Abstract. We prove exponential decay of correlations for a "reasonable" class of multi-dimensional dispersing billiards. The scatterers are required to be C 3 smooth, the horizon is finite, there are no corner points. In addition, we assume subexponential complexity of the singularity set.

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The fundamental theorem (also called the local ergodic theorem) was introduced by Sinai and Chernov in 1987, see [S-Ch(1987)] and an improved version in [K-S-Sz(1990)]. It provides sufficient conditions on a phase point under which some neighborhood of that point belongs to one ergodic component. This theorem has been instrumental in many studies of ergodic properties of hyperbolic dynamical systems with singularities, both in 2-D and in higher dimensions. The existing proofs of this theorem implicitly use the assumption on the boundedness of the curvature of singularity manifolds. However, we found recently ([B-Ch-Sz-T(2000)]) that, in general, this assumption fails in multidimensional billiards. Here the fundamental theorem is established under a weaker assumption on singularities, which we call Lipschitz decomposability. Then we show that whenever the scatterers of the billiard are defined by algebraic equations, the singularities are Lipschitz decomposable. Therefore, the fundamental theorem still applies to physically important models -among others to hard ball systems, Lorentz gases with spherical scatterers, and Bunimovich-Reháček stadia. 452In several papers that appeared, singularities were assumed -either explicitly or implicitly -to consist of smooth 1-codim submanifolds of the phase space. Often, Vol. 3, 2002 Multi-Dimensional Semi-Dispersing Billiards 457

New generations of Hoveyda and bis‐carbene type of ruthenium‐based olefin metathesis catalysts (10 and 12), containing cationic cyclic alkyl amino carbene (CAAC) ligands, have been synthetized. The catalysts show exceptional stability and activity in environmentally benign, protic media. Various olefin metatheses reactions of OH functionalized feedstock (e. g. RCM, ROMP CM) can be carried out at as low as 0.05 mol % catalyst loading in methanol, isopropanol, water or methanol/water solvent mixture, accomplishing the lowest applied catalyst loading reported so far in these media. The facile olefin metathesis of renewable feedstocks including phospholipids (23) and vegetable oils (20) in protic media has also been demonstrated.