Height-mapping based on Julia Sets

[Timeroot]

I know that all the interesting stuff with 3D fractals has essentially made people forget about height mapping, but I was wondering if Height mapping had ever been done based on the Julia Sets at that location. For example, Hausdorff Dimension of the Julia Set, Hausdorff Dimension of the border of the Julia Set, Area of the Julia Set, Lyapunov Exponent of the Periodic/Chaotic/Fixed Point, etc.

Some may have to be done "by hand", but probably with enough code it could be done by the computer. Area would probably be the easiest, Hausdorff Dimension-based ones would be second-easiest, and the Lyapunpov Exponent... I have no clue. For the Lyapunov Exponent, this might be easily estimated with just an orbit trap. This has the problem, however, for points very "deep" into the branches of the Julia Set. Thoughts/Criticisms/Links/Pictures/Formulas/Help/People telling me this list is too long?