标题： 关于开普勒台球的伯尔克霍夫猜想 显示英文标题

标题： On the Birkhoff conjecture for Kepler billiards

摘要： 我们研究开普勒台球系统的可积性——一种粒子在开普勒势影响下运动，并在严格凸平面区域边界处弹性反射的机械台球系统。 我们的主要结果表明，除了引力中心的一个可能位置外，高能情况下解析可积性仅当区域为椭圆且中心位于其一个焦点时才发生。 这为经典的伯克霍夫-波里茨基猜想的开普勒类似问题提供了部分肯定答案。 我们的方法基于高能 regime 中出现的混沌子系统所产生的符号动力学的构造。 根据边界几何配置和吸引中心的位置，我们通过跟踪穿孔的伯克霍夫型轨迹链，构造了三种类型的符号动力学。 这些构造产生了与伯努利移位共轭的子系统，意味着正的拓扑熵并排除了解析可积性。 我们进一步分析了第二类焦散点的概念，表明在实解析非椭圆区域中最多只有一个这样的点，而椭圆是唯一具有两个重合于其经典焦点的点的区域。 最后，我们证明了存在无限维的非椭圆区域族，它们具有第二类焦散点，并以数值模拟说明了这些情况下的混沌行为。 显示英文摘要

摘要： We investigate the integrability of Kepler billiards-mechanical billiard systems in which a particle moves under the influence of a Keplerian potential and reflects elastically at the boundary of a strictly convex planar domain. Our main result establishes that, except possibly for one location of the gravitational center, analytic integrability at high energies occurs only when the domain is an ellipse and the center is placed at one of its foci. This provides a partial affirmative answer to a Keplerian analogue of the classical Birkhoff-Poritsky Conjecture. Our approach is based on the construction of symbolic dynamics arising from chaotic subsystems that emerge in the high-energy regime. Depending on the geometric configuration of the boundary and the location of the attraction center, we construct three types of symbolic dynamics by shadowing chains of punctured Birkhoff-type trajectories. These constructions yield subsystems conjugated to Bernoulli shifts, implying positive topological entropy and precluding analytic integrability. We further analyze the notion of focal points of the second kind, showing that in real-analytic, non-elliptic domains there can be at most one such point, while ellipses are the only domains admitting two-coinciding with their classical foci. Finally, we demonstrate the existence of an infinite-dimensional family of non-elliptic domains possessing a focal point of the second kind, and conclude with numerical simulations illustrating chaotic behavior in such cases.