A general notion of algebraic entropy and the rank-entropy

Abstract

We give a general definition of a subadditive invariant i of Mod(R), where R is any ring, and the related notion of algebraic entropy of endomorphisms of R-modules, with respect to i. We examine the properties of the various entropies that arise in different circumstances. Then we focus on the rank-entropy, namely the entropy arising from the invariant ‘rank’ for Abelian groups. We show that the rank-entropy satisfies the Addition Theorem. We also provide a uniqueness theorem for the rank-entropy

A soft introduction to algebraic entropy

“…Remark 2.7 If in Definition 2.4 we replace the linear transformation φ : V → V by an endomorphism ψ : G → G of an Abelian group G, and the invariant "dimension" by the invariant "rank", we obtain the notion of rank-entropy, investigated in [16]. But, even if the algebraic entropy of linear transformations of Q-vector spaces and the rank-entropy of endomorphisms of Abelian groups are calculated in the same way (recall that rk…”

“…The proof of the Addition Theorem for the rank-entropy given in [16] was much simpler. Very recently, using ideas from both the above proofs and borrowing techniques typical of p-groups and of torsion-free groups, a very general Addition Theorem has been proved in [15] for suitable subcategories of modules over arbitrary rings, dealing with the algebraic entropies associated with length functions (see their definition below in this section).…”

“…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 usual, the more general the considered ring R is, the less specific are the results obtainable for R-modules.…”

“…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]). …”

On Adjoint Entropy of Abelian Groups

“…His work laid down the basic properties of this entropy and revealed the fundamental connection between the algebraic entropy of an endomorphism and the topological entropy of its adjoint under Pontryagin duality. In recent times these concepts have been re-examined and developed significantly -see for example, [7], [16] and [17]. The fundamental concept in algebraic entropy is the notion of the trajectory of a finite subgroup of a group G under an endomorphism φ of G; this essentially restricts the notion to torsion groups.…”