The Normal Form Of 3-dimensional Non-hyperbolic Mappings And Related Cohomology Equations

Normal form theory is an effective tool for studying the behavior of a system near a fixed point,which is of great significance to the study of dynamical system theory.Its idea originates from Poincare,which simplifies complicated problems by finding a simpler vector field or mapping consistent with the given vector field or mapping in the topological structure.Cohomology equation is often involved in solving the normal form problem.At the same time,this type of equation has important applications in other theories of dynamical systems.Therefore,this paper will mainly study the problem of normal form and cohomology equation.The famous Hartman-Grobman theorem proves that a hyperbolic system near its fixed point is topologically equivalent to its linear part.This theorem equates a hyperbolic nonlinear system with a linear system,which is the simplest normal form result.However,the study of non-hyperbolic systems is more complicated than that of hyperbolic systems.According to the current research results,we can see that there are satisfactory results for the unipotent case in two-dimensional spaces.However,so far there is still room for improvement in relevant conclusions of three-dimensional cases.Thus this paper will consider the normal form of the three-dimensional unipotent mappings.In addition,we will use the theory of functional analysis to find the smooth solutions of cohomology equations.The paper is divided into two parts:In the first part,we consider the normal form of three-dimensional mappings near the fixed point.In the classical theory,an effective method is to use approximate identical polynomial transformation to eliminate the higher-order terms of the system in turn and obtain the formal normal form.In this part,we will use this idea to further consider the normal form of three-dimensional mappings in unipotent cases,which is simpler than previous results.In the second part,we consider the problem of C1 and C2 solutions for the coho-mology equations.We will construct a special mapping through a kind of cohomology equations,which can be transformed into two general functional equations.In this way,the smooth solutions of the cohomology equations will be transformed into the smooth solutions of general functional equations.