THE MINKOWSKI QUESTION MARK, PSL(2;Z) AND THE MODULAR GROUP (EXPOSITORY)
THE MINKOWSKI QUESTION MARK, PSL(2;Z) AND THE MODULAR GROUP (EXPOSITORY)
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MINKOWSKI 问号、PSL(2;Z) 和模群（说明）
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发表时间：
2014
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通讯作者：
L. Vepstas
中科院分区：
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作者：
L. Vepstas
分形和连分数似乎在许多方面都有很深的联系。法里分数在两者中自然出现。这种关系的大部分可以解释为，两者都可以用无限二叉树来表示，而无限二叉树又描述了康托集的结构。无限二叉树可以看作模群PSL（2;Z）的一个子集。这个子集本质上是并元广群或并元么半群。它为许多分形的对称性和自相似性提供了自然的背景，包括那些与倍周期映射、锁相映射以及一般的各种动力系统相关的分形。本文的目的是提供一个简单的阐述对称性和它的衔接。在这个过程中，本文试图澄清一组相互关联的数论思想之间的关系：那些围绕模群，椭圆曲线和康托集。模群PSL（2;Z）（整数上2 × 2矩阵的一般线性群）是椭圆曲线的对称群，这一点早已为人们所熟知。这组之间的连接，有理数是常见的书籍和课程;连接到二元子集很少，如果有的话，提到-本文试图纠正这一缺点。同样地，康托集在数学的许多领域中扮演着重要的角色;除此之外，它是真实的数的超集，更一般地说，它是紧致度量空间的泛覆盖。它的对称性很少被讨论，但实际上是由PSL（2;Z）的非常相同的并矢子集描述的。本文展示了所有这些是如何相关的-无限二叉树，康托集，二进制数集（1和0的无限长字符串的集合），有理数，Farey和Stern-Brocot树，连分数，二次无理数的集合和Minkowski问号函数：这些都被示出为同一基础结构（具有二元分形自对称的结构）的相互关联的方面。本文是以说明性水平撰写的，高级本科生和所有研究生都应该很容易阅读。这篇论文永远处于未完成状态。虽然这个版本纠正了以前草案中的一些严重错误，但它肯定在许多方面仍然是误导和混乱的。尤其是后半部分，肯定包含错误和错误陈述！买者自负！XXX
Fractals and continued fractions seem to be deeply related in many ways. Farey fractions appear naturally in both. Much of this relationship can be explained by the fact that both can be represented with the infinite binary tree, which in turn describes the structure of the Cantor set. The infinite binary tree can be viewed as a certain subset of the modular group PSL(2;Z). The subset is essentially the dyadic groupoid or dyadic monoid. It provides the natural setting for the symmetry and self-similarity of many frac- tals, including those associated with period-doubling maps, with phase-locking maps, and with various dynamical systems in general. The aim of this text is to provide a simple exposition of the symmetry and its articulation. In the process, this paper attempts to clarify the relationships between a cluster of inter- related ideas from number theory: those surrounding the modular group, elliptic curves and the Cantor Set. It has long been widely known that the modular group PSL(2;Z) (the general linear group of 2 by 2 matrices over the integers) is the symmetric group of elliptic curves. Connections between this group, and the rational numbers are commonly presented in books and coursework; the connection to the dyadic subsets is rarely, if ever, mentioned - this paper attempts to correct this shortcoming. Likewise, the Cantor set plays an important role in many areas of mathematics; among others, it is a superset of the real numbers, and more general a universal cover for compact metric spaces. Its symmetry properties are rarely discussed, but are in fact described by the very same dyadic subsets of PSL(2;Z). This paper shows how all of these are related - the infinite binary tree, the Cantor set, the set of binary numbers (the set of infinitely long strings of 1's and 0's), the rational num- bers, the Farey and Stern-Brocot trees, continued fractions, the set of quadratic irrationals and the Minkowski Question Mark function: these are all shown to be inter-related aspects of the same underlying structure, a structure having dyadic fractal self-symmetry. This paper is written at an expository level, and should be readily accessible to ad- vanced undergraduates and all graduate students. XXX This paper is in a perpetual state of being unfinished. Although this version corrects a number of serious errors in the pre- vious drafts, it is surely still misleading and confusing in many ways. The second half, in particular must surely contain errors and mis-statements! Caveat emptor! XXX