Some open billiards with distinct top Lyapunov exponents

Abstract

We study open billiard flows in ${\mathbb{R}}^3$ generated by reflections in the exterior of at least three ellipsoids with $C^3$ boundaries. Each obstacle is congruent to $$\tilde{K}=\{(x,y,z)\in {\mathbb{R}}^3:x^2+y^2+(\beta z{)}^2\le r^2\},$$ with parameter $\beta >1$, small radii, and centers $O_i$ lying in the plane $\Gamma =\{z=0\}$. The configuration satisfies the no-eclipse condition (H). For any Gibbs measure on the non-wandering set of the billiard ball map, we prove that if $\beta >1/c_1^2$, for some constant $c_1>0$ determined by the geometry of the configuration, the two positive Lyapunov exponents are distinct and positive: ${\lambda }_1(\beta )>{\lambda }_2(\beta )>0$. Moreover, ${\lambda }_2$ is independent of $\beta$ and corresponds to the restriction of the billiard flow to the plane $\Gamma$.

10 pages