| Blaschke product is one important kind of inner functions.The properties of Blaschke products are closely related to the distribution of zero sequences,so we can get the relations between Blaschke products and convergence,con-nectivity,interpolation Blaschke products,one-component inner function.The topological conjugate classification of the upper half plane is given by Chang Wei and Hou Bingzhe in " Topologically Conjugate Classifications for Holomor-phic Automorphisms on Unit Disk and Upper Half Plane.",the corresponding Mobius transformation is considered in this paper,and whether all zero se-quences after n,n∈N iterations can constitute an infinite Blaschke product.Since the upper half plane is holomorphic conjugate with the unit disk,we can discuss it by using the classification in"Topologically Conjugate Classifica-tions for Holomorphic Automorphisms on Unit Disk and Upper Half Plane.".By mapping the holomorphic analytic automorphisms of the upper half plane to gi(z)corresponding to hyperbolic,parabolic and elliptic,through holo-morphic conjugate h(z),(?),i=1,2,3,it is mapped to holomorphic analytic automorphisms(?),where A(z)∈D,to find the corresponding zero point {an}.At this time,except for the zero sequences {an}n=1∞ of the elliptic,it can converge to a point on the boundary.Finally,it can be obtained that except for the elliptic,which zero sequences cannot convergence to the boundary of unit disk,so it cannot constitute the Blaschke product,the other two cases,named hyperbolic and parabolic,can constitute the interpolated Blaschke product,which is one-component interpo-lated Blaschke products.Similar problems have been considered in the article"One-component inner functions",but the general parabolic type has not been considered.Here,we give a proof of the general parabolic type through detailed calculation,we will also give a new proof of hyperbolic.This article mainly includes the following items:1.We introduce the background of the research problem,that is,the space of Hp,inner function,Blashcke product and so on.2.We introduce the related theorems of one-component inner function and interpolation function,and introduce an example of Blaschke product which is not absolutely convergent3.We prove in details that the Blaschke products generated by the zeros of hyperbolic and parabolic analytic automorphisms iterations has the properties of interpolation Blaschke products and one-component inner function. |
