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A dynamical version of the Kuratowski–Mycielski theorem and invariant chaotic sets

Published online by Cambridge University Press:  24 January 2018

JIAN LI Open the ORCID record for JIAN LI [Opens in a new window] ,
JIE LÜ  and
YUANFEN XIAO
JIAN LI
Affiliation:
Department of Mathematics, Shantou University, Shantou, 515063, Guangdong, China email lijian09@mail.ustc.edu.cn
JIE LÜ
Affiliation:
School of Mathematics, South China Normal University, Guangzhou 510631, China email ljie@scnu.edu.cn, xiaoyuanfen@foxmail.com
YUANFEN XIAO
Affiliation:
School of Mathematics, South China Normal University, Guangzhou 510631, China email ljie@scnu.edu.cn, xiaoyuanfen@foxmail.com
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Abstract

We establish a dynamical version of the Kuratowski–Mycielski theorem on the existence of ‘large’ invariant dependent sets. We apply this result to the study of invariant chaotic sets in topological dynamical systems, simplify many known results on this topic and also obtain some new results.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 11 , November 2019 , pp. 3089 - 3110
Copyright
© Cambridge University Press, 2018 

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References

Akin, E.. Lectures on Cantor and Mycielski sets for dynamical systems (Chapel Hill Ergodic Theory Workshops, Contemporary Mathematics, 356) . American Mathematical Society, Providence, RI, 2004, pp. 21–79.Google Scholar
Akin, E., Glasner, E., Huang, W., Shao, S. and Ye, X.. Sufficient conditions under which a transitive system is chaotic. Ergod. Th. & Dynam. Sys. 30(5) (2010), 1277–1310.10.1017/S0143385709000753Google Scholar
Balibrea, F., Guirao, J. L. G. and Oprocha, P.. On invariant 𝜀-scrambled sets. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 20(9) (2010), 2925–2935.10.1142/S0218127410027465Google Scholar
Banakh, T. and Zdomskyy, L.. Non-meager free sets for meager relations on Polish spaces. Proc. Amer. Math. Soc. 143(6) (2015), 2719–2724.10.1090/S0002-9939-2015-12419-7Google Scholar
Blanchard, F., Glasner, E., Kolyada, S. and Maass, A.. On Li–Yorke pairs. J. Reine Angew. Math. 547 (2002), 51–68.Google Scholar
Blanchard, F., Huang, W. and Snoha, L.. Topological size of scrambled sets. Colloq. Math. 110(2) (2008), 293–361.10.4064/cm110-2-3Google Scholar
Bruckner, A. M. and Hu, T.. On scrambled sets for chaotic functions. Trans. Amer. Math. Soc. 301(1) (1987), 289–297.10.1090/S0002-9947-1987-0879574-0Google Scholar
Doležal, M. and Kubiś, W.. Perfect independent sets with respect to infinitely many relations. Arch. Math. Logic 55(7–8) (2016), 847–856.10.1007/s00153-016-0498-3Google Scholar
Du, B.-S.. On the invariance of Li–Yorke chaos of interval maps. J. Difference Equ. Appl. 11(9) (2005), 823–828.Google Scholar
Foryś, M., Huang, W., Li, J. and Oprocha, P.. Invariant scrambled sets, uniform rigidity and weak mixing. Israel J. Math. 211(1) (2016), 447–472.10.1007/s11856-015-1278-1Google Scholar
Foryś, M., Oprocha, P. and Wilczyński, P.. Factor maps and invariant distributional chaos. J. Differential Equations 256(2) (2014), 475–502.Google Scholar
Huang, W. and Ye, X.. Devaney’s chaos or 2-scattering implies Li–Yorke’s chaos. Topology Appl. 117(3) (2002), 259–272.10.1016/S0166-8641(01)00025-6Google Scholar
Iwanik, A.. Independence and scrambled sets for chaotic mappings. The Mathematical Heritage of C. F. Gauss. World Science Publishers, River Edge, NJ, 1991, pp. 372–378.10.1142/9789814503457_0027Google Scholar
Kechris, A. S.. Classical Descriptive Set Theory (Graduate Texts in Mathematics, 156) . Springer, New York, 1995.10.1007/978-1-4612-4190-4Google Scholar
Kuratowski, K.. Applications of the Baire-category method to the problem of independent sets. Fund. Math. 81(1) (1973), 65–72.Google Scholar
Li, J., Tu, S. and Ye, X.. Mean equicontinuity and mean sensitivity. Ergod. Th. & Dynam. Sys. 35(8) (2015), 2587–2612.Google Scholar
Li, J. and Ye, X.. Recent development of chaos theory in topological dynamics. Acta Math. Sin. (Engl. Ser.) 32(1) (2016), 83–114.10.1007/s10114-015-4574-0Google Scholar
Li, T.-Y. and Yorke, J. A.. Period three implies chaos. Amer. Math. Monthly 82(10) (1975), 985–992.Google Scholar
Medini, A., Repovš, D. and Zdomskyy, L.. Non-meager free sets and independent families. Proc. Amer. Math. Soc. 145 (2017), 4061–4073.Google Scholar
Mycielski, J.. Independent sets in topological algebras. Fund. Math. 55 (1964), 139–147.10.4064/fm-55-2-139-147Google Scholar
Oprocha, P.. Coherent lists and chaotic sets. Discrete Contin. Dyn. Syst. 31(3) (2011), 797–825.10.3934/dcds.2011.31.797Google Scholar
Tan, F.. On an extension of Mycielski’s theorem and invariant scrambled sets. Ergod. Th. & Dynam. Sys. 36(2) (2016), 632–648.10.1017/etds.2014.76Google Scholar
Tan, F., Xiong, J. and Lü, J.. Dependent sets of a countable family of thick relations on a metric space. Southeast Asian Bull. Math. 31(6) (2007), 1077–1089.Google Scholar
Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79) . Springer, New York, 1982.Google Scholar
Yuan, D. and Lü, J.. Invariant scrambled sets in transitive systems. Adv. Math. (China) 38(3) (2009), 302–308.Google Scholar