论文标题
Birkhoff-Poritsky猜想中心对称台球桌子
The Birkhoff-Poritsky conjecture for centrally-symmetric billiard tables
论文作者
论文摘要
在本文中，我们证明了Birkhoff-Poritsky的中央对称$ C^2 $ -SMOOTH凸面平面台球的猜想。我们假设域$ \ MATHCAL a $在不变的曲线之间，$ 4 $ - 周期轨道和相缸的边界以$ C^0 $ -INVARIANT曲线侧向。在这个假设下，我们证明了台球曲线是椭圆形。对于原始的伯克霍夫 - 波利切斯基公式，我们表明，如果台球域边界的邻域具有$ c^1 $ - 平滑的叶子，则通过凸的旋转数（0; 1/4）的旋转数量（0; 1/4），那么边界曲线是椭圆形的。在第一个积分的语言中，如果是一个集体的语言。 $γ$承认$ c^1 $ - 平滑的首先是$ \ Mathcal a $的不可变化的梯度，然后曲线$γ$是一个椭圆形。曲线由$ 4 $ - 周期性的轨道组成；
In this paper we prove the Birkhoff-Poritsky conjecture for centrally-symmetric $C^2$-smooth convex planar billiards. We assume that the domain $\mathcal A$ between the invariant curve of $4$-periodic orbits and the boundary of the phase cylinder is foliated by $C^0$-invariant curves. Under this assumption we prove that the billiard curve is an ellipse. For the original Birkhoff-Poritsky formulation we show that if a neighborhood of the boundary of billiard domain has a $C^1$-smooth foliation by convex caustics of rotation numbers in the interval (0; 1/4] then the boundary curve is an ellipse. In the language of first integrals one can assert that {if the billiard inside a centrally-symmetric $C^2$-smooth convex curve $γ$ admits a $C^1$-smooth first integral with non-vanishing gradient on $\mathcal A$, then the curve $γ$ is an ellipse.} The main ingredients of the proof are : (1) the non-standard generating function for convex billiards discovered in \cite{BM}, \cite{B}; (2) the remarkable structure of the invariant curve consisting of $4$-periodic orbits; and (3) the integral-geometry approach initiated in B0, B1 for rigidity results of circular billiards. Surprisingly, we establish a Hopf-type rigidity for billiard in ellipse.
