1: GAIO/local dimension
2: Reading: infinite dimension/topology
3: Another Question
1: GAIO is the program I usually use (with Matlab) to create my "boxes" when I compute box-counting dimension. I'm trying to use GAIO's grid to compute local dimension of the attractor, in this case, taking the grid to a certain depth, choosing one box at that depth and only looking at that box when taking the grid to a larger (deeper) depth, so that I can compute further. Using GAIO should be more efficient than creating my own grid. Right now though, I can't seem to get it to work. I don't know the program well enough to stop the tree and select the box. In trying, however, I did find that using the crs (chain recurrent set) instead of mis (maximal invariant set) seems to give me a better covering of the (entire) attractor. I tried calculating the dimension with this method, and seemed to get a closer approximation than with the mis method. It still didn't seem to converge very quickly to the "true" value (which I guess is expected when using a box-covering with relatively large epsilon). This method probably doesn't guarantee an upper bound like the mis though (maybe). (Another one was rga, but I don't think it returned square boxes.) If I could insert figures into a GAIO grid, I could test the accuracy of my methods much more easily. Most of my day was spent trying to get Matlab to work. I read most of the GAIO manual that Prof Day gave me and so I learned some other things that GAIO can do, which might help me out later. The manual didn't seem to say how you would insert novel figures or models as a "Model" though.
2: I did do some reading today in a book called The Infinite-Dimensional Topology of Function Spaces by J. van Mill, online, and of this paper I found.
In the book: Most of the dimension theory had to do with topological dimension instead of fractal dimension. For topological dimension, dim(X x Y) <= dim(X) + dim (Y), by the way. An example of when equality does not hold in this case is the one-dimensional space E, with E x E one dimensional also, since E x E = E, where E is something called Erdös' space. The book had a lot of other information, but nothing I understood well enough to explain.
On Wikipedia and SpringerLink: the infinite product of the interval with itself is homeomorphic to the Hilbert Cube and is called a Tikhonov cube (I^ω). Surprisingly, there is also a cube I^c with c = the cardinality of the reals. I had assumed from the beginning I was working in a countably infinite dimensional space. I'm not sure how one is supposed to take the uncountable product of anything anyway, or if doing so is really legitimate.
I also found this interesting paper called "On the Mendès France Fractal Dimension of Strange Attractors: the Hénon Case" written by M. Piacquido, R. Hansen, and F. Ponta. The Mendès France version of fractal dimension is interesting because it deals with "zooming out" instead of "zooming in", like we normally do. It also only applies to curves, but can be adapted to other objects, as demonstrated in the paper. Anyway, it gave an interesting perspective, including on the construction of the fractals I'm dealing with. I may be mistaken, but it also appears that in the paper, the authors are considering something that could be considered the local dimension through measuring the Cantor set-like section of the Hénon attractor. I probably need to read over it more closely to know.
3: I did think of another question: I was wondering how the Cantor Sets present in the Hénon Attractor changed as the parameters change. How does the change in parameter affect the Cantor Set direction as opposed to how it affects the entire set? When using the Hausdorff dimension, you're supposed to look at each side's "epsilon", the scaling factor. Even though I'm using box-counting dimension, it would be interesting to know what kind of scaling factor we have as it is dependent on the parameters.
I need to find out how I can look at a Cantor Set present in the Hénon Attractor in Matlab. I'm not sure how to restrict my grid to a small box yet (as we know from (1)).