The Study Of Sensitive Dependence And Scattering In Dynamical Systems

There are three parts in this paper.In the first part(Chapter 1), we describe the origination and development of the theory of dynamical systems, and give a brief survey of the background and recent progress of complexity in dynamical systems.In the second part(Chapter 2), we study the sensitive dependence of semigroup actions. The main results are as follows: Weak-mixing implies sensitive dependence for measure-preserving maps and measure-preserving semiflows. We describe the relations of sensitive dependence with M systems, E systems, and give two sufficient conditions for semigroup actions to be sensitive dependence. Finally, we establish the connection between mixing and spatiotemporal chaos.In the third part(Chapter 3), we study the complexity of semigroup actions using complexity functions of open covers. The main results are as follows: The bounded-ness of complexity functions is equivalent to equicontinuity. Topological weak-mixing implies scattering, and the problem whether 2 scattering is equivalent to scattering is studied. We get a criterion for the scattering property, and characterize mild mixing and strong scattering using the complexity function of an open cover along some infinite nets.