Strong Shadowing Property And Point-wise Pseudo-Orbit Tracing Property In Topological Dynamical Systems

The pseudo orbit tracing property was generated along with the study and development of the stability in dynamical system,and had been one of the important properties. The Pseudo Orbit was not the genuine orbit, but the path under mapping iteration with error. The Pseudo Tracing Property closely related with the system's stable theory and the chaos theory, was played the vital role in qualitative theory of dynamic system. Based on demanding of theory and application, and different standard, related researchers proposed various Pseudo Tracing Properties. There were Average Tracing Property, Strong Shadowing Property, Limit Tracing Property, Point-wise Pseudo-Orbit Tracing Property and Sequence Tracing Property. In this paper, we mainly studied the problems about Strong Shadowing Property and Point-wise Pseudo-Orbit Tracing Property in dynamical systems. On condition of reading many literatures, materials, in predecessor's foundation the author analyzed, pondered and obtained the following conclusions and had given the detailed proof processes.⑴Strong Shadowing Property separately given by R.Easton and Pilyugin, which was one of very important pseudo orbit tracing property, was attended by many relative researchers. In this article, the author proved that the transfer homeomorphismσf on the inverse limit space generated by X had Strong Shadowing Property if there were Strong Shadowing Property and Lipschitz condition in continuous map f on X .⑵The lift system was an useful tool on the dynamic system which set up on special differential manifold such as n-dimensional anchor ring and so on when it was described. The studying about uniformity between lift system and the foundation system on some property was a very important question. In this paper, the system ( X , f ) was the lift system of system ( X , f ). The author proved that ( X , f ) had the Point-wise Pseudo-Orbit Tracing Property if and only if ( X , f ) had the Point-wise Pseudo-Orbit Tracing Property.⑶The Point-wise Pseudo-Orbit Tracing Property was the promotion about Peudo-Orbit Tracing Property. Systems ( X _∞, f_∞) was induced by systems { X_i ,φ_i , f_i }_i=1^∞. It was proved that f_∞had Point-wise Pseudo-Orbit Tracing Property if each f i had Point-wise Pseudo-Orbit Tracing Property and found an example to explain the inverse proposition which was not true. This conclusion promoted the conclusion of Pseudo-Orbit Tracing Property which was published on Math Yearly Journal by Li Si-min.