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The scale and tidy subgroups for endomorphisms of totally disconnected locally compact groups. (English) Zbl 1308.22002
Let \(G\) be a totally disconnected locally compact group and \(\alpha:G\to G\) a continuous endomorphism. In this paper \(\alpha\) is studied through its action on the family \(\mathcal B(G)\) of all compact open subgroups of \(G\), which forms a base of the neighborhoods of \(e_G\) by a theorem of van Dantzig. Indeed, the scale of \(\alpha\) is \[ s(\alpha)=\min\{[\alpha(U):\alpha(U)\cap U]:U\in\mathcal B(G)\}. \] This extends the same notion given by the author in previous papers, first for inner automorphisms and then for topological automorphisms of \(G\). Note that always \(s(\alpha)=1\) when \(G\) is compact.
A subgroup \(U\in\mathcal B(G)\) is minimizing for \(\alpha\) if the minimum in the definition of the scale is attained at \(U\), that is, \(s(\alpha)=[\alpha(U):\alpha(U)\cap U]\). Appropriately extending the so-called tidying procedure to the case of continuous endomorphisms, the author generalizes the structure theorem for minimizing subgroups, as follows. For \(U\in\mathcal B(G)\), let \[ U_+=\{x\in U:\exists \{x_n\}_{n\in\mathbb N}\subseteq U, x_0=x, \alpha(x_{n+1})=x_n \forall n\in\mathbb N\}, \]
\[ U_{++}=\bigcup_{n\in\mathbb N}\alpha^n(U_+)\;\text{and}\;U_-=\bigcap_{n\in\mathbb N}\alpha^{-n}(U). \] Then \(U\in\mathcal B(G)\) is minimizing for \(\alpha\) if and only if \(U=U_+U_-\), the subgroup \(U_{++}\) is closed and the sequence \(\{[\alpha^{n+1}(U_+):\alpha^n(U_+)]\}_{n\in\mathbb N}\) is constant.
Several properties of the scale, known for topological automorphisms, are extended to the general case. The same also occurs for the definition of various subgroups related to the scale function.

MSC:
22D05 General properties and structure of locally compact groups
54H11 Topological groups (topological aspects)
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