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Algebraic entropy for Abelian groups. (English) Zbl 1176.20057
The notion of entropy associated to some mathematical structures (maps) was first introduced by R. L. Adler, A. G. Konheim, M. H. McAndrew [Trans. Am. Math. Soc. 114, 309-319 (1965; Zbl 0127.13102)]. In this paper the authors note that it “can be tailored to fit mappings on other mathematical structures”, and they propose a notion of entropy associated to endomorphisms of Abelian groups. In the present paper this kind of entropy, called ‘algebraic entropy’, is studied.
Basic definitions and properties are listed in Section 1. It is proved that every infinite direct sum of non-zero Abelian \(p\)-groups admits an endomorphism (a shift map) of infinite algebraic entropy (Theorem 1.12). Moreover, if \(G\) is a reduced \(p\)-group which has an endomorphism of finite and strictly positive entropy then \(G\) has an infinite bounded summand (Theorem 1.19). Connections with ergodic theory are exhibited in Section 2. It is proved that an endomorphism of an Abelian \(p\)-group is recurrent if and only if it has zero entropy (Proposition 2.9). Using this result the authors observe that there exists a separable \(p\)-group which admits recurrent endomorphisms which are not strongly recurrent (Proposition 2.12).
A very important result, called the Addition Theorem, is proved in Section 3. It states that, for torsion groups, the entropy is additive with respect invariant subgroups (Theorem 3.1).
The rest of the paper is dedicated to study some important particular cases. It is proved that: there exists a standard essentially indecomposable group \(G\) which admits an endomorphism of infinite entropy (Theorem 4.4); every small endomorphism of a semi-standard \(p\)-group has zero entropy (Theorem 5.2). In the end of the paper it is proved that the algebraic entropy is the unique non-negative, real numerical invariant associated to the endomorphisms, which satisfies certain properties (Theorem 6.1).

MSC:
20K30 Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups
20K10 Torsion groups, primary groups and generalized primary groups
37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
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