TheK-property of four billiard balls

Abstract

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