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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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