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SOLVING THE BABYLONIAN PROBLEM OF QUASIPERIODIC ROTATION RATES

机译：解决QuaSiphyic旋转率的巴比伦问题

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摘要

A trajectory θ_n := F~n(θ_0); n = 0, 1, 2,... is quasiperiodic if the trajectory lies on and is dense in some d-dimensional torus T~d, and there is a choice of coordinates on T~d for which F has the form F(θ) = θ + ρmod 1 for all θ∈ T~d and for some ρ∈ T~d. (For d > 1 we always interpret mod1 as being applied to each coordinate.) There is an ancient literature on computing the three rotation rates for the Moon. However, for d > 1, the choice of coordinates that yields the form F(θ) = θ + ρmod 1 is far from unique and the different choices yield a huge choice of coordinatizations (ρ_1,..,ρ_d) of ρ, and these coordinations are dense in T~d. Therefore instead one defines the rotation rate ρΦ (also called rotation rate) from the perspective of a mapΦ, T~d→ S~1. This is in effect the approach taken by the Babylonians and we refer to this approach as the “Babylonian Problem”: determining the rotation rate ρΦ of the image of a torus trajectory - when the torus trajectory is projected onto a circle, i.e., determining ρΦ from knowledge of Φ(F~n(θ)). Of course ρΦ depends on Φ but does not depend on a choice of coordinates for T~d. However, even in the case d = 1 there has been no general method for computing ρΦ given only the sequence Φ(θ_n), though there is a literature dealing with special cases. Here we present our Embedding continuation method for general d for computing ρΦ from the image Φ(θn) of a trajectory, and show examples for d = 1 and 2. The method is based on the Takens Embedding Theorem and the Birkhoff Ergodic Theorem.
机译：轨迹θ_n：= f〜n（θ_0）; n = 0,1,2，...是QuaSiodic，如果轨迹在某些D维圆圈T〜D中致密，并且在T〜D上有一个坐标，因为f具有f（ θ）=θ+ρmod1，用于所有θ∈T~d和一些ρ∈t ~d。 （对于D> 1，我们始终将Mod1解释为应用于每个坐标。）计算月球三个旋转速率的古代文学。然而，对于D> 1，产生form f（θ）=θ+ρmod1的坐标的选择远非独特，不同的选择产生ρ的巨大选择（ρ_1，..，ρ_d）ρ，以及这些协调在t〜d中是密集的。因此，从Mapφ，T〜D→S〜1的角度来表示一个定义旋转速率ρφ（也称为旋转速率）。这实际上是巴比伦人所采取的方法，并将这种方法称为“巴比伦问题”：确定圆环轨迹的图像的图像的旋转率ρφ - 当圆环轨迹投射到圆上时，即确定ρφ从知识φ（f〜n（θ））。当然，ρφ取决于φ但不依赖于T〜D的坐标的选择。然而，即使在情况下，即使在D = 1中，对于仅给出序列φ（θ_n）的ρφ也没有一般方法，尽管存在特殊情况的文献。在这里，我们介绍了从轨迹的图像φ（θn）计算ρφ的一般d的嵌入的连续方法，并显示D = 1和2的示例。该方法基于嵌入定理和Birkhoff ergodic定理的方法。

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