7/2

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.

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.

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.

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.

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?

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 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.

6/19

1: TeX/Midterm
2: Today

1: The TeX workshop thing helped a lot. I know how to do slides now anyway, so I'll probably work on that next week for my midterm presentation (that I haven't yet picked a day for--should do that tomorrow I guess). I'm not sure how result-oriented (or introductory/broad) our presentation needs to be. Half the people have seen a lot of the introduction stuff and half haven't. But, thirty minutes gives you a lot of time to talk, so it's probably about half/half? Anyway, I've got plenty of introduction and not a lot of results, but I guess I could show some graphs from Matlab? Also, I'm not sure what part of my "research" I should emphasize, local or infinite. Or how one applies to the other. I haven't really proved much for either, and I haven't really found any sources for local dimension as I've defined it for myself.

2: Besides the workshop, today the only thing I really did was read (and read and read). Prof Day suggested a inf-dim space I could look at and I tried doing some things with it and didn't really get anywhere. Infinity is confusing (seriously). I almost finished rereading my topology textbook from 426. I don't think we ever went over "Jordan Curve Theorem" and homology in class though (at least I don't remember this stuff), so I'm not sure if I should bother finishing these sections (but I see stuff about homology a lot when I'm looking for sources, so maybe I should?). That kind of helped as a review of topologies and bases and open/closed and products and such. I also went back to look at the Conley Index stuff too which makes a little more sense I guess. I guess you use it to help identify and prove the existence of fixed and periodic points and other "invariant objects" in your set, but I'm not sure how I would use it (for this project), but in Prof Day's thesis, she talks about some sort of projection into a finite space that preserves certain properties of infinite dimensional objects. I don't know if I should look at that? Maybe not yet. I really didn't do anything with Matlab today--looked at some things but didn't really change or fix anything.

6/18

1: Remote Access
2: Matlab/Local

1: Thanks to Prof Phillips, I can now access GAIO (and look at the pictures) from my laptop. He got it to work and then I thought it wasn't working, but it was just a problem with my computer. All better. So that's good news.

2: I'm getting kind of weird graphs when dealing with local dimension, scaled and unscaled. I'm not sure if it's really looking the way I want it to look. I have to look at it again tomorrow I think. I wanted to think more about the theory and proving things to do with local dimension but I'm not really sure where to go with it right now. I really didn't get much at all done today anyway, so hopefully I'll get my thoughts more organized tomorrow. I went to the Matlab workshop (and I'm gonna go to the LaTeX one tomorrow), which was okay, but not too relevant to my project.

6/16

1: Visit to IT
2: Matlab
3: Inf Dim

1: So, I went to IT on Friday to see if they could help me with accessing GAIO remotely from my laptop. After I told them my problem, all I got (from everyone I talked to) was "Oh, the math department. Well, they got their own thing going on, so I don't know." And then they told me it would just probably be better to go to the lab to do my work. They did give me the name of someone in the math dept I can maybe talk to (I'm looking for where I wrote it down...). I also tried accessing it with XWin because I think that's how Prof Li was able to access it from his PC, but I can't figure out how to work it. So, for now, still doing GAIO from the lab.

2: I did do some work on Matlab today. I got some nice graphs (that I can't access right now since I can't access the lab, but I'll put up tomorrow). I've got a few ideas of things I can try tomorrow with local dimension. I tried using the "function" function with solve to find periodic points (so I could maybe find other points on the attractor) but it didn't work.

3: I'm not really sure what I should be doing with infinite dimensions. I found this book (Infinite Dimensional Dynamical Systems in Mechanics and Physics, by Roger Temam) that looks like it might be about the things I'm doing, but I'm not sure if I should go through and read/learn the stuff since it's very similar to what Prof Day told me I should be doing myself (dimensions of attractors, fractal (box counting) dimension in infinite dimensions). It all looks really complicated though, and I'm pretty sure I wouldn't be able to derive it all myself anyway.

Also, with the topology: I don't see how changing the topology of a set is going to change the box counting dimension. I'm pretty sure you can use open sets, closed sets or neither for your box as long as it's connected (maybe) and bounded with length equal to epsilon and epsilon going to zero. The impression I get from Hausdorff is that it is almost the same except that the "boxes" are less than or equal to epsilon with epsilon going to zero (sort of). That's why Haus = Box for regular Cantor Sets, since the epsilons are all equal.

6/12

1: Tex
2: Topology

1: So, today I took a break from Matlab and most of the day, I worked on entering my definitions and proofs that I have so far into Tex. Using Tex isn't as hard as I thought it would be. It's not so bad once you get used to it. I still am hoping they have some kind of Tex workshop to show us how to do the Powerpoint with Tex.

