Busemann Functions And Prime Ends

The notation of prime ends was first constructed by Caratheodory. This notation provides researchers a good tool to research relating problems. After the prime ends by Caratheodory, researchers describe the prime ends in some other ways and use the idea to other fields, such as studying the Julia set in complex dynamics. Busemann function is a continuous function on unbounded complete metric space. And we can apply Busemann function to discuss the properties of unbounded boundary at infinity. Since any simple connected domain with more than two boundary points on the complex plane can be endowed with Poincare metric, and the Poincare metric is a unbounded and complete metric on such domain, we can define the Busemann function in the sense of Poincare metric.In this article we discuss the prime ends by applying the theory and methods of Busemann function. In chapter two, we introduce the prime ends, some properties about Poincare metric, and Busemann function. In chapter three, we research the cor-responding relations between Busemann function and the horoball on a unit circle and boundary points. Moreover, we study their relations with prime ends.