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Periodic points and transitivity on dendrites

Published online by Cambridge University Press:  08 March 2016

GERARDO ACOSTA ,
RODRIGO HERNÁNDEZ-GUTIÉRREZ ,
ISSAM NAGHMOUCHI  and
PIOTR OPROCHA
GERARDO ACOSTA
Affiliation:
Instituto de Matemáticas, Universidad Nacional Autónoma de México, Ciudad Universitaria, D.F. 04510, Mexico email gacosta@matem.unam.mx
RODRIGO HERNÁNDEZ-GUTIÉRREZ
Affiliation:
UAP Cuautitlán Izcalli, Universidad Autónoma del Estado de México. Paseos de las Islas S/N, Atlanta 2da. sección. 54740, Cuautitlán Izcalli, México email rodrigo.hdz@gmail.com
ISSAM NAGHMOUCHI
Affiliation:
University of Carthage, Faculty of Sciences of Bizerte, Department of Mathematics, Jarzouna, 7021, Tunisia email issam.naghmouchi@fsb.rnu.tn, issam.nagh@gmail.com
PIOTR OPROCHA
Affiliation:
AGH University of Science and Technology, Faculty of Applied Mathematics, al. A. Mickiewicza 30, 30-059 Kraków, Poland National Supercomputing Centre IT4Innovations, Division of the University of Ostrava, Institute for Research and Applications of Fuzzy Modeling, 30. dubna 22, 70103 Ostrava, Czech Republic email oprocha@agh.edu.pl
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Abstract

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.

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Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 7 , October 2017 , pp. 2017 - 2033
Copyright
© Cambridge University Press, 2016 

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