Hyperbolic low-dimensional invariant tori and summations of divergent series

TL;DR

This paper investigates the regularity and summation of divergent series for low-dimensional hyperbolic invariant tori in quasi-integrable Hamiltonian systems, introducing a novel resummation criterion based on renormalization group methods.

Contribution

It establishes the existence and analytic structure of hyperbolic invariant tori under perturbations and develops a new summation method for their divergent asymptotic series.

Findings

Invariant tori exist under nonresonance conditions.

Tori are analytic inside a specific disc with a quadratic cusp.

A summation criterion for divergent series representing the tori is proposed.

Abstract

We consider a class of a priori stable quasi-integrable analytic Hamiltonian systems and study the regularity of low-dimensional hyperbolic invariant tori as functions of the perturbation parameter. We show that, under natural nonresonance conditions, such tori exist and can be identified through the maxima or minima of a suitable potential. They are analytic inside a disc centered at the origin and deprived of a region around the positive or negative real axis with a quadratic cusp at the origin. The invariant tori admit an asymptotic series at the origin with Taylor coefficients that grow at most as a power of a factorial and a remainder that to any order N is bounded by the (N+1)-st power of the argument times a power of $N!$. We show the existence of a summation criterion of the (generically divergent) series, in powers of the perturbation size, that represent the parametric equations of the tori by following the renormalization group methods for the resummations of perturbative series in quantum field theory