06-164 G. Del Magno, R. Markarian

On the Bernoulli property of planar hyperbolic billiards (684K, PDF) May 18, 06

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Abstract. We consider billiards in bounded non-polygonal domains of $ \R^{2} $ with piecewise smooth boundary. More precisely, we assume that the curves forming the boundary are straight lines or strictly convex inward curves (dispersing) or strictly convex outward curves of a special type (absolutely focusing). It follows from the work of Sinai, Bunimovich, Wojtkowski, Markarian and Donnay that these billiards are hyperbolic (non-vanishing Lyapunov exponents) provided that proper conditions are satisfied. In this paper, we show that if some additional mild conditions are satisfied, then not only these billiards are hyperbolic but are also Bernoulli (and therefore ergodic). Our result generalizes previous works and applies to a very large class of planar hyperbolic billiards. This class includes, among the others, the convex billiards bounded by straight lines and focusing curves studied by Donnay.

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