平移流和L形台球的长首中时
Long Hitting time for translation flows and L-shaped billiards
摘要：
我们认为流动方向$θ$上的平移表面和我们研究的渐近行为为$r\to 0$的轨道所需的时间击中$r$ -邻域的一个规定的点，或更准确地说，指数的相应的幂律，这是已知的首中时。对于平面环面，首中时的limsup等于方向$θ$的丢番图型。在高等亏格中，我们考虑了方向$θ$的丢番图型的广义几何概念，并寻求与首中时的关系。对于亏格为两个的曲面，证明了其首中时的limsup总是小于或等于丢番图型的平方。对于任何具有相同拓扑结构的正方形拼接曲面，丢番图型本身就是一个下界，并且两个界之间的任何值都可以实现，而且这对于满足特定拓扑假设的更大类的Origamis也成立。最后，对于所谓的Eierwald de Wollmilchsau折纸，首中时的limsup与丢番图类型之间的相等性存在。我们的结果适用于L形台球。
英文摘要：
We consider the flow in direction $θ$ on a translation surface and we study the asymptotic behavior for $r\to 0$ of the time needed by orbits to hit the $r$-neighborhood of a prescribed point, or more precisely the exponent of the corresponding power law, which is known as hitting time. For flat tori the limsup of hitting time is equal to the diophantine type of the direction $θ$. In higher genus, we consider a generalized geometric notion of diophantine type of a direction $θ$ and we seek for relations with hitting time. For genus two surfaces with just one conical singularity we prove that the limsup of hitting time is always less or equal to the square of the diophantine type. For any square-tiled surface with the same topology the diophantine type itself is a lower bound, and any value between the two bounds can be realized, moreover this holds also for a larger class of origamis satisfying a specific topological assumption. Finally, for the so-called Eierlegende Wollmilchsau origami, the equality between limsup of hitting time and diophantine type subsists. Our results apply to L-shaped billiards.
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