Non-integrability of cylindric billiards and transitive Lie group actions

摘要: A conjecture is formulated and discussed which provides a necessary sufficient condition for the ergodicity of cylindric billiards (this improves previous one second author). This requires that action Lie-subgroup ${\cal G}$ orthogonal group $SO(d)$ ($d$ being dimension billiard in question) be transitive on unit sphere $S^{d-1}$. If $C_1, \dots, C_k$ are scatterers billiard, then generated by embedded Lie subgroups G}_i$ $SO(d)$, where consists all transformations $g\in SO(d)$ ${\Bbb R}^d$ leave points generator subspace $C_i$ fixed ($1 \le i k$). In this paper we can prove necessity our also formulate some notions related to transitivity. For hard ball systems, show transitivity holds general: an arbitrary number $N\ge 2$ balls, masses $m_1, m_N$ $\nu \ge 2$. result implies stronger than Boltzmann–Sinai ergodic hypothesis systems. We note somewhat surprising characterization positive fundamental form evolution special manifold (wavefront), namely parallel beam light. Thus obtain new sufficiency orbit segment.