Looking at the standard tetration for \( 1 < \alpha < \eta \) I was wondering about something. Taking \( F(\alpha,z) = \,^z \alpha \) we first note that F is analytic in \( \alpha \). As we all know, bounded in z on the right half plane. It is monotone increasing on the real positive line, which leads us to a fixed point, let's call it \( \tau_\alpha \). I could show it, but I assume people also know that \( F_\alpha(\alpha,x) > 0 \).

Algebraically we can characterize \( \tau_\alpha \) by the equations

\( F(\alpha,\tau_\alpha) = \tau_\alpha \)

\( F(\alpha,F(\alpha,...k\,times ...F(\alpha,x) \to \tau_\alpha \) for all \( 0 \le x \le \tau_\alpha \)

I'm wondering if anyone has any information about the analycity of \( \tau_\alpha \) in \( \alpha \). This is rather important because if \( \tau_\alpha \) is analytic then by the functional identity

\( \tau'(\alpha) = \frac{F_\alpha(\alpha,\tau_\alpha)}{1 - F_x(\alpha,\tau_\alpha)} \)

and the fact \( F_\alpha(\alpha,x)>0 \) it follows that

\( 0<F_x(\alpha,\tau_\alpha) < 1 \)

and that the fixed point \( \tau_\alpha \) is geometrically attracting. This would instantly give a solution to pentation, and whats better, a solution to pentation with an imaginary period. Conversely, if \( 0<F_x(\alpha,\tau_\alpha) < 1 \) then necessarily \( \tau_\alpha \) is analytic in \( \alpha \) by the implicit function theorem.

All in all, I haven't been able to find results on tetrations fixed points, and whether they are analytic or not. I hope they are, but I can't be sure. This is bugging me because a positive answer would greatly simplify the construction of pentation, and hopefully will shed light on how to show pentations fixed points are geometrically attracting giving a nice solution for hexation, so on and so forth.

Algebraically we can characterize \( \tau_\alpha \) by the equations

\( F(\alpha,\tau_\alpha) = \tau_\alpha \)

\( F(\alpha,F(\alpha,...k\,times ...F(\alpha,x) \to \tau_\alpha \) for all \( 0 \le x \le \tau_\alpha \)

I'm wondering if anyone has any information about the analycity of \( \tau_\alpha \) in \( \alpha \). This is rather important because if \( \tau_\alpha \) is analytic then by the functional identity

\( \tau'(\alpha) = \frac{F_\alpha(\alpha,\tau_\alpha)}{1 - F_x(\alpha,\tau_\alpha)} \)

and the fact \( F_\alpha(\alpha,x)>0 \) it follows that

\( 0<F_x(\alpha,\tau_\alpha) < 1 \)

and that the fixed point \( \tau_\alpha \) is geometrically attracting. This would instantly give a solution to pentation, and whats better, a solution to pentation with an imaginary period. Conversely, if \( 0<F_x(\alpha,\tau_\alpha) < 1 \) then necessarily \( \tau_\alpha \) is analytic in \( \alpha \) by the implicit function theorem.

All in all, I haven't been able to find results on tetrations fixed points, and whether they are analytic or not. I hope they are, but I can't be sure. This is bugging me because a positive answer would greatly simplify the construction of pentation, and hopefully will shed light on how to show pentations fixed points are geometrically attracting giving a nice solution for hexation, so on and so forth.