Möbius transformations with cycles

This thesis consists of five chapters. The first chapter is devoted to the introduction section. In the second chapter, the basic required concepts for our study are given. In the third chapter, the basic definitions and theorems of Möbius transformations are considered. In the fourth chapter, basic definitions and theorems of 2-cycle Möbius transformations are discussed. In the fifth chapter, geometric interpretation of n-cycle Möbius transformations with the given theorems and definition and properties of n-cycle are given. In addition, the formation of 2-cycle and 3-cycle invariant circles is discussed, and the structure of invariant circles under n-cycles for n≥4 is investigated.