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Softcover ISBN:  9781470434922 
Product Code:  MEMO/257/1235 
List Price:  $81.00 
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AMS Member Price:  $48.60 
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Softcover ISBN:  9781470434922 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 257; 2019; 106 ppMSC: Primary 53; 70; Secondary 26
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 timeone 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

Chapters

1. Introduction

2. Presentation of the results

3. Stability theory for Gevrey nearintegrable maps

4. A quantitative KAM result—proof of Part (i) of Theorem

5. Coupling devices, multidimensional periodic domains, wandering domains

A. \texorpdfstring{Algebraic operations in ${\mathscr O}_k$}Algebraic operations in O

B. Estimates on Gevrey maps

C. Generating functions for exact symplectic $C^\infty $ maps

D. Proof of Lemma

Acknowledgements


Additional Material

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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 timeone 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.

Chapters

1. Introduction

2. Presentation of the results

3. Stability theory for Gevrey nearintegrable maps

4. A quantitative KAM result—proof of Part (i) of Theorem

5. Coupling devices, multidimensional periodic domains, wandering domains

A. \texorpdfstring{Algebraic operations in ${\mathscr O}_k$}Algebraic operations in O

B. Estimates on Gevrey maps

C. Generating functions for exact symplectic $C^\infty $ maps

D. Proof of Lemma

Acknowledgements