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February  2008, 22(1&2): 427-443. doi: 10.3934/dcds.2008.22.427

Algebro-geometric methods for hard ball systems

Domokos Szász 1,

1. 

Budapest University of Technology and Economics, Institute of Mathematics, Budapest, Egry J. u. 1, H–1111, Hungary

Received  March 2007 Revised  May 2007 Published  June 2008

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].
Keywords: algebro-geometric methods., Boltzmann--Sinai ergodic hypothesis, hyperbolicity, ergodicity, semi-dispersing billiards, Hard ball systems.
Mathematics Subject Classification: 37A60, 37D25, 37D5.
Citation: Domokos Szász. Algebro-geometric methods for hard ball systems. Discrete & Continuous Dynamical Systems, 2008, 22 (1&2) : 427-443. doi: 10.3934/dcds.2008.22.427
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