Periodic points and transitivity on dendrites

We study relations between transitivity, mixing and periodic points on dendrites. We prove that, when there is a point with dense orbit which is a cutpoint, periodic points are dense and there is a terminal periodic decomposition. We also show that it is possible that all periodic points except one...

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Bibliographic Details
Published in:Ergodic theory and dynamical systems 2017-10, Vol.37 (7), p.2017-2033
Main Authors: ACOSTA, GERARDO, HERNÁNDEZ-GUTIÉRREZ, RODRIGO, NAGHMOUCHI, ISSAM, OPROCHA, PIOTR
Format: Article
Language:English
Subjects:
Density
Graphs
Hyperspaces
Orbits
Topology
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Summary:We study relations between transitivity, mixing and periodic points on dendrites. We prove that, when there is a point with dense orbit which is a cutpoint, periodic points are dense and there is a terminal periodic decomposition. We also show that it is possible that all periodic points except one (and points with dense orbit) are contained in the (dense) set of endpoints. It is also possible that a dynamical system is transitive but there is a unique periodic point which, in fact, is the unique fixed point. We also prove that on almost meshed continua (a class of continua containing topological graphs and dendrites with closed or countable set of endpoints), periodic points are dense if and only if they are dense for the map induced on the hyperspace of all non-empty compact subsets.
ISSN:0143-3857
1469-4417
DOI:10.1017/etds.2015.137