×

Chaotic points of multifunctions. (English) Zbl 1504.37023

Summary: In this paper we will consider various kinds of chaotic points of multifunctions and show their application to the theory of infinite topological games.

MSC:

37B40 Topological entropy
37B02 Dynamics in general topological spaces
91A44 Games involving topology, set theory, or logic
91A06 \(n\)-person games, \(n>2\)
91A25 Dynamic games
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] L. Alsedà, J. Guaschi, J. Los, F. Mañosas and P. Mumbrú, Canonical representatives for patterns of tree maps, Topology 36 (1997), no. 5, 1123-1153. · Zbl 0887.58012
[2] L. Alsedà, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One, Advanced Series in Nonlinear Dynamics 5, World Scientific, River Edge, NJ, 1993. · Zbl 0843.58034
[3] G. Beer, The approximation of upper semicontinuous multifunctions by step multifunctions, Pacific J. Math. 87 (1980), no. 1, 11-19. · Zbl 0442.54014
[4] F. Blanchard, Topological chaos: what may this mean?, J. Difference Equ. Appl. 15 (2009), no. 1, 23-46. · Zbl 1253.37013 · doi:10.1080/10236190802385355
[5] J. Cao and W. B. Moors, A survey on topological games and their applications in analysis, RACSAM. Rev. R. Acad. Cienc. Exacts Fís. Nat. Ser. A. Mat. 100 (2006), no. 1-2, 39-49. · Zbl 1114.91024
[6] M. Čiklová, Dynamical systems generated by functions with connected \(G_{\delta}\) graphs, Real Anal. Exchange 30 (2004/05), no. 2, 617-637.
[7] J. Y. Chow, L. Seifert, R. Hérault, S. J. Y. Chia and M. C. Y. Lee, A dynamical system perspective to understanding badminton singles game play, Hum. Mov. Sci. 33 (2014), 70-84.
[8] J. Dvořáková, Chaos in nonautonomous discrete dynamical systems, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), no. 12, 4649-4652. · Zbl 1262.37013
[9] A. Fedeli, On chaotic set-valued discrete dynamical systems, Chaos Solitons Fractals 23 (2005), no. 4, 1381-1384. · Zbl 1079.37021
[10] R. Gu, Kato’s chaos in set-valued discrete systems, Chaos Solitons Fractals 31 (2007), no. 3, 765-771. · Zbl 1140.37305
[11] M. Kamenskii, V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, De Gruyter Series in Nonlinear Analysis and Applications 7, Walter de Gruyter, Berlin, 2001. · Zbl 0988.34001
[12] E. Korczak-Kubiak, A. Loranty and R. J. Pawlak, On focusing entropy at a point, Taiwanese J. Math. 20 (2016), no. 5, 1117-1137. · Zbl 1357.37028
[13] D. Kwietniak and P. Oprocha, Topological entropy and chaos for maps induced on hyperspaces, Chaos Solitons Fractals 33 (2007), no. 1, 76-86. · Zbl 1152.37306 · doi:10.1016/j.chaos.2005.12.033
[14] J. M. Lee, Introduction to Topological Manifolds, Second edition, Graduate Texts in Mathematics 202, Springer, New York, 2011. · Zbl 1209.57001
[15] J. Li and X. D. Ye, Recent development of chaos theory in topological dynamics, Acta Math. Sin. (Engl. Ser.) 32 (2016), no. 1, 83-114. · Zbl 1335.54039 · doi:10.1007/s10114-015-4574-0
[16] T. Y. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), no. 10, 985-992. · Zbl 0351.92021 · doi:10.2307/2318254
[17] X. Ma, B. Hou and G. Liao, Chaos in hyperspace system, Chaos Solitons Fractals 40 (2009), no. 2, 653-660. · Zbl 1197.37016
[18] M. Misiurewicz, Horseshoes for mappings of the interval, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), no. 2, 167-169. · Zbl 0459.54031
[19] J. von Neumann and O. Morgenstern, Theory of Games and Economic Behavior, Princeton University Press, Princeton, N.J., 1944. · Zbl 0063.05930
[20] J. C. Oxtoby, Measure and Category: A survey of the analogies between topological and measure spaces, Graduate Texts in Mathematics 2, Springer-Verlag, New York, 1971. · Zbl 0217.09201
[21] R. J. Pawlak, Distortion of dynamical systems in the context of focusing the chaos around the point, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 28 (2018), no. 1, 1850006, 13 pp. · Zbl 1384.37020
[22] R. J. Pawlak and A. Loranty, The generalized entropy in the generalized topological spaces, Topology Appl. 159 (2012), no. 7, 1734-1742. · Zbl 1243.54040 · doi:10.1016/j.topol.2011.05.043
[23] ____, On the local aspects of distributional chaos, Chaos 29 (2019), no. 1, 013104, 9 pp. · Zbl 1442.37055
[24] J. P. Revalski, The Banach-Mazur Game: History and Recent Developments, Institute of Mathematics and Informatics Bulgarian Academy of Sciences, Seminar notes, Pointe-a-Pitre, Guadeloupe, France, 2003-2004, avaliable at http://www1.univ-ag.fr/aoc/activite/revalski/.
[25] H. Román-Flores, A note on transitivity in set-valued discrete systems, Chaos Solitons Fractals 17 (2003), no. 1, 99-104. · Zbl 1098.37008
[26] S. Ruette, Chaos on the Interval, University Lecture Series 67, American Mathematical Society, Providence, RI, 2017. · Zbl 1417.37029
[27] B. Schweizer and J. Smítal, Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans. Amer. Math. Soc. 344 (1994), no. 2, 737-754. · Zbl 0812.58062
[28] R. Telgársky, Topological games: on the 50th anniversary of the Banach-Mazur game, Rocky Mountain J. Math. 17 (1987), no. 2, 227-276. · Zbl 0619.90110
[29] K. S. Wong and Z. Salleh, Topologically transitive and mixing properties of set-valued dynamical systems, Abstr. Appl. Anal. 2021, Art. ID 5541105, 7 pp. · Zbl 1482.37011
[30] X. Ye and G. Zhang, Entropy points and applications, Trans. Amer. Math. Soc. 359 (2007), no. 12, 6167-6186. · Zbl 1121.37020 · doi:10.1090/s0002-9947-07-04357-7
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.