  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.
