Linear and metric maps on trees via Markov graphs

Sergiy Kozerenko

Commentationes Mathematicae Universitatis Carolinae (2018)

  • Volume: 59, Issue: 2, page 173-187
  • ISSN: 0010-2628

Abstract

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The main focus of combinatorial dynamics is put on the structure of periodic points (and the corresponding orbits) of topological dynamical systems. The first result in this area is the famous Sharkovsky's theorem which completely describes the coexistence of periods of periodic points for a continuous map from the closed unit interval to itself. One feature of this theorem is that it can be proved using digraphs of a special type (the so-called periodic graphs). In this paper we use Markov graphs (which are the natural generalization of periodic graphs in case of dynamical systems on trees) as a tool to study several classes of maps on trees. The emphasis is put on linear and metric maps.

How to cite

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Kozerenko, Sergiy. "Linear and metric maps on trees via Markov graphs." Commentationes Mathematicae Universitatis Carolinae 59.2 (2018): 173-187. <http://eudml.org/doc/294841>.

@article{Kozerenko2018,
abstract = {The main focus of combinatorial dynamics is put on the structure of periodic points (and the corresponding orbits) of topological dynamical systems. The first result in this area is the famous Sharkovsky's theorem which completely describes the coexistence of periods of periodic points for a continuous map from the closed unit interval to itself. One feature of this theorem is that it can be proved using digraphs of a special type (the so-called periodic graphs). In this paper we use Markov graphs (which are the natural generalization of periodic graphs in case of dynamical systems on trees) as a tool to study several classes of maps on trees. The emphasis is put on linear and metric maps.},
author = {Kozerenko, Sergiy},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Markov graph; Sharkovsky's theorem; maps on trees},
language = {eng},
number = {2},
pages = {173-187},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Linear and metric maps on trees via Markov graphs},
url = {http://eudml.org/doc/294841},
volume = {59},
year = {2018},
}

TY - JOUR
AU - Kozerenko, Sergiy
TI - Linear and metric maps on trees via Markov graphs
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2018
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 59
IS - 2
SP - 173
EP - 187
AB - The main focus of combinatorial dynamics is put on the structure of periodic points (and the corresponding orbits) of topological dynamical systems. The first result in this area is the famous Sharkovsky's theorem which completely describes the coexistence of periods of periodic points for a continuous map from the closed unit interval to itself. One feature of this theorem is that it can be proved using digraphs of a special type (the so-called periodic graphs). In this paper we use Markov graphs (which are the natural generalization of periodic graphs in case of dynamical systems on trees) as a tool to study several classes of maps on trees. The emphasis is put on linear and metric maps.
LA - eng
KW - Markov graph; Sharkovsky's theorem; maps on trees
UR - http://eudml.org/doc/294841
ER -

References

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  1. Bernhardt C., 10.3934/dcds.2006.14.399, Discrete Contin. Dyn. Syst. 14 (2006), no. 3, 399–408. MR2171718DOI10.3934/dcds.2006.14.399
  2. Ho C. W., Morris C., 10.2140/pjm.1981.96.361, Pacific J. Math. 96 (1981), no. 2, 361–370. MR0637977DOI10.2140/pjm.1981.96.361
  3. Kozerenko S., Discrete Markov graphs: loops, fixed points and maps preordering, J. Adv. Math. Stud. 9 (2016), no. 1, 99–109. MR3495337
  4. Kozerenko S., Markov graphs of one-dimensional dynamical systems and their discrete analogues, Rom. J. Math. Comput. Sci. 6 (2016), no. 1, 16–24. MR3503059
  5. Kozerenko S., On disjoint union of M -graphs, Algebra Discrete Math. 24 (2017), no. 2, 262–273. MR3756946
  6. Kozerenko S., On the abstract properties of Markov graphs for maps on trees, Mat. Bilten 41 (2017), no. 2, 5–21. 
  7. Pavlenko V. A., Number of digraphs of periodic points of a continuous mapping of an interval into itself, Ukrain. Math. J. 39 (1987), no. 5, 481–486; translation from Ukrainian Mat. Zh. 39 (1987), no. 5, 592–598 (Russian). MR0916851
  8. Pavlenko V. A., Periodic digraphs and their properties, Ukrain. Math. J. 40 (1988), no. 4, 455–458; translation from Ukrainian Mat. Zh. 40 (1988), no. 4, 528–532 (Russian). MR0957902
  9. Pavlenko V. A., 10.1007/BF01074883, Cybernetics 25 (1989), no. 1, 49–54; translation from Kibernetika (Kiev) 25 (1989), no. 1, 41–44, 133 (Russian). MR0997003DOI10.1007/BF01074883
  10. Sharkovsky A. N., Co-existence of the cycles of a continuous mapping of the line into itself, Ukrain. Math. J. 16 (1964), no. 1, 61–71. MR1415876
  11. Straffin P. D. Jr., 10.1080/0025570X.1978.11976687, Math. Mag. 51 (1978), no. 2, 99–105. MR0498731DOI10.1080/0025570X.1978.11976687

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