(08/25/2009, 10:23 PM)jaydfox Wrote: At any rate, even though it "might" not be the same function, I see no reason it wouldn't be, so I've been assuming it's the same. To assume it is different is to go back to the old argument about uniqueness: it would necessarily introduce a cyclic shift, which would produce spectacularly different results near the primary singularities, and I don't see why this would be at all likely.
Well my thoughts behind that statement was like this:
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.
While matrix power sexp is defined also at the fixed point, the intuitive slog has of course a singularity at the fixed point and so we can not develop islog directly at the fixed point.
But the whole intuitive slog should converge to the whole regular slog at the fixed point when moving the development point to the fixed point.
This consideration would imply different islogs at different development points for bases less or equal than eta. And then I would just consider it quite unprobable that this should be different for bases greater eta.
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.