The Study Of Shadowing In Differentiable Dynamical Systems

There are three main parts in this paper.In the first part(Chapter 2), we study the limit shadowing property for continuous maps and continuous flows. Firstly, some basic properties of the limit shadowing are given; Secondly, we give the characterization of both lin-ear automorphisms and linear flows on Rⁿ with the limit shadowing property; Thirdly, as applications we prove that the hyperbolic endomorphisms on Tⁿ have the limit shadowing properties, Smale "horseshoes" have the same prop-erties on their invariant sets.In the second part(Chapter 3), we consider the Lipschitz shadowing and the inverse shadowing for C¹ endomorphisms. We show that near a hyperbolic set a C¹ endomorphism has the Lipschitz shadowing property, and a hyperbolic endomorphism has the inverse shadowing property with respect to a class of continuous methods. Moreover, each of these shadowing properties is also "uniform" with respect to C¹ perturbation.In the third part (Chapter 4), we consider the shadowing of C¹ random dynamical system. We define a type of hyperbolicity on the full measure invariant set which is given by the Oseledec's multiplicative ergodic theorem and prove that the system has the Lipschitz shadowing property on it.