Iterative Functional Equation And Invariant Curves Of The Mapping

Iteration is a general phenomenon in natural science and human life.Theory of iterative functional equation is a,immemorial history,abundant content, comprehensive application,branch of modern mathematics.Through years of development,iterative functional equation is closely related to differential equation,difference equation,integral equation and dynamical system,playing an important role in the study of experimental science and engineering.Iterative functional equation is the outcome of function compound and iteration,and,just like differential equation,is a special type of function equation.Accurately speaking,iterative functional equation is the identical equation formed by unknown function and compound operation.Since mathematicians like Babbage,Abel ets., iterative functional equation has formed a theory system with the development of iterative theory.In the introduction part of the thesis,the author introduces invariant curve of iterative equation.A brief introduction is also given about the achievements made in the field invariant curve in recent years.In this thesis,Chapter 2 discusses the characteristics of functional iteration, the concepts of iteration and dynamical system,the problems of iterative root,and,the codceptsof employing the method of Cauchy majorant seriesInvariant curves of the area preserving maps play an important role in the theory of periodic stability of discrete dynamical systems.It has an important effect on the existence of analytic invariant curves of a area mapping. In chapter 3,we are concerned with the existence of analytic invariant curves of a complex mapping.Firstly,we reduce the existence of analytic invariant curves to the existence of an iterative functional equation,Then we use the Schr(o|¨)der transformation to change the iterative functional equation to another without iterates of the unknown function.Further,we obtain the existence of analytic solutions of such an equation by means of majorant series,so we get an analytic invariant curves.Employing the method of majorant series,we need to discuss the eigenvalueαof the mapping at a fixed point.Besides the hyperbolic case 0＜|α|＜1,we focus on thoseαon the unit circle S1,i.e.,|α|=1.We discuss not only thoseαat resonance,i.e.at a root of the unity,but also thoseαnear resonance under the Brjuno condition.