Various Shadowing Properties On The Steinlein-Walther Hyperbolic Set

In this paper we give some kinds of stability of C1 map on a weak kind ofhyperbolic set, that is, Steinlein-Walther hyperbolic set, via the notions of strongshadowing and inverse shadowing,and hence obtain the generalized shadowing lem-mas.There are three parts in this paper.In chapter 1, we simply introduce the concepts of some kinds of shadowingproperties and some known results about them,and then give the notations andthe basic concepts, introduce the background and main results of this paper.In chapter 2, we construct a lemma and make use of it to prove the strongshadowing property on the Steinlein-Walther hyperbolic set.In chapter 3, we study the inverse shadowing property and strong inverse shad-owing property on the Steinlein-Walther hyperbolic set. Ifφis a C1 map of Banachspace,thenφhas the inverse shadowing property on the Steinlein-Walther hyper-bolic set with respect to a class of continuous methodθs, and its strong inverseshadowing property with respect to a class of continuous methodθss is proved fur-ther as well.