论文标题
凸台球和应用的非标准生成功能的本地最大化轨道
Locally maximizing orbits for the non-standard generating function of convex billiards and applications
论文作者
论文摘要
给定一个具有生成函数的圆柱体的确切符号地图$ t $ t $ h $满足所谓的负面扭曲条件，$ h_ {12}> 0 $，我们研究了本地最大化$ t $的轨道，即，是局部的构型，这些配置是动作函数$ \ sum_n h（q_n h（q_n h，q_n h，q_ q_ n oction y sumimima）。我们为配置提供了必要且充分的条件，以在局部最大化。使用它，我们考虑了$ t $相对于两组不同的符号坐标具有两个生成功能的情况。我们建议一种简单的几何条件，可以确保相对于这两个生成函数的局部最大化轨道的集合一致。作为主要应用程序，我们表明平面伯克霍夫台球的两个生成功能满足了这种几何条件。我们将其应用于以下结果：考虑一个集中对称曲线$γ$，伯克霍夫台球地图的旋转不变曲线$α$ $α$为$ 4 $ - 周期性轨道。我们证明，在$γ$及其“最佳近似”椭圆之间的某种$ l^2 $ distance可以从上面的界限，这是通过衡量$α$和相缸边界的本地最大化轨道填充的集合的补充的量度。此外，该估计值很清晰，为伯克霍夫（Birkhoff）猜想的中央对称曲线提供了有效版本。我们还获得了针对任意曲线$γ$的类似界限，这将一组本地最大化轨道的补充与$ l^2 $ distance的量相关，并在$γ$及其“最佳近似”圆圈之间进行$ l^2 $差异。
Given an exact symplectic map $T$ of a cylinder with a generating function $H$ satisfying the so-called negative twist condition, $H_{12}>0$, we study the locally maximizing orbits of $T$, that is, configurations which are local maxima of the action functional $\sum_n H(q_n,q_{n+1})$. We provide a necessary and sufficient condition for a configuration to be locally maximizing. Using it, we consider a situation where $T$ has two generating functions with respect to two different sets of symplectic coordinates. We suggest a simple geometric condition which guarantees that the set of locally maximizing orbits with respect to both of these generating functions coincide. As the main application we show that the two generating functions for planar Birkhoff billiards satisfy this geometric condition. We apply it to get the following result: consider a centrally symmetric curve $γ$, for which the Birkhoff billiard map has a rotational invariant curve $α$ of $4$-periodic orbits. We prove that a certain $L^2$-distance between $γ$ and its "best approximating" ellipse can be bounded from above in terms of the measure of the complement of the set filled by locally maximizing orbits lying between $α$ and the boundary of the phase cylinder. Moreover, this estimate is sharp, giving an effective version of a recent result on Birkhoff conjecture for centrally symmetric curves. We also get a similar bound for arbitrary curves $γ$, that relates the measure of the complement of the set of locally maximizing orbits with the $L^2$-distance between $γ$ and its "best approximating" circle.
