monograph guide

The purpose of this demo is to explain a connection between outer billiards on kites and the modular group. because it doesn't work for all irrational parameters.) To better appreciate this slideshow, you should first learn about the arithmetic graph. The applet at right allows you to compute the arithmetic graph for various parameters. These parameters all have the form T((r/s)+k) where T is an element of the modular group SL2(Z) that (as a linear fractional transformation) maps infinity to p/q. Here p/q is an odd/odd rational. The first vertical set of boxes on the applet (yellow/yellow) control the values of p/q. r/s is an odd/even rational. The second vertical set of boxes on the applet (yellow/orange) control the values of r/s. k is a positive integer. The magenta arrow keys on the applet control k. The choice of T is not unique, but we choose T so that it has positive entries, that are as small as possible.Modular Limit ConjectureLet G(k) be the arithmetic graph corresponding to the the quintuble (p,q,r,s;k). Then, the rescaled limit 1/k G(k) converges in the Hausdorff topology to a polygonal curve G which is not a straight line.You can test this out on the applet for choices of p,q,r,s less than 20, and k less than 30. Billiard King lets you test the conjecture more extensively. The Modular Limit Conjecture is an example of arenormalization phenomenonin dynamics. The theory developed in my monograph make some progress towards proving this conjecture, but I did not pursue the conjecture in the monograph. At present, I don't have a proof.