Algebraic entropy of shift endomorphisms on abelian groups

Abstract

For every finite-to-one map $λ:Γ\toΓ$ and for every abelian group $K$, the generalized shift $σ_λ$ of the direct sum $\bigoplus_ΓK$ is the endomorphism defined by $(x_i)_{i\inΓ}\mapsto(x_{λ(i)})_{i\inΓ}$. 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.