Let be the billiard map for a two-dimensional dispersing billiard without eclipse. We show that the nonwandering set for has a hyperbolic structure quite similar to that of the horseshoe. We construct a sort of stable foliation for each leaf of which is a -decreasing curve. We call the foliation a -stable foliation for . Moreover, we prove that the foliation is Lipschitz continuous with respect to the Euclidean distance in the so called -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 and why the -stable foliation turns out to be Lipschitz continuous.
Takehiko MORITA. "Construction of -stable foliations for two-dimensional dispersing billiards without eclipse." J. Math. Soc. Japan 56 (3) 803 - 831, July, 2004. https://doi.org/10.2969/jmsj/1191334087