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Hyperbolic billards. (English) Zbl 0794.58036

Proc. Int. Congr. Math., Kyoto/Japan 1990, Vol. I, 249-260 (1991).
[For the entire collection see Zbl 0741.00019.]
This article contains a review of the theory of billiards as it was developed between 1970 and 1990 by the author and other mathematicians, among which are Bunimovich, Chernov, Krámli, Simányi, Szász, Wojtkowski, etc. There are no proofs, but the author briefly explains the main difficulties and how they have been solved, and gives references to the original papers. He first states the theorem that every dispersing billiard is hyperbolic, and the one giving a sufficient condition for a semi-dispersing billiard to be hyperbolic. He then gives sufficient conditions for ergodicity in the dispersing or semi-dispersing case, before turning to the question of the existence of (countable) Markov partitions for two-dimensional hyperbolic billiards. Various deep statistical properties (central limit theorem, decay of time-correlation functions, properties of periodic orbits) can be deduced from this existence.

MSC:

37D99 Dynamical systems with hyperbolic behavior
37A99 Ergodic theory
82B05 Classical equilibrium statistical mechanics (general)
53C22 Geodesics in global differential geometry

Citations:

Zbl 0741.00019
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