The Scale and Tidy Subgroups for Endomorphisms of Totally Disconnected Locally Compact Groups

The scale of an endomorphism, $\alpha$, of a totally disconnected, locally compact group $G$ is the minimum index $[\alpha(U) : \alpha(U)\cap U]$, for $U$ a compact, open subgroup of $G$. A structural characterization of subgroups at which the minimum is attained is established. This characterization extends the notion of subgroup tidy for $\alpha$ from previously understood case when $\alpha$ is an automorphism to the case when $\alpha$ is merely an endomorphism.