On the Julia set, the iterating the function $f:z\mapsto z^2+c$ generates a sequence of points on the Julia set. Because roundoff errors would be expected to cause the sequence to fall into an attractive cycle, I used inverse iteration $f_i:z\mapsto \pm (z-c)^{1/2}$, where a random number generator was used to pick the sign, to generate large numbers of elements for the (arbitrarily picked) value of $c = -1$. The set looks like
I generated large (1,000,000) sets this way and For a set of random reference points in the set, I calculated the number of points within various radii of each reference point. The following graph is a log-log plot of the result:
Where the x axis is the radius and the y axis is the number of points within the radius. The radii ranged from 0.0001 to 0.0512 in factor of 2 steps. The slope of the line is about 0.855, suggesting a dimension less than 1, which is contrary to my intuition for this Julia set.
My guess is that I am studying a chaotic dynamic system, which results in sets that have a different dimension. From a naive statistical point of view, if I have N >>1 points, and the number of points in a ball of radius r goes as $r^{0.855}$ then the number of balls needed to cover the set should go as $r^{-0.855}$, suggesting that the Hausdorff dimension is 0.855.
Lutz comment on the density obtained via simple inverse iteration is certainly correct. There are, though, ways to modify the inverse iteration algorithm to obtain a more uniform distribution. You can find a description of these techniques with an implementation using Mathematica in this paper. The main algorithm described in that paper is built into Mathematica as of version 10 and can be accessed via a command like
The modified inverse iteration algorithm is also implemented in this web app, which I used to generate this image:
Performing box-counting on that image, I compute a fractal dimension of about 1.367.