Algebraic entropy of shift endomorphisms on abelian groups

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)↦(x_λ(i)). 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.