Analyticty of Normalizing Transformations for Area Preserving Maps

Abstract

It was proved by Poincaré ^[1] that hamiltonian systems with two degrees of freedom cannot have real analytic integrals of motion beyond the hamiltonian itself. His theorem excludes analyticity in open sets of real phase space, in agreement with KAM theory ^[2,3,4] on the existence of a Cantor set of invariant manifolds, which admit a C^∞ interpolation ^[5,6,7]. The normalizing transformations are given by asymptotic series, whose remainders can be made exponentially small according to Nekhoroshev ^[8]. We have analyzed a related problem, namely the analyticity properties of an area preserving map in the neighborhood of an elliptic fixed point, by considering its extension to C ². If the linear frequency is irrational then it is possible to conjugate, by a polynomial diffeomorphism of order N, the map with its normal form up to a remainder, analytic in a polydisc whose radius decreases with N, vanishing as N → ∞ ^[9,10]. By extending Moser’s proof ^[11] on the existence of invariant curves, we determine a domain of C ² where the remainder can be set to zero. In polar coordinates the analyticity domain of the normalizing transformation is given by a strip in the angle complex plane and, in the radius complex plane, by the complement of the disc with respect to a Cantor like set of measure exponentially small, which intersects the real and imaginay axis. This is consistent with Neishtadt estimates on the real axis ^[12] on the measure of KAM tori.