Group action on local dendrites
type: Item
instance of: scholarly article
title: Group action on local dendrites
zbMATH Open document ID: 1401.37017
author: Aymen Haj Salem
author: Hawete Hattab
published in: Topology and its Applications
publication date: 2018-08-24T00:00:00Z
review text: 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.
reviewed by: Mihai Turinici
Mathematics Subject Classification ID: 37B05
Mathematics Subject Classification ID: 37B20
Mathematics Subject Classification ID: 37B45
Mathematics Subject Classification ID: 54H20
zbMATH DE Number: 6924617
zbMATH Keywords: local dendrite
zbMATH Keywords: group action
zbMATH Keywords: minimal set
zbMATH Keywords: finitely generated group
MaRDI profile type: Publication
full work available at URL: j.topol.2018.08.002
OpenAlex ID: W2885773488
cites work: \(\omega\)-limit sets for monotone local dendrite maps
cites work: Topological size of scrambled sets for local dendrite maps
cites work: Q4040097
cites work: On recurrence in zero dimensional flows
cites work: Q4002072
cites work: Q5841789
cites work: Treelike structures arising from continua and convergence groups
cites work: Dendrites
cites work: Group Actions On R-Trees
cites work: Q5693497
cites work: Distral transformation groups
cites work: Groups acting on the circle.
cites work: Q3228119
cites work: Q4662006
cites work: Pointwise recurrent one-dimensional flows
cites work: Dendrite flows
cites work: The nonexistence of expansive commutative group actions on Peano continua having free dendrites
cites work: Classification of the group actions on the real line and circle
cites work: Minimal sets for group actions on dendrites
cites work: Universal Spaces for R-Trees
cites work: Q4004123
cites work: Q5845014
cites work: Recurrence and almost Periodicity in a Generative Transformation Group
cites work: Minimal group actions on dendrites
cites work: Fixed point properties of nilpotent group actions on 1-arcwise connected continua
cites work: TOPOLOGICAL TRANSITIVITY AND CHAOS OF GROUP ACTIONS ON DENDRITES
cites work: Q5837858
cites work: On dendritic sets
cites work: Recurrence, pointwise almost periodicity and orbit closure relation for flows and foliations
Wikidata QID: Q129394855
DOI: 10.1016/J.TOPOL.2018.08.002
description: scientific article; zbMATH DE number 6924617