Construction of K-stable foliations for two-dimensional dispersing billiards without eclipse
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- MORITA Takehiko
- Department of Mathematics Graduate School of Science Hiroshima University
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Description
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, \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+ and why the K-stable foliation turns out to be Lipschitz continuous.
Journal
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- Journal of the Mathematical Society of Japan
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Journal of the Mathematical Society of Japan 56 (3), 803-831, 2004
The Mathematical Society of Japan
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Details 詳細情報について
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- CRID
- 1390001205115168000
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- NII Article ID
- 10013358966
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- NII Book ID
- AA0070177X
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- ISSN
- 18811167
- 18812333
- 00255645
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- MRID
- 2071674
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- NDL BIB ID
- 7015172
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- Text Lang
- en
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- Data Source
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- JaLC
- NDL
- Crossref
- CiNii Articles
- KAKEN
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- Abstract License Flag
- Disallowed