What does the fractal dimension of a julia set tell us?

[fractal_dust]

Different julia sets have different dimension, and it seems possible to calculate in many cases. I know the fractal dimension is somewhat related to how crumpled and crinkled the shape is but are there more precise information we can find from the dimension of a fractal?

I also read that all the julia sets that come from outside the mandelbrot are equivalent to the cantor set.. as I understand it there should be a function which pushes the points around a bit to flatten the julia set into a cantor set, but it does not seem like anyone has explicitly found this function?

[Prokofiev]

Hello,
To know more about fractal dimension you can have a look at Wikipedia: http://en.wikipedia.org/wiki/Fractal_dimension. This article is a bit of a mess, but the main information is there.
The fractal dimension tells us also how its "measure" grows as its size grows.

All julia sets that come outside the Mandelbrot set are disconnected (That's one of the definitions of the Mandelbrot set, by the way). It means that they are made of disconected points, like the cantor set. I never encountered any function that could transform any Julia Set into the Cantor set.

[lkmitch]

The Julia set for 0+0i is just a disk and the fractal dimension of its boundary is 1.  So maybe the fractal dimension of the boundaries of filled-in Julia sets is related to the distance of c to the boundary of the Mandelbrot set.