Anahı ROJAS, Franco BARRAGAN, Sergıo MACÍAS

Conceptions on topological transitivity in products and symmetric products

Having a finite number of topological spaces Xi and functions fi : Xi → Xi , and considering one of the following classes of functions: exact, transitive, strongly transitive, totally transitive, orbit-transitive, strictly orbittransitive, ω-transitive, mixing, weakly mixing, mild mixing, chaotic, exactly Devaney chaotic, minimal, backward minimal, totally minimal, T T++ , scattering, Touhey or an F -system, in this paper, we study dynamical behaviors of the systems Xi, fi , ∏ Xi, ∏ fi , Fn ∏ Xi , Fn ∏ fi , and Fn Xi , Fn fi .

Keywords:

Topological transitivity, symmetric products, dynamical systems,

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[1] Akin E, Carlson JD. Conceptions of topological transitivity. Topology and its Applications 2012; 159 (12): 2815- 2830. doi: 10.1016/j.topol.2012.04.016
[2] Akin E, Glasner E. Residual properties and almost equicontinuity. Journal d’Analyse Mathématique 2001; 84 (1): 243-286. doi: 10.1007/bf02788112
[3] Barragán F. Induced maps on n-fold symmetric product suspensions. Topology and its Applications 2011; 158 (10): 1192-1205. doi: 10.1016/j.topol.2011.04.006
[4] Barragán F, Macías S, Rojas A. Conceptions of topological transitivity on symmetric products. Submitted for publication.
[5] Bilokopytov E, Kolyada SF. Transitive maps on topological spaces. Ukrainian Mathematical Journal 2014; 65 (9): 1293-1318. doi: 10.1007/s11253-014-0860-8
[6] Mangang KB. Li-Yorke chaos in product dynamical systems. Advances in Dynamical Systems and Applications 2017; 12 (1): 81-88.
[7] Borsuk K, Ulam S. On symmetric products of topological spaces. Bulletin of the American Mathematical Society 1931; 37 (12): 875-882. doi: 10.1090/s0002-9904-1931-05290-3
[8] Degirmenci N, Kocak S. Chaos in product maps. Turkish Journal of Mathematics 2010; 34 (4): 593-600. doi: ˘ 10.3906/mat-0807-51
[9] Hausdorff F. Grundzüge der Mengenlehre. Leipzig, 1914 (in German).
[10] Higuera G, Illanes A. Induced mappings on symmetric products. Topology Proceedings 2011; 37: 367-401.
[11] Hou B, Liao G, Liu H. Sensitivity for set-valued maps induced by M-systems. Chaos, Solitons & Fractals 2008; 38 (4): 1075-1080. doi: 10.1016/j.chaos.2007.02.015
[12] Illanes A, Nadler SB Jr. Hyperspaces: fundamentals and recent advances. Neww York, NY, USA: Marcel Dekker Inc., 1999.
[13] Li R, Zhou X. A note on chaos in product maps. Turkish Journal of Mathematics 2013; 37 (4): 665-675. doi: 10.3906/mat-1101-71
[14] Macías S. Topics on continua. 2nd Ed. Cham, Switzerland: Springer, 2018.
[15] Mai JH, Sun WH. Transitivities of maps of general topological spaces. Topology and its Applications 2010; 157 (5): 946-953. doi: 10.1016/j.topol.2009.12.011
[16] Munkres JR. Topology. Second Edition. Upper Saddle River, NJ, USA: Prentice Hall, 2000.
[17] Nadler SB Jr. Hyperspaces of sets. Monographs and Textbooks in Pure and Applied Math., Vol. 49. NY: Marcel Dekker, 1978.
[18] Touhey P. Yet another definition of chaos. The American Mathematical Monthly 1997; 104 (5): 411-414. doi: 10.1080/00029890.1997.11990658
[19] Vietoris L. Bereiche zweiter Ordnung. Monatshefte Für Mathematik Und Physik 1922; 32 (1): 258-280. doi: 10.1007/bf01696886
[20] Wu X, Wang J, Chen G. F -sensitivity and multi-sensitivity of hyperspatial dynamical systems. Journal of Mathematical Analysis and Applications 2015; 429 (1): 16-26. doi: 10.1016/j.jmaa.2015.04.009
[21] Wu X, Zhu P. Devaney chaos and Li-Yorke sensitivity for product systems. Studia Scientiarum Mathematicarum Hungarica 2012; 49 (4): 538-548. doi: 10.1556/sscmath.49.2012.4.1226

ISSN: 1300-0098
Yayın Aralığı: Yılda 6 Sayı
Yayıncı: TÜBİTAK

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