2: I reviewed definitions of topologies and bases and fractals and products, trying to get an idea of how to define an infinite dimensional fractal and infinite dimensional fractal dimension. Nothing came straight to mind, but I have some ideas of what I can look at. I'm still not sure how changing the topology of a space affects box-counting dimension. I tried looking at the box-counting dimension of the Cantor set in R with the discrete topology to help, but I don't see how changing the topology really gives you a different answer. I guess it's supposed to change your definition of a box? I can see how maybe a different metric might change things, but the topology?

I'll probably end up back on Matlab tomorrow, organizing all my graphs and doing some other things. Prof Day also suggested I try to prove something about local dimension and scaling for uniform fractals like the Cantor Set. I also need to visit IT tomorrow since I didn't today.

6/11

1: GAIO/Cantor Set
2: GAIO/Local Dimension
3: Infinite Dimensions
4: Other

1: So I did some more work with GAIO. I was able to plot the Middle Thirds Cantor Set and compute the approximate box dimension with my "insert" algorithm. I got something around 0.645 (true value is about .6309) for not very many iterations and not too deep, so probably for more iterations and higher depth, I'd get an even closer approximation.

2: I fixed my code from yesterday and looked at local dimension. Since I'm starting off with my epsilon so small, I'm getting a different kind of curve (of depth vs "dimension") than before that appears to go upwards towards the "true value" rather than downwards. I was thinking about rescaling my initial boxes to one each time I zoom in so that I can more easily compare my resulting graphs. I need to organize my graphs and look closer at the dimension of the entire attractor vs local dimension around points in the attractor (first actually finding points to use).

3: So, I think I can prove that the box dimension of this (different than before) set that lies in an infinite dimensional space has finite dimension using the box topology, but in this case, the boxdim(product of the sets) is LESS THAN the sum of the dimensions of the individual sets. Maybe it would behave better using the product topology though, if I can ever get a good definition of boxdim in infinite dimensions with the product topology together. If not, what topology on the set would give me the answer I want?

4: Other stuff: I need to work with LaTeX and type up some of my proofs/work. I need to visit IT and hopefully figure out how to access GAIO from my laptop. I have some reading I want to catch up on (I found some papers that might be interesting/relevant). Professor Day suggested that while she's gone, I meet with Prof Lutzer about the topological aspects of my project and Prof Phillips about the Matlab stuff.

6/10

1: GAIO/local dimension
2: CRS
3: infinite dimensions

1: Today, I spent a lot of time on Matlab trying to get my boxes inserted at a certain depth only within a certain neighborhood. Here's a picture!


The red boxes are the boxes I need to "count" for the local dimension of the area in the box. The big green boxes are just the mis-covering of the entire attractor at depth 6. For this picture, the iteration number is still small and the boxes are still large since it was just an experiment to try to get it right. It still needs some work, but I was able to at least figure out how to use "subdivide" and the flags (sort of). Tomorrow, I'm going to try to fix the redundancies that might be slowing the program down and clean it up a bit. Also, I can start to look at the dimension of the attractor versus the local dimension about a point (for example, the fixed point) as you get "more local" (smaller initial boxes to be further subdivided). I actually don't even have to use the "subdivide" command, and can probably remove that altogether. Anyway, in a perfect fractal, zooming in to look at the "local" dimension should return the same value (PD suggested I prove it and say something about the scaling). For something like the Hénon attractor, it may be the case. Or perhaps, as you move along the attractor, the "local" dimension changes? For this question, I should consider other points on the attractor (if I can find some!).

Also, I need to find some way to take the box counting dimension of the Cantor like set in the Hénon attractor. I think I might know some way of doing this, but I'm not sure if it will work. Also, I wouldn't be using square boxes, but let my epsilon be the x-radius, I think (if I'm taking the Cantor set parallel to the x-axis) and not divide horizontally. I think I remember seeing some way of doing that.

I'm hoping to find some way of using GAIO on my own computer, so I don't have to go to the computer lab to run the experiments. When using the SSH client, I am unable to view figures, which is pretty important, so I know what I'm doing is right. Also, I don't seem to have anyway of accessing ssh from my command prompt.

2: CRS = Chain Recurrent Set. Professor Day told me this could be an option instead of MIS (Maximal Invariant Set). Yesterday, I said it seemed to give a better covering than the mis method. Prof Day tried to explain it to me a little bit, but I really didn't understand it enough to remember. She gave me a paper I'm going to look at later to get an idea of the theory behind it. It might be useful later. Right now, I'm just using the "insert method" with calculating dimension.

