The Research Of The Asymptotically Stable Set Of Local Compact Metric Space

The topological space X is a locally compact space,which has a com-pact neighborhood for every x?X.In recent years,there are many schol-ars have studied dynamic properties of Euclidean space and compact space and have obtained a lot of meaningful results.This thesis mainly researches the asymptotically stable set of local compact metric space homeomorphis-m,which involves the properties of asymptotically stable set,no complete negative trajectory,attractor and so on.In this paper,the chapter two studies the asymptotically stable set of local compact metric space homeomorphism.Concretely,let X be a local compact metric space,f:X?X be a.homeomorphism.Then there are t-wo conclusions which are as follow(1)assume that K(?)X is an attractor and if there exists a compact neighborhood K of Q,such that f(Q)(?)Q and?n?0 fn(Q)(?)K.then K is asymptotically stable set.(2)?(x)is nonemp-ty set for every x of X.Assume that K(?)X is a compact strongly invariant set and there exists a compact neighborhood Q(?)K,such that Q\K contain-s no complete negative trajectory for every and is nonempty set,then K is asymptotically stableIn addition,the chapter three also mainly studies the correlation equiva-lence relationship of asymptotically stable set of local compact metric space Concretely,let X be a local compact metric space,f:X?X be a homeo morphism.There are two results which are as follow(1)if ?(x)is nonempty set for every x of X,and let K be a compact invariant subset,then there exists a Lyapunov function for f and K if and only if K is globally asymptomat ically stable.(2)if each bounded closed set is a compact set for X and let K(?)X be an attractor,then K is asymptotically stable with the same basin of attraction that K.Moreover,K is asymptotically stable if only if (?)=K.