无理三角形台球方向复杂度的下界

近等腰三角形中的台球

Billiards in Nearly Isosceles Triangles

证明了等腰三角形的任意充分小的扰动都有周期的台球路径.我们的证明涉及到与三角形台球有关的傅里叶级数的某些无限族的分析，并揭示了无理三角形台球中的一些自相似现象.我们的分析说明了一个令人惊讶的事实，即台球的三角形附近的维奇三角形是非常复杂的，即使台球的维奇三角形是非常好的理解。

有理多面体的台球复杂性

Billiard complexity in rational polyhedra

本文给出了立方体中有向台球复杂性的一个新的证明，这个证明在\cite{Ra}中得到了证明，在\cite{Ar.Ma.Sh.Ta}中得到了证明。我们的技巧为某些有理多面体给出了类似的定理。

带周期性障碍物台球的遍历方向

Ergodic directions for billiards in a strip with periodically located obstacles

我们研究了无限带$\R\times [0,h]$上的有向台球流的遍历方向集的大小，其中周期性地放置长度为$0<λ<h$的线性障碍。我们证明了遍历方向集总是不可数的。此外，如果$λ/h\in(0,1)$是有理数，则遍历方向集的豪斯多夫维数大于1/2。在这两种情况下（理性的和非理性的），我们明确地构造一些遍历方向集。

Polyhedral billiards, eigenfunction concentration and almost periodic control

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 polyh…

无穷保测度Z^d作用的方向常返性

Directional Recurrence for Infinite Measure Preserving Z^d Actions

本文定义了保无穷测度Z^d作用的方向递归，并证明了这两种定义是等价的。研究了递归方向集的结构，证明了它总是G_delta集。我们构造了一个没有递归方向的递归动作的例子，回答了Daniel J. Rudolph在2007年的一篇论文中提出的问题。我们还通过例子表明，它是可能的经常性的行动，不经常性的非理性方向，即使它的所有子行动是经常性的。

Weak mixing directions in non-arithmetic Veech surfaces

We show that the billiard in a regular polygon is weak mixing in almost every invariant surface, except in the trivial cases which give rise to lattices in the plane (triangle, square and hexagon). More generally, we study the problem of prevalence of weak mixing for the directional flow in an arbitrary non-arithmetic Veech surface, and show that the Hausdorff dimension of the set of non-weak mixing directions is not full. We also provide a necessary condition, verified for instance by the Veech surface corresponding to the billiard in the pentagon, for the set of non-weak mixing directions to…