Flexibility and analytic smoothing in averaging theory

TL;DR

This paper extends Nekhoroshev's stability estimates to H"older regular steep near-integrable Hamiltonian systems, establishing polynomial stability times and radii through a novel analytic smoothing approach and geometric analysis of resonant blocks.

Contribution

It introduces a new perturbation theory leveraging sharp analytic smoothing to handle H"older functions, extending stability estimates to broader classes of Hamiltonian systems.

Findings

Stability times are polynomial in inverse perturbation size.

Explicit stability exponents depend on regularity and steepness indices.

The approach applies to both steep and convex Hamiltonian systems.

Abstract

Using a new strategy, we extend the classical Nekhoroshev's estimates to the case of H\"older regular steep near-integrable hamiltonian systems, the stability times being polynomially long in the inverse of the size of the perturbation. We prove that the stability exponents can be taken to be $(\ell-1)/(2n\alpha_1...\alpha_{n-2})$ for the time of stability and $1/(2n\alpha_1...\alpha_{n-1})$ for the radius of stability, $\ell >n+1$ being the regularity and the $\alpha_i$'s being the indices of steepness. Our strategy consists in deriving a perturbation theory which exploits a sharp analytic smoothing theorem to approximate any H\"older function by an analytic one. In addition, an appropriate choice of the free parameters in the problem enables us to have a first grasp on the relation connecting the time and radius of stability to the threshold that the size of the perturbation must satisfy in order for the theorem to apply. Particular attention is payed to a geometric presentation of the construction of the so-called "resonant blocks", in order to shed a definitive light on the nature of the steepness condition. We also investigate the convex setting, using a similar approach.