• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 Math overflow question on fractional exponential iterations JmsNxn Ultimate Fellow Posts: 901 Threads: 111 Joined: Dec 2010 04/01/2018, 03:09 AM (This post was last modified: 04/01/2018, 06:22 AM by JmsNxn.) 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: $f(x)=\exp_{b+\delta}^{c}(x) - \exp^{c+\delta'}_b(x)$ changes sign infinitely often (given $\delta,\delta'<\epsilon$)? This reminds me of something. I've dealt with those limits before and felt discouraged at an ability to prove uniform convergence. Given holomorphic $f,g : \mathbb{D} \to \mathbb{D}$ where $f(0) = g(0) = 0$, when trying to find a function $\Psi:\mathbb{D}\to\mathbb{D}$ such that $\Psi(f(z)) = g(\Psi(z))$, the natural choice is $\Psi(z) =\lim_{n\to\infty} g^{-n}(f^{n}(z))$ (which never seems to work). But it sure does look nice. The only way this works, I found, is to assume $f'(0) = g'(0) = \lambda$ and take the Schroder function of both functions $h_0, h_1$ where $h_0(f(z)) = \lambda h_0(z)$ and $h_1(g(z)) = \lambda h_1(z)$ and then $\Psi(z) = h_1^{-1}(h_0(z))$ which works locally. Then the above limit for $\Psi$ is convergent. But we had to sacrifice a lot to get there.  Of course if we're working on a non simply connected set $H$ instead of $\mathbb{D}$ and we assumed that $f,g$ had no fixed points on this set, this could work. But tetration takes $\mathbb{C}/\{z \in (-\infty,-2)\}\to \mathbb{C}$, so it probably has fixed points (maybe this is provable). Which should guarantee a base change function $h$ is non extendable to $\mathbb{C}/\{z \in (-\infty,-2)\,\}$. This is kinda' helping me understand why these functions fail to be analytic. No conjugation can change the multiplier value and clearly $^ze$ will have a different multiplier at its fixed point as $^z 2$ 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 $\exp_{b+\delta}(x) < \exp_b^{1+\delta'}(x)$  then $\exp_{b+\delta}^c(x) < \exp_b^{c + \delta'}(x)$ 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. « Next Oldest | Next Newest »

 Messages In This Thread Math overflow question on fractional exponential iterations - by sheldonison - 03/28/2018, 07:57 PM RE: Math overflow question on fractional exponential iterations - by sheldonison - 03/29/2018, 10:13 PM RE: Math overflow question on fractional exponential iterations - by JmsNxn - 03/30/2018, 07:37 PM RE: Math overflow question on fractional exponential iterations - by sheldonison - 03/31/2018, 04:19 AM RE: Math overflow question on fractional exponential iterations - by JmsNxn - 04/01/2018, 03:09 AM

 Possibly Related Threads… Thread Author Replies Views Last Post Qs on extension of continuous iterations from analytic functs to non-analytic Leo.W 17 779 3 hours ago Last Post: JmsNxn Fibonacci as iteration of fractional linear function bo198214 23 247 08/09/2022, 04:49 AM Last Post: JmsNxn The iterational paradise of fractional linear functions bo198214 7 98 08/07/2022, 04:41 PM Last Post: bo198214 Describing the beta method using fractional linear transformations JmsNxn 5 71 08/07/2022, 12:15 PM Last Post: JmsNxn Apropos "fix"point: are the fractional iterations from there "fix" as well? Gottfried 12 542 07/19/2022, 03:18 AM Last Post: JmsNxn [question] Local to global and superfunctions MphLee 8 307 07/17/2022, 06:46 AM Last Post: JmsNxn [note dump] Iterations and Actions MphLee 23 3,411 07/15/2022, 04:08 PM Last Post: MphLee Slog(Exponential Factorial(x)) Catullus 19 1,092 07/13/2022, 02:38 AM Last Post: Catullus A random question for mathematicians regarding i and the Fibonacci sequence. robo37 1 3,972 06/27/2022, 12:06 AM Last Post: Catullus Question about tetration methods Daniel 17 672 06/22/2022, 11:27 PM Last Post: tommy1729

Users browsing this thread: 1 Guest(s)