Construction of K-stable foliations for two-dimensional dispersing billiards without eclipse
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Let T be the billiard map for a two-dimensional dispersing billiard without eclipse. We show that the nonwandering set Ω<SUP>+</SUP> for T has a hyperbolic structure quite similar to that of the horseshoe. We construct a sort of stable foliation for (Ω<SUP>+</SUP>, T) each leaf of which is a K-decreasing curve. We call the foliation a K-stable foliation for (Ω<SUP>+</SUP>, 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 Ω<SUP>+</SUP> and why the K-stable foliation turns out to be Lipschitz continuous.
- Journal of the Mathematical Society of Japan
Journal of the Mathematical Society of Japan 56(3), 803-831, 2004-07-01
The Mathematical Society of Japan