The Iteratively Degree-Preserving Planar Maps And The Solutions Of One Iterative Equation

Nonlinear science has become a hot topic in which its branch-iterative dynamical system(IDS) plays a key role. Results of IDS in chaos have brought a sccond surge on the overall human knowledge as had done the theory of relativity and quantum mechanics. Iterative equations, employing iteration as the basic operation and through intrinsic connections with dynamic system, differential equations, difference equations, and integral equations, have profound influence on natural sciences and technological development. The preface of this thesis summarizes the history of dynamic system and iterative equation, and poses the problems the thesis tries to solve.Chapter 2 provides some results on conjugacy, which is the most basic and important tool to extract certain characteristics of dynamic systems. We will introduce briefly some results on Schroder equation, the properties of conjugacy, and provide concrete examples on the applications of conjugacy in normalization of dynamic systems.In Chapter 3 is devoted to the study of iteratively degree-preserving planar polynomial maps. Computation of iterates of a polynomial map is difficult. Iteration increases the degree of a 1-D nonlinear polynomial map sharply. But an interesting phenomenon is that some examples of 2-D nonlinear polynomial maps whose iterates have degrees no greater than themselves do exist. To find this kind of polynomial maps is a work of biggish computational efforts and also...