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 Tetration below 1 Daniel Fellow Posts: 66 Threads: 29 Joined: Aug 2007 08/30/2007, 09:32 PM bo198214 Wrote:I have to admit I never considered the case $0, but I recently discussed it with Gianfranco. We know that ${}^{2n}x$ is no more injective if we allow this range. If we consider one rank below the ordinary exponentiation, we have a similar case there, that $b^x$ is no more defined for $-\infty, because $x^{2n}$ is not injective for $x<0$ so $b^{1/2}=\pm |b|^{1/2}$. This gives also trouble as then $b^n=b^{\frac{2n}{2}}=\pm b^n$. On the other hand for tetration ${}^{\frac{1}{2}} x$ is *not* the inversion of ${}^2 x$. So the rules may be a bit different here. So what do you think how tetration looks for $0? And what role plays the range $e^{-e}? My understanding is that $b = e^{-e}$ is the exceptional case here, it has a rationally neutral fixed point with a Lyapunov characteristic number of -1, see http://mathworld.wolfram.com/LyapunovCha...umber.html . It could be solved using Abel's functional equation for $f^{2 n}(x)$. The other values of $b$ have hyperbolic fixed points and could be solved using Schroeder's functional equation, Bell matrices or other equivalent methods. « Next Oldest | Next Newest »

 Messages In This Thread Tetration below 1 - by bo198214 - 08/29/2007, 06:14 PM RE: Tetration below 1 - by Daniel - 08/30/2007, 09:32 PM RE: Tetration below 1 - by jaydfox - 08/30/2007, 11:08 PM RE: Tetration below 1 - by jaydfox - 08/30/2007, 11:20 PM RE: Tetration below 1 - by Daniel - 08/31/2007, 12:47 AM RE: Tetration below 1 - by GFR - 09/02/2007, 01:30 PM RE: Tetration below 1 - by bo198214 - 09/02/2007, 01:40 PM RE: Tetration below 1 - by GFR - 09/02/2007, 05:38 PM RE: Tetration below 1 - by jaydfox - 09/03/2007, 03:58 PM RE: Tetration below 1 - by bo198214 - 09/03/2007, 04:07 PM RE: Tetration below 1 - by jaydfox - 09/03/2007, 04:36 PM RE: Tetration below 1 - by jaydfox - 09/05/2007, 11:24 PM RE: Tetration below 1 - by GFR - 09/06/2007, 12:01 AM RE: Tetration below 1 - by jaydfox - 09/06/2007, 03:28 AM RE: Tetration below 1 - by jaydfox - 09/06/2007, 07:21 AM RE: Tetration below 1 - by bo198214 - 03/26/2008, 04:51 PM

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