Abstract
The Local Ergodic Theorem (also known as the ‘Fundamental Theorem’) gives sufficient conditions under which a phase point has an open neighborhood that belongs (mod 0) to one ergodic component. This theorem is a key ingredient of many proofs of ergodicity for billiards and, more generally, for smooth hyperbolic maps with singularities. However, the proof of that theorem relies upon a delicate assumption (Chernov-Sinai Ansatz), which is difficult to check for some physically relevant models, including gases of hard balls. Here we give a proof of the Local Ergodic Theorem for two dimensional billiards without using the Ansatz.
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Chernov, N., Simányi, N. Upgrading the Local Ergodic Theorem for Planar Semi-dispersing Billiards. J Stat Phys 139, 355–366 (2010). https://doi.org/10.1007/s10955-010-9927-6
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DOI: https://doi.org/10.1007/s10955-010-9927-6