Algebraic Entropy of Shift Endomorphisms on Abelian Groups

Abstract

Abstract 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.