2015
DOI: 10.1016/j.jpaa.2014.09.033
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Intrinsic algebraic entropy

Abstract: 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… Show more

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Cited by 33 publications
(55 citation statements)
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“…Indeed, Corollary 2 and a limit-free formula as in [4,15] provide ent dim (β K ) = dim K (V − /β −1 K (V − )) = 1. Quite remarkably, φ-inert subspaces do not enrich the dynamics of linear flows like φ-inert subgroups do in the framework of abelian groups (see [9]). Indeed, one verifies that ent dim (φ) = ent dim (φ) for every φ : V → V , where ent dim (φ) := sup lim n→∞ dim K (T n (φ, F )) n | F ≤ V and dim K (F ) < ∞ , which is a classical entropy function for vector spaces and their endomorphisms (details about this entropy function can be found in [3]).…”
Section: Connection With Algebraic Entropymentioning
confidence: 99%
“…Indeed, Corollary 2 and a limit-free formula as in [4,15] provide ent dim (β K ) = dim K (V − /β −1 K (V − )) = 1. Quite remarkably, φ-inert subspaces do not enrich the dynamics of linear flows like φ-inert subgroups do in the framework of abelian groups (see [9]). Indeed, one verifies that ent dim (φ) = ent dim (φ) for every φ : V → V , where ent dim (φ) := sup lim n→∞ dim K (T n (φ, F )) n | F ≤ V and dim K (F ) < ∞ , which is a classical entropy function for vector spaces and their endomorphisms (details about this entropy function can be found in [3]).…”
Section: Connection With Algebraic Entropymentioning
confidence: 99%
“…We start with some results already proved in [6], or easily deducible from them. We provide their essential details, for the sake of completeness.…”
Section: -Inert and Fully Inert Subgroupsmentioning
confidence: 99%
“…In Section 2 we collect preliminary results on -inert subgroups, some of which are extracted from [6], and on fully inert subgroups. In Section 3 fully inert subgroups of divisible groups are investigated, as a preparation for the main results of the paper contained in Sections 4, 5 and 6, where we characterize inert groups as follows.…”
mentioning
confidence: 99%
“…As in [6] and [7], an endomorphism ϕ of an abelian group A (from now on always in additive notation) is said (right-) inertial iff:…”
Section: Introductionmentioning
confidence: 99%
“…Moreover, notice that in [7] the concept of inertial endomorphism is related to the investigation of the dynamical properties of an endomorphism of an abelian group.…”
Section: Introductionmentioning
confidence: 99%