The material treated in this book was brought together for a PhD course I taught at the University of Pisa in the spring of 1999. It is intended to be an introduction to small divisors problems.

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. Douady-Ghys' Theorem. Continued Fractions and the Brjuno Function 5. Siegel-Brjuno 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. Quasi-integrable Hamiltonian Systems 9. Nash-Moser's Implicit Function Theorem 10. From Nash-Moser's Theorem to KAM: Normal Form of Vector Fields on the Torus Appendices A1. Uniformization, Distorsion and Quasi-conformal Maps A2. Continued Fractions A3. Distributions, Hyperfunctions, Formal Series. Hypoellipticity and Diophantine Conditions