Group action on local dendrites
Some questions involving finite generated group actions in local dendrites are considered. The following are the main results. Theorem 1. Let \(G\) be a group acting by homeomorphisms on a local dendrite \(X\). Then, any minimal set \(M\) of \(G\) has one of the properties (a) \(M\) is a finite orbit; (b) \(M\) is a Cantor set; (c) \(M\) is a circle. Theorem 2. Let \((X,G)\) be a flow, where \(X\) is a local dendrite with countable set of endpoints and different from the circle, and \(G\) is a finitely generated group. Then, the following properties are equivalent: (a) \((G,X)\) is pointwise recurrent; (b) \((G,X)\) is pointwise almost periodic; (c) The orbit closure relation is closed; (d) \((G,X)\) is equicontinuous. Theorem 3. Let \((X,G)\) be a flow, where \(X\) is a local dendrite with countable set of endpoints, and \(G\) is a finitely generated group. Then, the following properties are equivalent: (a) \((G,X)\) is pointwise recurrent; (b) One of the two statements holds: (b-1) \(X=S^1\), and \((G,S^1)\) is a minimal flow conjugate to an isometric flow, or to a finite cover of a proximal flow; (b-2) \((G,X)\) is pointwise periodic. Further aspects are also discussed.
scientific article; zbMATH DE number 6924617
scientific article