On Gauss-Kuzmin Theorem And Related Problems For A Family Of Continued Fraction Expansions

In this dissertation,we are concerned with Gauss-Kuzmin theorems and relevant problems for the new continued fraction expansions related to a class of random Fibonacci type sequences.This type of continued fraction expansions were introduced by Chan in 2006,for a given positive integer l≥2 and any x∈[0,1),which is expressed as follows:where an(x)∈N={0,1,2,…} is called the n-th partial quotient or digit of x in this type of continued fraction expansions.This continued fraction expansion can be induced by the transformation τl:[0,1)→[0,1)as follows:where[u]denotes the integer part of a real u.Using Wirsing-type approach and the theory of random systems with complete connections(RSCC)Sebe and Lascu took up the GaussKuzmin-type problems for this type of continued fraction expansions in 2010 and 2013 respectively.For this work,on the one hand,we use Khintchine’s method to re-examine the Gauss-Kuzmin theorem of this new-type of continued fraction system;on the other hand,we obtain the two-dimensional Gauss-Kuzmin theorem of this type of continued fraction expansions.Including the first chapter of introduction and the second chapter of preliminaries,there are five chapters in the thesis.In Chapter three,for the one-dimensional case,we give an initial form of the GaussKuzmin theorem for τl,and then deduce the weak independence of partial quotient sequences(an)n∈N+under some given probability measure.An improvement on this Gauss-Kuzmin theorem is obtained by employing the properties of the Perron-Frobenius operator of τl under its invariant measure γl on the Banach space of Lipschitz continuous functions,that is,for the Lebesgue measure λ and any Borel set A(?)[0,1]we have|λ(τl-n(A))-γl(A)|<Cλ(A)qn,where 0<q<1 and C is an universal constant.In Chapter four,we consider the natural extension of this new continued fraction system.For any n∈N+ N+and(x,y)∈[0,1]2,letΦn(x,y)=λ({(χ,y)∈[0,1]2:τln(χ,y)∈[0,x]×[0,y]}),where λ is the Lebesgue measure on[0,1]2 and τl represents the natural extension of τl.We prove a Gauss-Kuzmin theorem for the natural extension τl,that is,for any(x,y)∈[0,1]2 and n≥ 2,we haveΦn(x,y)=γl(x,y)+O(δn),where 0<δ<1,γl is the extension of the probability measure γl.Then for a family of conditional probability measures,we conclude that the convergence rate of pertinent distribution function to its limit γl(x,y)is O(vn).where.Ultimately,in Chapter five,we summarize the main results of this dissertation,and raise some questions for further study.