Normal form \`a la Moser for diffeomorphisms and generalization of R\"ussmann's translated curve theorem to higher dimension

We prove a discrete time analogue of 1967 Moser’s normal form of real analytic perturbations of vector fields possessing an invariant, reducible, Diophantine torus; in the case of diffeomorphisms too, the persistence of such an invariant torus is a phenomenon of finite co-dimension. Under convenient non-degeneracy assumptions on the diffeomorphisms under study (torsion property for example), this co-dimension can be reduced. As a by-product we obtain generalizations of R"ussmann’s translated curve theorem in any dimension, by a technique of elimination of parameters.