About this Title
Nikolai Chernov, University of Alabama at Birmingham, Birmingham, AL and Roberto Markarian, Universidad de la República, Montevideo, Uruguay
Publication: Mathematical Surveys and Monographs
Publication Year 2006: Volume 127
ISBNs: 978-0-8218-4096-2 (print); 978-1-4704-1354-5 (online)
MathSciNet review: MR2229799
MSC: Primary 37D50; Secondary 37D45, 37N20
This book covers one of the most exciting but most difficult topics in the modern theory of dynamical systems: chaotic billiards. In physics, billiard models describe various mechanical processes, molecular dynamics, and optical phenomena.
The theory of chaotic billiards has made remarkable progress in the past thirty-five years, but it remains notoriously difficult for the beginner, with main results scattered in hardly accessible research articles. This is the first and so far only book that covers all the fundamental facts about chaotic billiards in a complete and systematic manner. The book contains all the necessary definitions, full proofs of all the main theorems, and many examples and illustrations that help the reader to understand the material. Hundreds of carefully designed exercises allow the reader not only to become familiar with chaotic billiards but to master the subject.
The book addresses graduate students and young researchers in physics and mathematics. Prerequisites include standard graduate courses in measure theory, probability, Riemannian geometry, topology, and complex analysis. Some of this material is summarized in the appendices to the book.
Graduate students and research mathematicians interested in mathematical physics, statistical mechanics, dynamical systems, and ergodic theory.
Table of Contents
- 1. Simple examples
- 2. Basic constructions
- 3. Lyapunov exponents and hyperbohcity
- 4. Dispersing billiards
- 5. Dynamics of unstable manifolds
- 6. Ergodic properties
- 7. Statistical properties
- 8. Bunimovich billiards
- 9. General focusing chaotic billiards