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Mathematical billiard systems demonstrate nonexistence of invariant curves, indicating significant complexity in their dynamics.
A mathematical billiard consists of a planar region as a table and a ball moving on the table and reflecting 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 of invariant 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 for Halpern’s billiard, which has a convergent billiard orbit.
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Shibayama et al. (2026) studied this question.
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