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

TL;DR

This paper develops a normal form theory for real analytic perturbations of diffeomorphisms with invariant tori, extending R"ussmann's translated curve theorem to higher dimensions through parameter elimination techniques.

Contribution

It introduces a discrete-time analogue of Moser's normal form for diffeomorphisms and generalizes R"ussmann's theorem to higher dimensions with reduced co-dimension.

Findings

Proves a normal form for perturbed diffeomorphisms with invariant tori.

Generalizes R"ussmann's translated curve theorem to higher dimensions.

Shows that under certain conditions, the co-dimension of invariant tori can be reduced.

Abstract

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.