Algebraic Entropy of Shift Endomorphisms on Abelian Groups

Published in: Quaestiones Mathematicae
Volume 32, issue 4, 2009 , pages: 529–550
DOI: 10.2989/QM.2009.
Author(s): Maryam AkhavinFaculty of Mathematical Sciences, Iran, Dikran DikranjanDipartimento di Matematica e Informatica, Italy, Anna, Giordano BrunoDipartimento di Matematica e Informatica, Italy, Arezoo HosseiniDepartment of Mathematics, Faculty of Science, Iran, Fatemah, Ayatollah Zadeh ShiraziFaculty of Mathematics, Statistics and Computer Science, College of Science, Iran


For every finite-to-one map λ : Γ → Γ and for every abelian group K, the generalized shift σλ of the direct sum ⊕Γ K is the endomorphism defined by (x i ) iεΓ ↦ (x λ(i)) iεΓ [3]. In this paper we analyze and compute the algebraic entropy of a generalized shift, which turns out to depend on the cardinality of K, but mainly on the function λ. We give many examples showing that the generalized shifts provide a very useful universal tool for producing counter-examples.

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