**Memoirs of the American Mathematical Society**

2019;
106 pp;
Softcover

MSC: Primary 53; 70;
Secondary 26

**Print ISBN: 978-1-4704-3492-2
Product Code: MEMO/257/1235**

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**Electronic ISBN: 978-1-4704-4953-7
Product Code: MEMO/257/1235.E**

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# Measure and Capacity of Wandering Domains in Gevrey Near-Integrable Exact Symplectic Systems

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*Laurent Lazzarini; Jean-Pierre Marco; David Sauzin*

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.

#### Table of Contents

# Table of Contents

## Measure and Capacity of Wandering Domains in Gevrey Near-Integrable Exact Symplectic Systems

- Cover Cover11
- Title page i2
- Chapter 0. Introduction 18
- Chapter 1. Presentation of the results 916
- Chapter 2. Stability theory for Gevrey near-integrable maps 2128
- Chapter 3. A quantitative KAM result—proof of Part (i) of Theorem D 3340
- 3.1. Elliptic islands in \A with a tuning parameter—Theorem F 3340
- 3.2. Theorem F implies Part (i) of Theorem D 3441
- 3.3. Overview of the proof of Theorem F 3542
- 3.4. Preliminary study near a q-periodic point 3542
- 3.5. Normalizations 5057
- 3.6. The invariant curve theorem 6471
- 3.7. Conclusion of the proof of Theorem F 6572

- Chapter 4. Coupling devices, multi-dimensional periodic domains, wandering domains 6976
- \appendixtocname 8390
- Appendix A. Algebraic operations in O 8390
- Appendix B. Estimates on Gevrey maps 9198
- Appendix C. Generating functions for exact symplectic 𝐶^{∞} maps 99106
- Appendix D. Proof of Lemma 2.5 103110
- Acknowledgements 107114
- Bibliography 109116
- Back Cover Back Cover1122