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Algebro-geometric methods for hard ball systems

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  • For the study of hard ball systems, the algebro-geometric approach appeared in 1999 --- in a sense surprisingly but quite efficiently --- for proving the hyperbolicity of typical systems (see [26]). An improvement by Simányi [22] also provided the ergodicity of typical systems, thus an almost complete proof of the Boltzmann--Sinai ergodic hypothesis. More than that, at present, the best form of the local ergodicity theorem for semi-dispersing billiards, [6] also uses algebraic methods (and the algebraicity condition on the scatterers). The goal of the present paper is to discuss the essential steps of the algebro-geometric approach by assuming and using possibly minimum information about hard ball systems. In particular, we also minimize the intersection of the material with the earlier surveys [29] and [20].
    Mathematics Subject Classification: 37A60, 37D25, 37D50.


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