群作用下乘积映射的渐进平均和利普希茨跟踪性

Asymptotic Average and Lipschitz Shadowing Property of the Product Map Under Group Action

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.

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.