1: GAIO/local dimension
3: infinite dimensions
1: Today, I spent a lot of time on Matlab trying to get my boxes inserted at a certain depth only within a certain neighborhood. Here's a picture!
The red boxes are the boxes I need to "count" for the local dimension of the area in the box. The big green boxes are just the mis-covering of the entire attractor at depth 6. For this picture, the iteration number is still small and the boxes are still large since it was just an experiment to try to get it right. It still needs some work, but I was able to at least figure out how to use "subdivide" and the flags (sort of). Tomorrow, I'm going to try to fix the redundancies that might be slowing the program down and clean it up a bit. Also, I can start to look at the dimension of the attractor versus the local dimension about a point (for example, the fixed point) as you get "more local" (smaller initial boxes to be further subdivided). I actually don't even have to use the "subdivide" command, and can probably remove that altogether. Anyway, in a perfect fractal, zooming in to look at the "local" dimension should return the same value (PD suggested I prove it and say something about the scaling). For something like the Hénon attractor, it may be the case. Or perhaps, as you move along the attractor, the "local" dimension changes? For this question, I should consider other points on the attractor (if I can find some!).
Also, I need to find some way to take the box counting dimension of the Cantor like set in the Hénon attractor. I think I might know some way of doing this, but I'm not sure if it will work. Also, I wouldn't be using square boxes, but let my epsilon be the x-radius, I think (if I'm taking the Cantor set parallel to the x-axis) and not divide horizontally. I think I remember seeing some way of doing that.
I'm hoping to find some way of using GAIO on my own computer, so I don't have to go to the computer lab to run the experiments. When using the SSH client, I am unable to view figures, which is pretty important, so I know what I'm doing is right. Also, I don't seem to have anyway of accessing ssh from my command prompt.
2: CRS = Chain Recurrent Set. Professor Day told me this could be an option instead of MIS (Maximal Invariant Set). Yesterday, I said it seemed to give a better covering than the mis method. Prof Day tried to explain it to me a little bit, but I really didn't understand it enough to remember. She gave me a paper I'm going to look at later to get an idea of the theory behind it. It might be useful later. Right now, I'm just using the "insert method" with calculating dimension.
Actually someone in our "mathematical family" who knows about this stuff (including CRS) is supposed to be visiting in 2 weeks to give a talk, so there's a chance I might be able to ask him about it in person.
3: Defining fractal dimension in infinite dimensional space: not so easy. I looked at the definitions of topological dimension (from book IDTFS) and box dimension (from Alligood's Chaos) to get ideas of where they came from. Box dimension is easy. It comes from the relation (1/eps)^dim = number of boxes, that seems present in regular dimension spaces, like the interval I, or I^2, or I^3 (all bounded spaces). For topological dimension, the definition has to do with "inessential" families of disjoint closed subsets of X. The internet definitions seem to have to do with coverings of the set.
In short, I'm not really sure how one is supposed to define "box" or box-counting dimension in infinite dimensional space. Infinity is tricky.
Question: if you have a bounded uncountable collection of points, is the fractal dimension necessarily positive? You can have Cantor set with arbitrarily small positive dimension and one with fractal dimension = 1. Can it be zero?