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 $$\tilde{K} = \{(x,y,z)\in \mathbb{R}^3 : x^2+y^2+(βz)^2 \leq r^2\},$$ with parameter $β>1$, small radii, and centers $O_i$ lying in the plane $Γ=\{z=0\}$. 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 $β>1/c_1^2$, for some constant $c_1>0$ determined by the geometry of the configuration, the two positive Lyapunov exponents are distinct and positive: $λ_1(β)>λ_2(β)>0$. Moreover, $λ_2$ is independent of $β$ and corresponds to the restriction of the billiard flow to the plane $Γ$.