One-Dimensional Dynamics

The aim of the book is to give an introduction to the main concepts in modern dynamics. The Herman Theorem on rigidity of circle di eomoprhisms is used to achieve this goal. Indeed, we reprove this sophisticated theorem with modern techniques involving renormalization and, in the meantime, discuss most of the important concepts used in dynamics. The one-dimensional context of the Herman Theorem reduces the technical di culties and allows to present tools and theories to students with modest background. The book can definitely be used on Master's level.

Chapter 1. Introduction. - Chapter 2. Topology of Circle Homeomorphisms. - Chapter 3. Topology of Circle Diffeomorphisms. - Chapter 4. Renormalization. - Chapter 5. The Three Ingredients for Convergence of Renormalization. - Chapter 6. The Collaboration of the Three Ingredients. - Chapter 7. Exponential Convergence. - Chapter 8. Geometry of Circle Diffeomorphisms. - Chapter 9. Non-Rigidity. - Chapter 10. The Renormalization Horseshoe.