Flexibility and analytic smoothing in averaging theory

Abstract

Using a new strategy, we extend the classical Nekhoroshev's estimates to the case of Hölder 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 ( − 1)/(2nα 1 ...α n−2 ) for the time of stability and 1/(2nα 1 ...α n−1 ) for the radius of stability, > n + 1 being the regularity and the α 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 Hölder 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. Contents 1. Introduction and main results 1 2. General setting and classical methods: a geometric framework 6 3. Functional setting 13 4. Analytic smoothing 15 5. Stability estimates in the convex case with the patchwork method 20 6. Stability estimates in the convex case with the periodic averaging method 27 7. Stability estimates in the steep case 32 Appendix A. Smoothing estimates 37 Appendix B. Analytic and arithmetic tools for the convex case 38 References 40