The Minkowski Question Mark, Gl(2,z) and the Modular Group (expository)

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 through the fact that a certain subset of the general linear group GL(2,Z) over the integers; called the “dyadic monoid”, “dyadic groupoid”, or “dyadic lattice”, is the natural symmetry of many fractals, 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. The core underlying idea is that many fractals can be represented as an infinite binary tree. Aside from fractals, binary numbers (or “dyadic numbers”) can also be represented as a binary tree. A string of 1’s and 0’s can be understood to be a choice of left and right movements through the branches of a tree; which gives it the structure of the Cantor set. The rational numbers also may be arranged into a binary tree, which is variously called a Farey tree or a Stern-Brocot tree. The Minkowski question mark function arises as the isomorphism between the tree of dyadic numbers, and the tree of rational numbers. Fractal self-similarity arises naturally, with the observation that a subtree of a binary tree is isomorphic to the tree itself. This paper is written at an expository level, and should be readily accessible to advanced undergraduates and all graduate students. XXXX This paper is unfinished. Although this version corrects a number of serious errors in the previous drafts, it is still misleading and confusing in many ways. The second half, in particular must surely contain errors and mis-statements! Caveat emptor! XXXX Expository