Algebraic entropies of commuting endomorphisms of torsion abelian groups
type: Item
instance of: scholarly article
title: Algebraic entropies of commuting endomorphisms of torsion abelian groups
zbMATH Open document ID: 1496.20086
DOI: 10.4171/RSMUP/55
author: Andrzej Biś
author: Dikran Dikranjan
author: Anna Giordano Bruno
author: Luchezar Stoyanov
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 2021-07-02T00:00:00Z
review text: A left semigroup action of a semigroup \(S\) on an abelian group \(A\) by group endomorphisms is defined by \(\alpha: S\times A \to A, (s, x) \mapsto \alpha(s)(x)\) with \(\alpha(st)=\alpha(s) \circ \alpha(t)\) and \(\alpha(s)(x+y)=\alpha(s)(x)+\alpha(s)(y)\) for every \(s, t\in S\) and \(x,y\in A\). The concept of the algebraic entropy of \(\alpha\), developed in [\textit{D. Dikranjan} et al., Trans. Am. Math. Soc. 361, No. 7, 3401--3434 (2009; Zbl 1176.20057)], has some nice properties, but also has some shortcomings. For instance, if \(S\) is finite and \(A\) is infinite then ent(\(\alpha)=\infty\). If \(S\) is a group and \(T\) is a subgroup of \(S\) there are many examples where the restricted action \(\alpha \upharpoonright_T\) has ent(\(\alpha \upharpoonright_T )=\infty\) and ent(\(\alpha\))=0. This led the authors to define \(\widetilde{\mathrm{ent}}^\Gamma(\alpha)\), the algebraic receptive entropy of \(\alpha\) with respect to the regular system \(\Gamma\) of \(S\). \par The authors will show in Section 2 that the algebraic receptive entropy naturally extends the algebraic entropy for \(\mathbb{N}\)-actions. Section 3 makes use of the correspondence between \(\mathbb{N}^m\)-actions on abelian groups \(A\) with prime exponent \(p\) and \(R_p\)-module structures on \(A\), where \(R_p=\mathbb{F}_p[X_1, \ldots , X_m]\). This leads to the computation of the algebraic entropy and the algebraic receptive entropy of \(R_P\) in Theorem 3.3. Section 4 includes the computations of the algebraic entropies. Theorem 4.1 shows that when \(\mathfrak{a}\) is a non-zero ideal of \(R_p\) and \(A=R_p/\mathfrak{a}\) is an infinite cyclic \(R_p\)-module then ent(\(A\))=0. This leads to Theorem 4.4 showing that ent(\(A)=\mathrm{rank}_{R_P}(A) \log p\) for an arbitrary \(R_p\) module \(A\). Proposition 4.7 is a characterization of \(\mathbb{N}^m\)-actions on torsion abelian groups with zero algebraic entropy using Bernoulli shifts. In Section 5, Theorem 5.1 establishes that the r…
Mathematics Subject Classification ID: 20K30
Mathematics Subject Classification ID: 20M10
Mathematics Subject Classification ID: 22A26
Mathematics Subject Classification ID: 22D05
zbMATH DE Number: 7367407
zbMATH Keywords: algebraic entropy
zbMATH Keywords: receptive entropy
zbMATH Keywords: regular system
zbMATH Keywords: abelian group
zbMATH Keywords: torsion abelian group
zbMATH Keywords: \(p\)-group
reviewed by: Ross P. Abraham
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 7367407