07/21/2019, 01:29 AM
The Abel functional equation and the Schroeder functional equation represent different symmetries. Saying that there can be a mathematical connection between the Abel and Schroeder functional equations doesn't mean that they coexist in the same system. Let's consider complex functions. Generally speaking, we will be looking at a system with an infinite set of hyperbolic fixed points, although complex conjugates can give pairs of parabolic fixed points along with infinite hyperbolic fixed points. The same for parabolically neutral fixed points. So while some folks like the properties of Abel's functional equation, the mathematical existence of these systems are rare.
I am striving for the ultimate generalization of my work (we probably all are). That means working with iterated functions in Banach or Frechet space. Representation theory ties symmetries and matrices together. Currently, the Tetration Forum focuses on real numbers as iterators while I'm looking at how to use the General Linear group.
At Wolfram's request, I'm working on a Mathematica application that can be migrated to the new Mathematica Notebook interface. Having this application will allow folks to see tetration based phenomena that is core to my work and that I have never seen a reference to. I hope this will be a picture that is worth a thousand words.
I am striving for the ultimate generalization of my work (we probably all are). That means working with iterated functions in Banach or Frechet space. Representation theory ties symmetries and matrices together. Currently, the Tetration Forum focuses on real numbers as iterators while I'm looking at how to use the General Linear group.
At Wolfram's request, I'm working on a Mathematica application that can be migrated to the new Mathematica Notebook interface. Having this application will allow folks to see tetration based phenomena that is core to my work and that I have never seen a reference to. I hope this will be a picture that is worth a thousand words.
Daniel