Thursday, June 5, 2008


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.

No comments: