This is just a partial answer, but it looks like your figures are not "pure" fractals, but a union of different fractals (with different fractal dimensions). If I am not mistaken, these are called multi-fractals. It looks like part of your figure is the [Dragon curve fractal.][1] This particular fractal is space-filling, so it would not surprise me if your set is actually everything, (unless I have misinterpreted something). Also, your dynamics looks a lot like some two-dimensional version of the iterations of something similar to the Collatz conjecture, which give rise to a fractal behaviour. Look at the corresponding wikipedia page, there is a "complexification" of the iterations, and I think something similar can be done in your case. I will update this post with a concrete example of this in a moment... [1]: http://en.wikipedia.org/wiki/Dragon_curve