Algebraic Entropy of Shift Endomorphisms on Abelian Groups

Abstract

For every finite-to-one map λ : Γ → Γ and for every abelian group K, the generalized shift σ_λ 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.