标题： 无穷远处的外辛折线映射 显示英文标题

标题： Outer symplectic billiard map at infinity

摘要： We show that the second iteration $T^2$ of the outer symplectic billiard map with respect to a convex domain $M$ in a symplectic vector space is approximated by an explicit Hamiltonian flow for points far away from $M$. More precisely, denote by $N$ the symplectic polar dual of the symmetrization $M\ominus M$ of $M$. 如果我们把$N$写成一个1次齐次函数$H$的单位水平集，那么$T^2$与将$H$的时间-2哈密顿流作用于点$x$所得到的结果之间的差异小于$c/|x|$，其中常数$c$仅依赖于$M$。 此外，我们证明如果一个轨道逃逸到无穷远，那么其到原点的距离在迭代次数中的增长速度不超过$\sqrt{k}$。 最后，我们证明一个$k$-周期轨道在$k$的意义上需要接近$M$。 显示英文摘要

摘要： We show that the second iteration $T^2$ of the outer symplectic billiard map with respect to a convex domain $M$ in a symplectic vector space is approximated by an explicit Hamiltonian flow for points far away from $M$. More precisely, denote by $N$ the symplectic polar dual of the symmetrization $M\ominus M$ of $M$. If we write $N$ as the unit level set of a 1-homogeneous function $H$, then the difference between $T^2$ and the time-2-Hamiltonian flow of $H$ applied to a point $x$ is smaller than $c/|x|$ for some constant $c$ depending only on $M$. Moreover, we show that if an orbit escapes to infinity, then its distance to the origin grows not faster than $\sqrt{k}$ in the number of iterations. Finally, we prove that a $k$-periodic orbit needs to be close, in terms of $k$, to $M$.