We prove a discrete time analogue of Moser’s normal form (1967) 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 codimension. Under convenient nondegeneracy assumptions on the diffeomorphisms under study (a torsion property for example), this codimension can be reduced. As a by-product we obtain generalizations of Rüssmann’s translated curve theorem in any dimension, by a technique of elimination of parameters.
"A normal form à la Moser for diffeomorphisms and a generalization of Rüssmann's translated curve theorem to higher dimensions." Anal. PDE 11 (1) 149 - 170, 2018. https://doi.org/10.2140/apde.2018.11.149