Research Of Stability Of Nonwandering Operator On Recurrent Set And Products Of 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 and semigroups and does further research to the nonwandering property of the operator and semigroups. Especially, in this paper we introduce recurrent set in infinite dimensional seperable Banach space and show that there exists a nonwandering operator on recurrent set by methods of functional analysis and the tool of pseudo orbit. Moreover, we prove that every infinite dimensional seperable Banach space with an unconditional basis supports a nonwandering operator on the recurrent set.Next, on the basis of recurrent set, we give the definition of R stability of the nonwandering operator withΩstability. And we show that the nonwandering operator is R-stable by using Axiom A. Using this property, we get some good results.In the end, we do further research to the nonwandering semigroups T(t)×S(t). Based on the definition of the nonwandering operator and semigroups in infinite dimensional seperable Banach space, we introduce the non-wandering criterion (NWC) and the recurrent non-wandering criterion (RNWC). Then it is proved that semigroups T(t)×S(t) is nonwandering if T(t) or S(t) satisfies the recurrent non-wandering criterion (RNWC).