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Tetration by Escape  Tetration by Period 
My name is Daniel Geisler, welcome to my web site dedicated to promoting research into the question of what lies beyond exponentiation. Tetration is defined as iterated exponentiation but while exponentiation is essential to a large body of mathematics, little is known about tetration due to its chaotic properties. The standard notation for tetration is \(^{1}a=a, ^{2}a=a^a, ^{3}a=a^{a^a},\) and so on. Mathematicians have been researching tetration since at least the time of Euler but it is only at the end of the twentieth century that the combination of advances in dynamical systems and access to powerful computers is making real progress possible.
The big question in tetration research is how can tetration be extended to complex numbers. How do you compute numbers like \(^{.5}2\), and \(^{\pi i}e\) ? This web site will show how to compute these and other problems. See the Tetration page for a one page overview of extending tetration to the complex numbers.
A question that goes back to Poincaré's time is, "do maps have flows?" For a smooth function \(f(x)\) and it's iterate, \(f^t(x)\), what degree does \(t \in \mathbb{N^+}\) imply \(t \in \mathbb{R}, \mathbb{C}\), and \(\mathbb{GL}(n)\)? Physics appears to do fine reconciling iterated functions and continuous time. Arnold \cite{Jackson} defines physical dynamical systems as measure preserving iterated functions acting on an initial state. The system under review is more general than any physical system as it has no constraint to be measure preserving.
Returning to the realm of mathematics and arithmetic, let \(f(x)\) and \(g(x)\) be functions in Banach space, then the composite \(f(g(x))\) can be constructed using Faà di Bruno's formula.
\( D^nf(g(x)) = \sum_{\pi(n)} \frac{n!}{k_1! \cdots k_n!} (D^kf)(g(x)) \left(\frac{Dg(x)}{1!}\right)^{k_1} \cdots \left(\frac{D^ng(x)}{n!}\right)^{k_n} \)
where \(\pi(n)\) denotes a partition of \(n\), usually denoted by \(1^{k_1}2^{k_2}\cdots n^{k_n}\), with \(k_1+2k_2+ \cdots nk_n=k\); where \(k_i\) is the number of parts of size \(i\). The partition function \(p(n)\) is a decategorized version of \(\pi(n)\), the function \(\pi(n)\) enumerates the integer partitions of \(n\), while \(p(n)\) is the cardinality of the enumeration of \(\pi(n)\). ^{[1]} ^{[2]}
Schroeder's Fourth Problem  OEIS A000311
The Schroeder tree graph of {{{1, 5}, 3}, {{2, 4}, 6}}
Code[edit]
f[0] = 0;
f[1] = D[f[g[x]], {x, 1}] /. g[x] > 0;
s[n_] := D[f[g[x]], {x, n}] /. g[x] > 0 /. Derivative[1][f][0]*Derivative[n][g][x] > 0 /. Derivative[1][g][x] > l^(t  1)
dyn[1] = l^t; dyn[n_] := dyn[n] = Simplify[Sum[s[n], t] /. D[g[x], {x, m_}] :> dyn[m] /. l > Derivative[1][f][0]]
Hier[n_] := Block[{l = 1}, Sum[s[n], t] /. D[g[x], {x, m_}] :> dyn[m] /. t > 1 /. f > Exp]
Table[Hier[i], {i, 2, 15}]
Out[]= {1,4,26,236,2752,39208,660032,12818912,282137824,6939897856,188666182784,5617349020544,181790703209728,6353726042486272}
<a href="http://tetration.org/Hyperoperators.pdf">Extending the Hyperoperators</a>
Just as the exponential function of invertible matrices can be computed, all hyperoperations can be defined with invertible matrices.