We introduce a large class of billiards with convex pieces of the boundary which have nonvanishing Lyapunov exponents.

Abstract. We discuss the Sinai method of proving ergodicity of a discontinuous Hamiltonian system with (non-uniform) hyperbolic behavior.

We show that in several cases preservation of cones leads to non-vanishing of (some) Lyapunov exponents. It gives simple and effective criteria for nonvanishing of the exponents, which is demonstrated on the example of the billiards studied by Bunimovich. It is also shown that geodesic flows on manifolds of non-positive sectional curvature can be treated from this point of view.

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We present new proofs of two results on the billiard ball problem by Rychlik [R] and Bialy [B]. IntroductionWe will give new proofs of two results on the billiard ball problem by Rychlik [7] and Bialy [1], The original proofs were based on variational considerations. In our approach the variational context is absent, the dynamical system takes the center stage. We hope the simplifications provided by our method will make possible some progress on the conjectures for which these results lend partial support.