THE MINKOWSKI QUESTION MARK, PSL(2,Z) AND THE MODULAR GROUP

ABSTRACT. 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