Some Studies Of Positive Upper Banach Density Recurrent Points

In this thesis, we introduce two new concepts of proper positive upper Banach density recurrent points minimal systems and proper positive upper Banach density recurrent points minimal semi-flows, and study the dynamical properties of such dynamical systems. The full thesis is divided into three chapters.In the first chapter,It is an introduction. In the introduction,review the historical background and some known achievements in topological dynamical systems.In the second chapter, we firstly introduce the new conception of proper positive upper Banach density recurrent points and proper positive upper Banach density recurrent points minimal system.Suppose that fX),( is a proper positive upper Banach density recurrent points minimal system, we prove that f is Takens-Ruelle chaotic; fX),( is an E- system, which implies that f is strongly ergodic and is totally ergodic sensitive. Furthermore, we point out that the property of proper positive upper Banach density return points minimal system is topological conjugation invariant property. At the end of this chapter, we give some equivalent definitions of positive upper Banach density recurrent points and prove that such a system is strongly ergodically chaotic.In the third chapter, we firstly introduce a new conception of proper positive upper Banach density recurrent points in flows and give an example to show that the set of weakly almost periodic points is proper contained in the set of positive upper Banach density recurrent points. Let FX),( be a proper positive upper Banach density recurrent points minimal semi-flow, we obtain that F is Takens-Ruelle chaotic; FX),( is an E- system, so F is ergodic. Furthermore, in the case of flows,we point out that the property of proper positive Banach density return points minimal system is a topological conjugation invariant property.