Group action with finite orbits on local dendrites. (English) Zbl 1489.37024
A dendrite is a locally connected metrizable continuum none of whose subsets is homeomorphic to the circle. A dendrite is a metrizable dendron – a continuum \(X\) whose every pair of distinct points \(x, y\) can be separated by a third point \(z\) in \(X\). A local dendron is a continuum whose every point has a neighbourhood which is a dendron. If \(G\) is a group acting continuously on a continuum \(X\), a closed nonempty subset \(A\) of \(X\) is called minimal if \(A\) is \(G\)-invariant and minimal in inclusion sense. The minimal set can only be a circle, a Cantor set, or a finite set (see [L. A. Beklaryan, Math. Notes 71, No. 3, 305–315 (2002; Zbl 1024.22012); translation from Mat. Zametki 71, No. 3, 334–347 (2002)]). A classification of the minimal sets of the group action on dendrites and local dendrites was published in [H. Marzougui and I. Naghmouchi, Proc. Am. Math. Soc. 144, No. 10, 4413–4425 (2016; Zbl 1418.37018); Math. Z. 293, No. 3–4, 1057–1070 (2019; Zbl 1436.37011)]. Furthermore, E. Shi and X. Ye [Proc. Am. Math. Soc. 145, No. 1, 177–184 (2017; Zbl 1368.37019)] proved that countable amenable group actions on dendrites have periodic point of order not exceeding 2. They proved also that the induced action of any amenable group on any minimal set in a dendrite is equicontinuous (see [E. Shi and X. Ye, Contemp. Math. 744, 175–180 (2020; Zbl 1446.37015)]).
The present paper generalizes this result in the following way: the action with finite orbit of a group \(G\) on a dendrite \(X\), or a local dendrite \(X\), restricted to an infinite minimal set of \(X\), is equicontinuous. The authors also consider a class of non-amenable groups whose action on a dendrite has a finite orbit. It follows that, in this case, the infinite minimal sets are Cantor sets (see [H. Marzougui and the second author, Proc. Am. Math. Soc. 144, No. 10, 4413–4425 (2016; Zbl 1418.37018)]).
The present paper generalizes this result in the following way: the action with finite orbit of a group \(G\) on a dendrite \(X\), or a local dendrite \(X\), restricted to an infinite minimal set of \(X\), is equicontinuous. The authors also consider a class of non-amenable groups whose action on a dendrite has a finite orbit. It follows that, in this case, the infinite minimal sets are Cantor sets (see [H. Marzougui and the second author, Proc. Am. Math. Soc. 144, No. 10, 4413–4425 (2016; Zbl 1418.37018)]).
Reviewer: K. T. Hallenbeck (Chester)
MSC:
| 37B45 | Continua theory in dynamics |
| 37B05 | Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) |
| 37B20 | Notions of recurrence and recurrent behavior in topological dynamical systems |
| 54H15 | Transformation groups and semigroups (topological aspects) |
| 54F50 | Topological spaces of dimension \(\leq 1\); curves, dendrites |
| 22A25 | Representations of general topological groups and semigroups |
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