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<p>A mathematical billiard consists of a planar region as a table and a ball moving on the table andreflecting from the boundary. By considering the motion as a discrete dynamical system,the billiard map is an area-preserving twist map.In this work, we show the result of nonexistence ofinvariant curves for two different billiard maps.The first result presents a sufficient condition for the nonexistence of invariant curves.In the second one, we prove the nonexistence of invariant curves near the boundary forHalpern’s billiard, which has a convergent billiard orbit.</p>

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Nonexistence of Invariant Curves for Some Billiard Systems

  • 期刊论文

  • 发表日期：

    2026年6月15日

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  • Mitsuru Shibayama,
  • Hajime Tateishi

摘要

A mathematical billiard consists of a planar region as a table and a ball moving on the table andreflecting from the boundary. By considering the motion as a discrete dynamical system,the billiard map is an area-preserving twist map.In this work, we show the result of nonexistence ofinvariant curves for two different billiard maps.The first result presents a sufficient condition for the nonexistence of invariant curves.In the second one, we prove the nonexistence of invariant curves near the boundary forHalpern’s billiard, which has a convergent billiard orbit.