The Dependence Of Metric Properties Of Generalized Continued Fraction On Parameter Function

The continued fraction of a number indicates the superiority of the fractional representation,therefore,it is widely used in many fields,such as number theory,probability theory,function approximation and computational coding,etc.With the Gauss transform,we can obtain the regular continued fraction expansion,namely regular continued fraction(RCF)expansion.Meanwhile,the general continued fraction expansion of a parameter is obtained by using a kind of transformation with parameters,namely generalized continued fraction(GCFw)expansion.Due to the variability of the parameter w,the GCF expansion has greatly enriched the number of continued fraction expansions.In this paper,the GCF expansion of the number is given by the method of function iteration,and make the definition of expansion more concise and easy to understand.Meanwhile,the form of the expansion is similar to that of regular continued fraction,so it is easier to compare and study.Some existing conclusions of GCF expansion are summarized,we also study the Hausdorff dimension of the Hirst set of GCF of several parameter w,and compare it with the Hirst set of regular continued fraction of Hausdorff,and find that there are two times the relationship,while others are not multiples.Finally,we discuss the asymptotic fraction and put forward some new problems.