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Null systems in the non-minimal case

Published online by Cambridge University Press:  17 June 2019

JIAHAO QIU
Affiliation:
Wu Wen-Tsun Key Laboratory of Mathematics USTC, Chinese Academy of Sciences and School of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, PR China email qiujh@mail.ustc.edu.cn, zjianjie@mail.ustc.edu.cn
JIANJIE ZHAO
Affiliation:
Wu Wen-Tsun Key Laboratory of Mathematics USTC, Chinese Academy of Sciences and School of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, PR China email qiujh@mail.ustc.edu.cn, zjianjie@mail.ustc.edu.cn
Corresponding

Abstract

In this paper, it is shown that if a dynamical system is null and distal, then it is equicontinuous. It turns out that a null system with closed proximal relation is mean equicontinuous. As a direct application, it follows that a null dynamical system with dense minimal points is also mean equicontinuous. Meanwhile, a distal system with trivial $\text{Ind}_{\text{fip}}$-pairs and a non-trivial regionally proximal relation of order $\infty$ are constructed.

Type
Original Article
Copyright
© Cambridge University Press, 2019

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References

Akin, E. and Glasner, E.. Residual properties and almost equicontinuity. J. Anal. Math. 84 (2001), 243286.CrossRefGoogle Scholar
Auslander, J.. On the proximal relation in topological dynamics. Proc. Amer. Math. Soc. 11 (1960), 890895.CrossRefGoogle Scholar
Auslander, J.. Minimal Flows and Their Extensions (North-Holland Mathematics Studies, 153). North-Holland, Amsterdam, 1988.Google Scholar
Dong, P., Donoso, S., Maass, A., Shao, S. and Ye, X.. Infinite-step nilsystems, independence and complexity. Ergod. Th. & Dynam. Sys. 33 (2013), 118143.CrossRefGoogle Scholar
Downarowicz, T. and Glasner, E.. Isomorphic extensions and applications. Topol. Methods Nonlinear Anal. 48 (2016), 321338.Google Scholar
Fomin, S.. On dynamical systems with a purely point spectrum. Dokl. Akad. Nauk SSSR 77 (1951), 2932 (in Russian).Google Scholar
Furstenberg, H.. Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton University Press, Princeton, NJ, 1981.CrossRefGoogle Scholar
García-Ramos, F.. A characterization of 𝜇-equicontinuity for topological dynamical systems. Proc. Amer. Math. Soc. 145(8) (2017), 33573368.CrossRefGoogle Scholar
García-Ramos, F.. Weak forms of topological and measure-theoretical equicontinuity: relationships with discrete spectrum and sequence entropy. Ergod. Th. & Dynam. Sys. 37(4) (2017), 12111237.CrossRefGoogle Scholar
García-Ramos, F. and Jin, L.. Mean proximality and mean Li–Yorke chaos. Proc. Amer. Math. Soc. 145(7) (2017), 29592969.CrossRefGoogle Scholar
García-Ramos, F., Li, J. and Zhang, R.. When is a dynamical system mean sensitive? Ergod. Th. & Dynam. Sys. 39 (2019), 16081636.CrossRefGoogle Scholar
Glasner, E., Gutman, Y. and Ye, X.. Higher order regionally proximal equivalence relations for general group actions. Adv. Math. 333 (2018), 10041041.CrossRefGoogle Scholar
Goodman, T. N. T.. Topological sequence entropy. Proc. Lond. Math. Soc. 3 (1974), 331350.CrossRefGoogle Scholar
Host, B., Kra, B. and Maass, A.. Nilsequences and a structure theorem for topological dynamical systems. Adv. Math. 224 (2010), 103129.CrossRefGoogle Scholar
Huang, W., Li, J., Thouvenot, J., Xu, L. and Ye, X.. Mean equicontinuity, bounded complexity and discrete spectrum. Ergod. Th. & Dynam. Sys. to appear.Google Scholar
Huang, W., Li, S., Shao, S. and Ye, X.. Null systems and sequence entropy pairs. Ergod. Th. & Dynam. Sys. 23 (2003), 15051523.CrossRefGoogle Scholar
Huang, W., Maass, A. and Ye, X.. Sequence entropy pairs and complexity pairs for a measure (paires d’entropie séquentielle et paires de complexiteé pour une mesure). Ann. Inst. Fourier (Grenoble) 54 (2004), 10051028.CrossRefGoogle Scholar
Huang, W. and Ye, X.. Combinatorial lemmas and applications to dynamics. Adv. Math. 220 (2009), 16891716.CrossRefGoogle Scholar
Kerr, D. and Li, H.. Independence in topological and C -dynamics. Math. Ann. 338 (2007), 869926.CrossRefGoogle Scholar
Kushnirenko, A. G.. On metric invariants of entropy type. Russian Math. Surveys 22 (1967), 5361.CrossRefGoogle Scholar
Li, J. and Tu, S.. On proximality with Banach density one. J. Math. Anal. Appl. 416 (2014), 3651.CrossRefGoogle Scholar
Li, J., Tu, S. and Ye, X.. Mean equicontinuity and mean sensitivity. Ergod. Th. & Dynam. Sys. 35 (2015), 25872612.CrossRefGoogle Scholar
Qiu, J. and Zhao, J.. A note on mean equicontinuity. J. Dynam. Differential Equations, to appear.Google Scholar
Shao, S. and Ye, X.. Regionally proximal relation of order d is an equivalence one for minimal systems and a combinatorial consequence. Adv. Math. 231 (2012), 17861817.CrossRefGoogle Scholar

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