| One defines two ways of constructing rational maps derived from other rational maps, in a characteristic-free context. The first introduces the Newton complementary dual of a rational map. One main result is that this dual preserves birationality and gives an involutional map of the Cremona group to itself that restricts to the monomial Cremona subgroup and preserves de Jonquières maps. The second construction is an iterative process yielding, in particular, infinite families of Cohen--Macaulay Cremona maps with prescribed dimension, codimension and degree. |