Iterations of functions of noncommutative variables

[hilbert2]

As most people interested in mathematics know, iterations of rational functions of complex variable can be used to produce fractals such as the Mandelbrot set...

What about iterating functions whose domain is a noncommutative ring? A ring can be a vector space of any dimensionality (or can it?), so that might be a way to produce 3D, 4D, 5D, etc. fractal sets... Has anyone thought about this before?

 

[Stephen Tashi]

"iterating functions on a non-commutative ring to produce fractals" is what I'd call a "pre-idea". It isn't precise enough to describe a mathematical idea. It's just some words strung together. However, "pre-ideas" can lead to useful ideas. We can begin by pretending it means something and trying to guess what that meaning is.

If we are going to iterate a set of functions (in the usual sense of the term "iterate") then they must have a range that is a subset of their domain. So we'd be iterating functions of the ring into itself. If the end product of the iteration is going to be a fractal that is a set of real n-dimensional vectors then you need some way to map the elements of the ring to elements of the vector space. But if you have that mapping then the functions from the ring into itself can be matched with functions of the vector space into itself. So we are essentially just iterating functions of the vectors space into itself. Why bother thinking about the ring?. It might be that expressing things in terms ofthe ring makes things simpler or gives some insights.

It would be best to create some concrete examples if you are serious about the "pre-idea". Usually "a set of functions" is too general a "given" to draw many conclusions. Special cases like linear transformations, polynomial functions etc. are needed to product interesting results. What special cases of functions of a ring onto itself are known to be interesting? I don't know enough about Algebra to answer that. Perhaps another forum member can tell us.

 

[hilbert2]

I have heard of "quaternion fractals", see http://paulbourke.net/fractals/quatjulia/ for example.. Those are four-dimensional, and one can also draw 2D or 3D projections of them with a computer. What I'm asking, is whether one can just form an arbitrary (not necessarily 4-dimensional) noncommutative ring, form a matrix representation of it for calculations, and expect to find something interesting by iterating functions defined on it? Or is there something special about the complex and quaternion number systems that makes them suitable for producing fractals?

 

[mfb]

There is nothing special about complex numbers or quaternions. Plots with complex numbers are just more convenient to draw on a screen or on paper, so most of the time you see a graph of something fractal it is probably made with complex numbers. In addition, you can use the standard operations (multiplication, addition, ...) on them for your functions.
Fractals exist as subsets in every dimension.