Date: Fri, 14 Apr 95 17:06:48 -0400
From: Jerry Bryan <BRYAN@wvnvm.wvnet.edu >
~~~ Subject: Repetitive Application of Elements of Q*

Recall that Q* has been defined as the set of representatives

```Q* = {F*, R*, L*, B*, U*, D*, F'*, R'*, L'*, B'*, U'*, D'*}
```

where * has been defined as a function selecting a representative
element of an M-conjugacy class.

I have done a little experimentation with cycles of the form X* ^ n.
As long as the X* are directly in Q*, the sequences are quite short,
and the final cycle is of length 2 in all cases. I found the latter
surprising initially, but with the wisdom of hindsight, it was
inevitable. Here is a table of lengths.

```Operation         Length
of
Sequence
```
```F*                11
U*                10
L*                 7
R*                 3
D*                 9
B*                 2
F'*                7
U'*               10
L'*                4
R'*                6
D'*                7
B'*                2
```

A couple of points of clarification:

1. As an example, for F*, we take i(F*)^n (that is, apply F*
to Start repeatedly). The sequence has 11 elements before it
repeats, then the 10-th and 11-th element repeat over and
over again. In all twelve cases, it is the last two elements
of the sequence which repeat.

2. In order to replicate my results, you would have to define
a representative element function exactly like mine. Every
choice of representative element function can be expected to
yield different results.

To take a little more interesting case, I tried i(F*D'*) ^ n. In this
case, there were 63 unique elements in the sequence, and then the
8-th through the 63-rd elements repeated. Hence, the final cycle had
56 elements rather than the 2 elements of the simpler cases.

I suppose I could try quite a few other cases, but I have no idea how
to predict how long the sequences or the terminal cycles might be.
All I know to expect for sure is for things to be quite ill-behaved.

``` = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan)                        (304) 293-5192
Associate Director, WVNET                            (304) 293-5540 fax