2:Dimension of product is sum of dimensions (etc)
1: For my specific example from yesterday of (Middle Thirds) x (Middle 1/2), my problem was calculating N(epsilon) for some epsilon box. I asked Professor Day and we worked out several examples and decided as long as you believe that the limit in the definition of dimension converges for any epsilon (which it should), you can use the definition to prove whatever it is I am supposed to prove.
We tried to see what was supposed to happen with N(epsilon) as epsilon goes to zero for the Middle 1/2 set using epsilon = 1/3. We ended up with an interesting result: sqrt(3) = lim as eps->0 of N(eps)^(1/n). We know that initially N(eps) looks like 2^n, but in the limit goes to sqrt(3)^n. Prof Day suggested (and I think this is right) that what happens as epsilon goes to zero is that (keeping in mind n is a positive integer) that N(eps) as a function appears to follow 2^n and at certain points drop (since the number of boxes gets relatively smaller for smaller epsilon since 4^n grows so much faster than 3^n) but still appears to "look like" the 2^n graph, and in the limit, it leads to sqrt(3)^n, which wouldn't be such a nice number for sets whose dimension is not rational. This exersize made everything make so much more sense.
2: To prove that the dimension of the product of two Cantor sets is equal to the sum of the dimensions of the two Cantor sets, I just used the definition. Boxdim = lim eps->0, (ln(N(eps)))/(ln(1/eps)). So the dimension of C1 and dimension C2 need number of boxes N1(eps) and N2(eps) respectively. Then you can use the fact that the sum of two logarithms is the logarithm of the product and the sum of two limits (when they both exist and are finite) is the limit of the sums to prove your answer. It should also hold for finite products.
Supposedly, the true answer is that the dimension of the product is greater than or equal to the sum of the dimensions, but I looked it up and this is true for Hausdorff dimension. It's different for box-dimension (see below). I found this great book (that is only avaliable through Swem online) called Fractal Geometry: Mathematical Foundations and Applications by Kenneth Falconer, who is apparently an expert on this stuff.
Heres something interesting the book says: First it says that the Hausdorff dimension of the product is greater than or equal to the sum of the (Haus) dimensions. Then it says the upper box dimension of the product is less than or equal to the sum of the (upper box) dimensions! The book also says that if the Hausdorff dimension is equal to the upper box dimension then equality holds. ALSO: If the two sets in question are subsets of R and one is a "uniform Cantor set" (which he defines in the book and are exactly the sets that I have been working with), then equality holds (for Hausdorff AND box dimension, since in this case Haus = box). I'm not exactly sure what the relationship is between box dimension and upper box dimension, but they're probably usually equal in "nice" Cantor sets like mine.
Also, something interesting that I had been wondering about last semester but had no idea what to do with: in the book, it finds the (Haus) dimension of the "Cantor target", which is the Middle Thirds Cantor set rotated about one of its axes. (As I hoped) The dimension turns out to be 1+dim(Cantor Set) = 1+ln(2)/ln(3).
Also interesting, the book even mentions (but doesn't say much) about the fact that the Hénon Map appears locally to be the product of an interval and a Cantor-like set. It also talks about taking the dimension of two-Dimensional fractals, which may be useful or interesting for finding the dimension of the Hénon map by exploiting its local topology.
The book also gives an example of when equality doesn't hold, but I haven't been able to really understand it quite yet (since I really don't understand the definition of Hausdorff dimension yet). I couldn't find an example of when box-dim didn't hold. I think when the limit exists and boxdim exists, it may just be the case that equality holds.
3: ALSO exciting (and relavent), I found in another book by the same author called Techniques in Fractal Geometry some information about (upper and lower) local dimension, which is also called pointwise dimension or Hölder exponent. It looks like it can be used in studying multifractals. The whole thing looks really interesting and I'd definitely like to read more about it.