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July, 2004 Construction of K-stable foliations for two-dimensional dispersing billiards without eclipse
Takehiko MORITA
J. Math. Soc. Japan 56(3): 803-831 (July, 2004). DOI: 10.2969/jmsj/1191334087


Let T be the billiard map for a two-dimensional dispersing billiard without eclipse. We show that the nonwandering set Ω+ for T has a hyperbolic structure quite similar to that of the horseshoe. We construct a sort of stable foliation for (Ω+,T) each leaf of which is a K-decreasing curve. We call the foliation a K-stable foliation for (Ω+,T). Moreover, we prove that the foliation is Lipschitz continuous with respect to the Euclidean distance in the so called (r,ϕ)-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+ and why the K-stable foliation turns out to be Lipschitz continuous.


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Takehiko MORITA. "Construction of K-stable foliations for two-dimensional dispersing billiards without eclipse." J. Math. Soc. Japan 56 (3) 803 - 831, July, 2004.


Published: July, 2004
First available in Project Euclid: 2 October 2007

zbMATH: 1055.37037
MathSciNet: MR2071674
Digital Object Identifier: 10.2969/jmsj/1191334087

Primary: 37D50
Secondary: 37D05 , 37D10

Keywords: billiard map , Lipschitz continuous invariant foliation

Rights: Copyright © 2004 Mathematical Society of Japan


Vol.56 • No. 3 • July, 2004
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