An Introduction To Small Divisors
S. Marmi (Dipartimento di Matematica e Informatica, University of, Udine)

TL;DR
This book provides an accessible introduction to small divisors problems, covering key theorems, concepts, and open questions in one and multi-dimensional settings, with applications to dynamical systems and KAM theory.
Contribution
It compiles and explains foundational results and recent developments in small divisors theory, including Yoccoz's theorems and their implications for dynamical systems.
Findings
Yoccoz's theorems on analytic linearization
Connections between small divisors and differentiability loss
Applications to Hamiltonian and quasi-integrable systems
Abstract
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. 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.…
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Taxonomy
TopicsMathematical Dynamics and Fractals · Quantum chaos and dynamical systems · Algebraic Geometry and Number Theory
