Nonwandering points of monotone local dendrite maps revisited. (English) Zbl 1403.37023
Summary: Let \(X\) be a local dendrite and let \(f : X \rightarrow X\) be a monotone map. Denote by \(P(f)\) and \(\Omega(f)\) the sets of periodic points and nonwandering points of \(f\), respectively. We show that \(\Omega(f) = \overline{P(f)}\), whenever \(P(f)\) is nonempty and \(\Omega(f)\) is the unique minimal set included in a circle which is either a Cantor set or a circle, whenever \(P(f)\) is empty. In the case where the set of endpoints of \(X\) is countable, we show that \(\Omega(f) = P(f)\) whenever \(P(f)\) is nonempty.
MSC:
| 37B20 | Notions of recurrence and recurrent behavior in topological dynamical systems |
| 37B45 | Continua theory in dynamics |
| 54H20 | Topological dynamics (MSC2010) |
References:
| [1] | Abdelli, H., ω-limit sets for monotone local dendrite maps, Chaos Solitons Fractals, 71, 66-72, (2015) · Zbl 1352.37123 |
| [2] | Abdelli, H.; Marzougui, H., Invariant sets for monotone local dendrites, Int. J. Bifurc. Chaos Appl. Sci. Eng., 26, (2016) · Zbl 1347.37051 |
| [3] | Abouda, H.; Naghmouchi, I., Monotone maps on dendrites and their induced maps, Topol. Appl., 204, 121-134, (2016) · Zbl 1353.37014 |
| [4] | Arevalo, D.; Charatonik, W. J.; Covarrubias, P. P.; Simon, L., Dendrites with a closed set of end points, Topol. Appl., 115, 1-17, (2001) · Zbl 0979.54035 |
| [5] | Katznelson, J. Auslander. Y., Continuous maps of the circle without periodic points, Isr. J. Math., 32, 375-381, (1979) · Zbl 0442.54011 |
| [6] | Block, L.; Cover, E.; Merlvey, I.; Nitecki, Z., Homoclinic and nonwandering point for maps of the circle, Ergod. Theory Dyn. Syst., 3, 521-532, (1983) · Zbl 0523.58030 |
| [7] | Blokh, A., On the limit behaviour of one-dimensional dynamical systems, Russ. Math. Surv., 37, 157-158, (1982) · Zbl 0511.28014 |
| [8] | Blokh, L., Continuous maps of the interval with finite nonwandering set, Trans. Am. Math. Soc., 240, 221-230, (1978) · Zbl 0389.54006 |
| [9] | Coven, E. M.; Nitecki, Z., Non-wandering sets of the powers of maps of the interval, Ergod. Theory Dyn. Syst., 1, 9-31, (1981) · Zbl 0477.58031 |
| [10] | Efremova, L. S.; Makhrova, E. N., The dynamics of monotone maps of dendrites, Sb. Math., 192, 807-821, (2001) · Zbl 1008.37009 |
| [11] | Huang, W.; Ye, X., Non-wandering sets of the powers of maps of a tree, Sci. China Ser. A, 44, 31-39, (2001) · Zbl 1019.37023 |
| [12] | Kuratowski, K., Topology, vol. 2, (1968), Academic Press New York · Zbl 1444.55001 |
| [13] | Mai, J.; Sun, T., The w-limit set of a graph map, Topol. Appl., 154, 2306-2311, (2007) · Zbl 1115.37044 |
| [14] | Mai, J.; Zhang, G.; Sun, T., Recurrent points and non-wandering points of graph maps, J. Math. Anal. Appl., 383, 553-559, (2011) · Zbl 1226.37024 |
| [15] | Makhrova, E. N.; Vaniukova, K. S., On the set of non-wandering points of monotone maps on local dendrites, NOMA’15 International Workshop on Nonlinear Maps and Applications, J. Phys. Conf. Ser., 692, (2016) |
| [16] | Mai, J. H.; Shao, S., The structure of graph maps without periodic points, Topol. Appl., 154, 2714-2728, (2007) · Zbl 1118.37023 |
| [17] | Mai, J. H.; Shao, S., Spaces of ω-limit sets of graph maps, Fundam. Math., 196, 91-100, (2007) · Zbl 1128.37027 |
| [18] | Nadler, S. B., Continuum theory: an introduction, Monogr. Textb. Pure Appl. Math., vol. 158, (1992), Marcel Dekker, Inc. New York · Zbl 0757.54009 |
| [19] | Naghmouchi, I., Dynamical properties of monotone dendrite maps, Topol. Appl., 159, 144-149, (2012) · Zbl 1242.37011 |
| [20] | Naghmouchi, I., Dynamics of homeomorphisms of regular curves, (2018) |
| [21] | Nikiel, J., A characterisation of dendroids with uncountably many end-points in the classical sense, Houst. J. Math., 9, 421-432, (1983) · Zbl 0537.54029 |
| [22] | Nitecki, Z., Maps of the interval with closed periodic set, Proc. Am. Math. Soc., 85, 542-546, (1982) |
| [23] | Sun, T. X.; Xi, H. J., Non-wandering sets of the powers of dendrite maps, Acta Math. Sin., Engl. Ser., 33, 3, 449-454, (2017) · Zbl 1364.37022 |
| [24] | Sun, T.; He, Q.; Liu, J.; Tao, C.; Xi, H., Non-wandering sets for dendrite maps, Qual. Theory Dyn. Syst., 14, 1, 101-108, (2015) · Zbl 1376.37087 |
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