Research Of Nonwandering Operators And Semigroups

The paper takes the method of differential dynamics and basic theories of the operator as tools on the basis of Hypercyclicity of operator, Chaos, do further research to the nonwandering property of the operator and semigroup.Especially, this paper studies weighted shifts (bilateral weighted forward or unilateral weighted backward) on sequence space l^p(1≤p≤∞) and characterizes their nonwandering property when their weighting sequences satisfy certain condition. Simultaneously, we identify that the perturbation of these operators by a small norm operator is still nonwandering; further, we can obtain the theory that every bounded linear operator can be decompositioned into nonwandering operators. This paper have also explored the nonwandering property of sum of operator, direct sum of operator,tensor products of operators and differentiation operators which can be viewed as weighted shifts on some space.On the other hand, we make certain study to the nonwandring property of bounded, unbounded operator semigroups ,and give out concrete application.By the theory of semigroups, we establish if {T_n}_n≥1 is a bounded linear operator sequence onBanach space X_n converging in the sense of Kato to a bounded operator T on theBanach space X , then their nonwandering property can be inherited by each other under the appropriate conditions. Several consequences are also obtained.In this paper, we attempt to extend the study of structure stability in finite dimensional space into infinite dimensional space .By giving out a concept of local structural stability, we prove that the system formed by nonwandering operator is locally structurally stable.