I'm a high school student doing my math IA on fractal dimensions.
According to this Wikepedia page, the logistic map's correlation dimension, Hausdorff dimension, and information dimension are all roughly 0.5. However, since the logistic map is made up of a bunch of lines, I assume its topological dimension is 1. How, then, can its fractal dimension be less than this?
I also found another page stating that the aforementioned Hausdorff dimension is actually that of the Feigenbaum attractor, which seems to me to be 0 dimensional. This makes more sense, but somewhat contradicts the first link.
What's going on here? Is the topological dimension of the logistic map 0 or 1, and is ~0.5 the dimension of the entire map or just the Feigenbaum attractor?
UPDATE #1: I realized that both Wikepedia pages are only talking about the dimension at a specific r value (which is 0-dimensional topologically). That should solve the above questions. Does anybody know if a fractal dimension of the entire bifurcation diagram has been found? To my knowledge there's nothing.