Dickson, Polynomial Mapping Fixed Point Number

Let OP, Ip be the valuation ring and prime ideal of the prime divisor P ofthe function field Ep and let Re = p/e be the residue ring of Op modulo Iep. In thispaper, we give a necessary and sufficient condition for those mappings of the ring Re that are induced by Dickson polynomials gk(t,a) to be permutation mappings. And we determine the number of fixed points of gk(t,a) by transforming this problem into the question of determining the numbers of the special equations over Re and some extension of Re,when gk(t,a) is a permutation mapping and ak-1 =1.