Dynamical Systems and Small Divisors: Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 13-20, 1998

207 Pages·2002·6.398 MB·English

by  Hakan Eliasson, Sergei Kuksin, Stefano Marmi, Jean-Christophe Yoccoz (auth.)

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Lecture Notes in Mathematics 1784
Editors:
J.-M. Morel, Cachan
F. Takens, Groningen
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Subseries:
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Tokyo
L. H. Eliasson S. B. Kuksin
S. Marmi J.-C. Yoccoz
Dynamical Systems
and Small Divisors
Lectures given at the C.I.M.E. Summer School
held in Cetraro, Italy, June 13 - 20, 1998
Editors: S. Marmi
J.-C. Yoccoz
FUlloozionl'
C.I.M.E.
, Springer
Editors and Authors
Stefano Marmi Hakan Eliasson
Scuola Normale Superiore Geometrie et Dynamique
Piazza dei Cavalieri 7 UFR. de Mathematiques
56126 Pisa, Italy Universite Paris 7
e-mail: [email protected] Case 7012, 2 place Jussieu
and 75251 Paris Cedex 05, France
Dipartimento di Matematica e-mail: hakane®math.jussieu.fr
e Informatica
Universita di Udine Sergei Kuksin
Via delle Scienze 206 Department of Mathematics
33100 Udine, Italy Heriot-Watt University
e-mail: [email protected] Riccarton, Edinburgh EH 14 4AS
United Kingdom
Jean-Christophe Yoccoz e-mail: [email protected]
College de France
3rued'Ulm
75005 Paris, France
e-mail:
[email protected]
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Die Deutsche Bibliothek -CIP-Einheitsaufnahme
Dynamical systems and small divisors: held in Cetraro, Italy, Juny 13 -20,
1998/ CIME. 1. H. Eliasson ... Ed.: S. Marmi ; J.-C. Yoccoz. -Berlin ;
Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris;
Tokyo: Springer, 2002
(Lectures given at the CIME summer school; 1998, summer school)
(Lecture notes in mathematics; Vol. 1784 : Subseries: Fondazione CIME)
ISBN 3-540-43726-6
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Preface
The C.I.M.E. Session "Dynamical Systems and Small Divisors" was held in
Cetraro (Cosenza, Italy) from June 13 to June 20, 1998.
Lecture series were held by:
• L.H. Eliasson (8 hours): Linear quasiperiodic systems
• S. Kuksin (3 hours): KAM theory and PDEs
• M.R. Herman (8 hours): Abstract methods in small divisors: Implicit func
tion theorems in Frechet spaces
• J.N. Mather (3 hours): Variational construction of orbits
• J.-C. Yoccoz (8 hours): Geometrical methods in small divisor problems
In addition there was a 3 hours long open problem session.
FUrthermore, the following participants gave a talk:
1. U. Bessi: A counterexample to KAM theorem
2. D. De Latte: Linearization of commuting holomorphic maps
3. G. Gentile: Scaling properties near resonances for the (semistandard and)
standard map
4. H. Ito: Integrable symplectic maps and their Birkhoff normal forms
5. R. Krikorian: Reducibility of linear quasiperiodic systems: a global den
sity result
6. P. Moussa: Regularity properties of Brjuno functions
7. D. Sauzin: Quasianalytic monogenic solutions of a cohomological equa-
tion
8. S. Smirnov: Weak expansion and geometry of Julia sets
9. L. Stolovitch: Singular complete integrability
10. M. Vittot: Perturbation method for Floquet hamiltonians with dense
point spectrum
This volume contains expanded versions of the lecture series of Eliasson,
Kuksin and Yoccoz and of the open problem session.
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-Arnol'd-Moser theory on the persistence of quasi-periodic solu
tions of Hamilton's equations for quasi-integrable Hamiltonian systems (both
finite and infinite-dimensional, like nonlinear wave equations). This is a very
natural situation with many applications to physics and astronomy. What all
these different problems have in common is roughly speaking what follows:
one can associate some "frequencies" to the orbits under investigation and
some arithmetical condition is needed to prove their existence and stability. In
VI
the smooth case one must impose a diophantine condition on the frequencies,
in the analytic (finitEKiimensional) case one can impose a weaker condition.
For one-dimensional analytic small divisor problems the optimal arith
metical condition to impose is now known: in the local case it is the same
Brjuno condition introduced some 30 years ago, whereas in the global case
one has to impose a more restrictive condition: the lectures by J.-C. Yoccoz
published in these notes deal with these cases.
