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.

収録刊行物

  • Journal of the Mathematical Society of Japan

    Journal of the Mathematical Society of Japan 56(3), 803-831, 2004-07-01

    一般社団法人 日本数学会

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各種コード

  • NII論文ID(NAID)
    10013358966
  • NII書誌ID(NCID)
    AA0070177X
  • 本文言語コード
    ENG
  • 資料種別
    ART
  • ISSN
    00255645
  • NDL 記事登録ID
    7015172
  • NDL 雑誌分類
    ZM31(科学技術--数学)
  • NDL 請求記号
    Z53-A209
  • データ提供元
    CJP書誌  NDL  J-STAGE 
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