Interactive Guide to the Monograph

by Rich Schwartz

Billiard King homepage
``` Introduction
The purpose of this guide is
to explain the main ideas in
my monograph using Java applets.
I discovered all the results in
the monograph using my Java program,
of the monograph would greatly benefit
from playing with Billiard King,
but Billiard King is pretty
complicated.  The applets in this
guide break Billiard King
into small pieces that are
easy to understand and operate.

Note: In a certain sense, the
monograph comes in 2 halves.
Parts I-IV comprise the first
half, and parts V-VI comprise
the second half.  The second
half has much sharper results.
(I added the second half in the
last 6 months.)  This exposition
really only deals with the first
half.

The Main Results
Here are the main results of the (first
half of) the monograph.

1.  Erratic Orbits Theorem
Outer billiards on any  irrational kite
has uncountably many erratic special orbits.

2.  Dichotomy Theorem
Relative to any irrational kite,
every special orbit is periodic or
else unbounded in both directions.

3.  Density Theorem
Relative to any irrational kite,
the set of periodic special orbits is
dense in the set of all special orbits.

4.  Hexagrid Theorem
(See below for the statement.)

5.  Master Picture Theorem
(See below for the statement.)

The Erratic Orbits Theorem gives a
robust positive solution to the
Moser-Neumann problem , posed
around 1960, which asks if one can
have unbounded orbits in an outer
billiards system.   The other results

This guide will focus on the Erratic
Orbits Theorem.  We will also discuss
the Hexagrid Theorem and the
Master Picture Theorem, because these
results are needed for the Erratic
Orbits Theorem.  We hope to later
expand the guide so that it also
discusses the Dichotomy Theorem and
the Density Theorem.

The Arithmetic Graph
The key idea in my monograph is
to study an object that I call
the  arithmetic graph .
The arithmetic graph is a graph
in the plane, with integer coordinate
vertices, that encodes some of the
dynamics of outer billiards on a
need to understand the arithmetic
graph. Click  here  to learn about it.

Ingredients for the EOT
The proof of the Erratic Orbits
Theorem has 3 main ingredients.

1.   Box Theorem
2.   Period Copying Theorem
3.   Geometric limits

The Hexagrid Theorem
The Box Theorem is a consequence
of the Hexagrid Theorem .
Roughly speaking, the H.T.
says that the (extended) arithmetic
graph is controlled by 6 infinite
families of parallel lines.  This
result reflects a similar kind of
quasi-periodicity one sees in
DeBruijn's  pentagrid method
for generating the Penrose tiling.

Master Picture Theorem
All the results mentioned above are
derived from our main structural result,
the Master Picture Theorem .  This
result gives a "formula" for
the arithmetic graph, in terms of certain
4-dimensional integer lattice polytope
partitions.  The Master Picture Theorem
seems likely to generalize to outer
billiards on polygons, though I have
not worked this out.

The Penrose Kite
Before I proved the Erratic
Orbits Theorem in general, I
proved it for the single example
of the Penrose kite, the kite
that arises in the  Penrose tiling
To learn about the E.O.T. for this
example, see the  Penrose kite demo .
You can understand half the
arithmetic graph, and the whole
arithmetic graph.

Modular Limits
I discovered another neat
property of outer billiards on
kites, which I call the
modular limit phenomenon .
This is a certain connection
between outer billiards and the
modular group related to
renormalization.  Here is the
modular limit demo .

```