发表机构
University of Rome Tor Vergata; Institute for Advanced Study（罗马托尔维加塔大学; 高等研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明在$C^1$-generic中心对称严格凸外弹子球中，存在无界轨道，肯定回答了Moser和Neumann的著名问题，关键利用刚性定理和Mather扩散机制。
AI 中文摘要
设$\mathcal H$为边界是$C^1$曲线的中心对称、严格凸、紧致平面体的支撑函数空间，赋予$C^1$拓扑。我们证明，对于$\mathcal H$的一个剩余子集，关于相应物体的外弹子球映射具有逃逸到无穷远的无界轨道。特别地，这个剩余集合包含任意$C^1$-接近圆盘或椭圆支撑函数的支撑函数。这就在严格凸$C^1$情形下，对Moser和Neumann关于无界外弹子球轨道存在性的著名问题给出了肯定回答。关键成分是一个刚性定理：如果$h\in\mathcal H$具有纯奇异的曲率半径测度，那么每个连续不变切图完全由周期点组成。然后我们证明，对于$\mathcal H$中的一般台面，这些周期不变图不存在，从而允许我们通过Mather在Birkhoff不稳定区域中的扩散机制构造逃逸轨道。
英文摘要
Let $\mathcal H$ be the space of support functions of centrally symmetric, strictly convex, compact planar bodies whose boundary is a $C^1$ curve, endowed with the $C^1$ topology. We prove that for a residual subset of $\mathcal H$ the outer billiard maps about the associated bodies admit unbounded orbits escaping to infinity. In particular, this residual set includes support functions arbitrarily $C^1$-close to those of disks or ellipses. This provides an affirmative answer, in the strictly convex $C^1$ setting, to a famous question of Moser and Neumann concerning the existence of unbounded outer-billiard orbits. The key ingredient is a rigidity theorem: if $h\in\mathcal H$ has a purely singular radius-of-curvature measure, then every continuous invariant tangent graph consists entirely of periodic points. We then show that for a generic table in $\mathcal H$ these periodic invariant graphs are absent, allowing us to construct escaping orbits via Mather's diffusing mechanism in a Birkhoff region of instability.
Comments47 pages, 7 figures
