Hmm, you seem to have constructed a formal Taylor Series; I don't really see a proof that the Taylor series converges though. Though yes it does, and is definitely provable using current analysis. As far as I can tell this tetration would be equivalent to,
\(
F(z) = \Psi^{-1}(\lambda^z A)\\
\)
For the Schroder function \( \Psi \) about a fixed point of \( e^{z} \); where \( A \) is some constant and \( \lambda \) is the multiplier. This would definitely be analytic; I agree. It will not be real valued on the real-line; as far as I can tell it should be entire because it's analytic in a neighborhood of zero and \( \exp^{\circ n} \Psi^{-1}(z) = \Psi^{-1}(\lambda^n z) \) where since \( |\lambda| > 1 \) this will cover the plane inside of \( \Psi^{-1} \), and the iterated exponential is always valid. And then the theory is to find a fixed point of \( F \); and construct a Schroder function again about this,
\(
\Psi_1(\lambda_1^z A_1)\\
\)
In essentially the same manner as Tetration. And you add the case that if we have a neutral fixed point we'll use the Abel-form of iteration. Presuming you are avoiding super-attracting fixed points as hyper-operators should have a non-zero derivative. Whereof you continue this train of construction to create a sequence of super-functions:
\(
F_n(F_{n+1}(z)) = F_{n+1}(z+1)\\
\)
I'm curious as to how you are choosing your fixed points? And I'm curious how you are choosing your value of \( A \); or rather what your initial conditions are, \( F_n(0) = ? \). Presumably you want that it equals \( 1 \); I'm not quite too sure how you're doing this though, but I assume it's by setting \( A = \Psi(1) \)--which I can't see anything wrong with in Tetration (as the entire complex plane is the Julia set of \( e^z \), and \( \Psi^{-1} \) is an entire function). But I don't see how you are guaranteed that \( \Psi_n(1) \) is finite for higher order hyper-operations--necessarily you would need to discuss the domain of \( \Psi \) or \( \Psi^{-1} \). And largely the reason it is working for tetration is because all the fixed points of \( e^z \) are repelling \( |\lambda| > 1 \) (implying an entire inverse Schroder function). As I see it you would need the fixed point you use from tetration to construct pentation to be repelling as well (so that it's inverse Schroder function is entire).
If you were to take the alternative route, that it's geometrically attracting, you are opening yourself up to branch-cuts because \( \Psi^{-1} \) will not be entire, and instead \( \Psi \) will be holomorphic on the immediate basin of attraction sending to \( \mathbb{C} \). Which typically means that \( \Psi^{-1} \) has branch cuts; otherwise if it were entire on \( \mathbb{C} \) it would only send to the immediate basin of the fixed point, which is impossible due to Picard, unless the immediate basin is the entire complex plane minus a point (which cuts out a very large resevoir of functions and essentially means we have 2 fixed points, one attractive and one repelling).
If the fixed point is neutral you open yourselves up to similar problems as the attractive case. When you construct the Abel function of a neutral fixed point it only converges in a petal of the fixed point (not the neighborhood), and so if the inverse Abel function \( g \) were entire, it again would send to only a petal of the complex plane, which is impossible due to Picard. Again, opening yourself up to branch-cuts.
Although I agree absolutely with your paper and the construction of the formal Taylor Series; which is very impressive (something I never would've thought would be feasible), I think you're avoiding the biggest question. What's the radius of convergence of these Taylor series? For Tetration it'll be infinite, but this isn't necessarily so. Even taking something like \( \alpha^\xi \) for \( 1 < \alpha < e^{1/e} \) your theory seems to be incomplete by avoiding where in fact the iteration will converge. Though yes, I do agree, that locally everything should be fine; so long as we are not on the Julia set (boundary of an immediate basin).
I should say, it would be very impressive if you managed to derive a manner of computing the radius of convergence--though this is largely the criticism of using Faa di Bruno's method; it doesn't plug too well into Cauchy's limsup formula (or any radius of convergence method). Though, I will say that you are absolutely correct if we were to restrict your construction to the theory of Sheafs. And in fact, seems like a very smart move. As I see it, you are more aptly talking about flows on sheafs rather than flows on functions (mostly because you are treating the Taylor Series as formal objects)--though proving local convergence almost everywhere does seem feasible. Which is again, very good because it means it's a sheaf with a non-zero radius of convergence. However, when discussing as a global thing; we'd like to have our hands on exactly where it converges.
I hope I don't come off as rude or anything (rereading I'm trying not to be harsh, but I may be coming off that way, I apologize if so); this is still very impressive. I had always thought the Taylor series approach to iteration was a null method but you've definitely convinced me otherwise. But I do think we'd need a stronger sense of where these things converge, rather than simply knowing that we are guaranteed local convergence.
Regards, James
Edit: I made a minor error when discussing the neutral fixed point by not referring to repelling petals and attracting petals; if the petal is repelling the superfunction will be entire (much like repelling fixed points), so substitute my above comment with the word attracting petal. Had to double check Milnor, haven't thought about neutral fixed points in a while, lol.
