy ent(−) [4,Theorem 6.1].…”