1: GAIO/Cantor Set
2: GAIO/Local Dimension
3: Infinite Dimensions
1: So I did some more work with GAIO. I was able to plot the Middle Thirds Cantor Set and compute the approximate box dimension with my "insert" algorithm. I got something around 0.645 (true value is about .6309) for not very many iterations and not too deep, so probably for more iterations and higher depth, I'd get an even closer approximation.
2: I fixed my code from yesterday and looked at local dimension. Since I'm starting off with my epsilon so small, I'm getting a different kind of curve (of depth vs "dimension") than before that appears to go upwards towards the "true value" rather than downwards. I was thinking about rescaling my initial boxes to one each time I zoom in so that I can more easily compare my resulting graphs. I need to organize my graphs and look closer at the dimension of the entire attractor vs local dimension around points in the attractor (first actually finding points to use).
3: So, I think I can prove that the box dimension of this (different than before) set that lies in an infinite dimensional space has finite dimension using the box topology, but in this case, the boxdim(product of the sets) is LESS THAN the sum of the dimensions of the individual sets. Maybe it would behave better using the product topology though, if I can ever get a good definition of boxdim in infinite dimensions with the product topology together. If not, what topology on the set would give me the answer I want?
4: Other stuff: I need to work with LaTeX and type up some of my proofs/work. I need to visit IT and hopefully figure out how to access GAIO from my laptop. I have some reading I want to catch up on (I found some papers that might be interesting/relevant). Professor Day suggested that while she's gone, I meet with Prof Lutzer about the topological aspects of my project and Prof Phillips about the Matlab stuff.