Some Studies Of M-Sensitivity In Topological Dynamical Systems

Let (X,f) be a compact system, that is (X,d) is a compact metric space,f:X?X is a continuous mapping. This thesis focuses on the study of sensitivity.The whole thesis is divided into four chapters.The first chapter is an introduction. In this chapter, some basic conceptions of dynamical systems and its historical background as well as some related known results were reviewed.In the second chapter, m- sensitivity and cofinite sensitivity are studied.Furthermore, the concept of m- confinite sensitivity is introduced, and for the cases of dynamical systems on lines, trees and finite sub-shifts, it is proved that confinite sensitivity and m- confinite sensitivity are equivalent for any m32. Moreover, it is shown in this chapter that a strong mixing system is m- confinite sensitivity for any m32.In the third chapter, by the properties of families and a sequence of continuous mappings satisfying some conditions, the equivalent conditions for a compact system to be m- sensitive, m- ergodic sensitive, m- Banach upper density sensitive and m- confinite sensitivity are obtained, respectively. After those, the relation between the sensitivity of a compact system and the transitivity of the product system with itself is studied.In the fourth chapter, by the idea of statistics, the new conceptions of m- mean sensitivity and m- equicontinuity are introduced, some equivalent conditions for a compact system to be m- mean sensitivity are gotten and the properties of m- mean sensitivity are studied.