Dan A. writes:
Mike Stay asks:
<< Let g(n,x) be the nth iterate of f(x), i.e. g(0,x) = x g(1,x) = f(x) g(2,x) = f(f(x)) etc.
If f is the log function, is there a nice function that behaves like g when n is integral, but allows for non-integer values (something like the gamma function wrt factorial)?
Mike, this is a question that has occupied some of us on math fun (alone and jointly) for many years!
Wow! It's nice to know I can at least ask good a question, though I didn't understand a lot of what you said. Looks like I'll have to spend a few hours on mathworld working it out. My interest stems from looking at differential cryptanalysis and Rijndael: you can define differences such that it's easy to predict the effect + and * have, but not inversion, or the effect that * and ^ have, but not addition. I was wondering if there was something in between that would give good results for * and less than awful results for + and inverse.
Usually the question has been posed about exp(x) or exp(z). There's a lot to say about it.
One thing is that there are infinitely many ways to define such an iteration, using the fact that R / (x ~ exp(x)) defines a real analytic structure on the circle. There are infinitely ways to express its equivalence g: R/(x~exp(x)) --> R/Z by a real analytic diffeomorphism g. Lift this equivalence to a diffeomorphism from h:R --> R, and that can be used to define a "flow" of exp(x) via exp_t(x) = h^-1 ( h(x) + t ). Integral t will give the expected integer iteration.
What's h here?
This is kind of an embarrassment of riches, since there doesn't seem to be a clear way to pick out the "right" g. (This was discovered by Bill Thurston the same weekend that I noticed it independently.)
If one wants to find a complex analytic embedding of exp in a flow near certain points in C, then there are infinitely many, but at least *discrete* choices to make, which will give such a thing. Alas, it doesn't seem to extend in any natural way to the reals. (This is from work I've done, though quite likely others have found the same thing; I'm not sure.)
That's OK, I guess, since I'm working in GF(2^8).
For each real c in (1, exp(1/e)) (and probably a complex neighborhood thereof) the function f(x) = c^x (for x positive real and some complex neighborhood of R+, there are only *two* methods that appear to be the rightest ways to embed f(x) in a complex analytic flow (a flow of analytic functions which is also analytic in the flow (time) parameter), but no apparent way to choose between these two. (This is from work I've done, and closely related if not overlapping things have been discovered by Rich Schroeppel. Dean Hickerson also did related research, which included demonstrating that the two methods mentioned are distinct, after I'd convinced myself they yielded identical answers.)
What are they?
There is little doubt that much of the above has been found by other people at another time, but I'm not aware of such research.
Maybe I'd better let it go at that for the time being.
--Dan A. -- Mike Stay (who's still feeling guilty for not having followed up on the last e-mail Dan sent me...) staym@clear.net.nz http://www.xaim.com/staym