双中心台球桌的解析刚性与符号动力学
Analytic rigidity and symbolic dynamics for two-centre billiards
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中文总结 AI 辅助
针对平面双中心台球桌，建立刚性-混沌二分性，证明非共焦椭圆边界的台球桌具混沌动力学，共焦椭圆边界则满足Birkhoff-Poritsky猜想的实解析形式。
中文摘要 AI 辅助
受Birkhoff-Poritsky猜想自然类比的启发，我们为平面双中心台球桌建立了严格的刚性-混沌二分性：所有能量下都可积的台球桌仅为与两中心共焦的椭圆。设Ω为含两中心连线的有界区域，边界为C¹类。对每个固定能量h≥0，若∂Ω非共焦椭圆，我们构造台球轨道，使其追踪碰撞-反射轨道的稳定与不稳定流形，并实现绕连线的任意指定足够大绕数序列。这产生了与可数字母表上全移位半共轭的不变集、具有任意指定有限行程的周期轨道，以及任意大拓扑熵的紧不变子系统。若∂Ω为实解析的，则固定能量相空间M_h上每个在台球映射下不变的实解析函数均为常数。这在整个非负能量范围内确立了双中心Birkhoff-Poritsky猜想的实解析形式。
英文摘要
We establish a sharp rigidity--chaos dichotomy for planar two-centre billiards, motivated by a natural analogue of the Birkhoff--Poritsky conjecture: the only tables integrable at every energy should be ellipses confocal with the two centres. Let $Ω$ be a bounded domain with $\mathcal C^1$ boundary containing the segment joining the centres. At every fixed energy $h\geq 0$, if $\partialΩ$ is not a confocal ellipse, we construct billiard trajectories that shadow the stable and unstable manifolds of the collision--reflection orbit and realise arbitrarily prescribed sequences of sufficiently large winding numbers around the segment. This yields an invariant set semiconjugate to the full shift on a countable alphabet, periodic trajectories with prescribed finite itineraries, and compact invariant subsystems with arbitrarily large topological entropy. If, in addition, $\partialΩ$ is real-analytic, every real-analytic function on the fixed-energy phase space $M_h$ that is invariant under the billiard map is constant. This establishes the real-analytic form of the two-centre Birkhoff--Poritsky conjecture throughout the non-negative-energy regime.
