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Geometry and billiards. (English) Zbl 1119.37001

Student Mathematical Library 30. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3919-5/pbk). xi, 176 p. (2005).
The mathematical theory of billiards involves several domains of mathematics: geometry, number theory, symbolic dynamics, ergodic theory, Hamiltonian mechanics and others. Billiards are also studied from the physical point of view, as a billiard trajectory describes the motion of elastic particles, of optic rays, and so on.
The book under review is a nice introduction to the theory of billiards, with an emphasis on the points of view of geometry and of geometric optics.
This book addresses natural and old questions on the subject such as the existence and the stability of periodic orbits, the existence of special families of such orbits and the asymptotic behaviour of the number of such orbits. It also addresses some more special questions which are equally important, like the existence and uniqueness of caustics (a caustic being a curve in the billiard table such that if a segment of a billiard trajectory is tangent to this curve, then it remains so after reflection) and the theory of dual billiards, which are discrete-time dynamical systems that were studied by J. Moser in the 1970s, where the motion takes place ouside of the billiard table and where the trajectory of a point is obtained by taking the reflection on the tangent line to the billiard table. In general, questions on billiards, even if they are natural and easily formulated, are hard to tackle, and several problems in billiard theory gave rise to rich theories. The results that are available depend of course on the shape and the regularity of the billiard table.
The book under review is very well written, with nice illustrations. The author presents the results very clearly, with interesting digressions and he mentions applications of billiards to various fields (probability theory, computing the decimals of \(\pi\), configuration spaces of linkages, Poncelet porism, Aubry-Mather theory, the four-vertex theorem, etc.).
I highly recommend this book as an introduction to the subject. The book can be used as a textbook for an advanced undergraduate or a graduate course. The theory that is developed contains related background on surface theory (Gauss-Bonnet, area, Euler formula, etc.), on symplectic geometry and on dynamics (Poincaré recurrence, etc.) that will be useful for students.
The book is divided into the following chapters: (1) Motivation: Mechanics and optics. (2) Billiards in the circle and the square. (3) Billiard ball map and integral geometry. (4) Billiards inside conics and quadrics. (5) Existence and non-existence of caustics. (6) Periodic trajectories. (7) Billiards in polygons. (8) Chaotic billiards. (9) Dual billiards.

MSC:

37-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to dynamical systems and ergodic theory
51-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to geometry
37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
37J10 Symplectic mappings, fixed points (dynamical systems) (MSC2010)
78A05 Geometric optics
70H05 Hamilton’s equations
82C05 Classical dynamic and nonequilibrium statistical mechanics (general)
00A09 Popularization of mathematics
00A05 Mathematics in general
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