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Introduction to the Ergodic Theory of Chaotic Billiards; Nikolai Chernov, Roberto Markarian
In the last three decades of the twentieth century, chaotic billiards became one of the most active and popular research areas in statistical mechanics. This started with a seminal paper by Ya. Sinai in 1970, where he developed a mathematical apparatus for the study of hyperbolic and ergodic properties for a large class of plane billiards. He also obtained an exact formula for the entropy of billiards. Sinai?s theory led to an outburst of papers in mathematics and physics journals devoted to various types of billiards on plane and space of any dimension.
The first one comprises Chapters I, II and III. It contains a brief exposition of the most important elements of ergodic theory. Chapter I is largely a transcription of the first section of the notes by Ricardo Mañe that he prepared for his course at the ICTP, Trieste in 1988. Chapter II contains a proof of Ergodic Theorem inspired by the text of Mañe. Chapter III focuses on smooth hyperbolic dynamics and covers the main results of the Pesin theory.
The second part of the book is formed by Chapter IV and is fully devoted to billiards. It starts with elementary properties of planar billiards, then goes on to multidimensional models, including Lorentz gases and hard ball systems. It covers mean free path formulas and bounds on the number of reflections. Sinai?s theory of planar dispersing billiards is outlined. Then other hyperbolic billiards are described in detail. Formulas for the entropy are derived. Overall, the material of Chapter IV constitutes a basic course in billiards, after which one should be able to read main research papers in the area.