 Math overflow question on fractional exponential iterations - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: Math overflow question on fractional exponential iterations (/showthread.php?tid=1189) Math overflow question on fractional exponential iterations - sheldonison - 03/28/2018 Dmytro Taranovsky asks in his post, https://mathoverflow.net/questions/283816/fractional-exponentiation-with-different-bases ... " ... Let a and b be real numbers above e^{1/e}, and c exp(1/e), then my conjecture is that the limit holds.  Unfortunately, Peter Walker's solution is also conjectured to be nowhere analytic; see: https://math.stackexchange.com/questions/2010406/peter-walkers-c-infty-conjectured-nowhere-analytic-slog (2) That for Kneser's solution, there are counter examples and the limit does not hold, even with the restriction that c1, and helps one understand why Walker's solution works. RE: Math overflow question on fractional exponential iterations - JmsNxn - 03/30/2018 It seems upsetting that the only tetrations that could satisfy this are non-analytic. I'm not prone to believe this, only because it doesn't look nice... I'm wondering if we can look at it the following way Then the question boils into whether, for all then I think we can show this is true when . By induction, first, for the induction step is obvious. Namely . Suppose the result holds for then (taking to mean asymptotically less than): Sadly, I can't think of anyway to generalize this to non-integral .... I'm thinking, a nice way to look at it from here is to look at root functions of the function. But then we'd need an implication , which looks like it could be true for monotonically growing unbounded functions. But that's probably too easy, we'd probably need a nice condition on the root functions for that to be true. EDIT: It appears I made a fruitful mistake in the above proof. The base induction step would have to be (1) , not the obvious one . This is the base step I should have used. The proof then says if this base step (1) holds the result holds for all natural , namely . And I think with some finesse we can show that this implies it's true for root functions of , which should leave for a proof where . Then perhaps a density argument may work on non rational . I'll work on this more later, but I think maybe we can reduce this entire problem into the condition that if for all we have then it follows that . ...We'll probably have to assume that is monotone non-decreasing in and unbounded, or at least, eventually monotone non-decreasing. RE: Math overflow question on fractional exponential iterations - sheldonison - 03/31/2018 (03/30/2018, 07:37 PM)JmsNxn Wrote: It seems upsetting that the only tetrations that could satisfy this are non-analytic. I'm not prone to believe this, only because it doesn't look nice... There's an old thread http://math.eretrandre.org/tetrationforum/showthread.php?tid=236 I started the thread in 2009, before I had written generic programs for analytic tetration for any base, and I was using an excel spreadsheet to approximate analytic tetration.  I estimated that as x gets arbitrarily large The 2009 thread continues on to discuss what I called "the wobble"... Credit needs to go to William Paulsen and Samuel Cowgill in their upcoming paper which discusses these issues more rigorously than I can.  But it was quickly clear in the 2009 thread that there is an inherent wobble when comparing tetration bases; this was apparent for bases a little bit bigger than eta=exp(1/e) using straightforward techniques on an excel spreadsheet.  The limit as x get arbitrarily large does not converge to a simple number like the 1.1282 estimate, but instead converges to a 1-cyclic function near that value.  For base(2) and for base(e), if you use Kneser's construction; then the 1-cyclic limit is graphed below. [attachment=1303] On this forum, other ideas like "the base change function" were discussed, where you use Peter Walker's idea to define tetration base(a) from tetration base(b).  For example, you could define tetration base(2) from Kneser's tetration base(e).  The relevant equations might look something like this.  But the "h" function below is conjectured to be nowhere analytic, even though Walker proved it is for the case in his paper.  Walker defined the base(e) slog from the Abel function for iterating .  This is mathematically conjugate (or exactly equivalent) to iterating base eta.  /* constant to guarantee slog_b(1)=0 */ RE: Math overflow question on fractional exponential iterations - JmsNxn - 04/01/2018 Okay, so the question and your intuition relates to the old base change formula and that it failed to be analytic. That makes sense, but is disappointing to think we're going to lose this property if we choose an analytic solution. So what this slog limit is saying is that for ''good'' analytic tetrations:  changes sign infinitely often (given )? This reminds me of something. I've dealt with those limits before and felt discouraged at an ability to prove uniform convergence. Given holomorphic  where , when trying to find a function such that , the natural choice is (which never seems to work). But it sure does look nice. The only way this works, I found, is to assume and take the Schroder function of both functions where and and then which works locally. Then the above limit for is convergent. But we had to sacrifice a lot to get there.  Of course if we're working on a non simply connected set instead of and we assumed that had no fixed points on this set, this could work. But tetration takes , so it probably has fixed points (maybe this is provable). Which should guarantee a base change function  is non extendable to . This is kinda' helping me understand why these functions fail to be analytic. No conjugation can change the multiplier value and clearly will have a different multiplier at its fixed point as will have at its fixed point. I'll have to read Walker's paper. The only work around I had to this was working with Schroder functions and when dealing with the real line where there are no fixed points I can't imagine a manner of getting a nice uniform convergence. I'm still wondering if I can prove that if   then which could then be a condition for tetration to be non-analytic.  Still seems like a lot of this is up in the air though. I apologize if this has me a bit scatter brained.