The Research On The Average Shadowing Property And The Limit Weak Inverse Shadowing Property In Discrete Dynamical Systems

Content: The shadowing property is a very important concept inves-tigated in dynamical systems. It has close relations with the stabilityof the systems and plays an important role in the general qualitativetheory of dynamical systems. On the basis of the need of the the-ory and application, people show the concepts of different shadowingproperties from different measured standard, such as the limit shad-owing property, the weak shadowing property, the average shadowingproperty, the inverse shadowing property and so on. People do muchresearch on various shadowing properties.In this paper, we discuss the shadowing property of discrete dynam-ical systems induced by a homeomorphism(continuous epimorphism)f, we main study the average shadowing property and the limit weakinverse shadowing property. There are three sections in this paper.In the first section, we introduce the concepts and the results of theshadowing property and other kinds of shadowing properties. In thesecond section, we research on the average shadowing property, intro-duce the concept and the results of other people about the averageshadowing property, prove some basic properties of the average shad-owing property, and discuss the average shadowing property on theinverse limit space. In the third section, we study the relations be-tween the limit weak inverse shadowing property and theΩ-stability,introduce some basic knowledge about theΩ-stability and the relatedconcepts, and prove the C~1 interior of the set of diffeomorphisms withthe limit weak inverse shadowing property coincides with the set ofΩ-stable diffeomorphisms.