We study dynamical properties of the billiard flow on convex polyhedra away from a neighbourhood of the non-smooth part of the boundary, called “pockets”. We prove there are only finitely many immersed periodic tubes missing the pockets and moreover establish a new quantitative estimate for the lengths of such tubes. This extends well-known results in dimension 2. We then apply these dynamical results to prove a quantitative Laplace eigenfunction mass concentration near the pockets of convex polyhedral billiards. As a technical tool for proving our concentration results on irrational polyhedra, we establish a control-theoretic estimate on a product space with an almost-periodic boundary condition. This extends previously known control estimates for periodic boundary conditions, and seems to be of independent interest.

中文翻译：

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多面体台球、本征函数浓度和近周期控制

我们研究了凸多面体上远离边界非光滑部分附近的台球流的动力学特性，称为“口袋”。我们证明只有有限多个浸没的周期管缺少口袋，而且建立了对此类管长度的新定量估计。这扩展了维度 2 中众所周知的结果。然后我们应用这些动力学结果来证明凸多面体台球袋附近的定量拉普拉斯本征函数质量浓度。作为证明我们在无理多面体上的集中结果的技术工具，我们建立了对具有几乎周期性边界条件的乘积空间的控制理论估计。这扩展了先前已知的周期性边界条件的控制估计，并且似乎是独立的兴趣。