Continued Fractions And Kleinian Groups

It is well-known that any continued fraction may be regarded as infinite compound of a sequence of Mo|¨bius transformations. This shows that these two research areas in complex analysis are closedly related. In the 20th century, the theory of continued fractions was developed increasingly by the works of Jones, Thron, Lisa, Andrews, Berndt etc. It has been applied in transcendental functions, control theory, asymptotic theory, dynamic system, q-series and so on. When Andrews and Berndt arranged Ramanijan's manuscript, they started their research concerning the problems in Ramanijan's manuscript. In recent years, by using the expression of Clifford matrices of higher dimensional Mo|¨bius transformations, Beardon started the study of Clifford continued fractions (that's, higher dimensional continued fractions), and obtained many interesting and improtant results. For example, he has generalized some famous results in the classic continued fractions such as Pringsheim Theorem, Hillam-Thron Theorem and Parabola Theorem to the case of Clifford continued fractions. In this way, he established the base for the further study of Clifford continued fractions. Also he raised sevaral related open problems. A Clifford matrix has the same form as the expressed matrix of a 2-dimensional Mo|¨bius transformation. Hence the expression of Clifford matrices of higher dimensional Mo|¨bius transformations provides some methods for the study of the higher dimensional Mo|¨bius transformations and groups. These interesting and significant studies stimulate our interest in the study of continued fractions and higher dimensional Mo|¨bius transformations and groups. Our research will focus on these areas. We mainly discuss Beardon's open problems in Clifford continued fractions, the properties of continued fractions of generalized Rogers-Ramanujan type and the discreteness of the normalizers of higher dimensional Kleinian groups. This dissertation is arranged as follows.In Chapter 1, we present the backgroud information about our research, and the statement and the significance of our main results.In Chapter 2, we introduce some basic notions, notations and properties about continued fractions, higher dimensional Mo|¨bius transformations, Clifford algebra, Clifford matrix, Clifford continued fractions, Rogers-Ramanujan continued fractions and continued fractions of generalized Rogers-Ramanujan type.In Chapter 3, we define value regions and element regions in Clifford continued fractions. By constructing some sequences of value regions and element regions, we establish Nested Region Theorem in Clifford continued fractions. As an application of our obtained results, we get a class of convergent Clifford continued fractions.In Chapter 4, we establish the three-term recurrence relations, Stern-Stolz Theorem and Pincherle Theorem in Clifford continued fractions. An application of the obtained Pincherle Theorem is also given. In the end, three properties of the minimal solutions of Clifford continued fractions are proved.In Chapter 5, we establish a sufficient condition for the convergence of Clifford continued fractions, and two applications are presented.In Chapter 6. we first discuss the convergent of continued fractions of generalized Rogers-Ramanujan type. Second, we get some equalities between continued fractions of generalized Rogers-Ramanujan type and q-series.In Chapter 7, we prove a necessary and sufficient condition for the normal-izers of higher dimensional Kleinian groups to be discrete.