仿射扭转映射的标准扰动
On Standard perturbations of the Affine twist map
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中文总结 AI 辅助
本文研究仿射扭转映射的标准扰动，证明有理旋转时竖直旋转集为单点，无理旋转时变为包含该无理数的非退化区间，表明构造单无理数旋转集的保面积扭转映射之难。
中文摘要 AI 辅助
对于任意 $t\in\mathbb{R}$，考虑仿射扭转映射 $\rm{Aff_t}:\mathbb{T}^2\to\mathbb{T}^2$，定义为 $$\rm{Aff_t}(x,y)=(x+y \text{ mod 1}, y+t \text{ mod 1}).$$ 映射 $\rm{Aff_t}$ 显然具有由水平曲线构成的不变叶理。若 $t$ 为有理数，则每条叶是周期的；若 $t$ 为无理数，则每条叶的轨道在环面上稠密。在两种情形下，$\rm{Aff_t}$ 到竖直圆柱的适当提升的竖直旋转集均缩减为 $\{t\}$。现在，对于 $k\in\mathbb{R}$，定义 $$f_{k,t}(x,y):=(x+y+k\sin (2\pi x)\text{ mod 1},y+k\sin (2\pi x)+t\text{ mod 1}).$$ 我们证明：1) 当 $t=p/q$（其中 $p$ 和 $q>0$ 为整数）时，KAM 理论蕴含存在常数 $k_{p/q}>0$，使得当 $|k|<k_{p/q}$ 时，$f_{k,p/q}$ 的适当提升的竖直旋转集恰为 $\{p/q\}$。2) 当 $t$ 为无理数时，对于任意 $k\neq 0$，$f_{k,t}$ 的适当提升的竖直旋转集是一个非退化区间，且 $t$ 位于其内部。换言之，构造竖直旋转集缩减为单个无理数的保面积扭转映射并不容易。根据 \cite{eujul} 的定理 A，这样的映射必须具有由水平坐标上的 Lipschitz 图构成的不变叶理。特别地，其所有迭代必须满足扭转条件。而这正是 $f_{k,t}$ 对所有非零 $k$ 值所不满足的。
英文摘要
For any $t\in\mathbb{R}$, consider the Affine twist map $\rm{Aff_t}:\mathbb{T}^2\to\mathbb{T}^2$ given by $$\rm{Aff_t}(x,y)=(x+y \text{ mod 1}, y+t \text{ mod 1}).$$ The map $\rm{Aff_t}$ clearly possesses an invariant foliation by horizontal curves. If $t$ is rational, each leaf is periodic, and if $t$ is irrational, the orbit of each leaf is dense in the torus. In both cases, the vertical rotation set of the proper lift of $\rm{Aff_t}$ to the vertical cylinder is reduced to $\{t\}.$ Now, for $k\in\mathbb{R},$ define $$f_{k,t}(x,y):=(x+y+k\sin (2πx)\text{ mod 1},y+k\sin (2πx)+t\text{ mod 1}).$$ We show that: 1) when $t=p/q$ for integers $p$ and $q>0$, KAM theory implies the existence of a constant $k_{p/q}>0$ such that for $|k|<k_{p/q}$, the vertical rotation set of an adequate lift of $f_{k,p/q}$ is just $\{p/q\}.$ 2) when $t$ is irrational, for any $k\neq 0$ the vertical rotation set of the adequate lift of $f_{k,t}$ is a non-degenerate interval which contains $t$ in its interior. In other words, it is not easy to build area-preserving twist maps whose vertical rotation sets are reduced to a single irrational number. From Theorem A of \cite{eujul}, such a map needs to have an invariant foliation by Lipschitz graphs over the horizontal coordinate. In particular, all its iterates must satisfy a twist condition. This is precisely what does not hold for $f_{k,t}$, for all non-zero values of $k$.
