While talking about tetration with my friend the following idea (re)occured.
Equation A
$$f(f(z)) = z , f(\exp(z)) = \exp(f(z)) $$
or variations of it like the weaker
Equation B
$$f(f(f(f(z)))) = z , f(\exp(\exp(z))) = \exp(\exp(f(z))) $$
Now clearly if f satisfies equation A it also satisfies equation B, but not necessarily the other way around.
What are analytic solutions $f(z)$ for some local region $D$ bounded by a jordan curve $C$ where $z,f(z),\exp(z),f(\exp(z)),\exp(f(z))$ are in $D$ ( $D$ being the interior part of the curve $C$) , that satisfy
$f(f(z)) = z , f(\exp(z)) = \exp(f(z)) $
?
without the trivial $f(z)=z$ ofcourse.
edit
One idea is to use Carleman matrices and Diagonalize.
Consider the Carleman matrix of size $n \times n$ for the exp :
$$M_n(\exp(z)) =A_n$$.
Now diagonalize
$$A_n = P_n \times D_n \times P_n^{-1}$$
Then we can define $f(z)$ by Carleman matrix of size $n \times n$ :
$$M_n(f(z)) =B_n$$.
$$B_n = P_n \times D*_n \times P_n^{-1}$$
where $D*_n$ is the diagonal matrix of size $n$ with all $-1$ on the diagonal.
This way we get
$$B_n^2 = D°$$
where $D°$ is the (diagonal) Carleman matrix of $id(z)$.
and
$$A_n B_n = B_n A_n$$
as desired.
Now let $n$ increase and hope for convergeance. And nonzero radius.
And ofcourse we assume diagonalization is always possible.
Might this work ? How to prove it ? What is the result ?
This method does seem to prefer the carleman interpretation of tetration ofcourse, which is not my bias but a consequence of the idea of using eigenvalues and matrices by lack of other " simple " ideas.
edit
Maybe look at it this way :
$$G(z^2) = \exp(G(z))$$
Then $G(z^{-1})$ is associated with $f(f(z)) = z$... Unless the case $G(z^{-1}) =G(z)$ for all $z$ or the unless for the solutions $z$ such that $G(z^{-1}) =G(z)$.
And ofcourse $z=0$ does not work there.
not an answer but an extended comment towards @mick's comment of Mar 16
Hmm, I like to see the "Carleman-matrix" mentioned - a long & intense hobby of mine for the understanding of problems on powerseries. But I want to mention two obstacles which are rarely, if at all, mentioned and/or considered.
First: the concept of Carlemanmatrices (working on problems of powerseries) is always meant to use matrices of two-dimensionally infinite size. Truncations to finite size, of course when approximate solutions by software are attempted, are not always smoothly extensible to the case of infinite size. A very nice example of unavoidable miscalculation has been brought to my knowledge by Ed Sandifer on an example of L. Euler, where he proposes a powerseries solution for the logarithm, based on increasing polynomials (which would today be expressed by increasing finite Carlemanmatrices), but which is systematically false, as Euler has shown by himself. (Ed Sandifer's "HEDI" pages,Dec 2007, "A False Logarithm series"). I've discussed this with my understanding of matters in this essays
There are other examples, where the extension from the "polynomial approximations" (finite-sized Carleman-matrix) to the "powerseries-solution" seem to work (for the alms of "tetration"), possibly the Ansatz of A. Robbins for the
slog()is here an example.The commonly known Schroeder/Koenigs-solution for the iterated exponential $\exp(z)-1$ and other series allowing triangular Carleman-matrices (and their diagonalization) is another example for the concept to be working. But the examples with the attempt to diagonalize Carleman-matrices for powerseries with constant term (not triangular matrices) - well, I don't know whether (or: how far) they are already safe...
Second, we find cases, where we need Integral expressions to formulate complete solutions for some power series problems. (The Ecallè-solution for the
slog()for instance uses an Integral) Eri Jabotinsky has been an exponent and early pioneer for the use of Carlemanmatrices. But for some cases he proposed to use extended Carlemanmatrices, which model then Laurent series or even twoway infinite powerseries. I've found one article where he introduced thus a Carlemanmatrix extended to index $-1$ - but it seems that he couldn't/didn't finish the complete model. To see better what I mean with this (as far I could understand the problem correctly) I have written a small essay with a proposition for the extension of the $\exp()$-series (and its Carlemanmatrix) to the $-1$-index; that small note does not solve the problem but puts it a bit more explicite than I could do this here. See "problem with Bell-matrix" at my homepage.I think a progress in this concept of Carleman-matrices for the problem of tetration should be made by answering/solving the two mentioned problems here.