Singularities and nonhyperbolic manifolds do not coincide

TL;DR

This paper proves that in billiard systems of hard balls on a flat torus, singularity manifolds do not coincide with non-hyperbolic manifolds, leading to the ergodicity and Bernoulli mixing properties of these systems.

Contribution

It establishes that singularities and nonhyperbolic manifolds do not coincide in such billiard systems, confirming the Boltzmann-Sinai Ergodic Hypothesis.

Findings

Proves non-coincidence of singularity and nonhyperbolic manifolds.

Establishes ergodicity and Bernoulli mixing for these billiard systems.

Abstract

We consider the billiard flow of elastically colliding hard balls on the flat $\nu$-torus ($\nu\ge 2$), and prove that no singularity manifold can even locally coincide with a manifold describing future non-hyperbolicity of the trajectories. As a corollary, we obtain the ergodicity (actually the Bernoulli mixing property) of all such systems, i.e. the verification of the Boltzmann-Sinai Ergodic Hypothesis.