随机动力系统的 Weyl 平均等度连续性和 Weyl 平均敏感性

数学物理学报 ›› 2024, Vol. 44 ›› Issue (6): 1595-1606.

随机动力系统的 Weyl 平均等度连续性和 Weyl 平均敏感性

连媛^1,^*(),刘红军²(),朱斌³()

¹太原师范学院数学与统计学院山西晋中 030619
²贵州师范大学数学科学学院贵阳 550025
³重庆工商大学数学与统计学院重庆 400067

收稿日期:2024-02-21 修回日期:2024-06-11 出版日期:2024-12-26 发布日期:2024-11-22
通讯作者: ^*连媛，Email: andrea@tynu.edu.cn
作者简介:刘红军, Email:hongjunliu@gznu.edu.cn;|朱斌, Email:binzhuctbu@ctbu.edu.cn
基金资助:
山西省基础研究(202103021223322)

Weyl Mean Equicontinuity and Weyl Mean Sensitivity of A Random Dynamical System

Lian Yuan^1,^*(),Liu Hongjun²(),Zhu Bin³()

¹College of Mathematics and Statistics, Taiyuan Normal University, Shanxi Jinzhong 030619
²School of Mathematical Sciences, Guizhou Normal University, Guiyang 550025
³School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067

Received:2024-02-21 Revised:2024-06-11 Online:2024-12-26 Published:2024-11-22
Supported by:
Fundamental Research Program of Shanxi Province(202103021223322)

摘要/Abstract

摘要：

该文首先介绍了无限可数离散顺从群作用的连续随机动力系统的 Weyl 平均等度连续性和 Weyl 平均敏感性,然后得到了当随机动力系统相应的斜积变换为极小作用时系统的 Weyl 平均等度连续性和 Weyl 平均敏感性之间具有二分法的结论.

关键词: 平均等度连续性, 平均敏感性, 随机动力系统

Abstract:

In this article, we introduce the concepts of Weyl-mean equicontinuity and Weyl-mean sensitivity of a random dynamical system associated to an infinite countable discrete amenable group action. We obtain the dichotomy result to Weyl-mean equicontinuity and Weyl-mean sensitivity of a random dynamical system when the corresponding skew product transformation is minimal.

Key words: Mean equicontinuity, Mean sensitivity, Random dynamical system

中图分类号:

O193

连媛, 刘红军, 朱斌. 随机动力系统的 Weyl 平均等度连续性和 Weyl 平均敏感性[J]. 数学物理学报, 2024, 44(6): 1595-1606.

Lian Yuan, Liu Hongjun, Zhu Bin. Weyl Mean Equicontinuity and Weyl Mean Sensitivity of A Random Dynamical System[J]. Acta mathematica scientia,Series A, 2024, 44(6): 1595-1606.

参考文献

[1]	Arnold L. Random Dynamical Systems. Berlin: Springer-Verlag, 1998
[2]	Akin E, Auslander J, Berg K. When is a transitive map chaotic? Convergence in Ergodic Theory and Probability. Berlin: De Gruyter, 1996: 25-40
[3]	Auslander J, Yorke J A. Interval maps, factors of maps, and chaos. Tohoku Math J, 1980, 32(2): 177-188
[4]	Baake M, Lenz D, Moody R V. Charaucterization of model sets by dynamical systems. Ergos Th and Dynam sys, 2007, 27(2): 341-382
[5]	Cong N D. Topological Dynamics of Random Dynamical Systems. Oxford: Oxford Univ Press, 1997
[6]	Crauel H. Random Probability Measures on Polish Spaces. London: Taylor and Francis, 2002
[7]	Danilenko A I. Entropy theory from the orbital point of view. Monatsh Math, 2001, 134(2): 121-141
[8]	Dooley A, Zhang G. Local Entropy Theory of a Random Dynamical System. Providence RI: Amer Math Soc, 2015
[9]	Fuhrmann G, Gr?ger M, Lenz D. The structure of mean equicontinuous group actions. Israel Journal of Mathematics, 2022, 247: 75-123
[10]	García-Ramos F. Weak forms of topological and measure theoretical equicontinuity: relationships with discrete spectrum and sequence entropy. Ergos Th and Dynam Sys, 2017, 37(4): 1211-1237
[11]	Glasner S, Maon D. Rigidity in topological dynamics. Ergods Th and Dynam Sys, 1989, 9: 309-320
[12]	Glasner E, Weiss B. Sensitive dependence on initial conditions. Nonlinearity, 1993, 6(6): 1067-1075
[13]	Kakutani S. Random ergodic theorems and Markoff processes with a stable distribution. Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability. California: University of California Press, 1951, 247-261
[14]	Kifer Y. Ergodic Theory of Random Transformations. Boston: Birkh?user,986
[15]	Kerr D, Li H. Ergodic Theory:Independence and Dichotomies. Switzerland: Springer, 2016
[16]	Liu P D, Qian M. Smooth Ergodic Theory of Random Dynamical Systems. Berlin: Springer-Verlag, 1995
[17]	Li J, Tu S, Ye X. Mean equicontinuity and mean sensitivity. Ergod Th Dynam Sys, 2015, 35(8): 2587-2612
[18]	Ledrappier F, Walters P. A relativized variational principle for continuous transformations. J London Math Soc, 1977, 16(2): 568-576
[19]	Ornstein D S, Weiss B. Entropy and isomorphism theorems for actions of amenable groups. J Analyse Math, 1987, 48: 1-141
[20]	Rudolph D J, Weiss B. Entropy and mixing for amenable group actions. Ann of Math, 2000, 151(3): 1119-1150
[21]	Rokholin V A. On the fundamental ideas of measure theory. Matematicheskii Sbornik, 1949, 67(1): 107-150
[22]	Rokholin V A. Selected topics from the metric theory of dynamical systems. Uspekhi Matematicheskikh Nauk, 1949, 4(1): 57-128
[23]	Schenk-Hoppé K R. Random dynamical systems in economics. Stochastics Dynamics, 2001, 1(1): 63-83
[24]	Thouvenot J P. Quelques properties des systemes dynamiques que se decomposent en un produit de deux systemes dont l'un est un schema de Bernoulli. Israel J Math, 1975, 21: 177-207
[25]	Ulam S M, Von Neumann J. Random ergodic theorems. Bull Amer Math Soc, 1945, 51:660
[26]	Weiss B. Actions of Amenable Groups, Topics in Dynamics and Ergodic Theory. Cambridge: Cambridge Univ Press, 2003, 310: 226-262
[27]	Zhu B, Huang X, Lian Y. The systems with almost Banach mean equicontinuity for Abelian group actions. Acta Mathematica Scientia, 2022, 42(3): 919-940