The Limit Quasi-Shadowing Property

This paper studies three kinds of shadowing properties on the quasi-partially hyperbolic pseudoorbit in differential dynamical systems: Limit quasi-shadowing, pL - quasi-shadowing and asymptotic quasi-shadowing property. The article also investigates the Lp -, limit and asymptotic quasi-shadowing properties for partially hyperbolic diffeomorphisms which are dynamically coherent.This paper is organized as the following three parts.The paper firstly introduces the historical background and research status of differential dynamical system. It then recalls the concept of quasi-hyperbolic pseudoorbit and generalizes the concept of quasi-hyperbolic pseudoorbit to quasi-partially hyperbolic pseudoorbit. Finally gives the definitions of limit quasi-shadowing property, Lp - quasi-shadowing property and asymptotic quasi-shadowing property which are weaker than limit shadowing property, pL - shadowing property and asymptotic shadowing property respectively. This is the first three chapters of this thesis.The second part mainly studies the shadowing property for quasi-partially hyperbolic pseudoorbits. Firstly, the part proves two propositions. Using the analytic method, it then proves that a quasi-partially hyperbolic pseudoorbit can be limit, pL - and asymptotic quasi-shadowed by a points sequence. The Theorems of Section 4.2 are some main results of this paper.The third part investigates the shadowing property for partially hyperbolic diffeomorphisms which are dynamically coherent. Firstly, this part recalls the definitions of partially hyperbolic diffeomorphism and dynamically coherence. Using the geometric method, it then proves that a partially hyperbolic diffeomorphism which is dynamically coherent has limit, Lp - and asymptotic quasi-shadowing properties. The main result is theorem 5.2.1.