Actually someone in our "mathematical family" who knows about this stuff (including CRS) is supposed to be visiting in 2 weeks to give a talk, so there's a chance I might be able to ask him about it in person.

3: Defining fractal dimension in infinite dimensional space: not so easy. I looked at the definitions of topological dimension (from book IDTFS) and box dimension (from Alligood's Chaos) to get ideas of where they came from. Box dimension is easy. It comes from the relation (1/eps)^dim = number of boxes, that seems present in regular dimension spaces, like the interval I, or I^2, or I^3 (all bounded spaces). For topological dimension, the definition has to do with "inessential" families of disjoint closed subsets of X. The internet definitions seem to have to do with coverings of the set.

In short, I'm not really sure how one is supposed to define "box" or box-counting dimension in infinite dimensional space. Infinity is tricky.

Question: if you have a bounded uncountable collection of points, is the fractal dimension necessarily positive? You can have Cantor set with arbitrarily small positive dimension and one with fractal dimension = 1. Can it be zero?

6/9

1: GAIO/local dimension
2: Reading: infinite dimension/topology
3: Another Question

1: GAIO is the program I usually use (with Matlab) to create my "boxes" when I compute box-counting dimension. I'm trying to use GAIO's grid to compute local dimension of the attractor, in this case, taking the grid to a certain depth, choosing one box at that depth and only looking at that box when taking the grid to a larger (deeper) depth, so that I can compute further. Using GAIO should be more efficient than creating my own grid. Right now though, I can't seem to get it to work. I don't know the program well enough to stop the tree and select the box. In trying, however, I did find that using the crs (chain recurrent set) instead of mis (maximal invariant set) seems to give me a better covering of the (entire) attractor. I tried calculating the dimension with this method, and seemed to get a closer approximation than with the mis method. It still didn't seem to converge very quickly to the "true" value (which I guess is expected when using a box-covering with relatively large epsilon). This method probably doesn't guarantee an upper bound like the mis though (maybe). (Another one was rga, but I don't think it returned square boxes.) If I could insert figures into a GAIO grid, I could test the accuracy of my methods much more easily. Most of my day was spent trying to get Matlab to work. I read most of the GAIO manual that Prof Day gave me and so I learned some other things that GAIO can do, which might help me out later. The manual didn't seem to say how you would insert novel figures or models as a "Model" though.

2: I did do some reading today in a book called The Infinite-Dimensional Topology of Function Spaces by J. van Mill, online, and of this paper I found.

In the book: Most of the dimension theory had to do with topological dimension instead of fractal dimension. For topological dimension, dim(X x Y) <= dim(X) + dim (Y), by the way. An example of when equality does not hold in this case is the one-dimensional space E, with E x E one dimensional also, since E x E = E, where E is something called Erdös' space. The book had a lot of other information, but nothing I understood well enough to explain.

On Wikipedia and SpringerLink: the infinite product of the interval with itself is homeomorphic to the Hilbert Cube and is called a Tikhonov cube (I^ω). Surprisingly, there is also a cube I^c with c = the cardinality of the reals. I had assumed from the beginning I was working in a countably infinite dimensional space. I'm not sure how one is supposed to take the uncountable product of anything anyway, or if doing so is really legitimate.

I also found this interesting paper called "On the Mendès France Fractal Dimension of Strange Attractors: the Hénon Case" written by M. Piacquido, R. Hansen, and F. Ponta. The Mendès France version of fractal dimension is interesting because it deals with "zooming out" instead of "zooming in", like we normally do. It also only applies to curves, but can be adapted to other objects, as demonstrated in the paper. Anyway, it gave an interesting perspective, including on the construction of the fractals I'm dealing with. I may be mistaken, but it also appears that in the paper, the authors are considering something that could be considered the local dimension through measuring the Cantor set-like section of the Hénon attractor. I probably need to read over it more closely to know.

3: I did think of another question: I was wondering how the Cantor Sets present in the Hénon Attractor changed as the parameters change. How does the change in parameter affect the Cantor Set direction as opposed to how it affects the entire set? When using the Hausdorff dimension, you're supposed to look at each side's "epsilon", the scaling factor. Even though I'm using box-counting dimension, it would be interesting to know what kind of scaling factor we have as it is dependent on the parameters.

I need to find out how I can look at a Cantor Set present in the Hénon Attractor in Matlab. I'm not sure how to restrict my grid to a small box yet (as we know from (1)).

6/6 (Friday!)

