Frchet空间上的非游荡算子的遗传超循环分解

The Hereditarily Hypercyclic Decomposition of Nonwandering Operators in Infinite Dimensional Fréchet Space

混沌现象并非仅仅局限于非线性映射或算子 ,在无穷维空间中 ,某些线性映射或线性算子也有可能是混沌的 ,这是一个奇特的现象 ,这也使得混沌学的研究内容更为丰富 无穷维可分Fr啨ｃｈｅｔ空间上的非游荡算子是一类具有混沌特征的线性算子 ,因而研究这类算子具有重要的意义 线性算子混沌要求其具有拓扑传递性 ,事实上拓扑传递性与超循环是一致的 ,而遗传超循环是更强的超循环 笔者首先给出超循环算子、混沌算子、遗传超循环算子以及非游荡算子的定义 ,列举了一个具体的非游荡算子 ,事实上文中列举的非游荡算子是线性混沌算子 。

Chaotic phenomena are not only restricted to nonlinear mappings or operators. In infinite dimensional space, certain linear mappings or operators do behave chaotically, which is a strange phenomenon, and will make the field of chaos richer. In infinite dimensional separable space, we will introduce the nonwandering operators, which are linear ones of chaotic character. So the research of this sort of operators is of great importance. Chaotic linear operators must satisfy the topological transitivity. In fact, topological transitivity is equivalent to hypercyclicity, while the hereditarily hypercyclicity is much stronger. Firstly, we give the definitions of hypercyclic operators, chaotic operators, hereditarily hypercyclic operators and nonwandering operators, then present a concrete nonwandering operator, which is actually a linear chaotic operator. Finally, we complete the hereditarily hypercyclic decomposition on the compact set of nonwandering operators in infinite dimensional separable Fréchet space.