| Befor years, some views and methods became one of the means in the researching of concrete operators, for example, the so called concepts of hypercyclicand cyclic are correspondings to the concept of limits set and so on in operator theorey. However, Herro and other people clearified the target to study the dynamic properties.As an important operator, unilateral shiifts are the representative study with rich dynamic properties. Within ten years,the study of its dynamic propertieshas become more and more important, becoming a target with own independent tools and methods. This paper is an short introduction to the recent progress in this field.In the preface, we first introduced the dynamics and topological transitiity.Chaos dynamical systems have received a great deal of attention in recent years. A discrete dynamical system is simily a continuous function f, denoted as < X,f >. Forx∈X, the orbit of x under f is orb(f,x) = {x, f(x), f2(x),…}, where fn= f o f o…f is the nth iterate of f. In dynamicalsystems one is interested in understanding dynamical properties, i.e., the eventual or asymptotoc behavior of an iterative process,its orbits have received a great deal of attention in recent years. ([3],[4]). the main is chaos, for the definition of chaos, the popular is introduced by Devaney.It is often said the chaos cannot appear in linear systems. However, Nathan S. Feldman fist studied the chaos of weighted operator shifts, giving a very surprsing result.Theorem : If B is a Backward shift on l2,then 2B is chaotic. whereB(x0,x1,x2…)= (x1,x2,…). For transitivity is equivalent to Hypercyclic,and chaotic in some sense is dependent on transitivity,if we want to consider the chaotic of an operator,we should first study its Hypercyclicity.So we introduced the works about how to decide which operator is hypercyclic and the existance of the hypercyclic operator. However,it is too difficult to study the dynamical properties of an operator. Noting that each operator is in connection with the weighted shift operator through the famous Von Neumann-Wold theorem,we can study the dynamical properties of weighted shift operator.In section 3,we introduce the development of Hypercyclicity for weighted shift operator.Salas,K. G. GRosse-ERDMANN(Hagen)gave the Hypercyclic properties of the general weighted shift operator,and next Feldman,Munmum Hazarika and S. C. Arora gave the Hypercyclic criterion on operator weighted shifts according to the Ki-tai/Gethner. Shapiro/Salas Hypercyclic criterion.Having the previous works,we introduce the chaotic criterion of weighted shift operator with the following main works:NathanS.Feldman considered the general case of Backward shifts.If 1≤n≤∞, and Bn = B(?) B (?)... (?) B, then 2Bn is chaos on l2(Hn).In 2000, K-G. Grosse-ERdmann(Hagen) gave a compelete characterion of weighted shifts operators on F-suquence space, showing this theorem:Theorem : X is a F -suquence space, where {en}n∈N is an unconditionalbasis , let T : X→X is a unilateral shiift, with weight sequence {an}n∈N, then the follwwing are equivalent:(i) T is chaos;(ii) T is hypercyclic, and has a non-trivial periodic point;(iii) Thas a non-trivial periodic point;(iv) the series (?) en converges in X.Then, FELIX MARTINEZ-GIMENEZ studied f(Bω) on Kothe echelon, where Bωis a Backward shift, f(z) = (?)fjzj.And , for every injective weighted shifts is unitarily equivalent to a multiplicationby z, in 1990, GILLE GODEFROY considered the chaotic propertyof weighted shiifts in view of multiplication by z, showing that chaos is equivalent to having a nontrival periodic point.However, does the operator weighted shifts as the natural extenstion of weighted shiifts act like them ? In 2007,pu yu,cui studied the an operator weighted shifts, gaining the conditions when it is chaotic.Theorem : (1) The weight sequence is upper triangle andThen, S - {Wk} is chaotic (?)|λ1|> 1 and |λ2| > 1.(2) The weight sequence is upper triangle andLeta2 =μ1ω2+v2a1,…,an=μ1·μ2…μ(n-1)·ωn +vn·a(n- 1), Then, S is chaotic (?) seriesall converge. Corollary : S- {Wk}, whose weight is a second order matrix, S- {Wk} is transitive (?)it is weakly mixing .Theorem : (1) S- {Wk} , with weight sequence W1 = W2=…= Wn = W.Then S- {Wk} is chaotic (?) |λ| > 1.(2) If the weight sequence is W1 = W2 =…= Wn = W.,where W is a Jordan matrix compomsed by a Jordan component, W = B1(?) B2(?)…Bl,λi is the eigenvalue of Bi,i = 1,2,…,l, Then S- {Wk} is chaotic (?) |λi| > 1,1 = 1,2,…,l.In operator theory, the similarity is an equivalent relation, then people classified operator via the similarity.In dynamical system, topological conjugacyis also very important, obviously, "similar" is a stronger relationship than topological conjugacy. Mover, it is easy to see that such properties as , havinga dense set of periodic points, transitivity, mixing ...,are preserved under topological conjugacy, so as "chaotic".In the last,we condiserd the application of dynamical system in the classificationof weighted shiift operators.In 2007, Bing zhe, Hou and yang, cao considered the classfication of weighted operator shifts with sequence constant via topological conjugacy.Theorem: Suppose 0 < p <∞,λ,ω∈C, thenλBp andωBp topologicallyconjugate if and only ifΧ(|λ|)=Χ(|ω|). withΧdenoted as:Theorem: Suppose 0 < p, q <∞,λ,ω==== C,thenλBp andωBq topologicallyconjugate if and only ifΧ(|λ|)=Χ(|ω|). Seeing from the above results,the dynamical system indeed plays a greatrole in the study of operator particularly weighted shift operator.
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