Measure and Capacity of Wandering Domains in Gevrey near-Integrable Exact Symplectic Systems.

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.

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.

A wandering domain for a diffeomorphism \Psi of \mathbb A^n=T^*\mathbb T^n is an open connected set W such that \Psi ^k(W)\cap W=\emptyset for all k\in \mathbb Z^*. The authors endow \mathbb A^n with its usual exact symplectic structure. An integrable diffeomorphism, i.e., the time-one map \Phi ^h of a Hamiltonian h: \mathbb A^n\to \mathbb R which depends only on the action variables, has no nonempty wandering domains. The aim of this paper is to estimate the size (measure and Gromov capacity) of wandering domains in the case of an exact symplectic perturbation of \Phi ^h, in the analytic or Gevrey category. Upper estimates are related to Nekhoroshev theory; lower estimates are related to examples of Arnold diffusion. This is a contribution to the "quantitative Hamiltonian perturbation theory" initiated in previous works on the optimality of long term stability estimates and diffusion times; the emphasis here is on discrete systems because this is the natural setting to study wandering domains.

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