标题： 带有奇异不变曲线的击球运动 显示英文标题

标题： Billiards with singular invariant curves

摘要： 我们研究环面上一类特殊的辛扭转映射，即轨道数为$1/2$的不变曲线的正则性，这类映射包括台球映射。 我们构造了接近圆的严格凸光滑桌面对，这些桌子具有奇异的（即不是$C^1$）不变曲线。 我们的方法依赖于经典弦构造的修改，允许对奇点的位置进行精确控制：它们形成一个离散集，其闭包可以包含$\mathbb{S}^1$的几乎任何闭子集。 每个奇点对应一个双曲的$2$-周期轨迹，且在这些点上，不变曲线在两侧具有不同的导数。 类似的构造产生了常宽桌子的扰动，这些扰动具有轨道数为$1/2$的不变曲线。 显示英文摘要

摘要： We investigate the regularity of invariant curves of rotation number $1/2$ for a special class of symplectic twist maps of the annulus, billiard maps. We construct strictly convex smooth tables close to the circle having singular (i.e. not $C^1$) invariant curves. Our method relies on a modification of the classical string construction and allows precise control over the location of singularities: they form a discrete set whose closure can contain virtually any closed subset of $\mathbb{S}^1$. Each singularity corresponds to a hyperbolic $2$-periodic trajectory and the invariant curves admit distinct one-sided derivatives at these points. An analogous construction yields perturbations of constant-width tables with invariant curves of rotation number $1/2$.