1: topologies
2: local dimension

1: So, yesterday, I guess I was using the box topology to build my infinite dimensional "boxes". Prof Day suggested I use the product topology instead. However, I think open sets in this topology must be the infinite product of sets Vi with only a finite number of these sets not equal to the real line. But then is the complement of my fractal open? Does that mean that my fractal isn't closed? But I know my fractal is compact since I know the infinite product of compact sets is compact. But wouldn't the infinite product of the unit interval should also be compact?

2: As for local dimension, I'm probably gonna scrap the "official" definition and just work with my own definition. I'm gonna try out some Matlab experiments on the Hénon attractor. I've gotta try to work out some things with GAIO.

6/5

1: infinite dimensional set
2: local dimension
3: overall project

1: Let's say I build my infinite dimensional set. Middle 1/2 x Middle 7/8 x Middle 31/32 x ... x Middle ((2^(2k+1) - 1)/(2^(2k+1))) x ... is my set. My infinite dimensional boxes look like the product of intervals [xj, xj + ε] for j in J (some index). Well, letting eps = 1, we get N = 1. Then for any epsilon less than 1/2, we need an uncountable number of boxes (at least 2^inf). If I build my set as I let epsilon go to zero, I again get N=1 for eps=1. But then for epsilon=1/4^n, boxdim = limit as n--> infinity, of ln(2^(n(n+1)/2))/ln(4^n) which is limit as n-->infinity of (n^2 + 1)/4n which is infinity.

However just because I'm getting infinity as the box dimension (if I'm even doing this right--how do you know how to count infinite dimensional boxes?) doesn't mean the Hausdorff dimension will be infinity. In general Boxdim >= Hausdorff dimension. For regular Cantor sets, the Hausd Dim usually equals the Box Dim, but maybe not for the infinite product. I looked up the definition and tried to understand it, but I'm still not 100% on how to calculate Hausdorff dimension, especially in more than one dimension, let alone infinite dimensions. I'm not sure if I'm going to go in that direction anyway since I usually work in box-counting dimension, but I guess it would be interesting to know whether you can get some kind of finite dimension. So, so far the object seems to have topological dimension of zero and boxdim of infinity (maybe).

2: As for local dimension, I don't think it is defined to be what I thought it was. For example, in some finite-dimensional sets, the local dimension can be equal to infinity. The definition is confusing and will probably take me a little longer than expected to understand. I'm not so sure it's what I want anyway though. I wanted "local" dimension to be defined as the dimension of the set contained in an epsilon ball about a point in the set, the dimension of a subset about a point (Maybe it's called something else?) , so that the dimension of the epsilon ball is less than or equal to the dimension of the entire set. And then I was going to observe whether local dimension changes as you move along the attractor (if that is even a feasible task).

3: I'm not so sure where my project is going. It doesn't really have a clear direction right now. I'm hoping I can sort that out before next week.

6/4

1:N(epsilon)
2:Dimension of product is sum of dimensions (etc)
3:Local dimension!

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.

6/3

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.

6/2

For my research project I am studying dimension of fractals and chaotic attractors. It is sort of a continuation of my project from Math 410. For that project, I worked with Matlab on a way to compute the dimension of the Hénon attractor and observe how dimension changes as parameters change.

There are a few different directions I can take my summer/honors project in:

1.) I can continue working with Matlab, trying to better approximate dimension of other attractors for certain parameters.

2.) I can look at "local" dimension, which should be a lower bound on the dimension of the entire object. Also, there is the idea that we should be able to somehow exploit the fact that an attractor locally looks like the product of a submanifold and a Cantor set to better approximate dimension of the entire object. I should be able to find some verification that the dimension of a subset is less than or equal to the dimension of the entire set. I think from there, it should be true that: dim(entire set) >= dim(Cantor set coordinate) + 1. If true, this would give us a lower bound, if we can get a good approximation of the dimension of the Cantor set or even part of it.

3.) I can use methods from algebraic topology to obtain rigorous results and approach the problem from 2 directions. Professor Day suggested using methods from Conley Index Theory and symbolic dynamics.

4.) I can construct a fractal in infinite dimensional space as the infinite product of Cantor Sets, which has a known dimension, and apply what we know from this object to compute dimensions (if finite) or other topological properties of fractal-like structures in infinite dimensional space whose dimension is unknown and topology may be difficult to study. Professor Day mentioned that I might want to look into studies done with multi-fractals for this part.

So, today, I'm considering my options, looking for papers on local dimension on MathSciNet (and references), and reading through some papers Professor Day sent me (two about strange attractors in population dynamics, one about Conley Index theory, and one about GAIO--I need to find out how to define a Model that isn't Hénon). Also, getting Tex installed when I can find it.