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Billiards. (English) Zbl 0833.58001
Panoramas et Synthèses. 1. Paris: Société Mathématique de France. vi, 142 p. (1995).
This is an excellent survey of the theory of mathematical billiards, which is a very active research area. In the book, the author attempts to make the exposition as geometrical as possible.
The survey consists of five chapters.
Chapter 1, General theory and mathematical background, provides a necessary mathematical background for the rest of the book, including the area form invariant under the billiard transformation in the two- dimensional case, the billiard transformation of the space of rays in the plane, symplectic geometry, discontinuities of the billiard transformation.
Chapter 2, Convex billiards, concerns convex smooth billiards (elliptic case) in the plane. The author introduces the classical results by Jacobi on integrability of the geodesic flow in ellipsoids, Birkhoff’s conjecture that the integrable plane billiards are the ones in ellipses, existence of Birkhoff’s periodic trajectories of the billiard ball, the relation between the set of length of periodic trajectories and the spectrum of the Laplace operator in the billiard table with the Dirichlet boundary condition, the existence and nonexistence of invariant circles of the billiard transformation.
Chapter 3, Billiards in polygons, deals with billiards in polygons and polyhedra (parabolic case). The author treats in succession the method of unfolding billiard trajectories, encoding billiard trajectories with irrational slopes in a cube according to the faces they reflect in, a criterion for stability of periodic billiard trajectories in polygons, rational billiard polygons, recent strong results on rational billiards and the relation between the dynamics of point masses and billiards.
Chapter 4, Dual billiards, discusses the less-known topics of dual billiard transformations, which are of particular interest to the author. This chapter gives basic features of dual billiards, an approximation of the dual billiard dynamics far away from a dual billiard, the dynamical proof of the classical Poncelet’s theorem, polygonal dual billiards and higher-dimensional dual billiards.
Chapter 5, Hyperbolic billiards, gives a very brief treatment of chaotic billiards (hyperbolic case). The author starts with two examples: a torus automorphism and the geodesic flow on a surface of constant negative curvature, and then introduces dispersing billiards, chaotic billiards with convex arcs, and Boltzmann’s hypothesis and an application of Poincaré’s recurrence theorem to the billiard under a curve lower- asymptotic to the horizontal axis.
The book is clearly written and replete with biographical notes containing almost 200 references up to 1994.
Reviewer: Wang Duo (Beijing)

37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
37Cxx Smooth dynamical systems: general theory
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37A99 Ergodic theory
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
37D99 Dynamical systems with hyperbolic behavior
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory, local dynamics
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry