Algebraic entropy of generalized shifts on direct products

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Abstract

For a set $\Gamma$, a function $\lambda:\Gamma\to \Gamma$ and a non-trivial abelian group $K$, the generalized shift $\sigma_\lambda:K^\Gamma\to K^\Gamma$ is defined by $(x_i)_{i\in \Gamma}\mapsto (x_{\lambda(i)})_{i\in\Gamma}$. In this paper we compute the algebraic entropy of $\sigma_\lambda$; it is either zero or infinite, depending exclusively on the properties of $\lambda$.