6/23

1: Matlab
2: Presentation
3: Other

1: I had a little trouble with Matlab not doing what I want it to do. I was trying to look at the dimension of the Cantor Set-cross section of the Henon Map but it was not subdividing the way I wanted it to. I only wanted it to subdivide in one dimension. So I'm tried it the way I measured the MidThirds Cantor Set. It sort of works. I got kind of a weird result. Like part of a sine curve. I'm trying smaller and smaller slices, which gives me closer approximation of what I want, but it's also taking longer and longer, since I'm only looking at points with in a small range (but I also only have to look at a smaller number of points altogether, which is good when you get to the inserting-method). For thinner and thinner slices, I seem to get something closer to what I think the "true" value is, but I'm also getting more and more choppy looking graphs (less like curves) and one that doesn't seem to really be leveling off towards the value (or any value... sort of).

Also, something sort of weird, when I look at the actual points plotted, they seem to lie along a line, in each section with the same slope, which was kind of weird.

2: I was going to work on my slides for my presentation, but the back-light on my monitor went out so I.T. has my laptop (and I don't think I'm getting it back today). I did a sort-of outline on paper though.

3: So, Prof Day sent me a link to this article about the dimension of the boundary of the Mandelbrot set (=2, by the way). I found the actual paper on mathscinet and on the 1st page, he says "Theorem A. H-dim(boundary M) = 2. Moreover for any open set U which intersects boundary of M, H-dim(boundaryM intersect U) = 2." This sort of sounds like local dimension, but I was wondering at first how he went about proving the local dimension is equal to the dimension of the entire thing, but it turns out it has to do with the dimension around the point having a dimension of at least the dimension of the Julia Set corresponding to that point and the set of points whose Julia sets have dimension=2 being dense in the set of points on the boundary of M. So, I'm not sure that helps me with local dimension of the Henon attractor since I don't really know anything about the points lying on the Henon attractor except that they map to other points on the Henon attractor. Unlike the Mandelbrot set, it doesn't lie in "parameter space", but rather (like a Julia set I guess) lies in x-y plane (Julia in complex plane). It was still interesting though. I didn't actually read the whole paper, but it might be interesting to see how they know that there exist all these Julia Sets of dimension 2.

I haven't met with Prof Lutzer yet. I emailed him and everything, but I'm not sure if he's actually here this week?