Ok. So I think I fixed my problem from yesterday and I think I got something that I want and it's better than what I had. I just need to go over it again to make sure I didn't make any other mistakes and then check up on the second part of it, which I think is right though and then type it all up on Tex. I'm pretty happy with what I have right now though. It's still taking a while typing it all up on Tex, since I have to rephrase it into something comprehensible and put it in proof-format.

Also, I got the email confirmation from the Charles Center today, so all is well.

## Wednesday, July 2, 2008

## Tuesday, July 1, 2008

### 7/1

Yesterday, I talked with Prof Lutzer about certain fractals lying in infinite dimensional space that I had been considering and attempting to compute the dimension of. Some of them are kind of interesting, but today, I realized that I made a big mistake with one of them and I tried fixing it today but haven't gotten too far. Prof Lutzer and I talked about grids and infinite products and I don't think we were really on the same page most of the time, but I am grateful that he met with me.

Right now, I'm trying to type some of my problems up on TeX so that I can send them to Prof Day, but in doing so, I'm noticing some possible problems. Hopefully I can work some of them out and redo what I thought I did earlier with my first fractal and maybe my second one was already done correctly but I want to make sure. So, it's taking some time.

I haven't done anything with local box-counting dimension yet. I think it would probably be a good idea to prove that the supremum of the set of local dimensions around x for all x in X is the dimension of X (if it's true, which I think it is).

So tomorrow, I'm going back to fix what I messed up the first time and trying to type things up on Tex. Also, I need to visit the Charles Center to make sure they got my honors request sheet, b/c they never sent me the confirm email.

Right now, I'm trying to type some of my problems up on TeX so that I can send them to Prof Day, but in doing so, I'm noticing some possible problems. Hopefully I can work some of them out and redo what I thought I did earlier with my first fractal and maybe my second one was already done correctly but I want to make sure. So, it's taking some time.

I haven't done anything with local box-counting dimension yet. I think it would probably be a good idea to prove that the supremum of the set of local dimensions around x for all x in X is the dimension of X (if it's true, which I think it is).

So tomorrow, I'm going back to fix what I messed up the first time and trying to type things up on Tex. Also, I need to visit the Charles Center to make sure they got my honors request sheet, b/c they never sent me the confirm email.

## Wednesday, June 25, 2008

### 6/25

1: Tex/Presentation

2: Inf Dim

1: Good news! I'm almost done with my slides. I still have to insert pictures (somehow), but Prof Day sent me some suggestions on how to resize pictures and insert them where you want them, so that should help. I'm not sure if my slides are okay as far as how much I have "proven". I'm going to send them to Prof Day and see if she has anything to say about them, but they're probably okay.

2: I met with Prof Lutzer today. That was actually pretty helpful. He seems to understand what I'm talking about most of the time and he was helpful in talking about infinite dimensional spaces. Most of the stuff I had seen before, but it was still pretty helpful as a review. I'm meeting with him again on Monday (and he's coming to my presentation to "learn" he says--the pressure's on!).

Also, I sent Prof Day what I could about the boxes associated with the product topology for the fractal lying in . It might be something, but I'm not sure.

2: Inf Dim

1: Good news! I'm almost done with my slides. I still have to insert pictures (somehow), but Prof Day sent me some suggestions on how to resize pictures and insert them where you want them, so that should help. I'm not sure if my slides are okay as far as how much I have "proven". I'm going to send them to Prof Day and see if she has anything to say about them, but they're probably okay.

2: I met with Prof Lutzer today. That was actually pretty helpful. He seems to understand what I'm talking about most of the time and he was helpful in talking about infinite dimensional spaces. Most of the stuff I had seen before, but it was still pretty helpful as a review. I'm meeting with him again on Monday (and he's coming to my presentation to "learn" he says--the pressure's on!).

Also, I sent Prof Day what I could about the boxes associated with the product topology for the fractal lying in . It might be something, but I'm not sure.

## Tuesday, June 24, 2008

### 6/24

1" TeX

2: Inf Dim

3: Tomorrow

1: I am having lots of trouble with Tex. Inserting pictures where I want and what size I want so it's not hanging off the frame and placing my text where I want it. So my slides are nowhere near done, but I think I know what I'm going to present. Sort of. I am still not sure how I'm gonna relate the infinite dimensional stuff to everything else or what I'm gonna say about it.

2: Maybe I might have an idea of what I can do with my fractal in the infinite dimensional space? Maybe. Not really sure if it's gonna work.

3: So tomorrow I'm meeting with Prof Lutzer. I'm going to try finish up my slides and maybe work on some of the local dim stuff.

2: Inf Dim

3: Tomorrow

1: I am having lots of trouble with Tex. Inserting pictures where I want and what size I want so it's not hanging off the frame and placing my text where I want it. So my slides are nowhere near done, but I think I know what I'm going to present. Sort of. I am still not sure how I'm gonna relate the infinite dimensional stuff to everything else or what I'm gonna say about it.

2: Maybe I might have an idea of what I can do with my fractal in the infinite dimensional space? Maybe. Not really sure if it's gonna work.

3: So tomorrow I'm meeting with Prof Lutzer. I'm going to try finish up my slides and maybe work on some of the local dim stuff.

## Monday, June 23, 2008

### 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?

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?

## Friday, June 20, 2008

### 6/20 (Part 2)

So, I was able to find the local dimension around a point not equal to the fixed-point! :)

### 6/20

1: Matlab

2: Inf

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 HÃ©non 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.

2: Inf

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 HÃ©non 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.

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