08/30/2007, 09:32 PM

bo198214 Wrote:I have to admit I never considered the case , but I recently discussed it with Gianfranco.

We know that 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 is no more defined for , because is not injective for so . This gives also trouble as then .

On the other hand for tetration is *not* the inversion of . So the rules may be a bit different here.

So what do you think how tetration looks for ? And what role plays the range ?

My understanding is that 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 . The other values of have hyperbolic fixed points and could be solved using Schroeder's functional equation, Bell matrices or other equivalent methods.