Some Studies On The Stability Of Measures Of Topological Dynamical Systems

Topologically stable measure is an important concept in topological dynamical systems.In this thesis,we carry out the researches on the topologically stable measures of semi-discrete dynamical systems and non-autonomous dynamical systems.The detailed arrangements are as follows:The first chapter is to describe the development backgrounds of dynamical systems and the recent researches on topologically stable measures.Also,the source of the considered problems in this thesis and the relative concepts of semi-discrete dynamical systems are introduced.The second chapter is to introduce the concept of topologically stable measures of semi-discrete dynamical systems and explore some basic properties of topologically stable measures.Then in this chapter,the dynamics of semi-discrete dynamical systems which can be approximated by semi-discrete dynamical systems with a certain dynamical properties and admit topologically stable measures are studied,and it is proved that the original semi-discrete dynamical system possesses the same dynamics to those tending to it,such as sensitivity and transitivity.In the third chapter,under a certain conditions,the complexity of non-autonomous dynamical systems with topologically stable measures is investigated.It is showed that the non-autonomous dynamical system with topologically stable measures admits the same dynamics to those non-autonomous systems which are sufficiently closed to it.The fourth chapter is a summary and prospect of this thesis.This chapter presents some problems for us to consider in future.