Modular Limit Phenomenon

Billiard King homepage
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 


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 Conjecture 
 Let 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 a  renormalization phenomenon  in 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.