Strong Shadowing Property And Strong Inverse Shadowing Property In Dynamical Systems

Content: In this paper, we consider the strong shadowing propertyof the dynamical system generated by a continuous map on a compactmetric space. A system is said to have the strong shadowing propertyif there exists real trajectory for any strong pseudo trajectory, suchthat the sum of their one-step errors is small enough. The strongshadowing property is an important concept in the stability theoryof differential dynamical system, and plays a significant role in someparts of the computational mathematics.There are three parts in this paper.In chapter 1, the introduction, we simply introduce the develop-ment of the shadowing property, and give some basic concepts andmain results.In chapter 2, we study the strong shadowing property for a contin-uous map f on a compact metric space X. In section 2.1, it is provedtwo basic properties of the strong shadowing property: if f satisfiesthe Lipschitz conditions, then f has the strong shadowing property ifand only if for any k∈Z⁺, f^k has the strong shadowing property;f: X→X is homeomorphism, if f, f^-1 all satisfy the Lipschitz con-ditions, then f has the strong shadowing property if and only if f^-1has the strong shadowing property. In section 2.2, it is proved thatthe lift system ((?), (?))has the strong shadowing property if and onlyif (X, f) has the strong shadowing property.In chapter 3, we study the strong inverse shadowing property fora structurally stable diffeomorphism f. In section 3.2, it is proved that if diffeomorphism f is structurally stable, then f has the stronginverse shadowing property with respect to classes of continuous meth-odsΘ_sc(f) andΘ_ss(f).