dimension-set for all Julia-fractals/Mandelbrot set

Question

You know the Julia-sets have all different not integer dimensions. I think it would be very interesting to have a map like the Mandelbrot-set, that shows the magnitude of the dimension of the Julia-set for that parameter $c$. So that a higher dimension creates a darker point.

Does anybody know if it already exists or what it would look like?

I thought that I could learn to program it because I think it will be very interesting. So a programme needs to create the Julia set, then to calculate its dimension (I think box-counting) and then to give the point a magnitude. Do you think I could create it or is it too hard for a beginner?

Hope you can help me and thanks a lot! Ami

Answer 1

Although this does not completely answer your question, nonetheless you might take a look at some of the known methods for calculating Hausdorff dimensions, found in the following papers:

• Garnett, Lucy, "A computer algorithm for determining the Hausdorff dimension of certain fractals." Math. Comp. 51 (1988), no. 183, 291–300.
• McMullen, Curtis T. "Hausdorff dimension and conformal dynamics. III. Computation of dimension." Amer. J. Math. 120 (1998), no. 4, 691–721.

In particular, McMullen's paper plots the actual graph of the dimension of the Julia set of $z^2+c$ for the real values $c \in [-1,1/2]$, together with a complete numerical table of his data behind that plot.