- As the author of http://www.tetration.org
I have been deeply involved in extending tetration to the complex numbers as well as matrices. You may be interested to know that Philippe Flajolet's research and your writing have had the biggest impact on my research since the turn of the millennia. Back in 2006 I noticed an article while searching on “tetration” with Google - it was the Wikipedia article on tetration. I made what I felt were significant improvements to the article, but after struggling for a year with people who didn’t follow Wikipedia’s policy of no unpublished personal research I didn’t want my name associated with the article.
Here are some things to consider regarding extending tetration. For links on who has done what regarding tetration see http://tetration.org/Links/index.html
Extending tetration is just an example of a much broader and important problem, that of fractional iteration. If you conquer fractional iteration you can not only extend tetration, but also pentation and all the higher operators of the Ackermann function. I met with Stephen Wolfram in 1986 several times and discussed my work with extending tetration. Wolfram was very interested because at the time he was trying to unify different dynamical systems into a single system that could mathematically model any dynamical system. There are two dynamical systems deemed capable of representing the dynamical systems in physics - PDEs and iterated functions. The problem is that iterated functions are discrete while physics given every appearance of being continuous. Therefore a theory of fractional iteration would provide a new inroad into understanding physics. See
R. Aldrovandi and L. P. Freitas,
Continuous iteration of dynamical maps,
J. Math. Phys. 39, 5324 (1998)
for a great paper on continuous/fractional iteration with an application to the Naiver Stokes equation. My derivation of fractionally iterated functions is experimentally consistent with Aldrovandi and Freitas, but I haven’t proven their mathematical equivalence.
What combinatorial structure is associated with iterated functions. Look at a simpler problem and ask what combinatorial structure is associated with composite functions? The answer is set partitions or Bell numbers. Aldrovandi and Freitas paper follows this line of thought and use Bell matrices in their paper. Riordan’s book Combinatorial Identities and the chapter on Bell polynomials gives further insights. Bell polynomials provide a way of representing arbitrary functions in terms of the derivitives of composite functions. The combinatorial structure associated with Bell polynomials has a number of names: Schroeder’s Fourth Problem, hierarchies total partitions and phylogenetic trees. See http://oeis.org/A000311
for more information. My own work focuses on the derivatives of iterated functions which are Bell polynomials because iterated functions are trivially composite functions. See http://tetration.org/Combinatorics/index.html
to view how Schroeder’s Fourth Problem is recovered from the coefficients of the derivatives of iterated functions.
Tetration is based on iterated functions which are based on complex dynamics. Working alone for decades it has become very important for me to find flaws in my extension of tetration. The research community mostly consists of I. N. Galidakis, and myself working as individuals and researchers associated with the Tetration Forum including Kouznetsov http://math.eretrandre.org/tetrationforum/
. My point of difference with the Tetration Forum and Kouznetsov is that they are proposing solutions to the extension of tetration that are easily seen as being inconsistent with complex dynamics. In fact they claim that fixed points are irrelevant to extending tetration. According to complex dynamics -
Lennart Carleson, Theodore W. Gamelin
In reviewing Chapter 2 - the Classification of Fixed Points, one can see that because of the conjugacy of fixed points that there can be no “simple” single equation to extending tetration. In the classification of fixed points there are hyperbolic, super-attracting, parabolic rationally neutral, rationally neutral and irrationally neutral fixed points. Each type of fixed point leads to a different type of tetration. For example e ↑ (1/e) is the sole parabolic rationally neutral fixed point for tetration, so e ↑ (1/e) must have it’s own separate equation for extending tetration. Not only is my work consistent with the Classification of Fixed Points, it provides an algebraic explanation of the Classification of Fixed Points that is much simpler than the topologically based proofs in Complex Dynamics.
While I have been able to fractionally iterate any complex function and therefore tetration, there is a major problem everyone researching tetration faces, that of convergence. In my mind all the real action in tetration is focused on people showing that they have a convergent solution. A couple of years ago the issue was quite active on MathOverflow. See http://mathoverflow.net/questions/tagged/fractional-iteration