The Boltzmann-Sinai Ergodic Hypothesis in Two Dimensions (Without Exceptional Models)

We consider the system of $N$ ($\ge 2$) elastically colliding hard balls of masses $m_1,\mathrm{.}\mathrm{.}\mathrm{.},m_N$ and radius $r$ in the flat unit torus ${\mathbb{T}}^{\nu }$, $\nu \ge 2$. In the case $\nu =2$ we prove (the full hyperbolicity and) the ergodicity of such systems for every selection $(m_1,\mathrm{.}\mathrm{.}\mathrm{.},m_N;r)$ of the external geometric parameters, without exceptional values. In higher dimensions, for hard ball systems in ${\mathbb{T}}^{\nu }$ ($\nu \ge 3$), we prove that every such system (is fully hyperbolic and) has open ergodic components.