Some stronger forms of mean sensitivity

Quanquan Yao; Yuanlin Chen; Peiyong Zhu; Tianxiu Lu; Quanquan Yao; Yuanlin Chen; Peiyong Zhu; Tianxiu Lu

doi:10.3934/math.2024054

AIMS Mathematics

2024, Volume 9, Issue 1: 1103-1115. doi: 10.3934/math.2024054

Previous Article Next Article

Research article

Some stronger forms of mean sensitivity

1.
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China
2.
College of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong 643000, China

Received: 29 August 2023 Revised: 21 November 2023 Accepted: 28 November 2023 Published: 06 December 2023
MSC : 37B45, 37B55, 54H20

The equivalence between multi-transitive mean sensitivity and multi-transitive mean $ n $-sensitivity for linear dynamical systems was demonstrated in this study. Furthermore, this paper presented examples that highlighted the disparities among mean sensitivity, multi-transitive mean sensitivity, and syndetically multi-transitive mean sensitivity.
- backward shift,
- multi-transitive mean $ n $-sensitivity,
- multi-transitive mean sensitivity
Citation: Quanquan Yao, Yuanlin Chen, Peiyong Zhu, Tianxiu Lu. Some stronger forms of mean sensitivity[J]. AIMS Mathematics, 2024, 9(1): 1103-1115. doi: 10.3934/math.2024054

Related Papers:

Abstract

The equivalence between multi-transitive mean sensitivity and multi-transitive mean $ n $-sensitivity for linear dynamical systems was demonstrated in this study. Furthermore, this paper presented examples that highlighted the disparities among mean sensitivity, multi-transitive mean sensitivity, and syndetically multi-transitive mean sensitivity.

