On the characterization of circle-preserving maps

In this research, consists of six chapters. The first chapter is devoted to the introduction section. In the second chapter, some required preparatory notions recalled. In the third chapter, Möbius tranformations properties were studied. In the fourth chapter, non-degenerate maps and degenerate circle preserving maps were studied. İn the fifth chapter, praperties of sphere-preserving maps are considered. In the final chapter, characterizations of sphere-preserving maps and degenerate sphere-preserving maps were examined.