In his lectures Hakan Eliasson addresses the problem of reducibility of lin
ear quasiperiodic skew-products. Originating from the pioneering works of
Dinaburg-Sinai and Riissmann, who constructed Floquet solutions for the
continuous Schrodinger equation with a quasiperiodic potential in the so
called weak coupling regime (the potential is "small" w.r.t. the kinetic part),
many authors have since then studied more general situations (strong cou
pling, discrete case, more general groups, etc.). The review of Eliasson at the
ICM 1998 could serve as a general introduction to this topic.
The contribution of S. Kuksin is devoted to a proof of KAM theorem of
persistence of finite-dimensional tori under small Hamiltonian perturbations
of Lax-integrable Hamiltonian PDEs. Here the model problem is provided by
Korteweg-de Vries equation Ut = -Uxxx + 6uux where the spatial variable x
varies on the one-dimensional torus. Also other problems can be treated by
the same method: sine-Gordon equation, (cp4 )-equation, etc.. The study of
nonlinear Hamiltonian PDEs is a rapidly growing research subject: excellent
reviews of recent work are provided by the addresses of Bourgain at the ICM
1994 and Kuksin at the ICM 1998.
Finally, the open problems proposed by the speakers have been collected
in the last contribution. They deal both with finite and infinite-dimensional
small divisor problems. As far as possible we have tried to make this text
self-contained.
We are grateful to all speakers and partecipants for their essential contri
bution to the success of this C.I.M.E. session. Special thanks go to Roberta
Fabbri who helped us in the organization and to Raphael Krikorian who held
several tutorials on the material of Herman's lectures.
Stefano Marmi and Jean-Christophe Yoccoz
Contents
Perturbations of linear quasi-periodic system . . . . . . . . . . . . . . . . . 1
L. H. Eliasson
1 Introduction................................................. 1
2 Covariance and normal form matrices. . . . . . . . . . . . . . . . . . . . . . . . . .. 13
3 Block splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 17
4 Quadratic convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 18
5 Block clustering ............................................. 25
6 Transversality of resultants ................................... 29
7 A Perturbation theorem ...................................... 34
8 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 42
A Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 46
B Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 56
KAM-persistence of finite-gap solutions ...................... 61
Sergei B. Kuksin
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 61
1 Some analysis in Hilbert spaces and scales ...................... 61
2 Symplectic structures and Hamiltonian equations .. . . . . . . . . . . . . .. 67
3 Lax-integrable Hamiltonian equations
and their integrable subsystems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 74
4 Finite-gap manifolds and theta-formulas. . . . . . . . . . . . . . . . . . . . . . .. 80
5 Linearised equations and their Floquet solutions . . . . . . . . . . . . . . . .. 86
6 Linearised Lax-integrable equations ............................ 96
7 Normal form ................................................ 104
8 The KAM theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
9 Examples ................................................... 117
Analytic linearization of circle diffeomorphisms ............... 125
Jean-Christophe Yoccoz
1 Introduction ................................................. 125
2 Arithmetics ................................................. 127
3 The cr theory for r 20 ....................................... 137
4 Analytic case ................................................ 146
5 Appendix: Estimates of moduli of annular domains ............... 168
VIII
Some open problems related to small divisors ................. 175
S. Marmi, J.-O. Yoccoz
o
Introduction ................................................. 175
lOne-Dimensional Small Divisor Problems (On Holomorphic Germs
and Circle Diffeomorphisms) .................................. 175
2 Finite-Dimensional Small Divisor Problems ...................... 178
3 KAM Theory and Hamiltonian Systems ......................... 181
4 Linear Quasiperiodic Skew-Products, Spectral Theory and Hamil-
tonian Partial Differential Equations ........................... 184