Various Shadowing Properties On The Quasi-hyperbolic Orbit

In this paper,we consider two kinds of shadowing properties on the quasihyperbolic orbit:limit shadowing property and strong shadowing property,the paper also proves a structurally stable diffeomorphism has the strong inverse shadowing property with respect to two classes of continuous methods.This paper consists of three chapters:In chapter I,we simply introduce the development of dynamical systems,give the basic concepts and the main results of this paper.In chapterâ…¡,we consider two shadowing properties on the quasi-hyperbolic orbit.In section 2.1,we prove the limit shadowing property on a quasi-hyperbolic orbit for a diffeomorphism on a closed C^∞manifold M:Let f∈Diff(M),assume that A is a closed invariant set of f and there is a continuous invariant splitting T_ΛM = E⊕F onΛ.For anyλ∈(0,1) there exists L＞0,d₀＞0 such that for any d∈(0,d₀]and anyλ-quasi-hyperbolic d-limit pseudoorbit {x_i,n_i}_i=-∞^∞,there exists a point x∈M which Ld-limit shadows {x_i,n_i}_i=-∞^∞.In section 2.2,we prove the strong shadowing property on a quasi-hyperbolic orbit for a diffeomorphism on a closed C^∞manifold M.In chapterâ…¢,we prove a structurally stable diffeomorphism which has the strong inverse shadowing property with respect to two classes of continuous meth-ods.