Algebraic entropy of shift endomorphisms on abelian groups

Abstract

For every finite-to-one map $\lambda :\Gamma \to \Gamma$ and for every abelian group $K$, the generalized shift ${\sigma }_{\lambda }$ of the direct sum ${⨁}_{\Gamma }K$ is the endomorphism defined by $(x_i{)}_{i\in \Gamma }\mapsto (x_{\lambda (i)}{)}_{i\in \Gamma }$. 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 $\lambda$. We give many examples showing that the generalized shifts provide a very useful universal tool for producing counter-examples.

15 pages