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Discontinuity, Nonlinearity, and Complexity 9(4) (2020) 541--552 | DOI:10.5890/DNC.2020.12.007

E. N. Makhrova

Lobachevsky State University of Nizhni Novgorod, Gagarin ave., 23, Nizhni Novgorod, 603095, Russia

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Abstract

Let $X$ be a dendrite, $f:X\to X$ be a monotone map. In the article the relationship between a structure of a dendrite $X$ and dynamical properties of $f$ is studied. Namely the relation between a structure of $X$ and a structure of the sets of periodic points of $f$, non-wandering points of $f$, $\omega$-limit sets of trajectories is established. The structure of dendrites on which there exist monotone pointwise chain recurrent maps is characterized. The structure of dendrites on which there exist monotone maps with homoclinic points is described.

References

1. [1]	Kuratovski, K. (1968), Topology, vol. 2, Academic Press, New York.

2. [2]	Charatonik, J.J. (1991), Monotone mappings of universal dendrites, Topology and its Application, 38, 163-187.

3. [3]	Charatonik, J.J. and Charatonik, W.J. (1997), Monotone homogeneity of dendrites, Math. Univ. Carolinae, 38, 361-370.

4. [4]	Efremova, L.S. and Makhrova, E.N. (2001), The dynamics of monotone maps of dendrites, Matem. Sbornik, 192, 15-29.

5. [5]	Charatonik, J.J. (2004), Dendrites and monotone mappings, Mathematica Pannonica, 15(1), 115-125.

6. [6]	Wright, E.P. (2009), Monotone preimages of dendrites, Topology and its Applications, 156, 663-668.

7. [7]	Naghmouchi, I. (2011), Dynamics of monotone graph, dendrite and dendroid maps, Int. J. Bifurcation and Chaos, 21, 3205-3215.

8. [8]	Naghmouchi, I. (2012), Dynamical properties of monotone dendrite maps, Topology and its Applications, 159, 144-149.

9. [9]	Abouda, H. and Nagmouchi, I. (2016), Monotone maps on dendrites and their induced maps, Topology and its Application, 204, 121-134.

10. [10]	Martinez-de-la-Vega, V. and Mouron, C. (2016), Monotone classes of dendrites, Canadian Journal of Ma\-thematics, 68(3), 675-697.

11. [11]	Gordh, G.R. and Lum, L. (1976), Monotone retracts and some characterizations of dendrites, Amer. Math. Soc., 59, 1561-1568.

12. [12]	Nadler, S.B. (1992), Continuum Theory, Marcel Dekker, New York.

13. [13]	Kocan, Zd., Korneka-Kurkova, V., and Malek, M. (2011), Entropy, horseshoes and homoclinic trajectories on trees, graphs and dendrites, Ergodic theory $\&$ Dynam. Sys., 31, 165-175.

14. [14]	Ar$\rm\acute{e}$valo, D., Charatonic, W.J., Covarrubias, P.P., and Sim$\rm\acute{o}$n, L. (2001), Dendrites with a closed set of end points, Topology and its Applications, 115, 1-17.

15. [15]	Peitgen, H.O. and Richter, P. (1986), The Beauty of Fractals, Springer-Verlag, Heidelberg.

16. [16]	Agronsky, S.J. and Ceder, J.G. (1991/1992), What sets can be $\omega$-limit sets in $E^n$, Real Analysis Exchange, 17, 97-109.

17. [17]	Balibrea, F. and Garc$\rm\acute{i}$a-Guirao, J.L. (2005), Continua with empty interior as $\omega$-limit sets, Applied General Topology, 6, 195-205.

18. [18]	Gollo, L., Kinouchi, O., and Copelli, M. (2009), Active dendrites enhance neuronal dynamic range, Computational Biology, \textbf(5), e1000402.

19. [19]	Ayres, W.L. (1930), Some generalizations of the Scherrer fixed-point theorem, Fund. Math., 16, 332-336.

20. [20]	Mai, J. and Shi, E. (2009), $\overline{R}=\overline{P}$ for maps of dendrites $X$ with $Card(End(X))

21. [21]	Sibirskii, K.S. (1975), Introduction to topological dynamics, Springer Netherlands.

22. [22]	Zhang, G. and Zeng, F. (2004), Pointwise chain recurrent maps of the tree, Bull. Austral. Math. Soc., 69, 63-68.

23. [23]	Block, L. (1978), Homoclinic points of mappings of the interval, Proceedings of the AMS, 72, 576-580.