Are tetrations fixed points analytic?

[JmsNxn]

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.

[sheldonison]

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

Yes it is analytic. There is the well known equation for the fixed point using the LambertW function.
\( \alpha^{\tau_\alpha}={\tau_\alpha} \;\;\;\; {\tau_\alpha} =\frac{W(-\ln(\alpha))}{-\ln(\alpha)} \)

[JmsNxn]

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

Yes it is analytic. There is the well known equation for the fixed point using the LambertW function.
\( \alpha^{\tau_\alpha}={\tau_\alpha} \;\;\;\; {\tau_\alpha} =\frac{W(-\ln(\alpha))}{-\ln(\alpha)} \)

Are you talking about exps fixed points, or tetration's fixed points? You wrote exp, and I'm pretty sure that's the equation for exps fixed points. I'm interested in, \( ^{\tau_\alpha} \alpha = \tau_\alpha \) not \( \alpha^{\tau_\alpha} = \tau_\alpha \). I'm well aware exps fixed points are analytic, I'm interested in the fact that if we keep on increasing the hyper operator index in the bounded case that the hyper operators always have an analytic fixed point function and hence geometrically attracting fixed points.