From Iterated Functions

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.