Dynamical Systems and Small Divisors: Lectures Given at the Cime Summer School, Held in Cetraro

Stimson, S. Kuksin, Jean-Christophe Yoccoz (Editor), S., Kuksin R. J., Yoccoz J. Ch. (Ed)

Many problems of stability in the theory of dynamical systems face the difficulty of small divisors. The most famous example is probably given by Kolmogorov-Arnold-Moser theory in the context of Hamiltonian systems, with many applications to physics and astronomy. Other natural small divisor problems arise considering circle diffeomorphisms or quasiperiodic Schroedinger operators. In this volume Hakan Eliasson, Sergei Kuksin and Jean-Christophe Yoccoz illustrate the most recent developments of this theory both in finite and infinite dimension. A list of open problems (including some problems contributed by John Mather and Michel Herman) has been included.

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Table of contents :
1.1 Examples……Page 1
1.2 Two basic difficulties……Page 7
1.3 Results and References……Page 9
1.4 Description of the paper……Page 11
2.1 Covariance……Page 13
2.2 Normal Form Matrices……Page 14
3 Block splitting……Page 17
4 Quadratic convergence……Page 18
5 Block clustering……Page 25
5.1 Choice of parameters……Page 27
6 Transversality of resultants……Page 29
6.1 Choice of parameters……Page 32
7 A Perturbation theorem……Page 34
7.1 Choice of parameters……Page 38
8.1 Discrete Schrödinger Equation……Page 42
8.2 Discrete Linear Skew-Products……Page 44
Gevrey classes……Page 46
Estimates of eigenvalues……Page 47
Orthogonalization……Page 49
Subspaces and Angles……Page 50
Block splitting……Page 53
A numerical lemma……Page 55
Estimates of preimages……Page 56
Transversality of products of functions……Page 57
References……Page 59
1.1 Smooth and analytic maps……Page 61
1.2 Scales of Hilbert spaces and interpolation……Page 63
1.3 Differential forms……Page 65
2.1 Basic definitions……Page 67
2.2 Symplectic transformations……Page 70
2.3 Darboux lemmas……Page 72
Appendix. Time-quasiperiodic solutions……Page 73
3 Lax-integrable Hamiltonian equations and their integrable subsystems……Page 74
3.1 Examples of Hamiltonian PDEs……Page 75
3.2 Lax-integrable equations……Page 77
3.3 Integrable subsystems……Page 78
4.1 Finite-gap manifolds for the KdV equation……Page 80
4.2 The Its–Matveev theta-formulas……Page 81
4.4 Sine-Gordon equation under Dirichlet boundary conditions……Page 84
5.1 The linearised equation……Page 86
5.2 Floquet solutions……Page 87
5.3 Complete systems of Floquet solutions……Page 89
5.4 Lower-dimensional invariant tori of finite-dimensional systems and Floquet’s theorem……Page 95
6.1 Abstract situation……Page 96
6.2 Linearised KdV equation……Page 98
6.3 Higher KdV-equations……Page 102
6.4 Linearised SG equation……Page 103
7.1 A normal form theorem……Page 104
7.2 Examples……Page 110
8.1 The main theorem and related results……Page 111
8.2 Reduction to a parameter-depending case……Page 113
8.3 A KAM-theorem for parameter-depending equations……Page 115
8.4 Completion of the Main Theorem’s proof (Step 4)……Page 116
9.1 Perturbed KdV equation……Page 117
9.2 Higher KdV equations……Page 119
9.4 KAM-persistence of lower-dimensional invariant tori of nonlinear finite-dimensional systems……Page 120
References……Page 121
1 Introduction……Page 125
2.1 Introduction……Page 127
2.2 Continued Fractions……Page 128
2.4 Brjuno function and condition $cal B$……Page 130
2.5 Condition $cal H$……Page 131
2.6 $mathbb Z^2$-actions by translations and continued fractions……Page 136
3.1 The $mathcal{C}^0$ theory……Page 137
3.2 Equicontinuity and topological conjugacy……Page 139
3.3 The Denjoy theory……Page 140
3.5 The Schwarzian derivative……Page 142
3.6 Partial renormalization……Page 145
4.2 Local Theorem 1.2: big strips……Page 146
4.3 Local Theorem 1.3: small strips……Page 153
4.4 Global Theorem: complex Denjoy estimates……Page 154
4.5 Global Theorem: proof of linearization……Page 157
4.6 Global Theorem: Construction of nonlinearizable diffeomorphisms……Page 160
5.2 First kind of moduli estimates……Page 168
5.3 Second kind of moduli estimates……Page 170
References……Page 172
1.1 Linearization of the quadratic polynomial. Size of Siegel disks……Page 175
1.2 Herman rings. Differentiable conjugacy of diffeomorphisms of the circle……Page 176
1.3 Gevrey classes……Page 177
2.1 Linearization of germs of holomorphic diffeomorphisms of $(mathbb{C}^n, 0)$……Page 178
2.3 $mathbb{Z}^k$-actions……Page 179
2.4 Diffeomorphisms of compact manifolds……Page 180
3.1 Twist maps……Page 181
3.3 $n$-body problem……Page 183
4.1 Reducibility of skew-products……Page 184
4.2 Spectral theory and integrated density of states……Page 185
4.3 Nonlinear Hamiltonian PDEs……Page 186
References……Page 188