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Entropy on abelian groups. (English) Zbl 1368.37015
The authors introduce the concept of algebraic entropy for endomorphisms of arbitrary abelian groups, thus modifying existing notions of entropy. The basic properties of their algebraic entropy are given, as well as various examples. The main result of this paper is an addition theorem, showing that the algebraic entropy is additive with respect to invariant subgroups. The authors give several applications of the addition theorem. Among them a uniqueness theorem for the algebraic entropy in the category of all abelian groups and their endomorphisms. Furthermore, they point out the connection of algebraic entropy with the Mahler measure and Lehmer problem in number theory.

MSC:
37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
37B40 Topological entropy
20K30 Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups
11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
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