\(
F(z) = \Psi^{-1}(\lambda^z A)\\
\)
For the Schroder function \( \Psi \) about a fixed point of \( e^{z} \); where \( A \) is some constant and \( \lambda \) is the multiplier. This would definitely be analytic; I agree. It will not be real valued on the real-line; as far as I can tell it should be entire because it's analytic in a neighborhood of zero and \( \exp^{\circ n} \Psi^{-1}(z) = \Psi^{-1}(\lambda^n z) \) where since \( |\lambda| > 1 \) this will cover the plane inside of \( \Psi^{-1} \), and the iterated exponential is always valid. And then the theory is to find a fixed point of \( F \); and construct a Schroder function again about this,
\(
\Psi_1(\lambda_1^z A_1)\\
\)
In essentially the same manner as Tetration. And you add the case that if we have a neutral fixed point we'll use the Abel-form of iteration. Presuming you are avoiding super-attracting fixed points as hyper-operators should have a non-zero derivative. Whereof you continue this train of construction to create a sequence of super-functions:
\(
F_n(F_{n+1}(z)) = F_{n+1}(z+1)\\
\)
I'm curious as to how you are choosing your fixed points? And I'm curious how you are choosing your value of \( A \); or rather what your initial conditions are, \( F_n(0) = ? \). Presumably you want that it equals \( 1 \); I'm not quite too sure how you're doing this though, but I assume it's by setting \( A = \Psi(1) \)--which I can't see anything wrong with in Tetration (as the entire complex plane is the Julia set of \( e^z \), and \( \Psi^{-1} \) is an entire function). But I don't see how you are guaranteed that \( \Psi_n(1) \) is finite for higher order hyper-operations--necessarily you would need to discuss the domain of \( \Psi \) or \( \Psi^{-1} \). And largely the reason it is working for tetration is because all the fixed points of \( e^z \) are repelling \( |\lambda| > 1 \) (implying an entire inverse Schroder function). As I see it you would need the fixed point you use from tetration to construct pentation to be repelling as well (so that it's inverse Schroder function is entire).
If you were to take the alternative route, that it's geometrically attracting, you are opening yourself up to branch-cuts because \( \Psi^{-1} \) will not be entire, and instead \( \Psi \) will be holomorphic on the immediate basin of attraction sending to \( \mathbb{C} \). Which typically means that \( \Psi^{-1} \) has branch cuts; otherwise if it were entire on \( \mathbb{C} \) it would only send to the immediate basin of the fixed point, which is impossible due to Picard, unless the immediate basin is the entire complex plane minus a point (which cuts out a very large resevoir of functions and essentially means we have 2 fixed points, one attractive and one repelling).
If the fixed point is neutral you open yourselves up to similar problems as the attractive case. When you construct the Abel function of a neutral fixed point it only converges in a petal of the fixed point (not the neighborhood), and so if the inverse Abel function \( g \) were entire, it again would send to only a petal of the complex plane, which is impossible due to Picard. Again, opening yourself up to branch-cuts.
Although I agree absolutely with your paper and the construction of the formal Taylor Series; which is very impressive (something I never would've thought would be feasible), I think you're avoiding the biggest question. What's the radius of convergence of these Taylor series? For Tetration it'll be infinite, but this isn't necessarily so. Even taking something like \( \alpha^\xi \) for \( 1 < \alpha < e^{1/e} \) your theory seems to be incomplete by avoiding where in fact the iteration will converge. Though yes, I do agree, that locally everything should be fine; so long as we are not on the Julia set (boundary of an immediate basin).
I should say, it would be very impressive if you managed to derive a manner of computing the radius of convergence--though this is largely the criticism of using Faa di Bruno's method; it doesn't plug too well into Cauchy's limsup formula (or any radius of convergence method). Though, I will say that you are absolutely correct if we were to restrict your construction to the theory of Sheafs. And in fact, seems like a very smart move. As I see it, you are more aptly talking about flows on sheafs rather than flows on functions (mostly because you are treating the Taylor Series as formal objects)--though proving local convergence almost everywhere does seem feasible. Which is again, very good because it means it's a sheaf with a non-zero radius of convergence. However, when discussing as a global thing; we'd like to have our hands on exactly where it converges.
I hope I don't come off as rude or anything (rereading I'm trying not to be harsh, but I may be coming off that way, I apologize if so); this is still very impressive. I had always thought the Taylor series approach to iteration was a null method but you've definitely convinced me otherwise. But I do think we'd need a stronger sense of where these things converge, rather than simply knowing that we are guaranteed local convergence.
Regards, James
Edit: I made a minor error when discussing the neutral fixed point by not referring to repelling petals and attracting petals; if the petal is repelling the superfunction will be entire (much like repelling fixed points), so substitute my above comment with the word attracting petal. Had to double check Milnor, haven't thought about neutral fixed points in a while, lol.