Algebraic entropies of commuting endomorphisms of torsion abelian groups
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 receptive algebraic entropy of an infinite cyclic \(R_P\)-module \(A =R_p/\mathfrak{a}\) is infinite when \(0\not= \mathfrak{a}\) is a principal ideal in \(R_p\). When \(m=2\) Theorem 5.5 and Corollary 5.7 allow the computation of the receptive algebraic entropy of \(A\) whether \(\mathfrak{a}\) is principal or not. In Corollary 5.8, keeping \(m=2\), we have for an infinite finitely generated \(R_p\)-module \(A\) the following are equivalent: (a) ent(\(A\))=0; (b) \(0< \widetilde{\mathrm{ent}}(A) <\infty\); rank\(_{R_p}(A)=0\).
scientific article; zbMATH DE number 7367407