Function Analysis Window

This window implements a fairly advanced feature of McBilliards. In brief, it lets you analyze certain universal formulas which arise in the Taylor series expansions of the defining functions for infinite families of orbit tiles . Many infinite families of orbit tiles we encounter converge up to scale to a limiting shape. At least for some examples of interest to us, this asymptotic convergence is a consequence of a certain kind of universal formula we see in these Taylor series expansions. We think this is a very wide ranging phenomenon for triangular billiards, but have yet to explore it systematically. This window lets the user explore the universality phenomenon from an algebraic point of view. In order to operate this window, and understand the documentation below, you should be pretty familiar with the following other windows: You really need to use all of these components when operating this window. For some of the features of this window (described at the end) you should also know a tiny bit about Fourier series and modular arithmetic.

COMMUNICATION:

On the most basic level, this window lets you store various pairs of the form (W,V), where W is a word and V is a pair of vertices on the unfolding of W. You could think of this feature of the window as something like short-term memory. The blue strip running across the window, near the top, stores the pairs you have saved. The green buttons at the top left control the memory system. Here is how it works: FUNCTION DISPLAY

The next most basic thing this window does is let you see the evaluations of the functions associated to the pairs you have stored. Here we explain how this is managed: POLYNOMIAL FITTING

If you just store some pairs at random, there will be no relations between the numbers on the list L_ab. However, often in McBilliards we see infinite families of orbit tiles. If the pairs (W_j,V_j) are chosen carefully and in relation to these tiles, then the numbers L_ab might be the consecutive values of a polynomial P_ab. That is, the jth member of L_{ab} is P_ab(j). Typically, the degree of P_{ab} is about a+b.
AN EXAMPLE

Here we will guide you through an example which shows off the sort of formulas we have been suggesting exist. Ultimately we are going to do our calculations at the point (1/3,1/3) in parameter space. You will be doing some searching and plotting near this point. We will describe a family of 5 consecutive tiles. Once you see the pictures, you should be able to easily guess how the family continues.

The Example Here will describe the first 5 tiles in an infinite family. notation (x,N,i,j) denotes an orbit tile associated to a word of length N, containing the point (x,x). (We are looking near the isosceles line.) The pair i,j denotes the vertex pair (a_i,b_j). In general, the pair (x,N) is not enough information to specify the tile uniquely. However, it works fine in our example. Also, the choices of x we make are not canonical. We just pick x to be some smallish fraction in the tile of interest to us. Here is the family you should find and load into this window: Notice that the word length and the vertex numbers all belong to arithmetic progressions. This sort of thing is the hallmark of a family of tiles. Once you have several terms in the family, you can use the subword search in tandem with the subword selection feature of the word window to rapidly generate additional tiles in the sequence.

Computing Things Now perform the following steps: If you have done all these steps, you will see entirely integers along the top lines of the display boxes, making a geometric progression. This sort of pattern persists for all the partial derivatives, and you can see more of it by collecting more pairs.

Universality We have given a very opaque description of the vertex pairs of interest to us. Once you have plotted the orbit tiles in our example, you can generate the vertex pairs in a more natural way: Turn on the compute leaders option on the unfolding window and then choose a parameter point very near the ``bottom'' edges of the tiles, the ones whose slopes are all about -1. You will then see the vertex pairs generated automatically. You could repeat this same exercise using a different sequence of edges in the tiles. You just have to take care that the edges you select are all ``the same one'' throughout the family. You will get the same formulas for your new example if you repeat the above calculation. You can do similar things at other points. (For instance, the points (1/n,1/n) have similar families converging to them, for n=4,5,6...) You can also do this away from the isosceles line. The simple formula highlighted above seems to work (when suitably generalized) for a fairly general kind of infinite family of orbit tiles. We hope to exposit this phenomenon in a forthcoming paper.

PIX MODE AND FOURIER TRANSFORMS

The defining function F and all its partial derivatives have the general form

A1 sin(B1 x+C1 y) + ... + An sin(Bn x + Cn y)

Where the pairs (Bj,Cj) are integers determined by the combinatorics of the pair (W,V). If you click on the pix button (on the magenta panel) you can see a picture of these integer squares. The number Aj is drawn in the box in the (Bj,Cj) position. The number is white if positive and black if negative. You select which function is displayed by using the dx and dy and term keys. The triples (Aj,Bj,Cj) give rise to a compactly supported integer valued function defined on the integer lattice. This function is essentially the Fourier transform of the corresponding defining function.

The pix mode draws similar pictures of the Fourier transforms for the defining functions P and Q and their partial derivatives. This time, these functions have the general form
A1 E(B1 x+C1 y) + ... + An E(Bn x + Cn y)
Where E(x)=exp(ix).

THE MODULAR TRANSFORM

One key step to understanding the universality phenomenon discussed above is to understand a kind of mod N integral of the Fourier transform. We call the resulting object a modular transform .

Let (x,y) be the grabbed coordinate. Let (p1/q1,p2/q2) be the rational approximation to (x,y). In cases of interest we will have x=p1/q1 and y=p2/q2. The green panel at the top has a modular transform option. We will suppose that the options are set to When the actual values option is selected on the green panel, each line of each display box shows some value H(x,y) where H is some partial derivative of either P,Q, or F. When the modular transform is on, we compute a function [H](A,B,k) according to the following recipe: Assuming that we can simplify our notation by writing [H](k)=[H](A,B,k). We call [H] the modular transform of H. It is not hard to see, once all the definitions are unravelled, that there are constants C1,...,CN such that

H(x,y)=C1 H[1]+...+CN H[N]

The constants C1,...,CN only depend on the integers p1,q1,p2,q2. We call the above equation the modular transform equation .

Once we have computed [H](k) in place of H we can do the polynomial fitting just as above. In other words, we can compute the polynomial growth rates of our modular transforms. These growth rates are the (algebraic) key to understanding the universality phenomenon.