Symmetries of Period-Doubling Maps
Symmetries of Period-Doubling Maps
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倍周期图的对称性
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发表时间：
2004
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通讯作者：
L. Vepstas
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作者：
L. Vepstas
自相似的概念是分形概念的核心，但产生自相似的实际对称群很少被命名，而且似乎分形几乎从来没有根据它们的对称性来研究。然而，在数学和物理的其他分支中，对称性为理解系统提供了一种强大的机制，这是众所周知的。在本文中，我们证明了倍周期映射的对称群是模群PSL（2，Z）的单（半群）。为了巩固这一论断，我们提出了一个显式的、精确可解的分形曲线，即Takagi或Blancmange曲线，作为模群（单阵）的三维表示下的变换。通过将生成Blancmange曲线的三角形状替换为多项式，我们发现生成的曲线在n + 2维的单形表示下变换，其中n是多项式的度。我们还发现Blancmange曲线的（定义不清的）导数本质上是Cantor函数的（逆），从而也证明了Cantor集上的半群对称性。实际上，任何拓扑共轭映射在三维表示下都会发生变换。然后我们展示了所有的周期加倍映射是如何证明单群对称的，这本质上是单群的二进表示的结果。本文还回顾了Georges deRham在1958年构造的Koch雪花、Levy c曲线、Peano空间填充曲线和Minkowski问号函数作为一类具有一元对称曲线的特例。左右对称的Levy c曲线和左右对称的Koch曲线分别属于另一个不相等的三维表示。本文是探讨实数、模群和分形之间关系的一组章节的一部分。它的结构也有些糟糕，至少有一部分是作为研究结果的日记来写的。人们普遍注意到，法雷数在某些分形中自然出现，最著名的是在曼德尔布罗集合中。例如，图1显示了如何计数由Farey数设置的Mandelbrot的芽。为什么Farey数字适合这样的计算，原因就不太清楚了。然而，可以说，许多分形现象，特别是周期加倍图，都有一个
The concept of self-similarity is central to the notion of a fractal, but the actual symmetry group that generates that self-similarity is rarely named, and it seems fractals are almost never studied according to their symmetries. Yet, in other branches of mathematics and physics, it is well understood that symmetry provides a powerful mechanism for understanding systems. In this paper, we identify the symmetry group of period-doubling maps as being a monoid (semigroup) of the modular group PSL(2,Z). To anchor this assertion, we work out an explicit, exactly-solvable fractal curve, the Takagi or Blancmange Curve, as transforming under the three-dimensional representation of the (monoid of the) modular group. By replacing the triangular shape that generates the Blancmange curve with a polynomial, we find that the resulting curve transforms under the n + 2 dimensional representation of the monoid, where n is the degree of the polynomial. We also find that the (ill-defined) derivative of the Blancmange curve is essentially the (inverse of the) Cantor function, thus demonstrating the semigroup symmetry on the Cantor Set as well. In fact, any topologically conjugate map will transform under the three-dimensional representation. We then show how all period-doubling maps can demonstrate the monoid symmetry, which is essentially an outcome of the dyadic representation of the monoid. This paper also includes a review of Georges deRham’s 1958 construction of the Koch snowflake, the Levy C-curve, the Peano space-filling curve and the Minkowski Question Mark function as special cases of a curve with the monoid symmetry. The left-right symmetric Levy C-curve and the left-right symmetric Koch curve are shown each belong to another, inequivalent three dimensional representation. This paper is part of a set of chapters that explore the relationship between the real numbers, the modular group, and fractals. Its also a somewhat poorly structured, written at least partly as a diary of research results. 1 Symmetries of Period-Doubling Maps It has been widely noticed that Farey numbers appear naturally in certain fractals, most famously in the Mandelbrot set. For example figure 1 shows how to count the buds of the Mandelbrot set by the Farey numbers. The reason why the Farey numbers are appropriate for such counting is somewhat more opaque. However, it can be said that many fractal phenomena, and in particular, period-doubling maps, have an