在线阅读 --自然科学版 2019年6期《群作用下乘积映射的渐进平均和利普希茨跟踪性》
群作用下乘积映射的渐进平均和利普希茨跟踪性--[在线阅读]
冀占江1,2, 张更容3, 涂井先1,2
1. 梧州学院 大数据与软件工程学院, 广西 梧州 543002;
2. 梧州学院 广西高校图像处理与智能信息系统重点实验室, 广西 梧州 543002;
3. 湖南第一师范学院 数学与计算科学学院, 湖南 长沙 410205
起止页码: 473--478页
DOI: 10.13763/j.cnki.jhebnu.nse.2019.06.004
摘要
跟踪性在理论和应用中有着重要的意义,给出了拓扑群作用下乘积空间中G-渐进平均跟踪性和G-利普希茨跟踪性的概念,结合乘积映射和零密度集的性质,研究了乘积映射f×g与分映射f和g在这些跟踪性方面的关系,得到如下结论:1)乘积映射f×g具有G-渐进平均跟踪性当且仅当f具有G1-渐进平均跟踪性,g具有G2-渐进平均跟踪性;2)乘积映射f×g具有G-利普希茨跟踪性当且仅当f具有G1-利普希茨跟踪性,g具有G2-利普希茨跟踪性.这些结论弥补了拓扑群作用下乘积空间中渐进平均跟踪性和利普希茨跟踪性理论的缺陷.

Asymptotic Average and Lipschitz Shadowing Property of the Product Map Under Group Action
JI Zhanjiang1,2, ZHANG Gengrong3, TU Jingxian1,2
1. School of Data Science and Software Engineering, Wuzhou University, Guangxi Wuzhou 543002, China;
2. Guangxi Colleges and Universities Key Laboratory of Image Processing and Intelligent Information System, Wuzhou University, Guangxi Wuzhou 543002, China;
3. Mathematics and Computational Science, Hunan First Normal University, Hunan Changsha 410205, China
Abstract:
The shadowing property is significant both in of theory and application.In this paper,we introduce the concept of G-asymptotic average shadowing property and G-Lipschitz shadowing property in the product space under the action of a topological group.By means of properties of the product map and zero density sets,we study the relationship of these shadowing propertes between product mapping f×g and sub mapping f,g.We obtain the following result:1) The product map f×g has the G-asymptotic average shadowing property if and only if the map f has the G1-asymptotic average shadowing property and the map g has the G2-asymptotic average shadowing property; 2) The product map f×g has the G-Lipschitz shadowing property if and only if the map f has the G1-Lipschitz shadowing property and the map g has the G2-Lipschitz shadowing property.These results enrich the theory of asymptotic average shadowing property and Lipschitz shadowing property in the product space under the action of topological group.

收稿日期: 2019-01-05
基金项目: 国家自然科学基金(11461002);湖南省自然科学基金(2018JJ2074);广西壮族自治区自然科学基金(2018JJB170034);广西高校中青年教师科研基础能力提升项目(2019KY0681);梧州学院校级重点项目(2017B004);梧州学院校级科研项目(2017C001)

参考文献:
[1]WALTERS P.On the Pseudo-orbit Tracing and Its Relationship to Stability[M].Berlin:Springer-verlag,1978.doi:10.1007/bfb0101795
[2]杨润生.伪轨跟踪性与混沌[J].数学学报,1996,39(3):382-386. YANG R S.Pseudo-orbit-tracing and Chaos[J].Acta Mathematica Sinica,1996,39(3):382-386.
[3]GU R B,SHENG Y Q,XIA Z J.The Average Shadowing Property and Transitivity for Continuous Flows[J].Chaos Solitons and Fractals,2005,23:989-995.doi:10.1016/j.chaos.2004.06.059
[4]顾荣宝,盛业青.关于渐进的伪轨跟踪性[J].安微大学学报(自然科学版),2003,27(3):1-5.doi:10.3969/j.issn.1000-2162.2003.03.001 GU R B,SHENG Y Q.On the Asymptotic Pseudo Orbit Tracing Property[J].Journal of Anhui University(Natural Science Edition),2003,27(3):1-5.
[5]冀占江.乘积空间与拓扑群作用下逆极限空间的动力学性质[D].南宁:广西大学,2014. JI Zhanjiang.Dynamical Property of Product Space and the Inverse Limit Space of a Topological Group Action[D].Nanning:Guangxi University,2014.
[6]李思敏.逆极限空间的伪轨跟踪性[J].数学年刊A辑(中文版),2001(4):479-482.doi:10.3321/j.issn:1000-8134.2001.04.012 LI Simin.Pseudo Orbit Tracking of the Inverse Limit Space[J].Chinese Annals of Mathematics,2001(4):479-482.
[7]李明军,曾凡平.序列伪轨跟踪性与拓扑可迁[J].广西大学学报(自然科学版),2000,25(2):137-140.doi:10.3969/j.issn.1001-7445.2000.02.016 LI Mingjun,ZENG Fanping.Sequence Pseudo-orbit-tracing Property and Topological Transitivity[J].Journal of Guangxi University(Natural Science Edition),2000,25(2):137-140.
[8]吴志湖,陈尔明.逆极限空间的逐点伪轨跟踪性[J].华侨大学学报(自然科学版),2009,30(5):593-595.doi:10.11830/issn.1000-5013.2009.05.0593 WU Zhihu,CHEN Erming.Pointwise Pseudo-orbit Tracing Property for the Inverse Limit Space[J].Journal of Huaqiao University(Natural Science),2009,30(5):593-595.
[9]赵俊玲.弱跟踪性的一些性质[J].广西师范大学学报(自然科学版),2004,22(3):40-44.doi:10.3969/j.issn.1001-6600.2004.03.009 ZHAO Junling.Some Properties of the Weak Shadowing Property[J].Journal of Guangxi University(Natural Science Edition),2004,22(3):40-44.
[10]冀占江.群作用下逆极限空间上移位映射的G非游荡点与G链回归点[J].湖南师范大学自然科学学报,2018,41(4):13-14.doi:10.7612/j.issn.2096-5281.2018.06.012 JI Zhanjiang J.G-Nonwandering Points and G-Chain Recurrent Points of the Shift Map in the Inverse Limit Space of A Topological Group Action[J].Journal of Natural Science of Hunan Normal University,2018,41(4):13-14.
[11]EKTA S,TARUN D.Consequences of Shadowing Property of G-Spaces[J].Mathematical Analysis,2013,7(12):579-588.doi:10.12988/ijma.2013.13056