Singularities and non-hyperbolic manifolds do not coincide (Q2839491)

This paper completes the proof of the Boltzmann-Sinai ergodic hypothesis for hard ball systems on the \(n\)-torus \(\mathbb{R}^n/\mathbb{Z}^n\), for \(n\geq 2\), without any assumed additional hypotheses or exceptional models.NEWLINENEWLINE The Boltzmann-Sinai hypothesis states that a gas with \(N\geq 2\) hard balls (having not too large a radius) on the \(n\)-torus \((n\geq 2)\) is ergodic, provided that certain necessary reductions have been made. Those reductions are: the total energy is fixed, the total momentum is zero, and the center of mass is restricted to a certain discrete lattice in the torus. The ``not too large a radius'' restriction is needed to make sure that the interior of the configuration space is connected.NEWLINENEWLINE The missing piece that this paper addresses is to prove the Chernov-Sinai Ansatz: No open piece of the singularity manifold can coincide precisely with the codimension-one manifold that describes future non-hyperbolicity of the trajectories. The proof is by contradiction, and is done first for \(n=2\), then \(n\geq 3\).