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.