Some Studies On The Oppenheim Continued Fraction Expansions

The Oppenheim continued fractions are a class of continued fraction expansion-s depending on parameter function,first studied by Fan,Wang and Wu[14],which extend the classical regular continued fraction and the so-called Engel continued frac-tion(see[23])formally.This thesis is mainly concerned with some distribution and metric properties of the Oppenheim continued fractions,it turns out that this kind of continued fractions share many common distribution and metric properties for various parameter functions h.Under certain necessary hypothesis on the parameter function,a Gauss-Kuzmin-Levy type theorem for the Oppenheim continued fractions is established,from which the nearly ?-mixing property of the digit sequence {dn(x)}n?2 is derived.The extreme value distribution in the Oppenheim continued fraction expansions is also considered,it turns out that a certain sequence related to {dn(x)}n?1 satisfies the Frechet law in the classical extreme value theory.Besides,some limit theorems in[14]are reproved by using the aforementioned distribution and mixing properties.This thesis is organized as follows:In Chapter 1,some backgrounds on the representation of real numbers are briefly introduced,in the meantime,the definition of the Oppenheim continued fraction ex-pansions is given.Chapter 2 is devoted to the introduction of some basic arithmetic properties of the Oppenheim continued fraction expansions.Besides,an inequality on the mathematical expectation of 1/dn(x)which will be extensively used in the next two chapters is proved.Chapter 3 is mainly concerned with the distribution properties of the Oppenheim continued fractions.Under certain hypothesis on the parameter function,a Gauss-Kuzmin-Levy type theorem as well as the nearly ?-mixing property of {dn(x)}n?1 are established.The Prechet law in classical extreme value theory is found to be satisfied in the realm of the Oppenheim continued fraction expansions.Besides,the sequence{Rn-1(x)} is shown to be uniformly distributed mod 1 for almost every x,which has the flavor of renormalization of algorithms introduced by Hubert and Lacroix(see[25]),In Chapter 4 some limit theorems in[14]are reproved by using the distribution and mixing properties established in Chapter 3.A new result being proved is that the strong law of large numbers for the sequence {Rn(x)}(for the definition see Section 3.3)does not hold.In the last chapter some discussions on the parameter function h are made.In one direction,the ergodic properties of the homogeneous Oppenheim continued fraction map are investigated.In another direction,what will happen when the hypothesis in Chapter 3 and 4 does not,hold is discussed by virtue of an example.