Contents
Chapter 0. Introduction and Some Salient Features of the Model Hamil-
tonian 1
Chapter 1. Symplectic Geometry and the Splitting of Invariant Manifolds 9
§1.1. Symplectic geometry: a short reminder 10
§1.2. Hyperbolic invariant manifolds 13
§1.3. Angles of Lagrangian planes: the symplectic viewpoint 16
§1.4. Angles of Lagrangian planes: the Euclidean viewpoint 19
§1.5. Symplectic isomorphisms, angles and splitting forms 22
§1.6. The splitting of Lagrangian submanifolds 26
§1.7. Lagrangian submanifolds in a cotangent bundle 28
§1.8. Hyperbolic tori and normally hyperbolic invariant manifolds 31
§1.9. The perturbative setting 36
§1.10. Lagrangian intersections and homoclinic trajectories 40
§1.11. The splitting of the invariant manifolds of hyperbolic tori 46
Chapter 2. Estimating the Splitting Matrix Using Normal Forms 51
§2.1. Resonant normal forms 53
§2.2. Computations in the vicinity of a resonant surface 59
§2.3. Splitting in a perturbative setting, variance and stability 61
§2.4. General exponential estimates for the splitting matrix 67
§2.5. Persistence of tori, invariant manifolds and homoclinic trajectories 77
§2.6. Splitting and stability 80
Chapter 3. The Hamilton-Jacobi Method for a Simple Resonance 91
§3.1. Notation and assumptions 92
§3.2. Formal solutions and the Hamilton-Jacobi algorithm 93
§3.3. Convergence and domains of analyticity 99
§3.4. Exponential closeness of the invariant manifolds 106
§3.5. Linear versus nonlinear splitting 115
§3.6. Some variants and possible generalizations 120
§3.7. A short historical tour and some concluding remarks 126
Appendix. Invariant Tori With Vanishing or Zero Torsion 133
Bibliography 141
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