# Hard ball systems are completely hyperbolic

### Abstract

We consider the system of $N$ ($\ge 2$) elastically colliding hard balls with masses $m+1,\ldots,m_N$, radius $r$, moving uniformly in the flat torus $\mathbb{T}_L^\nu = \mathbb{R}^\nu/L \cdot \mathbb{Z}^\nu$, $\nu \ge 2$. It is proved here that the relevant Lyapunov exponents of the flow do not vanish for almost every $(N+1)$-tuple $(m_1,\ldots,m_N;L)$ of the outer geometric parameters.

Nándor Simányi

Domokos Szász