Geometric Limits

Monograph guide
Billiard King homepage

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.


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.