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Hi,

I finally wrote a blog post about a class of dense Julia sets I've been exploring lately:
http://algorithmic-worlds.net/blog/blog.php?Post=20130428
Julia sets are associated to conformal maps of the sphere to itself. Very roughly, a point belong to the Julia set if its iterations under the conformal map behaves in a chaotic way. It happens that certain Julia sets cover the whole plane, i.e. all the orbits are chaotic. This does not occur for the familiar Julia sets z -> z^2 + c, but does occur for more general rational maps. Dense Julia sets produce dense fractal patterns, just like the Ducks-Kaliset type algorithms, which I find much more appealing than usual 2d fractals. See my galleries for many examples of such patterns:
http://algorithmic-worlds.net/expo/expo.php

One way to construct maps whose Julia set is dense is to use the fact that the sphere admits branched coverings by a torus. The maps of the sphere which are covered by "affine expanding maps" of the torus are called Lattès map, and they necessarily have dense Julia sets. The blog post provides more explanations. Here is a reference for the mathematically inclined:
http://arxiv.org/abs/math/0402147

You can check many pictures of Lattès Julia sets in this collection:
http://algorithmic-worlds.net/expo/expo.php?Collection=Lattes&CollSearch=0
Under each picture, you can find the formula of the corresponding conformal map.

Best,

Sam