求解准周期自转率的巴比伦问题
Solving the Babylonian Problem of quasiperiodic rotation rates
摘要：
轨迹$u_n := F^n(u_0), n = 0,1,2, \dots $是准周期的，如果轨迹位于某个$d$维环面上并且在其中是稠密的，并且在环面$\mathbb{T}$上有一个坐标选择，其中对于所有$θ\in\mathbb{T}$和某些$ρ\in\mathbb{T}$，$F$具有$F(θ) = θ+ ρ\bmod1$的形式。有一个古老的文献计算三个旋转率PDMATHAMT的月球。有一种文献是关于确定矢量$ρ$的坐标的，称为$F$的旋转速率。(For $d>1$我们总是将$\bmod1$解释为应用于每个坐标。）然而，即使在$d=1$的情况下，也没有仅给定轨迹$u_n$来计算$ρ$的通用方法，尽管有处理特殊情况的文献。在这里，我们提出了我们的嵌入延拓方法计算的$ρ$从轨迹的一些组件。它基于Takens嵌入定理和Birkhoff遍历定理。旋转速率通常被称为“旋转数”，两者都是指圆的旋转速率。然而，$ρ$的坐标取决于$\mathbb{T}$的坐标的选择。我们探讨了$ρ$可以产生的各种可能的旋转速率。我们用维度$d=1$和$2$的例子来说明我们的想法。
英文摘要：
A trajectory $u_n := F^n(u_0), n = 0,1,2, \dots $ is quasiperiodic if the trajectory lies on and is dense in some $d$-dimensional torus, and there is a choice of coordinates on the torus $\mathbb{T}$ for which $F$ has the form $F(θ) = θ+ ρ\bmod1$ for all $θ\in\mathbb{T}$ and for some $ρ\in\mathbb{T}$. There is an ancient literature on computing three rotation rates $ρ$ for the Moon. %There is a literature on determining the coordinates of the vector $ρ$, called the rotation rates of $F$. (For $d>1$ we always interpret $\bmod1$ as being applied to each coordinate.) However, even in the case $d=1$ there has been no general method for computing $ρ$ given only the trajectory $u_n$, though there is a literature dealing with special cases. Here we present our Embedding Continuation Method for computing some components of $ρ$ from a trajectory. It is based on the Takens Embedding Theorem and the Birkhoff Ergodic Theorem. Rotation rates are often called "rotation numbers" and both refer to a rate of rotation of a circle. However, the coordinates of $ρ$ depend on the choice of coordinates of $\mathbb{T}$. We explore the various sets of possible rotation rates that $ρ$ can yield. We illustrate our ideas with examples in dimensions $d=1$ and $2$.
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