I'm curious about any literature regarding a tetration function: \(\text{tet}_b(z)\), such that it is your typical tetration function for \(b \in \mathfrak{S}\), the Shell-Thron region, and in the complex plane uses traditional Schroder mechanisms. But then there exists a branch discontinuity at \(b \in (\eta,\infty)\). I'm currently encountering this, and I'm wondering if anyone has any literature on this. Paulsen has little description of this, he just points out his solution has a branching problem at \(\eta\), but he describes the branch cut along \((1,\eta)\), rather than the other way around.

Theoretically this is entirely possible, because the fixed points move continuously from the Shell-thron region \(|\log(y)| \le 1\) to \(|\log(y)| \ge 1\), excluding the branching point at \(e\), where as soon as you grow from here the fixed point pairs force a discontinuous choice. Largely because we must have real valued solutions, and any perturbation which is holomorphic must continue to be real valued.

To be clear, I'm looking for a tetration solution: \(\text{tet}_b(z)\) such that this function is holomorphic for \(b \in \mathbb{C}/B\) where \(B = (-\infty,e^{-e}] \cup [\eta,\infty)\). I've developed enough numerical evidence to suggest this should be a function, but I'm not entirely sure as of yet if it's a viable construction.

Any help or comments is greatly appreciated.

Theoretically this is entirely possible, because the fixed points move continuously from the Shell-thron region \(|\log(y)| \le 1\) to \(|\log(y)| \ge 1\), excluding the branching point at \(e\), where as soon as you grow from here the fixed point pairs force a discontinuous choice. Largely because we must have real valued solutions, and any perturbation which is holomorphic must continue to be real valued.

To be clear, I'm looking for a tetration solution: \(\text{tet}_b(z)\) such that this function is holomorphic for \(b \in \mathbb{C}/B\) where \(B = (-\infty,e^{-e}] \cup [\eta,\infty)\). I've developed enough numerical evidence to suggest this should be a function, but I'm not entirely sure as of yet if it's a viable construction.

Any help or comments is greatly appreciated.