1. Population papers
2. Building infinite dimensional set
3. Conley Index
1.) I read the first papers on population dynamics and skimmed the second. The first paper is "Chaotic dynamics of a nonlinear density dependent population model" by Ilie Ugarcovici and Howard Weiss. It introduced me to the nonlinear Leslie model, which (the authors remind us several times) gives us the largest known variety of complicated dynamics at this low dimension, complete with self-intersecting global stable and unstable manifolds(!) and strange attractors. Also interesting was the difference between the two- and three-dimensional models, which seem to have reversed dynamics from one another for large values of the parameter. The attractors in the paper might be interesting to consider at some point. In the three-dimensional case, the authors mention that the attractor becomes "thicker" at a parameter value in a discontinuous way, causing its dimension to "jump".
The paper says on pg 1707: "Figure 26 shows the existence of 'fat' strange attractors obtained for larger values of the parameter f." I don't think they actually mean that the attractor became a fat fractal, which is a fractal with positive Lebesque measure and integer dimension (equal to space in which it exists). If it is though, I think the problem would be a lot more interesting. How does an attractor suddenly become "fat", with positive (in this case) volume?
2.) After meeting with Prof Day this morning, I tried working on the set in infinite dimensions. The problem is to prove that the infinite product of Cantor Sets, with dimensions whose infinite sum converges to a finite number, has finite dimension. It should work since the dimension of the Cartesian product of two objects is usually the sum of the dimensions of the objects, except in a few cases (which I should probably look up). I started by showing that you can construct a Cantor set of whatever dimension you like (between 0 and 1). That part wasn't so hard. The dimension of a regular Cantor set, in which you remove the middle 1/x for any x>1 is: ln(2)/ln(2x/x-1). Second, I tried to show that the dimension of two arbitrary Cantor sets is the sum of the two dimensions. Not so easy. So I tried a specific example.
Before, I had done (Middle-Thirds Cantor set) x (the Interval) and (the Middle-Thirds) x (the Middle Thirds). These were simple. I had also done the (Middle 1/2 Cantor set) x (Middle 7/8 Cantor Set) which wasn't as easy since I had to figure out to skip ahead one iteration on the middle-halves to get an accurate answer (and accurate covering). So today, I tried (Middle-Thirds) x (Middle -1/2). Still no luck. I'm thinking about using upper and lower box-counting dimension? I don't know what those are specifically but I know I saw them somewhere last semester. Box-counting dimension is suddenly not as simple as it was yesterday. I can't seem to get the right N(eps). By the way, epsilon is usually the side of the box in question (in 1-D it's an interval, in 2-D it's a square, etc.) and N(epsilon) is the number of epsilon boxes it takes to cover the set. Both are needed to calculate box-counting dimension. However, if you over-cover in the limit (epsilon to zero) you get the wrong answer.
Here's a list of things that don't work:
1.) Using the sqrt(Area) of the rectangle for epsilon
2.) Using the diameter of the rectangle for epsilon
3.) Using the average of the sides for epsilon
4.) Using fractions of boxes (for epsilons of 1/3^n anyway)
5.) Using smaller boxes for low iterations (what I did first--I tried boxes of with epsilon = 1/12, and let N be 4*12)
I worked backwards to see what I should get. I should/could get N = 12^n with eps = 1/9 (Maybe I could find a fractal with this dimension and show they have the same dimension?), or the equivalent. If I take the Hausdorff defn, thats 4 copies of the originial shrunk by a factor of "3.4069", but I don't really know how Hausdorff is calculated. It might be an option though, since box-dim and Hausdorff often coincide. I would prefer to get an answer with the box counting dimension, but I'm not sure if that's even possible. How do you choose your epsilon, and then how do you calculate N(epsilon)?
3.) I read the paper on the Conley Index. I think I sort of got the concepts (when it was a function--not "flow") but I'm still not really sure how it is supposed to be used.
P.S. I got Tex to work! So at least I accomplished something...
P.P.S As a note of interest: This site: http://www.mathreference.com/top-ms,cset.html and wikipedia both say that the Cantor set is homeomorphic to the Cartesian product of two Cantor sets (Cantor Dust), even though they do have different dimensions. Both, however, have topological dimension of zero. I guess fractal dimension is not a topological invariant.