Problems On Linear Li-Yorke Chaos Under The Sense Of Weak Topology

Chaos theory,relativity and quantum mechanics are the most important discoveries in physics in 20th century.Chaos theory created a new paradigm of scientific modeling and provided us with new ideas for understanding the laws of the physical world and the development of human society.Therefore,the theoretical and experimental research of chaotic phenomena has been the main direction in nonlinear scientific research in the past half century.In this thesis,we summarize some properties of Li-Yorke chaos,Li-Yorke chaos under the sense of weak topology,hypercyclic vector,hypercyclic vector under the sense of weak topology,mean Li-Yorke chaos and mean Li-Yorke chaos under the sense of weak topology with respect to some linear functional x*.We explore the relevant conclusions of Li-Yorke chaos under the sense of weak topology by the method in the proof of Theorem 10.24 in Chapter 10 of[1],and we first gives a class of shift operators and proves that this type of operator is Li-Yorke chaotic under the sense of weak topologybut the scrambled set under the sense of weak topology is not necessa.rily the scrambled set under the sense of the ordinary norm;we study whether or not some results on Li-Yorke chaos in[2]are still valid for the Li-Yorke chaos under the sense of weak topology.Finally,we prove that some theorems in[46]concerning the relationship between mean Li-Yorke chaos and absolute mean irregular vectors are still valid for mean Li-Yorke chaos under the sense of weak topology.This thesis is divided into four chapters:Chapter 1 introduces the research background and structure of the article.Chapter 2 introduces some definitions and theorems of Li-Yorke chao,s,Li-Yorke chaos under the sense of weak topology,hypercyclic vector,hypercyclic vector under the sense of weak topology,mean Li-Yorke chaos and mean Li-Yorke chaos under the sense of weak topology.Chapter 3 is divided into four parts:(1)The proof of theorem 10.24 in[1]is supplemented and perfected.(2)According to the proof method in the first section,it is proved that the scrambled set under the sense of weak topology is not necessarily the scrambled set under the sense of norm topology.(3)It is proved that the theorem related to Li-Yorke chaos in[2]is no longer valid for Li-Yorke chaos under the sense of weak topology.(4)We show that some results in[46]on mean Li-Yorke chaos are still true for the mean Li-Yorke chaos for x*under the sense of weak topology.Chapter 4 summarizes the paper and the outlook for future research work.