AI 中文总结
该研究针对解析平面区域的台球类映射，证明了若两区域的Mather β-函数在KAM丢番图数正测度集上重合，则其在所有KAM丢番图数集上重合，且对应Marvizi-Melrose不变量也重合。
AI 中文摘要
在其开创性论文中，Kac提出了著名的问题：“能否‘听出鼓的形状’”，即欧氏空间中有界区域的等距类是否由其拉普拉斯谱唯一确定。拉普拉斯谱与对应台球的长度谱密切相关。对于凸有界平面区域，每个台球周期轨道不仅可关联其长度，还可关联其旋转数，每个周期轨道的长度与旋转数对构成的集合称为标记长度谱。利用该区域的标记长度谱，可关联其极小作用函数，即Mather的β-函数，记为β_Ω。通过唯一拟解析延拓，我们证明以下刚性问题：若两个解析平面区域中两个台球的Mather β-函数在KAM丢番图数的正测度集上重合，则它们在所有KAM丢番图数的集合上重合；特别地，若两个区域的Mather β-函数在KAM丢番图数的正测度集上重合，则这两个区域的Marvizi-Melrose不变量重合。
英文摘要
In his seminal paper, Kac famously asked whether "one can hear the shape of a drum" - that is whether the isometry class of a bounded domain in a Euclidean space is uniquely determined by the spectrum of its Laplace spectrum. The Laplace spectrum is closely related to the length spectrum of the associated billiard. For a convex bounded planar domain, to each billiard periodic orbit one can associate not only its length but also its rotation number. The set of pairs of length and rotation number of each periodic orbit is called the marked length spectrum. Using the marked length spectrum of the domain one can associate with it its minimal action function also known as Mathers $β$-function denoted by $β_Ω$. Via unique quasianalytic continuation, we prove the following rigidity problem: knowing that Mather's $β$-functions of two billiards in two analytic planar domains coincide on a set of positive measure of KAM diophantine numbers implies that they coincide on a set of all KAM diophantine numbers. In particular, knowing that Mather's $β$-functions of two domains coincide on a positive measure set of KAM diophantine numbers implies that the Marvizi-Melrose invariants of these domains coincide.
