Construction of K-stable foliations for two-dimensional dispersing billiards without eclipse

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Author(s)

Abstract

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

  • 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

References:  15

Codes

  • NII Article ID (NAID)
    10013358966
  • NII NACSIS-CAT ID (NCID)
    AA0070177X
  • Text Lang
    ENG
  • Article Type
    ART
  • ISSN
    00255645
  • NDL Article ID
    7015172
  • NDL Source Classification
    ZM31(科学技術--数学)
  • NDL Call No.
    Z53-A209
  • Data Source
    CJP  NDL  J-STAGE 
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