Algebraic entropy for amenable semigroup actions. (English) Zbl 1453.20074
The authors extend the classical algebraic entropy for endomorphisms of abelian groups. They introduce two notions of algebraic entropy for actions of cancellative right amenable semigroups \(S\) on discrete abelian groups \(A\) by endomorphisms (in the classical case \(S=\mathbb{N}\)). This notions extend algebraic entropy introduced by M. Weiss [Math. Syst. Theory 8, No. 3, 243–248 (1974–1975; Zbl 0298.28014)] for endomorphisms of torsion abelian groups and the algebraic entropy \(h_{\mathrm{alg}}\) introduced by D. Dikranjan [Adv. Math. 298, 612–653 (2016; Zbl 1368.37015)]. The fundamental properties of the algebraic entropy are investigated. The algebraic entropy are computed in several examples. The authors prove the next (additional) theorem: “let \(S\stackrel{\alpha}{\curvearrowright} A\) be a left action of a cancellative right amenable monoid \(S\) on a torsion abelian group \(A\) and let \(B\) be an \(\alpha\)-invariant subgroup of \(A\), and denote by \(\alpha_B\) and \(\alpha_{A/B}\) the induced actions of \(S\) on \(B\) and on \(A/B\), respectively, then \(\mathrm{ent}(\alpha) = \mathrm{ent}(\alpha_B) + \mathrm{ent}(\alpha_{A/B})\)”. The authors prove the bridge theorem, which connects the algebraic entropy with the topological entropy of the dual action by means by Pontryagin duality. Some open questions are formulated in the paper.
Reviewer: Nikolay I. Kryuchkov (Ryazan)
MSC:
| 20K30 | Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups |
| 20M20 | Semigroups of transformations, relations, partitions, etc. |
| 37A35 | Entropy and other invariants, isomorphism, classification in ergodic theory |
| 37B40 | Topological entropy |
| 43A07 | Means on groups, semigroups, etc.; amenable groups |
Keywords:
semigroup action; topological entropy; algebraic entropy; amenable semigroup; group endomorphism; amenable monoidReferences:
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