The Properties And Generalizations Of Inner Function And Winding Number

This thesis mainly studies the properties of inner functions,Blaschke prod-ucts are special inner functions. By the equivalent class division for inner functions, we get some relationships of two exceptional sets of inner functions.This thesis consists of seven chapters, the detailed arrangement is as fol-lows.In the first chapter, we introduce some research background, recent devel-opment and the significance of inner functions and winding number. Besides, We introduce some main results of this thesis.The second chapter focuses on the properties of two exceptional sets E1,E2 of inner functions, as well as the properties of infinite Blaschke products, which are special inner functions.In the third chapter, we introduce definitions and properties of the Mobius transformation of inner functions, as well as indestructible Blaschke product-s. Besides, We get the sufficient and necessary condition of indestructible Blaschke products.In the fourth chapter, by the equivalent class division for inner functions, we study some important properties of destructible Blaschke product, that is to say, E2 is not emptyIn the fifth chapter, We firstly introduce some basic concepts about com-pressed iteration function systems and geometric properties of attractor K with strong separation conditions satisfied, secondly, We give an example of bounded and hydrophobic complete set with a positive measure.In the sixth chapter, We generalize properties of winding number, We get some new properties of winding number under composite mapping of fractional linear transformation and polynomial. Besides, We also get the relationship between winding number and topological degree of analytic functionThe seventh chapter is the biggest innovation in this thesis, We get a sufficient condition for the establishment of K.Stephenson conjecture and give some examples.