Measure and Capacity of Wandering Domains in Gevrey Near-integrable Exact Symplectic Systems
A wandering domain for a diffeomorphism We first prove that the measure (or the capacity) of these wandering domains is exponentially small, with an upper bound of the form The second part of the paper is devoted to the construction of near-integrable Gevrey systems possessing wandering domains, for...
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| Main Authors: | , , |
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| Format: | eBook Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
2019
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| Edition: | 1 |
| Series: | Memoirs of the American Mathematical Society |
| Subjects: | |
| ISBN: | 9781470434922, 147043492X |
| ISSN: | 0065-9266, 1947-6221 |
| Online Access: | Get full text |
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Table of Contents:
- Introduction -- Presentation of the results -- Stability theory for Gevrey near-integrable maps -- A quantitative KAM result—proof of Part (i) of Theorem -- Coupling devices, multi-dimensional periodic domains, wandering domains -- \texorpdfstring{Algebraic operations in <inline-formula content-type="math/mathml"> O k {\mathscr O}_k </inline-formula>}Algebraic operations in O -- Estimates on Gevrey maps -- Generating functions for exact symplectic <inline-formula content-type="math/mathml"> C ∞<!-- ∞ --> C^\infty </inline-formula> maps -- Proof of Lemma -- Acknowledgements
- 4.3. Proof of Theorem C (lower bounds for wandering domains in \Aⁿ) -- 4.3.1. Overview of the proof -- 4.3.2. Standard maps with wandering discs in \A -Proof of Proposition 4.6 -- 4.3.3. Proof of Theorem C' -- \appendixtocname -- Appendix A. Algebraic operations in O -- Appendix B. Estimates on Gevrey maps -- B.1. Reminder on Gevrey maps and their composition -- B.2. A lemma on the flow of a Gevrey near-integrable Hamiltonian -- B.3. Proof of Proposition 1.7 -- B.4. Gevrey bump fuctions -- Appendix C. Generating functions for exact symplectic ^{∞} maps -- Appendix D. Proof of Lemma 2.5 -- D.1. Set-up -- D.2. Diffeomorphism property -- D.3. Study of the inverse map -- Acknowledgements -- Bibliography -- Back Cover
- Cover -- Title page -- Chapter 0. Introduction -- Chapter 1. Presentation of the results -- 1.1. Perturbation theory for analytic or Gevrey near-integrable maps-Theorem A -- 1.2. Wandering sets of near-integrable systems-Theorems B and C -- 1.3. Specific form of our examples and elliptic islands-Theorem D -- 1.4. Further comments -- Chapter 2. Stability theory for Gevrey near-integrable maps -- 2.1. Embedding in a Hamiltonian flow -Theorem E -- 2.2. Proof of Theorem E in the Gevrey non-analytic case -- 2.2.0. Overview -- 2.2.1. First step: finding a generating function -- 2.2.2. Second step: constructing a Hamiltonian isotopy -- 2.2.3. Completion of the proof of Theorem E -- 2.3. Proof of Theorem A (Nekhoroshev Theorem for maps) -- 2.4. Proof of Theorem B (upper bounds for wandering sets) -- Chapter 3. A quantitative KAM result-proof of Part (i) of Theorem D -- 3.1. Elliptic islands in \A with a tuning parameter-Theorem F -- 3.2. Theorem F implies Part (i) of Theorem D -- 3.3. Overview of the proof of Theorem F -- 3.4. Preliminary study near a q-periodic point -- 3.4.1. Localization -- 3.4.2. Local form -- 3.4.3. The Taylor expansion of the q iteration of G -- 3.5. Normalizations -- 3.5.1. Notations and statements -- Birkhoff normal form -- Herman normal form -- 3.5.2. Proof of Proposition 3.16 -- 3.5.3. Proof of Proposition 3.17 -- 3.5.4. Proof of Proposition 3.18 -- 3.6. The invariant curve theorem -- 3.7. Conclusion of the proof of Theorem F -- Chapter 4. Coupling devices, multi-dimensional periodic domains, wandering domains -- 4.1. Coupling devices -- 4.2. Proof of Part (ii) of Theorem D (periodic domains in \Aⁿ⁻¹) -- 4.2.1. Overview of the method -- 4.2.2. A -periodic polydisc for a near-integrable system of the form Φ^{ }∘ ^{ } in \A -- 4.2.3. A -periodic polydisc for a near-integrable system in \Aⁿ⁻² -- 4.2.4. Applying Corollary 4.2