[I'm sorry that I keep following up my own post, but after having spent too long on this problem, I seem to have been rather too eager to get it all finished with, and I spotted another error just now - when I was in bed trying to forget all about maths, and concentrate on a good ghost story!]
On Sat, 20 Nov 2004 23:12:28 +0000, I wrote:
>Suppose p \in S_oo. Then for all sufficiently large n, >the vector A^n(p) is in the positive octant, P, and has >u-coordinate < 1 (because A multiplies u-coordinates by >the positive factor m - 1 < 1). > >[Damn! ...
... and blast! Because quite apart from the gap in the proof, there is a trivial error here:
Because the (u, v, w) system is not normalised, the u- coordinate of q is not q.u (as I hastily assumed), but (q.u)/|u|^2, so the condition for n to be "large enough" should have been that the u-coordinate of q be < 1/|u|^2 (which of course is still O.K., because the choice of the value 1 was arbitrary in the first place), so that we still have q.u < 1.