Geometric Limits

This demonstration will illustrate
the geometric limit argument that
finishes the proof of the erratic
orbits theorem.  This demo
requires the most background.
Before trying to read this, you should
already have read about the box 
phenomenon and the copy phenomenon.

 canonical approximating sequence 
Any irrational parameter A in (0,1)
has an approximation by odd rationals
A(1),A(2),A(3)... such that 

1. Each consecutive pair (A(n),A(n+1))
is a Farey pair.

2. The diophantine constant of the pair
(A(n),A(n+1)) is at least 1 for all n.

3. The diophantine constant of the pair
(A(n),A(n+1)) is at least 2 for infinitely
many n.  

I give a proof in my monograph.  The proof
is similar to the usual proof concerning
the rational approximants in the continued
fraction expansion.

Each canonical sequence starts out with
the value 

(k-2)/k for some k=3,5,7...

You can select from the first few values
of k by clicking on the leftmost blue
arrow key on the  applet  at right.
(After the first click, the calculation
of the other rationals is activated.)

By clicking the other blue arrow keys,
you modify an initial sequence of
rationals.  The values of the arrow
keys indicate the Diophantine constants
of consecutive rationals.  For instance,
if the first two arrow keys have values
5 and 2, then the first two fractions
are 3/5 and 7/11.  One can check that
this is a Farey pair with Diophantine
constant 2.

If you set the arrow key to a 1, it will
alter the way the fractions are placed.
We will explain this below.

 Plotting 
The red arrow key changes the fraction
on the list that you are focusing on, and
furthermore plots the corresponding
arithmetic graph.

The applet will not consider any
fractions where the denominator is
larger than 999.  We make this
cutoff so that the pictures can
be rendered quickly.  If the rational
is too big, it is not listed.

You can scale the picture by using
the left and right mouse buttons.
If you don't have a 3-button mouse,
or if your mouse doesn't interact
well with the applet, you can use
the mouse emulator.

Copying Rules 
Once you set the Diophanine constants,
the boxes displaying the rationals
sort themselves out into  levels .
Our period copying results translate
into the following easy statement:

 Copying rule 
 One period of the arithmetic graph
corresponding to the last term on any
level is copied by one period of the
arithmetic graphs correspondont to all
subsequent terms.

In the monograph, we only prove this
statement for sufficiently deep levels.
However, one can see that the statment
holds exactly as it is.  The weaker
statement suffices to prove our main
theorem.  The purpose of the applet
is to illustrate this copying rule.

 Geometric Limits 
Given the copying rule, it is useful
to pass to a subsequence consisting
of the last arithmetic graphs on each
level.  Call these graphs G(1),G(2)...
These graphs have two important
properties:  

I. One period of G(n) is copied
by the one period of G(n+1) for all n.

II. The maximum distance G(n) rises
up from the bottom of the box R(n)
that contains it is at least 2^n.
Here R(n) is as in the discussion
of the Box Phenomenon.

The number of difference scales on
which G(n) oscillates tends to
infinity as n tends to infinity.
Given the way that our graphs copy each
other, we can take a limit.
The limiting graph corresponds to
an unbounded orbit on the irrational
kite K(A).  Our monograph does this
argument very carefully.