Chaotic billiards / Nikolai Chernov, Roberto Markarian.

Format:

Book

Author/Creator:

Chernov, Nikolai, 1956- author.

Markarian, Roberto, author.

Series:

Mathematical surveys and monographs ; v. 127.

Mathematical surveys and monographs, 0076-5376 ; volume 127

Language:

English

Subjects (All):

Dynamics.

Chaotic behavior in systems.

Ergodic theory.

Measure theory.

Probabilities.

Billiards.

Physical Description:

1 online resource (330 p.)

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2006]

Language Note:

English

Summary:

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.

Contents:

""Contents""; ""Preface""; ""Symbols and notation""; ""Chapter 1. Simple examples""; ""1.1. Billiard in a circle""; ""1.2. Billiard in a square""; ""1.3. A simple mechanical model""; ""1.4. Billiard in an ellipse""; ""1.5. A chaotic billiard: pinball machine""; ""Chapter 2. Basic constructions""; ""2.1. Billiard tables""; ""2.2. Unbounded billiard tables""; ""2.3. Billiard flow""; ""2.4. Accumulation of collision times""; ""2.5. Phase space for the flow""; ""2.6. Coordinate representation of the flow""; ""2.7. Smoothness of the flow""; ""2.8. Continuous extension of the flow""

""2.9. Collision map""""2.10. Coordinates for the map and its singularities""; ""2.11. Derivative of the map""; ""2.12. Invariant measure of the map""; ""2.13. Mean free path""; ""2.14. Involution""; ""Chapter 3. Lyapunov exponents and hyperbohcity""; ""3.1. Lyapunov exponents: general facts""; ""3.2. Lyapunov exponents for the map""; ""3.3. Lyapunov exponents for the flow""; ""3.4. Hyperbohcity as the origin of chaos""; ""3.5. Hyperbohcity and numerical experiments""; ""3.6. Jacobi coordinates""; ""3.7. Tangent lines and wave fronts""; ""3.8. Billiard-related continued fractions""

""3.9. Jacobian for tangent lines""""3.10. Tangent lines in the collision space""; ""3.11. Stable and unstable lines""; ""3.12. Entropy""; ""3.13. Proving hyperbolicity: cone techniques""; ""Chapter 4. Dispersing billiards""; ""4.1. Classification and examples""; ""4.2. Another mechanical model""; ""4.3. Dispersing wave fronts""; ""4.4. Hyperbolicity""; ""4.5. Stable and unstable curves""; ""4.6. Proof of Proposition 4.29""; ""4.7. More continued fractions""; ""4.8. Singularities (local analysis)""; ""4.9. Singularities (global analysis)""; ""4.10. Singularities for type B billiard tables""

""4.11. Stable and unstable manifolds""""4.12. Size of unstable manifolds""; ""4.13. Additional facts about unstable manifolds""; ""4.14. Extension to type B billiard tables""; ""Chapter 5. Dynamics of unstable manifolds""; ""5.1. Measurable partition into unstable manifolds""; ""5.2. u-SRB densities""; ""5.3. Distortion control and homogeneity strips""; ""5.4. Homogeneous unstable manifolds""; ""5.5. Size of H-manifolds""; ""5.6. Distortion bounds""; ""5.7. Holonomy map""; ""5.8. Absolute continuity""; ""5.9. Two growth lemmas""; ""5.10. Proofs of two growth lemmas""

""5.11. Third growth lemma""""5.12. Size of H-manifolds (a local estimate)""; ""5.13. Fundamental theorem""; ""Chapter 6. Ergodic properties""; ""6.1. History""; ""6.2. Hopf's method: heuristics""; ""6.3. Hopf's method: preliminaries""; ""6.4. Hopf's method: main construction""; ""6.5. Local ergodicity""; ""6.6. Global ergodicity""; ""6.7. Mixing properties""; ""6.8. Ergodicity and invariant manifolds for billiard flows""; ""6.9. Mixing properties of the flow and 4-loops""; ""6.10. Using 4-loops to prove K-mixing""; ""6.11. Mixing properties for dispersing billiard flows""

""Chapter 7. Statistical properties""

Notes:

Description based upon print version of record.

Includes bibliographical references (pages 309-314) and index.

Description based on print version record.

Description based on publisher supplied metadata and other sources.

ISBN:

1-4704-1354-X

OCLC:

879576508