Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-27T03:55:49.511Z Has data issue: false hasContentIssue false
  • English
  • Français

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 [email protected]
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 [email protected]
ISSAM NAGHMOUCHI
Affiliation:
University of Carthage, Faculty of Sciences of Bizerte, Department of Mathematics, Jarzouna, 7021, Tunisia email [email protected], [email protected]
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 [email protected]
Get access Rights & Permissions [Opens in a new window]

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.

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alsedá, Ll., del Ro, M. A. and Rodríguez, J. A.. A splitting theorem for transitive maps. J. Math. Anal. Appl. 232 (1999), 359–375.Google Scholar
Akin, E. and Kolyada, S.. Li–Yorke sensitivity. Nonlinearity 16 (2003), 1421–1433.Google Scholar
Alseda, L., Llibre, J. and Misiurewicz, M.. Combinatorial Dynamics and Entropy in Dimension One (Advanced Series in Nonlinear Dynamics, 5) , 2nd edn. World Scientific, River Edge, NJ, 2000.CrossRefGoogle Scholar
Banks, J.. Regular periodic decompositions for topologically transitive maps. Ergod. Th. & Dynam. Sys. 17 (1997), 505–529.CrossRefGoogle Scholar
Banks, J.. Chaos for induced hyperspace maps. Chaos, Solitons and Fractals 25 (2005), 681–685.Google Scholar
Banks, J., Brooks, J., Cairns, G., Davis, G. and Stacey, P.. On Devaney’s definition of chaos. Amer. Math. Monthly 99 (1992), 332–334.Google Scholar
Baldwin, S.. Entropy estimates for transitive maps on trees. Topology 40(3) (2001), 551–569.Google Scholar
Barge, M. and Martin, J.. Chaos, periodicity and snakelike continua. Trans. Amer. Math. Soc. 28 (1985), 355–365.CrossRefGoogle Scholar
Blanchard, F.. Topological chaos: what may this mean? J. Difference Equ. Appl. 15 (2009), 23–46.Google Scholar
Blokh, A. M.. On the connection between entropy and transitivity for one-dimensional mappings. Russian Math. Surveys 42 (1987), 165–166.Google Scholar
Blokh, A. M.. Dynamical systems on one-dimensional branched manifolds I (in Russian). Teor. Funktsii Funktsional. Anal. i Prilozhen. No. 46 (1986), 8–18; Engl. transl. J. Soviet Math. 48 (1990), 500–508.Google Scholar
Blokh, A. M.. Recurrent and periodic points in dendritic Julia sets. Proc. Amer. Math. Soc. 141 (2013), 3587–3599.Google Scholar
Block, L. S. and Coppel, W. A.. Dynamics in One Dimension (Lecture Notes in Mathematics, 1513) . Springer, Berlin, 1992.CrossRefGoogle Scholar
Charatonik, J. and Charatonik, W.. Dendrites (Aportaciones Mathematicas: Comunicaciones, 22) . Sociedad Matemática Mexicana, México, 1998, pp. 227–253.Google Scholar
Coven, E. M. and Hedlund, G. A.. P̄ = R̄ for maps of the interval. Proc. Amer. Math. Soc. 79 (1980), 316–318.Google Scholar
Devaney, R. L.. An Introduction to Chaotic Dynamical Systems (Studies in Nonlinearity) . Reprint of the second (1989) edition. Westview Press, Boulder, CO, 2003.Google Scholar
Dirbák, M., Snoha, L’. and Špitalský, V.. Minimality, transitivity, mixing and topological entropy on spaces with a free interval. Ergod. Th. & Dynam. Sys. 33 (2013), 1786–1812.Google Scholar
Furstenberg, H.. Disjointness in ergodic theory, minimal sets and a problem in diophantine approximation. Math. Syst. Theory 1 (1967), 1–49.Google Scholar
Gengrong, Z., Fanping, Z. and Xinhe, L.. Devaney’s chaotic on induced maps of hyperspace. Chaos Solitons Fractals 27 (2006), 471–475.Google Scholar
Glasner, E. and Weiss, B.. Sensitive dependence on initial conditions. Nonlinearity 6 (1993), 1067–1075.Google Scholar
Harańczyk, G., Kwietniak, D. and Oprocha, P.. A note on transitivity, sensitivity and chaos for graph maps. J. Difference Equ. Appl. 17 (2011), 1549–1553.CrossRefGoogle Scholar
Harańczyk, G., Kwietniak, D. and Oprocha, P.. Topological structure and entropy of mixing graph maps. Ergod. Th. & Dynam. Sys. 34 (2014), 1587–1614.CrossRefGoogle Scholar
Hernández-Gutiérrez, R., Illanes, A. and Martínez-de-la-Vega, V.. Uniqueness of hyperspaces for Peano continua. Rocky Mountain J. Math. 43 (2013), 1583–1624.Google Scholar
Hoehn, L. and Mouron, C.. Hierarchies of chaotic maps on continua. Ergod. Th. & Dynam. Sys. 34 (2014), 1897–1913.Google Scholar
Illanes, A. and Nadler, S. B. Jr. Hyperspaces Fundamentals and Recent Advances (Monographs and Textbooks in Pure and Applied Mathematics, 216) . Marcel Dekker, New York, 1999.Google Scholar
Kato, H.. A note on periodic points and recurrent points of maps of dendrites. Bull. Aust. Math. Soc. 51 (1995), 459–461.Google Scholar
Kočan, Z., Kurková, V. and Málek, M.. On the existence of maximal 𝜔-limit sets for dendrite maps. Commun. Nonlinear Sci. Numer. Simul. 17 (2012), 3169–3176.Google Scholar
Kolyada, S. and Snoha, L’.. Some aspects of topological transitivity—a survey. Grazer Math. Ber. 334 (1997), 3–35.Google Scholar
Mai, J. H. and Shi, E.. R = P for maps of dendrites X with Card(End(X)) < c . Int. J. Bifur. Chaos 19(4) (2009), 1391–1396.Google Scholar
Mai, J. H. and Shao, S.. Spaces of 𝜔-limit sets of graph maps. Fund. Math. 196 (2007), 91–100.Google Scholar
Mai, J. H. and Shao, S.. R = R ∪ P for graph maps. J. Math. Anal. Appl. 350 (2009), 9–11.CrossRefGoogle Scholar
Nadler, S. B. Jr. Continuum Theory: An Introduction. Marcel Dekker, New York, 1992.Google Scholar
Sharkovsky, A. N.. Coexistence of cycles of a continuous map of a line into itself (in Russian). Ukrän. Mat. Zh. 16 (1964), 61–71; Engl. transl. Int. J. Bifur. Chaos Appl. Sci. Engrg. 5 (1995), 1263–1273.Google Scholar
Sharkovsky, A. N.. Fixed points and the center of a continuous mapping of the line into itself. (Ukrainian) Dopov. Akad. Nauk Ukr. RSR 1964 (1964), 865–868.Google Scholar