We study the dynamics of the action of an automaton group on the set of infinite words, and more precisely the discontinuous points of the map which associates to a point its set of stabilizers — the singular points. We show that, for any Mealy automaton, the set of singular points has measure zero. Then we focus our attention on several classes of automata. We characterize those contracting automata generating groups without singular points, and apply this characterization to the Basilica group. We prove that potential examples of reversible automata generating infinite groups without singular points are necessarily bireversible. We also provide some conditions for such examples to exist. Finally, we study some dynamical properties of the Schreier graphs in the boundary.

中文翻译：

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双可逆和收缩自动机群的边界动力学

我们研究了自动机群对无限词集的作用的动力学，更准确地说，是地图的不连续点与它的一组稳定器（奇异点）相关联。我们证明，对于任何 Mealy 自动机，奇异点集的测度为零。然后我们将注意力集中在几类自动机上。我们描述了那些没有奇异点的收缩自动机生成组，并将这种描述应用于大教堂组。我们证明了生成没有奇异点的无限群的可逆自动机的潜在例子必然是双可逆的。我们还为此类示例的存在提供了一些条件。最后，我们研究了边界上的 Schreier 图的一些动力学性质。