The $K$-property of four billiard balls

摘要: A further step is achieved toward establishing the celebrated Boltzmann-Sinai ergodic hypothesis: for systems of four hard balls on ν-torus (ν>2) it shown that, submanifold phase specified by trivial conservation laws, system aK-flow. All parts our previous demonstration providing analogous result three are simplified and strengthened. The main novelties are: (i) refinement geometric-algebraic methods used earlier helps us to bound codimension arising implicitly given set degeneracies even if we can not calculate their exact dimension that was possible three-billiards. As a matter fact, this part arguments, where understanding new ideas necessary before attacking general problem; (ii) In “pasting” proof, which sophisticated version Hopf's classical device, arguments so hoped they work in case, too. This balls, particular, Transversal Fundamental Theorem which, one hand, simpler more suitable applications than and, other as have discovered earlier, tool prove global ergodicity semi-dispersing billiards; (iii) verification Chernov-Sinai ansatz essentially idea proof also promises case.