Hamiltonian Perturbation Theory for Ultra-Differentiable Functions.

Format:

Book

Author/Creator:

Bounemoura, Abed.

Contributor:

Féjoz, Jacques.

Series:

Memoirs of the American Mathematical Society

Memoirs of the American Mathematical Society ; v.270

Language:

English

Subjects (All):

Hamiltonian systems.

Physical Description:

1 online resource (102 pages)

Edition:

1st ed.

Place of Publication:

Providence : American Mathematical Society, 2021.

Summary:

"Some scales of spaces of ultra-differentiable functions are introduced, having good stability properties with respect to infinitely many derivatives and compositions. They are well-suited for solving non-linear functional equations by means of hard implicit function theorems. They comprise Gevrey functions and thus, as a limiting case, analytic functions. Using majorizing series, we manage to characterize them in terms of a real sequence M bounding the growth of derivatives. In this functional setting, we prove two fundamental results of Hamiltonian perturbation theory: the invariant torus theorem, where the invariant torus remains ultra-differentiable under the assumption that its frequency satisfies some arithmetic condition which we call BRM, and which generalizes the Bruno-Russmann condition; and Nekhoroshev's theorem, where the stability time depends on the ultra-differentiable class of the pertubation, through the same sequence M. Our proof uses periodic averaging, while a substitute for the analyticity width allows us to bypass analytic smoothing. We also prove converse statements on the destruction of invariant tori and on the existence of diffusing orbits with ultra-differentiable perturbations, by respectively mimicking a construction of Bessi (in the analytic category) and Marco- Sauzin (in the Gevrey non-analytic category). When the perturbation space satisfies some additional condition (we then call it matching), we manage to narrow the gap between stability hypotheses (e.g. the BRM condition) and instability hypotheses, thus circumbscribing the stability threshold. The formulas relating the growth M of derivatives of the perturbation on the one hand, and the arithmetics of robust frequencies or the stability time on the other hand, bring light to the competition between stability properties of nearly integrable systems and the distance to integrability. Due to our method of proof using width of regularity as a regularizing parameter, these formulas are closer to optimal as the the regularity tends to analyticity"-- Provided by publisher.

Contents:

Cover

Title page

Introduction

1. High regularity in perturbation theory

2. Ultra-differentiable functions

3. Functions C and Ω

4. An arithmetic condition for ultra-differentiable functions

5. KAM type results

6. Hamiltonian normal forms and Nekhoroshev type results

Chapter 1. Estimates on ultra-differentiable functions

1. Majorant series and ultra-differentiable functions

2. Properties of majorant series

3. Derivatives

4. Products

5. Compositions

6. Flows

7. Inverse functions

Chapter 2. Application to KAM theory

1. Statement of the KAM theorem with parameters

2. Approximation by rational vectors

3. KAM step

4. Iterations and convergence

5. Proof of Theorem A

6. Proof of Theorem F

7. Proof of Theorem B

Chapter 3. Application to Hamiltonian normal forms and Nekhoroshev theory

1. Periodic averaging

2. Stability in the linear case: proof of Theorem H

3. Diffusion in the linear case: proof of Theorem I

4. Stability in the non-linear case: proof of Theorem J

5. Stability in the quasi-convex case: proof of Theorem K

6. Diffusion in the quasi-convex case: proof of Theorem L

7. Stability in the steep case: proof of Theorem M

Appendix A. On the moderate growth condition

Bibliography

Back Cover.

Notes:

Description based on publisher supplied metadata and other sources.

Includes bibliographical references.

ISBN:

9781470465261

1470465264

OCLC:

1256821513