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An Introduction To Small Divisors



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This is an introduction to small divisors problems. The material treated in this book was brought together for a PhD course I tought at the University of Pisa in the spring of 1999. Here is a Table of Contents: Part I One Dimensional Small Divisors. Yoccoz's Theorems 1. Germs of Analytic Diffeomorphisms. Linearization 2. Topological Stability vs. Analytic Linearizability 3. The Quadratic Polynomial: Yoccoz's Proof of the Siegel Theorem 4. DouadyGhys' Theorem. Continued Fractions and the Brjuno Function 5. SiegelBrjuno Theorem, Yoccoz's Theorem. Some Open Problems 6. Small Divisors and Loss of Differentiability Part II Implicit Function Theorems and KAM Theory 7. Hamiltonian Systems and Integrable Systems 8. QuasiIntegrable Hamiltonian Systems 9. NashMoser's Implicit Function Theorem 10. From NashMoser's Theorem to KAM: Normal Form of Vector Fields on the Torus Appendices A1. Uniformization, Distorsion and Quasiconformal maps A2. Continued Fractions A3. Distributions, Hyperfunctions. Hypoellipticity and Diophantine Conditions 

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