Okay, so the question and your intuition relates to the old base change formula and that it failed to be analytic. That makes sense, but is disappointing to think we're going to lose this property if we choose an analytic solution.
So what this slog limit is saying is that for ''good'' analytic tetrations: \( f(x)=\exp_{b+\delta}^{c}(x) - \exp^{c+\delta'}_b(x) \) changes sign infinitely often (given \( \delta,\delta'<\epsilon \))?
This reminds me of something.
I've dealt with those limits before and felt discouraged at an ability to prove uniform convergence. Given holomorphic \( f,g : \mathbb{D} \to \mathbb{D} \) where \( f(0) = g(0) = 0 \), when trying to find a function \( \Psi:\mathbb{D}\to\mathbb{D} \) such that \( \Psi(f(z)) = g(\Psi(z)) \), the natural choice is \( \Psi(z) =\lim_{n\to\infty} g^{-n}(f^{n}(z)) \) (which never seems to work). But it sure does look nice.
The only way this works, I found, is to assume \( f'(0) = g'(0) = \lambda \) and take the Schroder function of both functions \( h_0, h_1 \) where \( h_0(f(z)) = \lambda h_0(z) \) and \( h_1(g(z)) = \lambda h_1(z) \) and then \( \Psi(z) = h_1^{-1}(h_0(z)) \) which works locally. Then the above limit for \( \Psi \) is convergent. But we had to sacrifice a lot to get there.
Of course if we're working on a non simply connected set \( H \) instead of \( \mathbb{D} \) and we assumed that \( f,g \) had no fixed points on this set, this could work. But tetration takes \( \mathbb{C}/\{z \in (-\infty,-2)\}\to \mathbb{C} \), so it probably has fixed points (maybe this is provable). Which should guarantee a base change function \( h \) is non extendable to \( \mathbb{C}/\{z \in (-\infty,-2)\,\} \).
This is kinda' helping me understand why these functions fail to be analytic. No conjugation can change the multiplier value and clearly \( ^ze \) will have a different multiplier at its fixed point as \( ^z 2 \) will have at its fixed point.
I'll have to read Walker's paper. The only work around I had to this was working with Schroder functions and when dealing with the real line where there are no fixed points I can't imagine a manner of getting a nice uniform convergence.
I'm still wondering if I can prove that if
\( \exp_{b+\delta}(x) < \exp_b^{1+\delta'}(x) \)
then
\( \exp_{b+\delta}^c(x) < \exp_b^{c + \delta'}(x) \)
which could then be a condition for tetration to be non-analytic.
Still seems like a lot of this is up in the air though. I apologize if this has me a bit scatter brained.
So what this slog limit is saying is that for ''good'' analytic tetrations: \( f(x)=\exp_{b+\delta}^{c}(x) - \exp^{c+\delta'}_b(x) \) changes sign infinitely often (given \( \delta,\delta'<\epsilon \))?
This reminds me of something.
I've dealt with those limits before and felt discouraged at an ability to prove uniform convergence. Given holomorphic \( f,g : \mathbb{D} \to \mathbb{D} \) where \( f(0) = g(0) = 0 \), when trying to find a function \( \Psi:\mathbb{D}\to\mathbb{D} \) such that \( \Psi(f(z)) = g(\Psi(z)) \), the natural choice is \( \Psi(z) =\lim_{n\to\infty} g^{-n}(f^{n}(z)) \) (which never seems to work). But it sure does look nice.
The only way this works, I found, is to assume \( f'(0) = g'(0) = \lambda \) and take the Schroder function of both functions \( h_0, h_1 \) where \( h_0(f(z)) = \lambda h_0(z) \) and \( h_1(g(z)) = \lambda h_1(z) \) and then \( \Psi(z) = h_1^{-1}(h_0(z)) \) which works locally. Then the above limit for \( \Psi \) is convergent. But we had to sacrifice a lot to get there.
Of course if we're working on a non simply connected set \( H \) instead of \( \mathbb{D} \) and we assumed that \( f,g \) had no fixed points on this set, this could work. But tetration takes \( \mathbb{C}/\{z \in (-\infty,-2)\}\to \mathbb{C} \), so it probably has fixed points (maybe this is provable). Which should guarantee a base change function \( h \) is non extendable to \( \mathbb{C}/\{z \in (-\infty,-2)\,\} \).
This is kinda' helping me understand why these functions fail to be analytic. No conjugation can change the multiplier value and clearly \( ^ze \) will have a different multiplier at its fixed point as \( ^z 2 \) will have at its fixed point.
I'll have to read Walker's paper. The only work around I had to this was working with Schroder functions and when dealing with the real line where there are no fixed points I can't imagine a manner of getting a nice uniform convergence.
I'm still wondering if I can prove that if
\( \exp_{b+\delta}(x) < \exp_b^{1+\delta'}(x) \)
then
\( \exp_{b+\delta}^c(x) < \exp_b^{c + \delta'}(x) \)
which could then be a condition for tetration to be non-analytic.
Still seems like a lot of this is up in the air though. I apologize if this has me a bit scatter brained.