AI 中文总结
研究平面辛台球的一般性质，证明对于\(C^\infty\)强凸域剩余集，周期轨迹相关性质，还证弗兰克斯引理等，得出\(C^\infty\)强凸辛台球一般有正拓扑熵的结论。
AI 中文摘要
我们研究平面辛台球的一般性质，它是由P.阿尔伯斯和S.塔巴奇尼科夫引入的经典伯克霍夫台球的辛类似物。我们证明，对于\(C^\infty\)强凸域的一个剩余集，周期辛台球轨迹处于一般位置且非退化。我们还证明了辛台球的弗兰克斯引理，并推断稳定双曲周期点形成一个由双曲周期点集的闭包给出的双曲集。我们进一步表明，对于\(C^\infty\)强凸域的一个剩余集，椭圆周期点是稳定的，双曲周期点有横向同宿点。基于这些结果，我们得出结论——一般来说——\(C^\infty\)强凸辛台球具有正拓扑熵。
英文摘要
We study generic properties of planar symplectic billiards, a symplectic analogue of classical Birkhoff billiards introduced by P. Albers and S. Tabachnikov. We prove that, for a residual set of $C^\infty$ strongly convex domains, periodic symplectic billiard trajectories are in general position and non-degenerate. We also prove a Franks' lemma for symplectic billiards and deduce that stably hyperbolic periodic points form a hyperbolic set given by the closure of the set of hyperbolic periodic points. We further show that, for a residual set of $C^\infty$ strongly convex domains, elliptic periodic points are stable and hyperbolic periodic points have transverse homoclinic points. Based on these results, we conclude that --generically-- $C^\infty$ strongly convex symplectic billiard has positive topological entropy.
Comments26 pages, comments are welcome
