The Classification Of D4-equivariant Mapping Germs Under The Left-right Equivalence

Since Golubitsky and Schaeffer have introduced the idea of appling the method of singularity theory and group theory to study the thought of bifur-cation problems,this has made bifurcation theory rapidly development.In the bifurcation theory,an important research topic is the study of variable bifur-cation problem under the equivalent of the standard form,to meet what kind of condition,bifurcation problem is equivalent to the given standard form,and gives a classification and recognition.This requires that we must analysis the characteristics of the tangent space and Power single tangent space to the orbit of standard form under the equivalent group.So we need to carefully analysis and discussion algebraic properties of equivalent bifurcation problem.In this paper,we study the algebraic properties of D4-equivariant mapping germs which have the compact Lie group D4 as the symmetry group,and no bifurcation parameter under the left-right equivalence group.We give the Hilbert basis of D4-invariant function germs and the generating elements of the model constituted by D4-equivariant mapping germs.According to this,we obtain the generating elements of the Power single tangent space and tangent space and space under the left-right equivalence group.Then we obtain the codimension after discussing and analysing the tangent space.We also classify D4-equivariant mapping germs of the topological codimension no more than 2 under the action of group D4 and obtain the corresponding conclusion.