Let $X$ be a locally compact zero-dimensional space, let $S$ be an equicontinuous set of homeomorphisms such that $1 \in S = S^{-1}$, and suppose that $\overline{Gx}$ is compact for each $x \in X$, where $G = \langle S \rangle$. We show in this setting that a number of conditions are equivalent: (a) $G$ acts minimally on the closure of each orbit; (b) the orbit closure relation is closed; (c) for every compact open subset $U$ of $X$, there is $F \subseteq G$ finite such that $\bigcap_{g \in F}g(U)$ is $G$-invariant. All of these are equivalent to a notion of recurrence, which is a variation on a concept of Auslander–Glasner–Weiss. It follows in particular that the action is distal if and only if it is equicontinuous.

中文翻译：

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等维连续性，轨道闭合和零空间上群动作的不变紧开集

令$ X $为局部紧凑的零维空间，令$ S $为等连续的同胚集合，使得S中的$ 1 = S ^ {-1} $，并假定$ \ overline {Gx} $是紧凑的对于X $中的每个$ x \，其中$ G = \ langle S \ rangle $。我们在这种情况下证明了许多条件是等效的：（a）$ G $对每个轨道的闭合作用最小；（b）轨道关闭关系已关闭；（c）对于$ X $的每个紧凑开放子集$ U $，都有$ F \ subseteq G $有限元，使得$ \ bigcap_ {g \ in F} g（U）$是$ G $不变的。所有这些都等同于复发概念，这是对Auslander–Glasner–Weiss概念的变体。特别是，当且仅当动作是等连续的时，动作才是远端的。