Construction of K-stable foliations for two-dimensional dispersing billiards without eclipse

Abstract

Let $T$ be the billiard map for a two-dimensional dispersing billiard without eclipse. We show that the nonwandering set ${\Omega }^{+}$ for $T$ has a hyperbolic structure quite similar to that of the horseshoe. We construct a sort of stable foliation for $({\Omega }^{+},T)$ each leaf of which is a $K$-decreasing curve. We call the foliation a $K$-stable foliation for $({\Omega }^{+},T)$. Moreover, we prove that the foliation is Lipschitz continuous with respect to the Euclidean distance in the so called $(r,\varphi )$-coordinates. It is well-known that we can not always expect the existence of such a Lipschitz continuous invariant foliation for a dynamical system even if the dynamical system itself is smooth. Therefore, we keep our construction as elementary and self-contained as possible so that one can see the concrete structure of the set ${\Omega }^{+}$ and why the $K$-stable foliation turns out to be Lipschitz continuous.