Some Geometric Properties Of Pseudo Hyperbolic Metric And Pseudo Metric On Unit Disk

M(?)bius transformation is an important part of conformal transformation.It is widely used in number theory,equation,statistical mechanics and other fields.Pseudo hyperbolic metric is obtained through the absolute value of M(?)bius transformation.It has a wide range of applications in characterizing operators on analytic function spaces and symbolic functions of operators.Pseudo hyperbolic disks can be represented by Euclidean disks one by one.In this thesis,the Euclidean disks in the unit disk can be expressed as pseudo hyperbolic disks one by one,and the necessary and sufficient conditions for the separation of special pseudo hyperbolic disks are obtained by using this property.In addition,we define the ratio of Euclidean metric and pseudo hyperbolic metric as pseudo metric,and explore the geometric properties of pseudo metric in the contour of Euclidean unit circle.This thesis is divided into four parts,which respectively introduce the research status of pseudo hyperbolic metric and pseudo metric on unit disk,as well as the relevant properties and conclusions of pseudo hyperbolic metric,pseudo hyperbolic disk and pseudo-hyperbolic metric,and the calculation process.At the end of the thesis,the conclusion is summarized and prospected.The details are as follows: Chapter 1 introduces the research status of M(?)bius transformation and pseudo hyperbolic metric,and gives the research ideas and main contents of this thesis.Chapter 2 gives the definition of M(?)bius transformation and pseudo hyperbolic metric,and introduces some geometric properties of pseudo-hyperbolic metric on the unit disk,so as to deeply explore the relationship between Euclidean disk and pseudo hyperbolic disk.Chapter 3 explores the separation of two special pseudo hyperbolic disk arrays.They are respectively the problem of the separation of the pseudo hyperbolic disk array when the radius of the pseudo-hyperbolic disk is fixed and the center of the pseudo hyperbolic disk approaches 1 along the positive half axis of the axis at a certain rate.When the radius of the pseudo-hyperbolic disk and the center of the pseudo hyperbolic disk change at a certain rate at the same time,the problem of the separation of the corresponding pseudo hyperbolic disk array.Chapter 4 gives the definition of pseudo metric,and some geometric properties of pseudo metric in the unit disk are given,and the contours of pseudo-measures in the Euclidean unit disk are explored.