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From Iterated Functions

Overview[edit]

     Escape.gif
     Period.gif
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 <a href="Tetration/index.html">Tetration</a> page for a one page overview of extending tetration to the complex numbers.

Schroeder's Fourth Problem - OEIS A000311
Trees.gif
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] -&gt; 0;

s[n_] := D[f[g[x]], {x, n}] /. g[x] -&gt; 0 /. Derivative[1][f][0]*Derivative[n][g][x] -&gt; 0 /. Derivative[1][g][x] -&gt; l^(t - 1)

dyn[1] = l^t; dyn[n_] := dyn[n] = Simplify[Sum[s[n], t] /. D[g[x], {x, m_}] :&gt; dyn[m] /. l -&gt; Derivative[1][f][0]]

Hier[n_] := Block[{l = 1}, Sum[s[n], t] /. D[g[x], {x, m_}] :&gt; dyn[m] /. t -&gt; 1 /. f -&gt; Exp]
Table[Hier[i], {i, 2, 15}]
Out[]= {1,4,26,236,2752,39208,660032,12818912,282137824,6939897856,188666182784,5617349020544,181790703209728,6353726042486272}

Tetration research: 1986 - 1991[edit]

In 1986 I had several conversations on extending tetration to the complex numbers with Stephen Wolfram. He suggested that I write my research up and he would edit and publish it in his journal. Unfortunately for me Wolfram had moved on to establishing Mathematica. <a href="http://tetration.org/1990.pdf" target="_blank">Algebraic Exponential Dynamics</a> is the article I submitted to Wolfram in 1990.

All Maps Have Flows  & All Hyperoperators Operate on Matrices

In 1986 Stephen Wolfram introduced me to the question of whether all maps are flows. Given the fifteen-year-old mathematics on Tetration.org, I have a simple proof that all maps are flows, that they are two different views of the same thing. Consider the Taylor series of an arbitrary smooth iterated function and it's representation as the combinatorial structure total partitions, the recursive version of set partitions. Each enumerated combinatorial structure has a symmetry associated with it. Let's say we want to consider \(C_2\), just remove all combinatorial structures inconsistent with \(C_2\). Because I can define \(GL(n)\) as the domain and the iterant, through representation theory, that if I can compute with matrices, I can compute within any symmetry.

<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.

Introduction

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