08/26/2009, 08:11 PM

(08/26/2009, 05:46 PM)bo198214 Wrote: Well my thoughts behind that statement was like this:The answer can be found by considering the intuitive slog first (because I don't really see how to get to the intuitive slog from the regular slog by reasoning alone, but perhaps you or sheldon can help me here?).

The matrix power method applied to fixed points is the regular iteration at that fixed point.

Also I would assume the change of function to be continuous in the development point.

Say we have to real fixed points x0 and x1 and going from x0 to x1 as development point of the matrix power sexp. The regular iteration at x0 would slowly transform into the regular iteration at x1, which we know are different (except for fractional linear functions).

As I consider the intuitive iteration as kinda opposite to the matrix power iteration. I would guess (though this is not supported by numerics yet) that the closer to the fixed point I choose my development point x0 the closer is the resulting islog to the regular slog at the fixed point. (To show that the islog to base sqrt(2) is different from the rslog one could consider the half-iterate isexp(1/2+islog(x)) and show that it has a singularity at 2 which is not true for the regular slog). If not so I would wonder how the islog decides which fixed point to take for being regular at.

...

Well if you have different evidence of any point in my chain of conclusion I would like to hear. My understanding of the complex behaviour of the islog does not suffice that I can support your statement of the spectacular different behaviour at the fixed points. Perhaps you can illustrate with some pictures or thought experiments.

Start with the islog, developed at 0. Create a straight line, from the origin to the upper primary fixed point, and call this line L1. If we calculate the islog of all the points of L1, we will trace out a curve that goes up and slightly to the left, that eventually straightens into a straight line, with slope equal to about -4.2034 for base e. Call this image L2.

If we travel along L2, and perform the isexp function, we should get back an image of the original line, L1.

However, if, at some point on the image L2, we instead travel in a straight line in the positive real direction (to the right), then as we perform the sexp operation (to get a new image L3), you will have something that resembles the regular sexp developed at the fixed point. But in order to get back to "the action", which I consider anything farther than a unit's distance from the fixed point, you must travel very far to the right. As you do so, you wind around the fixed point a great many times, and this puts you well outside the principal branch of the islog. Indeed, the islog in this particular branch will look pretty much exactly like the rslog.

Note that in the limit as you travel up L1 (perfoming the islog to get an image L2, then picking a point on L2 and performing the isexp while going to the right to get L3) all the way to the fixed point, you get exactly the regular sexp, developed at the fixed point. So the regular sexp at the upper fixed point is what lies "beyond" the top of the sexp developed from the inverse of the intuitive slog. The regular sexp at the lower fixed point is what lies "beyond" the bottom of the intuitive sexp.

The reason that the regular sexp and the intuitive sexp seem different is that, if you followed my instructions, you were travelling up and to the left of the origin as you approached the fixed point. When we think of the regular sexp, however, we view it from somewhere that is closer to "the action" (more than one unit distance from the fixed point), so it's developed probably from somewhere up and to the right, so that as we trace out in the imaginary direction, down towards the origin, we get to "the action" well before we get back to the principal branch of the islog. Thus we can't directly see the relation, going from regular to intuitive. But the relation is obvious (to me anyway) in the other direction.

It is easy to "get to" the regular sexp from the intuitive sexp, by this limiting behavior, and hence to the rslog from the islog. But I haven't been able to reverse the process, at least not in any thought experiment. Perhaps you or Sheldon can help me figure out a way to get "back"?

~ Jay Daniel Fox