| Since Li and Yorke proposed the exact definition of chaos in 1975, chaos theory flourished as a new discipline. Although it’s widely used in the project, but its theoretical research is seriously lagging behind. Especially the basic research in mathematics of chaos. This paper will study on this question. We focus on the chaotic properties of Li-Yorke chaos and distributional chaos of Linear operators in two spaces(Banach space and hyperspace), and we gain the following results:We give a study on the relation between topological mixing and DC3 of Linear operator on plspace. Firstly we gain some properties of DC3. By using the results, we prove: topological mixing doesn’t imply DC3 for weighted backward operator space.We study the properties of distributional chaotic operator on Banach space. Firstly we discuss on distributional 2-type chaos. We gain five equivalent propositions of distributional 2-type chaos. Then we study the product properties of distributional 1-type chaotic operators. Through the research on distributional 3-type chaos, we get the conclusion that if one bounded linear operator is DC3, it’s norm must greater than 1 on Banach space. Finally, using the results, we prove: the three type distributionally chaos is equivalent in backward operator space.In this section we study on the properties of Li-Yorke chaos and distributional chaos in hyperspace. Firstly, we propose the definition of operator hypernorm in hyperspace, and verify that it’s well defined. Then we gain four equivalent proposions of Li-Yorke chaos and one criterion of distributional chaos in hyperspace. |
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