Planar Quadratic Degree-Preserving Maps and Their Iteration

Abstract.

Usually it is a difficult work to compute a general iterate of a polynomial map because iteration increases the degree of a 1-dimensional nonlinear polynomial map sharply. However, we can find examples of higher dimensional nonlinear polynomial map which preserve their degrees under iteration. This motivates us to identify those degree-preserving maps because it is possible to give an elementary expression of general iterate for such a map. In this paper we first find all 2-dimensional degree-preserving maps of degree 2 by pseudo-division elimination and algebraically partition the class of degree-preserving maps into 11 subclasses. Then we prove the density and pathwise connectedness of the degree-preserving class. Furthermore, we give the eventual periodicity of those subclasses under iteration so as to trace the procedure of iteration in the degree-preserving class. Finally, we display the computation of general iterate with some examples of degree-preserving maps.