References

[1]	K. G. Grosse-Erdmann, A. P. Manguillot, Linear chaos, Springer, 2011. https://doi.org/10.1007/978-1-4471-2170-1
[2]	L. Luo, B. Hou, Some remarks on distribution chaos for bounded linear operators, Turk. J. Math., 39 (2015), 251–258. https://doi.org/10.3906/mat-1403-41 doi: 10.3906/mat-1403-41
[3]	Y. Huang, Functional analysis: an introduction, 2 Eds., Science Press, 2019.
[4]	J. Li, S. Tu, X. Ye, Mean equicontinuity and mean sensitivity, Ergod. Theory Dyn. Syst., 35 (2015), 2587–2612. https://doi.org/10.1017/etds.2014.41 doi: 10.1017/etds.2014.41
[5]	L. S. Banños, F. García-Ramos, Mean equicontinuity and mean sensitivity on cellular automata, Ergod. Theory Dyn. Syst., 41 (2020), 3704–3721. https://doi.org/10.1017/etds.2020.108 doi: 10.1017/etds.2020.108
[6]	F. García-Ramos, B. Marcus, Mean sensitive, mean equicontinuous and almost periodic functions for dynamical systems, Discrete Contin. Dyn. Syst., 39 (2019), 729–746. https://doi.org/10.3934/dcds.2019030 doi: 10.3934/dcds.2019030
[7]	J. Li, T. Yu, On mean sensitive tuples, J. Differ. Equations, 297 (2021), 175–200. https://doi.org/10.1016/j.jde.2021.06.032
[8]	X. Wu, S. Liang, X. Ma, T. Lu, S. A. Ahmadi, The mean sensitivity and mean equicontinuity in uniform spaces, Int. J. Bifurcat. Chaos, 30 (2020), 2050122. https://doi.org/10.1142/S0218127420501229 doi: 10.1142/S0218127420501229
[9]	F. García-Ramos, J. Li, R. Zhang, When is a dynamical system mean sensitive? Ergod. Theory Dyn. Syst., 39 (2019), 1608–1636. https://doi.org/10.1017/etds.2017.101
[10]	J. Li, C. Liu, S. Tu, T. Yu, Sequence entropy tuples and mean sensitive tuples, Ergod. Theory Dyn. Syst., 2023, 1–20. https://doi.org/10.1017/etds.2023.5
[11]	J. Li, X. Ye, T. Yu, Mean equicontinuity, complexity and applications, Discrete Contin. Dyn. Syst., 41 (2021), 359–393. https://doi.org/10.3934/dcds.2020167
[12]	T. K. S. Moothathu, Stronger forms of sensitivity for dynamical systems, Nonlinearity, 20 (2007), 2115–2126. https://doi.org/10.1088/0951-7715/20/9/006 doi: 10.1088/0951-7715/20/9/006
[13]	X. Yang, T. Lu, W. Anwar, Transitivity and sensitivity for the $p$-seriodic discrete system via Furstenberg families, AIMS Math., 7 (2022), 1321–1332. https://doi.org/10.3934/math.2022078 doi: 10.3934/math.2022078
[14]	J. Li, P. Oprocha, X. Wu, Furstenberg families, sensitivity and the space of probability measures, Nonlinearity, 30 (2017), 987–1005. https://doi.org/10.1088/1361-6544/aa5495 doi: 10.1088/1361-6544/aa5495
[15]	R. Li, A note on stronger forms of sensitivity for dynamical systems, Chaos Solitons Fract., 45 (2012), 753–758. https://doi.org/10.1016/j.chaos.2012.02.003 doi: 10.1016/j.chaos.2012.02.003
[16]	F. Tan, R. Zhang, On $\mathscr{F}$-sensitive pairs, Acta Math. Sci., 31 (2011), 1425–1435. https://doi.org/10.1016/S0252-9602(11)60328-7 doi: 10.1016/S0252-9602(11)60328-7
[17]	X. Wu, J. Wang, G. Chen, $\mathscr{F}$-sensitivity and multi-sensitivity of hyperspatial dynamical systems, J. Math. Anal. Appl., 429 (2015), 16–26. https://doi.org/10.1016/j.jmaa.2015.04.009 doi: 10.1016/j.jmaa.2015.04.009
[18]	X. Wu, R. Li, Y. Zhang, The multi-$\mathbb{F}$-sensitivity and ($\mathbb{F}_{1}$, $\mathbb{F}_{2}$)-sensitivity for product systems, J. Nonlinear Sci. Appl., 9 (2016), 4364–4370. https://doi.org/10.22436/jnsa.009.06.76 doi: 10.22436/jnsa.009.06.76
[19]	W. Huang, D. Khilko, S. Kolyada, A. Peris, G. Zhang, Finite intersection property and dynamical compactness, J. Dyn. Differ. Equations, 30 (2018), 1221–1245. https://doi.org/10.1007/s10884-017-9600-8 doi: 10.1007/s10884-017-9600-8
[20]	W. Huang, D. Khilko, S. Kolyada, G. Zhang, Dynamical compactness and sensitivity, J. Differ. Equations, 260 (2016), 6800–6827. https://doi.org/10.1016/j.jde.2016.01.011 doi: 10.1016/j.jde.2016.01.011
[21]	N. C. Bernardes, A. Bonilla, A. Peris, X. Wu, Distributional chaos for operators on Banach spaces, J. Math. Anal. Appl., 459 (2018), 797–821. https://doi.org/10.1016/j.jmaa.2017.11.005 doi: 10.1016/j.jmaa.2017.11.005
[22]	J. Xiong, Chaos in a topologically transitive system, Sci. China Ser. A, 48 (2005), 929–939. https://doi.org/10.1007/BF02879075 doi: 10.1007/BF02879075
[23]	S. Shao, X. Ye, R. Zhang, Sensitivity and regionally proximal relation in minimal systems, Sci. China Ser. A, 51 (2008), 987–994. https://doi.org/10.1007/s11425-008-0012-4 doi: 10.1007/s11425-008-0012-4
[24]	X. Ye, R. Zhang, On sensitive sets in topological dynamics, Nonlinearity, 21 (2008), 1601–1620. https://doi.org/10.1088/0951-7715/21/7/012 doi: 10.1088/0951-7715/21/7/012
[25]	J. Li, X. Ye, T. Yu. Equicontinuity and sensitivity in mean forms, J. Dyn. Differ. Equations, 34 (2022), 133–154. https://doi.org/10.1007/s10884-021-09945-9 doi: 10.1007/s10884-021-09945-9
[26]	N. C. Bernardes, A. Bonilla, A. Peris, Mean Li-Yorke chaos in Banach spaces, J. Funct. Anal., 278 (2020), 108343. https://doi.org/10.1016/j.jfa.2019.108343 doi: 10.1016/j.jfa.2019.108343
[27]	H. N. Salas, Hypercyclic weighted shifts, Trans. Amer. Math. Soc., 347 (1995), 993–1004. https://doi.org/10.2307/2154883

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)