Construction of Kstable foliations for twodimensional dispersing billiards without eclipse
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Abstract
Let T be the billiard map for a twodimensional 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 Kdecreasing curve. We call the foliation a Kstable 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 wellknown 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 selfcontained as possible so that one can see the concrete structure of the set Ω<SUP>+</SUP> and why the Kstable foliation turns out to be Lipschitz continuous.
Journal

 Journal of the Mathematical Society of Japan

Journal of the Mathematical Society of Japan 56(3), 803831, 20040701
The Mathematical Society of Japan