1: I've been working on Matlab today, and I think I fixed what was wrong from Wednesday, so I have this graph now:
The blue looks at the "local" dim in a box at depth 4. The green at depth 6. The red at depth 8. The black at depth 10. The solid lines--I used the "actual" radius. The dotted lines--I used the "scaled" radius. I took them all to depth 30, but they all start at different depths (since they all start with different size boxes--although for the scaled versions I set the initial size to 1).
It's what I expect sort of. It makes sense for dotted to give a higher approximation and for the black to terminate higher than the blue (since there is greater point-density at the same depth for black since I'm using a smaller initial box and the same number of points). I'm still not completely happy with it. I don't quite understand everything going on in the graph yet.
I think next, I should just consider the scaled versions and let my new "depth" be scaled as well, so that for blue, actual depth of 4 would be a scaled depth of 1 and 6 would be 2 and 8 would be 3 and then for green, the actual depth of 6 would be 1 and 8 would be 2, etc. That way, the graphs line up and I can compare them. I also might do a for-loop instead of a while-loop so that I'm not using more points for black (even though that is an advantage normally) just to see if my local dimension is actually changing as I zoom in or if I'm just getting a different value because I have more points. But doing this still won't really verify anything since I'll just have relatively less points for black, but I still want to look at it.
So, how accurate are these estimates? They're low. Here's another picture.
The yellow is the local dim unscaled and black is local dim scaled. Red--CRS, Blue--MIS, Green-insert.
But, I'm not sure if the local dimension is low because the dimension of the subset is actually lower or if it's a scaling-thing. I think I might also try rescaling the other radii (for the other methods) to see what I get. The way I'm thinking about it I guess (if this is right?) is that if I scale all initial boxes to one, then it will make more sense when I compare them. I think I should try finding the "local" dimension of the Mid3rd Cantor Set, scaled and unscaled to make sure I'm right, since I'm not sure (unless there's a way to prove that for boxdim converges faster if the initial epsilon is scaled to one? -- I'm not sure how that would work though).
Another thing: these are all for a = 1.28, b= .3 about the fixed point. I need to find another way of choosing another point (like the K_1 point of the iteration or something) and also the fixed point isn't always on the attractor (like for a = .9, b = .4). I was gonna try rand, but I wasn't really getting that to work either.
Another thing I tried, but I still haven't gotten, is measuring the dimension of a Cantor-slice of the Hénon Attractor.
Another thing: I know that C is homeomorphic to C x C, but C has lower dimension. I was wondering: are any some of the Henon attractors homeomorphic to each other (obviously a strange attractor isn't going to be homeomorphic to the periodic point attractor--I was thinking about the strange attractors specifically)? Is that even useful? Is it easier to work with lower-dimension attractors? I know sometimes dimension has some actual physical interpretation for systems including bounds on certain things. If you have to attractors that are homeomorphic with different dimension, can we use either dimension as a "bound" on whatever physical interpretation on either system. This might actually be a kind of stupid question though. They're all 1-D manifolds already. So they all have the same topological dimension. Are they already homeomorphic anyway?
2: I don't know really what I'm gonna say about infinite dimensional objects for my midterm or if I should even bring it up since I'm not sure I could answer any questions about it.