Dynamical systems and small divisors: Lectures CIME school

Hakan Eliasson, Sergei Kuksin, Stefano Marmi, Jean-Christophe Yoccoz, Stefano Marmi, Jean-Christophe Yoccoz3540437266, 9783540437260

Authors illustrate the most recent developments of the dynamical systems theory both in finite and infinite dimensions. Softcover.

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Table of contents :
1.1 Examples……Page 2
1.2 Two basic difficulties……Page 8
1.3 Results and References……Page 10
1.4 Description of the paper……Page 12
2.1 Covariance……Page 14
2.2 Normal Form Matrices……Page 15
3 Block splitting……Page 18
4 Quadratic convergence……Page 19
5 Block clustering……Page 26
5.1 Choice of parameters……Page 28
6 Transversality of resultants……Page 30
6.1 Choice of parameters……Page 33
7 A Perturbation theorem……Page 35
7.1 Choice of parameters……Page 39
8.1 Discrete Schrödinger Equation……Page 43
8.2 Discrete Linear Skew-Products……Page 45
Gevrey classes……Page 47
Estimates of eigenvalues……Page 48
Orthogonalization……Page 50
Subspaces and Angles……Page 51
Block splitting……Page 54
A numerical lemma……Page 56
Estimates of preimages……Page 57
Transversality of products of functions……Page 58
References……Page 60
1.1 Smooth and analytic maps……Page 62
1.2 Scales of Hilbert spaces and interpolation……Page 64
1.3 Differential forms……Page 66
2.1 Basic definitions……Page 68
2.2 Symplectic transformations……Page 71
2.3 Darboux lemmas……Page 73
Appendix. Time-quasiperiodic solutions……Page 74
3 Lax-integrable Hamiltonian equations and their integrable subsystems……Page 75
3.1 Examples of Hamiltonian PDEs……Page 76
3.2 Lax-integrable equations……Page 78
3.3 Integrable subsystems……Page 79
4.1 Finite-gap manifolds for the KdV equation……Page 81
4.2 The Its–Matveev theta-formulas……Page 82
4.4 Sine-Gordon equation under Dirichlet boundary conditions……Page 85
5.1 The linearised equation……Page 87
5.2 Floquet solutions……Page 88
5.3 Complete systems of Floquet solutions……Page 90
5.4 Lower-dimensional invariant tori of finite-dimensional systems and Floquet’s theorem……Page 96
6.1 Abstract situation……Page 97
6.2 Linearised KdV equation……Page 99
6.3 Higher KdV-equations……Page 103
6.4 Linearised SG equation……Page 104
7.1 A normal form theorem……Page 105
7.2 Examples……Page 111
8.1 The main theorem and related results……Page 112
8.2 Reduction to a parameter-depending case……Page 114
8.3 A KAM-theorem for parameter-depending equations……Page 116
8.4 Completion of the Main Theorem’s proof (Step 4)……Page 117
9.1 Perturbed KdV equation……Page 118
9.2 Higher KdV equations……Page 120
9.4 KAM-persistence of lower-dimensional invariant tori of nonlinear finite-dimensional systems……Page 121
References……Page 122
1 Introduction……Page 126
2.1 Introduction……Page 128
2.2 Continued Fractions……Page 129
2.4 Brjuno function and condition $cal B$……Page 131
2.5 Condition $cal H$……Page 132
2.6 $mathbb Z^2$-actions by translations and continued fractions……Page 137
3.1 The $mathcal{C}^0$ theory……Page 138
3.2 Equicontinuity and topological conjugacy……Page 140
3.3 The Denjoy theory……Page 141
3.5 The Schwarzian derivative……Page 143
3.6 Partial renormalization……Page 146
4.2 Local Theorem 1.2: big strips……Page 147
4.3 Local Theorem 1.3: small strips……Page 154
4.4 Global Theorem: complex Denjoy estimates……Page 155
4.5 Global Theorem: proof of linearization……Page 158
4.6 Global Theorem: Construction of nonlinearizable diffeomorphisms……Page 161
5.2 First kind of moduli estimates……Page 169
5.3 Second kind of moduli estimates……Page 171
References……Page 173
1.1 Linearization of the quadratic polynomial. Size of Siegel disks……Page 176
1.2 Herman rings. Differentiable conjugacy of diffeomorphisms of the circle……Page 177
1.3 Gevrey classes……Page 178
2.1 Linearization of germs of holomorphic diffeomorphisms of $(mathbb{C}^n, 0)$……Page 179
2.3 $mathbb{Z}^k$-actions……Page 180
2.4 Diffeomorphisms of compact manifolds……Page 181
3.1 Twist maps……Page 182
3.3 $n$-body problem……Page 184
4.1 Reducibility of skew-products……Page 185
4.2 Spectral theory and integrated density of states……Page 186
4.3 Nonlinear Hamiltonian PDEs……Page 187
References……Page 189